Sign of the pulsating wave speed for the bistable competition–diffusion system in a periodic habitat.

Introduction and main results

This paper is concerned with the speeds of pulsating waves for two-species competition–diffusion systems with bistable structures in periodically varying media. The existence and qualitative properties of pulsating waves have been established recently. In this work, assuming that the two species share the same diffusion rates, we study the sign of wave speeds by comparing the reactions and competitions. We first give a criterion when the speed is zero, and then provide some sufficient conditions ensuring the speed has a strict sign. We also show that the spatial heterogeneity has a significant consequence on the sign and gives rise to some new phenomena. More precisely, the pulsating waves in two opposite directions may have different speeds. Particularly, from the ecological point of view, our result indicates that the invasion of a new species may succeed in one direction but fail in the opposite one. Finally, we show the presence of multiple stationary waves which is in contrast with the uniqueness of non-stationary waves.

In this paper, we study the following competition–diffusion system

1.1 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{t}u=d_1{\partial }_{\mathit{xx}}u+f_1(x,u)-k_1(x)uv\hfill & \text{in}(0,\infty )\times \mathbb{R},\hfill \\ {\partial }_{t}v=d_2{\partial }_{\mathit{xx}}v+f_2(x,v)-k_2(x)uv\hfill & \text{in}(0,\infty )\times \mathbb{R}.\hfill \end{array}\right)\end{array}$$

Graph

For each $i=1,2$ , the diffusion rate $d_i$ is a positive constant, the competition coefficient $k_i:\mathbb{R}\to \mathbb{R}$ is positive, of class $C^{0,\alpha }(\mathbb{R})$ (with $0<\alpha <1$ ), and L-periodic for some $L>0$ , that is, $k_i(x+L)=k_i(x)$ for all $x\in \mathbb{R}$ , and the reaction $f_i:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ , $(x,s)\mapsto f_i(x,s)$ is continuous, L-periodic in x, of class $C^{0,\alpha }$ in x locally uniformly in s, and of class $C^{1,1}$ in s uniformly in x. For each $x\in \mathbb{R}$ , we also assume the following conditions on $f_i(x,·)$ :

• (H1)

• $f_i(x,0)\equiv f_i(x,1)\equiv 0$

• ,

• ${\partial }_s{f}_i(x,0)>0$

• and

• $f_i(x,s)/s$
• is decreasing in
• $s>0$
• ; furthermore, the kinetic problem
• $$\begin{array}{c}\hfill \frac{\mathit{du}}{\mathit{dt}}=f_1(x,u)-k_1(x)uv,\frac{\mathit{dv}}{\mathit{dt}}=f_2(x,v)-k_2(x)uv\end{array}$$

Graph

• has a unique positive equilibrium.

Models like (1.1) are usually used to describe the evolution of interaction and diffusion of two competition species in mathematical ecology. A typical example satisfying (H1) is the Lotka–Volterra competition system of type

1.2 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}u=d_1{\partial }_{\mathit{xx}}u+a_1(x)u(1-u)-k_1(x)uv,\hfill \\ {\partial }_{t}v=d_2{\partial }_{\mathit{xx}}v+a_2(x)v(1-v)-k_2(x)uv,\hfill \end{array}\right)\end{array}$$

Graph

where $a_i:\mathbb{R}\to \mathbb{R}$ , $i=1,2$ , are the growth rates of species, which are assumed be positive, of class $C^{0,\alpha }(\mathbb{R})$ and L-periodic. The periodic dependency of $f_i$ (or $a_i$ ) and $k_i$ on the spatial variable x characterizes a typical spatial heterogeneity of environments. Propagation phenomena of scalar reaction–diffusion equations in periodically varying habitats have been extensively investigated earlier (see e.g., [[4], [9]–[11], [15], [29], [37]] and references therein), and taking into account such a spatial heterogeneity in the study of competition–diffsuion system (1.1) has also attracted many recent attentions [[14], [17], [21], [35]].

Under the assumption (H1), it is clear that system (1.1) has a trivial steady state (0, 0) which is unstable, and two semi-trivial steady states (1, 0) and (0, 1). It is also easily checked that (0, 1) and (1, 0) are the only two semi-trivial equilibria of (1.1) (see, e. g. [[5], [21]]). In this paper, we will assume that the competition between the two species u and v is strong, and investigate the pulsating wave of (1.1) connecting (0, 1) and (1, 0) (which is an entire solution of (1.1) describing the transition between the steady states (0, 1) and (1, 0); the precise definition will be recalled later). To make the assumption precisely, let us first introduce some notations. For $i=1,2$ , denote by ${\lambda }_{\ast ,i}^{-}$ the principal eigenvalue of the linear problem

1.3 $$\begin{array}{c}\hfill d_i{\psi }^{\prime \prime }+({\partial }_s{f}_i(x,0)-k_i(x))\psi =\lambda \psi \text{in}\mathbb{R},\psi >0\text{in}\mathbb{R},\psi \text{is}\text{L-periodic.}\\ \hfill \end{array}$$

Graph

We assume that

• (H2)

• ${\lambda }_{\ast ,i}^{-}<0$

• for

• $i=1,2$

• .

Clearly, (H2) holds if, in particular, ${\partial }_s{f}_i(·,0)\le ,\not\equiv k_i(·)$ . The assumptions (H1) and (H2) ensure that (1, 0) and (0, 1) are two linearly stable steady states of (1.1) in the space $C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ , where

$$\begin{array}{c}\hfill C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2):=\left\{({\varphi }_1,{\varphi }_2),\in ,C^2,(\mathbb{R};{\mathbb{R}}^2),:,{\varphi }_i,(·+L),=,{\varphi }_i,(·),,,\text{for},,,i,=,1,,,2\right\}.\end{array}$$

Graph

Indeed, the linearized operators of the elliptic part of (1.1) at the steady states (1, 0) and (0, 1) have a negative eigenvalue; then the stability of (1, 0) and (0, 1) follows directly from Mora's theorem [[31]].

Next, we recall the definition of pulsating waves of (1.1). By a pulsating wave (U, V)(t, x) connecting (1, 0) and (0, 1) in the direction $e\in \{\pm 1\}$ , we mean that it is an entire solution of (1.1) satisfying

$$\begin{array}{c}\hfill U(t,x)={\mathrm{\Phi }}_1(xe-c(e)t,x),V(t,x)={\mathrm{\Phi }}_2(xe-c(e)t,x),\end{array}$$

Graph

where $c(e)\ne 0$ and the functions ${\mathrm{\Phi }}_i$ , $i=1,2$ , are L-periodic in the second variable, and satisfy

$$\begin{array}{c}\hfill ({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)(-\infty ,·)=(1,0),({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)(+\infty ,·)=(0,1),\end{array}$$

Graph

with the convergences taking place in the topology of $L^{\infty }(\mathbb{R})$ . The constant c(e) is called the speed of the wave and $e\in \{\pm 1\}$ is its direction. An equivalent definition is that (U, V)(t, x) is an entire solution satisfying

1.4 $$\begin{array}{c}\hfill (U,V)\left(t,+,\frac{\mathit{mLe}}{c(e)},,,x,+,m,L\right)=(U,V)(t,x)\text{for}(t,x)\in {\mathbb{R}}^2,m\in \mathbb{Z},\end{array}$$

Graph

with $(U,V)(t,x)\to (1,0)$ as $xe-c(e)t\to -\infty$ and $(U,V)(t,x)\to (0,1)$ as $xe-c(e)t\to \infty$ . On the other hand, we say that $(U,V)(t,x)=(U,V)(x)$ is a stationary pulsating wave in the direction e if it is a classical solution of (1.1) and satisfies

$$\begin{array}{c}\hfill (U,V)(x)\to (1,0)\text{as}xe\to -\infty \text{and}(U,V)(x)\to (0,1)\text{as}xe\to +\infty ,\end{array}$$

Graph

where all the convergences are understand to be locally uniformly with respect to x. In such a situation, we say that the wave speed is zero.

Since (1, 0) and (0, 1) are two stable steady states of (1.1), there exists at least one steady state in $C_{\mathit{per}}(\mathbb{R};{(0,1)}^2)$ (see e.g., [[19]]). Yet, the spatial heterogeneity may give rise to the presence of multiple steady state in $C_{\mathit{per}}(\mathbb{R};{(0,1)}^2)$ , and this possibility may prevent the existence of pulsating wave connecting the two extreme steady states (1, 0) and (0, 1). On the other hand, if system (1.1) admits a bistable structure between (0, 1) and (1, 0), the existence of pulsating waves has been proved recently [[14], [17], [20], [35]]. More precisely, the bistable structure assumption is stated as follows:

• (H3) Any steady state
• $(\tilde{u},\tilde{v})\in C_{\mathit{per}}(\mathbb{R};{(0,1)}^2)$
• of (1.1) is linearly unstable.

Here, an L-periodic steady state $(\tilde{u},\tilde{v})$ is said to be linearly stable if the linearized operator of the elliptic part of (1.1) around $(\tilde{u},\tilde{v})$ , restricted in the space $C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ , has an eigenvalue with a positive real part.

It is worthy to point out that a sufficient condition for the bistable structure (i.e, (H2) and (H3) are satisfied) was provided by Girardin [[21]] in the study of segregation phenomena induced by large competitions. More precisely, the author considered system (1.1) with $k_1$ and $k_2$ being constants and proportional (i.e., $k_1/k_2$ is equal to a fixed constant), and obtained the bistable structure for all large $k_1$ when the period L is small. Furthermore, the limit of the wave speed c(e) as $k_1\to \infty$ as well as its sign were determined in a subsequent paper [[22]]. We also mention that, regardless of the quantities of the competition rates, system (1.1) admits a bistable structure when L is small provided that the homogenized system is of the bistable type. This will be addressed in a future paper [[12]]. In this work, we will provide some other sufficient conditions for the bistable structure (see Theorem 1.5 below).

For clarity, we summarize some known results concerning the existence and qualitative properties of pulsating wave in the following theorem.

Theorem 1.1

Let (H1)–(H2) hold. For each $e\in \{\pm 1\}$ , the following statements hold true:

• If (H3) is satisfied, then system (1.1) admits a pulsating wave connecting (1, 0) and (0, 1) in the direction e.
• The speed c(e) of pulsating wave is unique.
• If there exists a pulsating wave (U, V)(t, x) with speed $c(e)\ne 0$ , then (U, V) is unique up to time shifts, and U(t, x) is increasing (resp. decreasing) in $t\in \mathbb{R}$ while V(t, x) is decreasing (resp. increasing) in $t\in \mathbb{R}$ if $c(e)>0$ (resp. $c(e)<0$ ). Furthermore, (U, V) is globally asymptotic stable in the sense that for any $(u_0,v_0)\in C(\mathbb{R};{[0,1]}^2)$ satisfying
• 1.5 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{lim\; sup}_{xe\to -\infty }{u}_0(x)\ge 1-\delta ,{lim\; inf}_{xe\to +\infty }{u}_0(x)\le \delta ,\hfill \\ {lim\; sup}_{xe\to +\infty }{v}_0(x)\ge 1-\delta ,{lim\; inf}_{xe\to -\infty }{v}_0(x)\le \delta ,\hfill \end{array}\right)\end{array}$$

Graph

• for some small $\delta >0$ , there exists $t_{\ast }\in \mathbb{R}$ such that
• $$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\underset{x\in \mathbb{R}}{sup}\left(|u(t,x)-U,(t+t_{\ast },x),|+|v,(t,x),-,V,(t+t_{\ast },x),|\right)=0,\end{array}$$

Graph

• where (u, v)(t, x) is the solution of the Cauchy problem of (1.1) with initial data $(u_0,v_0)$ .

We refer to [[14], [21], [35]] for the existence of pulsating waves. The proofs share the same idea in the sense that they transform system (1.1) into a cooperation system, and then apply the abstract theory on the existence of bistable waves developed in [[17]] for monotone semiflows. The uniqueness of waves speeds, and the monotonicity, uniqueness and global stability of non-stationary waves were proved in [[14]] which handled a more general model with the presence of advection term.

In this paper, we are particularly interested in the sign of wave speeds. As a matter of fact, this problem has great biological significance as it decides which species eventually wins the competition. More precisely, given a direction $e\in \{\pm 1\}$ , it is known from Theorem 1.1 (iii) that if $c(e)>0$ , then the species u will be the winner and the species v will go to extinction eventually provided that the initial population density satisfies (1.5). On the other hand, if $c(e)<0$ , the outcome is then contrary to the above.

Let us first mention an important case where the functions $k_1$ , $k_2$ are two positive constants, and the functions $f_1$ , $f_2$ do not depend on x, and satisfy the assumption (H1) and $0<f_i^{\prime }(0)<k_i$ for $i=1,2$ . It is well known [[8], [20]] that, for this homogeneous equation

1.6 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}u=d_1{\partial }_{\mathit{xx}}u+f_1(u)-k_{1}uv,\hfill \\ {\partial }_{t}v=d_2{\partial }_{\mathit{xx}}v+f_2(v)-k_{2}uv,\hfill \end{array}\right)\end{array}$$

Graph

there exists a unique speed c and a traveling wave $(u,v)(t,x)=({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)(xe-ct)$ such that $0<{\mathrm{\Phi }}_i<1$ in $\mathbb{R}$ for $i=1,2$ , and $({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)(-\infty )=(1,0)$ , $({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)(+\infty )=(0,1)$ . The front $({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)$ is unique up to shifts, globally asymptotic stable and ${\mathrm{\Phi }}_1(·)$ , $-{\mathrm{\Phi }}_2(·)$ are decreasing functions (no matter whether the wave is stationary or not). Moreover, since system (1.6) is invariant under the spatial reflection $x\to -x$ , the front $({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)$ and the speed c are independent of the direction $e\in \{\pm 1\}$ . The sign of c has been studied in the case where $f_i(s)=a_{i}s(1-s)$ with $0<a_i<k_i$ for $i=1,2$ . More precisely, when $d_1=d_2$ , it is known from [[23], Theorem 1.1] (see also [[24]]) that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}c>0\hfill & {\displaystyle \text{if}{a}_1\ge a_2,k_1\le k_2,\text{and}\frac{a_1}{k_1}\ne \frac{a_2}{k_2};}\hfill \\ c<0\hfill & {\displaystyle \text{if}{a}_1\le a_2,k_1\ge k_2,\text{and}\frac{a_1}{k_1}\ne \frac{a_2}{k_2};}\hfill \\ c=0\hfill & \text{if}{a}_1=a_2,k_1=k_2.\hfill \end{array}\right)\end{array}$$

Graph

A key ingredient in the proof is the monotone dependence of the speed with respect to the coefficients, that is, c is increasing in $a_1$ and $k_2$ , while it is decreasing in $a_2$ and $k_1$ [[26]]. Therefore, the case when $d_1=d_2$ is completely understood. However, when $d_1\ne d_2$ , the situation is much more complicated. Under various restrictions on the coefficients $d_i$ , $a_i$ and $k_i$ , some criteria about the sign of the speed were obtained in [[23]] by using integration arguments, and in [[30]] by the method of sub and supersolutions. The results in [[30]] were also extended to the spatially periodic case recently [[35]]. Yet, the restrictions provided by [[35]] are not easy to check, as they involve the quantities of the principal eigenvalues of some linear operators in the space $C_{\mathit{per}}(\mathbb{R};\mathbb{R})$ and the corresponding positive eigenfunctions.

In this work, we focus our attention on the case where $d_1=d_2$ , and investigate the sign of wave speeds by comparing the reactions $f_i(x,s)$ and the competition rates $k_i(x)$ , $i=1,2$ . It turns out that even in this simple case, the presence of spatial periodicity makes the problem significantly more difficult than in the spatially homogeneous case. We expect that some new phenomena will occur, inspired by the larger complexity of the dynamics in periodic media than in homogeneous media for the following scalar equation

1.7 $$\begin{array}{c}\hfill {\partial }_{t}u=d{\partial }_{\mathit{xx}}u+f(x,u)\text{in}\mathbb{R}\times \mathbb{R},\end{array}$$

Graph

where $d>0$ , $f(x,·)$ is L-periodic in x, and $f(x,0)\equiv f(x,1)\equiv 0$ . Indeed, it is known from [[11], [15]–[17]] that similar existence, uniqueness and stability results to Theorem 1.1 hold for equation (1.7) provided that it admits a bistable structure in the sense that 0 and 1 are two linearly stable steady states and any L-periodic steady states strictly between 0 and 1 are linearly unstable. On the other hand, in sharp contrast with the results in homogeneous media (see [[2]]), the stationary pulsating waves of (1.7) may be not unique up to L-periodic shifts [[10]], and the wave speeds in two opposite directions may be different [[9]]. We will show that, for our competition model (1.1), similar phenomena will be observed in the spatially periodic media. It should be pointed out that the speed of non-stationary pulsating wave of (1.7) has the same sign as that of ${\int }_0^1{\int }_0^{L}f(x,u)dxdu$ [[11], [15]], while there is no such characterization for our competition model. Therefore, it is more challenging to handle system (1.1).

Now, we list the main results of our paper. As mentioned above, we only consider the case where $d_1=d_2$ . By rescaling the variables, we may assume without loss of generality that

$$\begin{array}{c}\hfill d_1=d_2=1,\end{array}$$

Graph

and for convenience, we drop the notations $d_1$ and $d_2$ in (1.1). We begin with the following observation.

Proposition 1.2

Let (H1) and (H2) hold. Assume that for each $e\in \{\pm 1\}$ , system (2.1) admits a pulsating wave connecting (1, 0) and (0, 1) with speed c(e). Then, the following statements hold true:

• If $k_1(x)\le k_2(-x)$ and $f_1(x,·)\ge f_2(-x,·)$ in [0, 1] for all $x\in \mathbb{R}$ , then $c(\pm 1)\ge 0$ ;
• If $k_1(x)\ge k_2(-x)$ and $f_1(x,·)\le f_2(-x,·)$ in [0, 1] for all $x\in \mathbb{R}$ , then $c(\pm 1)\le 0$ .

Consequently, if $f_1(x,·)\equiv f_2(-x,·)$ in [0, 1] and $k_1(x)\equiv k_2(-x)$ , then $c(\pm 1)=0$ .

The above proposition partially extends [[23], Theorem 1.1] to the spatially periodic case. Yet, it does not provide any information about when the wave speed is nonzero. This problem is addressed in the following theorem under some stronger conditions.

Theorem 1.3

Let all the assumptions in Proposition 1.2 hold. Assume further that there exist a positive constant $k_{\ast }$ and a $C^{1,1}$ function $f_{\ast }:\mathbb{R}\to \mathbb{R}$ , $u\mapsto f_{\ast }(u)$ satisfying (H1) (that is, $f_{\ast }(s)/s$ is decreasing in $s>0$ and the problem $du/dt=f_{\ast }(u)-k_{\ast }uv$ , $dv/dt=f_{\ast }(v)-k_{\ast }uv$ has a unique positive equilibrium) such that

1.8 $$\begin{array}{c}\hfill k_1(x)\le k_{\ast }\le k_2(x)\text{and}{f}_1(x,·)\ge f_{\ast }(·)\ge f_2(x,·)\text{for all}x\in \mathbb{R},\end{array}$$

Graph

and that $0<f_{\ast }^{\prime }(0)<k_{\ast }$ . Then, $c(\pm 1)>0$ provided that one of the following conditions holds:

• $k_1(x)$ is not identically equal to $k_2(x)$ ;
• There exists $x_{\ast }\in \mathbb{R}$ such that $f_1(x_{\ast },s)>f_2(x_{\ast },s)$ for all $s\in (0,1)$ ;
• There exists $u_{\ast }\in (0,1)$ such that $f_1(x,u_{\ast })>f_2(x,u_{\ast })$ for all $x\in \mathbb{R}$ .

Clearly, by permuting the roles of $f_1(x,u)$ and $f_2(x,u)$ and those of $k_1(x)$ and $k_2(x)$ in the above theorem, one obtains $c(\pm 1)<0$ .

Theorem 1.3 is proved by constructing a suitable subsolution of (1.1) which forces the species to propagate with a positive speed (see Sect. 3 below). Notice that if $k_1(x)\equiv k_2(x)\equiv k_{\ast }$ and $f_1(x,·)\equiv f_2(x,·)\equiv f_{\ast }(·)$ in [0, 1], then (1.1) is reduced to a spatially homogeneous system, and Proposition 1.2 in particular implies that it admits a stationary traveling wave. The extra conditions listed in (a)–(c) are used to exclude this possibility. It is natural to ask whether the condition (b) or (c) can be relaxed by the assumption that $f_1(x,s)$ is not identically equal to $f_2(x,s)$ in $\mathbb{R}\times [0,1]$ . In the special case where system (1.1) is of the Lotka–Volterra type (1.2), the answer is affirmative, as stated in the following corollary.

Corollary 1.4

Let (H2) hold for system (1.2). Assume that for any $e\in \{\pm 1\}$ , system (1.2) admits a pulsating wave connecting (1, 0) and (0, 1) with speed c(e). Assume further that there exist two positive constants $0<a_{\ast }<k_{\ast }$ such that

1.9 $$\begin{array}{c}\hfill k_1(x)\le k_{\ast }\le k_2(x)\text{and}{a}_1(x)\ge a_{\ast }\ge a_2(x)\text{for all}x\in \mathbb{R}.\end{array}$$

Graph

Then, $c(\pm 1)>0$ provided that either $k_1(x)\not\equiv k_2(x)$ or $a_1(x)\not\equiv a_2(x)$ in $\mathbb{R}$ .

On the other hand, for general reactions $f_i(x,s)$ satisfying (H1), the situation becomes more complicated. Actually, the assumption that $f_1(x,s)\not\equiv f_2(x,s)$ in $\mathbb{R}\times [0,1]$ cannot guarantee that the wave speeds are nonzero simultaneously in two opposite directions. The possibility that the speed is nonzero in one direction while it is zero in the opposite direction may happen, as the following theorem shows.

Theorem 1.5

Assume that $k_1=k_2$ is a positive constant. There exist functions $f_1(x,s)$ (L-periodic in x) and $f_2(s)$ (x-independent) satisfying (H1) and $f_1(x,s)\ge ,\not\equiv f_2(s)$ for $x\in \mathbb{R}$ , $s\in [0,1]$ , such that system (1.1) is of the bistable type in the sense that (H2)–(H3) hold, and that

$$\begin{array}{c}\hfill c(1)=0<c(-1),\end{array}$$

Graph

where $c(\pm 1)$ are the pulsating wave speeds of (1.1).

The above theorem in particular implies that the wave speeds in two opposite directions can be different, which means that, due to the spatial heterogeneity, the shape of propagation may be not symmetric.

From the ecological point of view, Theorem 1.5 also describes some new and interesting phenomena: The conclusion $c(1)=0<c(-1)$ means that the species u can invade from right to left successfully and then will be the winner of the competition; On the other hand, u cannot invade from left to right. In the case where the advection exists, such a result can be imagined easily, but what we consider here is the case without the presence of advection term. Furthermore, the conditions $k_1=k_2,d_1=d_2$ represent that u and v have the same abilities of dispersal and competition, and the condition $f_1(x,s)\ge ,\not\equiv f_2(s)$ means that the growth rate of u is larger than that of v. Intuitively, the species u can be viewed as a stronger competitor, and it is expected u will win the competition in the sense of propagation eventually. However, Theorem 1.5 shows that the propagation of u from left to right fails. In earlier works on scale reaction–diffusion equations in continuous or discrete media, it is found that the propagation failure (also called wave-block phenomena) can be caused by the small diffusion [[28]], or by the large variation of the diffusion [[36]], or by the slow spatial oscillation of the reaction [[6], [10]]. Theorem 1.5 implies that the propagation failure can appear in the general competition systems, more exactly, it can appear only in one direction.

It is easily checked that if $f_1(x,·)$ is replaced by $f_1(-x,·)$ , then one reaches a situation where $c(1)>0=c(-1)$ . Moreover, similarly as above, by permuting the roles of $f_1$ and $f_2$ , one can construct examples such that the wave speed is negative in one direction while it is zero in the opposite direction.

Theorem 1.5 is proved by perturbing a homogeneous competition system in the u-component by a non-symmetric periodic function. Similar perturbation idea has been used for the scalar equation (1.7) in [[9]]. Indeed, by adapting the strategy in [[9]], it is also possible to construct examples such that $c(\pm 1)$ are simultaneously non-positive or non-negative, but they are dramatically different (i.e., $c(1)/c(-1)$ can be any non-negative number). As the main issue of this paper is concerned with the sign of wave speeds, this general result is not achieved here. Finally, we point out that the problem whether there exists a possibility such that $c(1)/c(-1)$ is a negative number is still unclear. This possibility does not happen in the scalar case (1.7) (see [[9], [11]]). However, the answer for the competition model (1.1) in spatially periodic media is far from clear.

Let us continue to consider the case where $k_1=k_2$ is a positive constant. In view of Proposition 1.2 and [[23], Theorem 1.1], it is natural to ask whether the speeds $c(\pm 1)$ have a strict sign if $f_1(x,·)-f_2(-x,·)$ has a fixed strict sign in (0, 1) for all $x\in \mathbb{R}$ . Such a result does not hold in general. Actually, once the condition on $f_i$ in (1.8) is violated, the slowly oscillating media will lead to propagation failure in both directions, as stated in the following theorem.

Theorem 1.6

Let $k_1=k_2$ be a positive constant. Assume, in addition to (H1), that the functions ${\partial }_s{f}_i(·,0)$ , ${\partial }_s{f}_i(·,1)$ are constants, that $0<{\partial }_u{f}_i(·,0)<k_i$ , and that there exist $\overline{x}$ , $\underline{x}$ in $\mathbb{R}$ such that

1.10 $$\begin{array}{c}\hfill f_1(\overline{x},·)\le ,\not\equiv f_2(\overline{x},·)\text{and}{f}_1(\underline{x},·)\ge ,\not\equiv f_2(\underline{x},·)\text{in}[0,1].\end{array}$$

Graph

Then, there exists $l_{\ast }>0$ such that for any $l>l_{\ast }$ and any $e\in \{\pm 1\}$ , the following system

1.11 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}u={\partial }_{\mathit{xx}}u+f_1(x/l,u)-k_{1}uv,\hfill \\ {\partial }_{t}v={\partial }_{\mathit{xx}}v+f_2(x/l,u)-k_{2}uv,\hfill \end{array}\right)\end{array}$$

Graph

admits two stationary pulsating waves (U, V)(x) and $(\tilde{U},\tilde{V})(x)$ connecting (1, 0) and (0, 1) in the direction e such that (U, V) is not identically equal to $(\tilde{U},\tilde{V})$ up to any lL-periodic shift.

Clearly, for any $l>0$ , any lL-periodic shift of a stationary pulsating wave of (1.11) remains a stationary wave. Theorem 1.6 implies that when l is large, the stationary pulsating waves of (1.11) are not unique up to lL-periodic shifts, which is in contrast with the uniqueness of non-stationary pulsating waves stated in Theorem 1.1 (iii). Moreover, it is known from the uniqueness of wave speeds (Theorem 1.1 (ii)) that, under the assumptions of Theorem 1.6, system (1.11) admits no pulsating wave with nonzero speed for all large l. Let us also mention that the reason of the existence of stationary pulsating waves in the above theorem is different from that of Theorem 1.1 (i). Indeed, since l is large, the media oscillate very slowly, and hence, they are nearly locally homogeneous (see [[10], Theorem 1.7]). The stationary pulsating waves are then obtained by constructing suitable super and subsolutions which makes use of some homogeneous traveling waves, despite whether (1.11) admits a bistable structure is unknown.

Outline of this paper. In Sect. 2, we transform system (1.1) into a cooperation–diffusion system and recall the comparison principle for cooperation systems. This section contains the proof of Proposition 1.2. Section 3 is devoted to the proofs of Theorems 1.3 and 1.5. Part of these proofs shares the same arguments. In Sect. 4, we give the proof of Theorem 1.6 on the existence of multiple stationary pulsating waves.

Preliminaries

In this section, we assume that (H1) and (H2) hold for system (1.1), and construct a pair of sub and supersolutions of (1.1) which will play an important role in showing our main results. At the end of the section, we will give the proof of Proposition 1.2.

For the convenience of using the comparison arguments, let us first transform the competition–diffusion system (1.1) into a cooperation–diffusion system. More precisely, by a change of variables $\tilde{u}=u$ and $\tilde{v}=1-v$ , and dropping the tilde for convenience, we obtain

2.1 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{t}u={\partial }_{\mathit{xx}}u+f_1(x,u)-k_1(x)u(1-v)\hfill & \text{in}(0,\infty )\times \mathbb{R},\hfill \\ {\partial }_{t}v={\partial }_{\mathit{xx}}v-f_2(x,1-v)+k_2(x)u(1-v)\hfill & \text{in}(0,\infty )\times \mathbb{R}.\hfill \end{array}\right)\end{array}$$

Graph

Cleary, the constant steady states (0, 0), (1, 0) and (0, 1), become (0, 1),

$$\begin{array}{c}\hfill {\mathbf{e}}_1:=(1,1)\text{and}{\mathbf{e}}_0:=(0,0),\end{array}$$

Graph

respectively, and ${\mathbf{e}}_1$ , ${\mathbf{e}}_0$ are two linearly stable steady states of (2.1) in the space $C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ . It is also easily seen that, for any $e\in \{\pm 1\}$ , (U, V) is a pulsating wave of (1.1) connecting (1, 0) and (0, 1) with speed c(e) if and only if $(U,1-V)$ is a pulsating wave of (2.1) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ in the same direction with the same speed.

We define an order relation in $C(\mathbb{R};{\mathbb{R}}^2)$ by $(u_1,v_1)\ge (u_2,v_2)$ if $u_1(x)\ge u_2(x)$ and $v_2(x)\ge v_1(x)$ for all $x\in \mathbb{R}$ . We say $(u_1,v_1)>(u_2,v_2)$ if $(u_1,v_1)\ge (u_2,v_2)$ but $(u_1,v_1)$ is not identically equal to $(u_2,v_2)$ ; and $(u_1,v_1)\gg (u_2,v_2)$ if $u_1(x)>u_2(x)$ and $v_2(x)>v_1(x)$ for all $x\in \mathbb{R}$ .

Definition 2.1

A pair of functions $(u,v)\in C^{1,2}((0,\infty )\times \mathbb{R};{\mathbb{R}}^2)$ is said to be a supersolution (resp. subsolution) of (2.1) if

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{t}u-{\partial }_{\mathit{xx}}u-f_1(x,u)+k_1(x)u(1-v)\ge 0(resp.\le 0)\hfill & \text{in}(0,\infty )\times \mathbb{R},\hfill \\ {\partial }_{t}v-{\partial }_{\mathit{xx}}v+f_2(x,1-v)-k_2(x)u(1-v)\ge 0(resp.\le 0)\hfill & \text{in}(0,\infty )\times \mathbb{R}.\hfill \end{array}\right)\end{array}$$

Graph

Moreover, a supersolution or a subsolution is said to be strict if it is not a solution.

The following lemma follows from the comparison principle for cooperation systems.

Lemma 2.2

Let $(u^{+},v^{+})$ (resp. $(u^{-},v^{-})$ ) be a supersolution (resp. subsolution) of (2.1) with initial data $(u_0^{+},v_0^{+})\in C(\mathbb{R};{\mathbb{R}}^2)$ (resp. $(u_0^{-},v_0^{-})\in C(\mathbb{R};{\mathbb{R}}^2)$ ). Suppose that either ${\mathbf{e}}_1\ge (u^{+},v^{+})(t,·)\ge {\mathbf{e}}_0$ or ${\mathbf{e}}_1\ge (u^{-},v^{-})(t,·)\ge {\mathbf{e}}_0$ holds for all $t\ge 0$ . Then, the following statements hold true:

• If $(u_0^{+},v_0^{+})\ge (u_0^{-},v_0^{-})$ , then $(u^{+},v^{+})(t,·)\ge (u^{-},v^{-})(t,·)$ for any $t>0$ ;
• If ${\mathbf{e}}_1\gg (u_0^{+},v_0^{+})>(u_0^{-},v_0^{-})\gg {\mathbf{e}}_0$ , then $(u^{+},v^{+})(t,·)\gg (u^{-},v^{-})(t,·)$ for any $t>0$ .

Proof

Without loss of generality, we assume that ${\mathbf{e}}_1\ge (u^{+},v^{+})(t,·)\ge {\mathbf{e}}_0$ for all $t\ge 0$ , since the other case can be treated similarly. Define

$$\begin{array}{cc}\hfill w_1(t,x)=& u^{+}(t,x)-u^{-}(t,x)\text{and}{w}_2(t,x)\hfill \\ \hfill =& v^{+}(t,x)-v^{-}(t,x)\text{for}t\ge 0,x\in \mathbb{R}.\hfill \end{array}$$

Graph

It is easily checked that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {\partial }_t{w}_1\ge {\partial }_{\mathit{xx}}{w}_1+\left({\int }_0^1,{\partial }_s,f_1,(x,u^{-}+\eta (u^{+}-u^{-})),d,\eta ,-,k_1,(1-v^{-})\right)w_1+k_1{u}^{+}{w}_2,}\hfill \\ {\displaystyle {\partial }_t{w}_2\ge {\partial }_{\mathit{xx}}{w}_2+\left({\int }_0^1,{\partial }_s,f_2,(x,v^{-}+\eta (v^{-}-v^{+})),d,\eta ,-,k_2,u^{-}\right)w_2+k_2(1-v^{+})w_1,}\hfill \end{array}\right)\end{array}$$

Graph

in $(0,\infty )\times \mathbb{R}$ . Since $k_1{u}^{+}\ge 0$ and $k_2(1-v^{+})\ge 0$ , statement (i) follows directly from the standard comparison principle for cooperation systems (see e.g., [[33]]).

Now, we turn to prove statement (ii). Since ${\mathbf{e}}_1\gg (u_0^{+},v_0^{+})>(u_0^{-},v_0^{-})\gg {\mathbf{e}}_0$ , by statement (i) and the strong maximum principle for cooperation systems (see, e.g., [[33], Chapter 3, Theorem 13]), we have

$$\begin{array}{c}\hfill (u^{+},v^{+})(t,·)>(u^{-},v^{-})(t,·),(u^{+},v^{+})(t,·)\gg {\mathbf{e}}_0,\text{and}{\mathbf{e}}_1\gg (u^{-},v^{-})(t,·)\end{array}$$

Graph

for all $t>0$ . Next, we claim that $u^{+}>u^{-}$ in $(0,\infty )\times \mathbb{R}$ . Assume by contradiction that there exists $(t_0,x_0)\in (0,\infty )\times \mathbb{R}$ such that $u^{+}(t_0,x_0)=u^{-}(t_0,x_0)$ . By the strong maximum principle for cooperation systems again, we have $u^{+}(t,·)\equiv u^{-}(t,·)>0$ for all $0\le t\le t_0$ . One then infers that $v^{+}(t,x)>v^{-}(t,x)$ for all $0<t\le t_0$ , $x\in \mathbb{R}$ (otherwise, by using the strong maximum principle again, one would get that $(u^{+},v^{+})$ is identically equal to $(u^{-},v^{-})$ , which would be impossible). This implies that $u^{+}(t,x)(1-v^{+}(t,x))<u^{-}(t,x)(1-v^{-}(t,x))$ in $(0,t_0)\times \mathbb{R}$ . Remember that $w_1(t,x)=u^{+}(t,x)-u^{-}(t,x)$ . It then follows that

$$\begin{array}{c}\hfill \left(\begin{array}{cc}{\partial }_t{w}_1\hfill & {\displaystyle \ge {\partial }_{\mathit{xx}}{w}_1+\left({\int }_0^1,{\partial }_s,f_1,(x,u^{-}+\eta (u^{+}-u^{-})),d,\eta \right)}\hfill \\ \hfill & w_1-(k_1{u}^{+}(1-v^{+})-k_1{u}^{-}(1-v^{-}))\hfill \\ \hfill & {\displaystyle >{\partial }_{\mathit{xx}}{w}_1+\left({\int }_0^1,{\partial }_s,f_1,(x,u^{-}+\eta (u^{+}-u^{-})),d,\eta \right)w_1}\hfill \end{array}\right)\end{array}$$

Graph

in $(0,t_0)\times \mathbb{R}$ , which is a contradiction with $w_1\equiv 0$ . Thus, we have $u^{+}(t,x)>u^{-}(t,x)$ for all $t>0$ , $x\in \mathbb{R}$ . Proceeding similarly as above, we can conclude that $v^{+}>v^{-}$ in $(0,\infty )\times \mathbb{R}$ . This completes the proof of Lemma 2.2. $\square$

To present our sub and supersolutions of (2.1), we need to introduce a few more notations. Recall from (H2) that for $i=1,2$ , ${\lambda }_{\ast ,i}^{-}<0$ is the principal eigenvalue of (1.3). Let ${\lambda }_{\ast ,i}^{+}$ denote the principal eigenvalue of the following problem

2.2 $$\begin{array}{c}\hfill {\psi }^{\prime \prime }+{\partial }_s{f}_i(x,1)\psi =\lambda \psi \text{in}\mathbb{R},\psi >0\text{in}\mathbb{R},\psi \text{is}\text{L-periodic.}\end{array}$$

Graph

Since $f_i(s,x)/s$ is decreasing in $s>0$ and $f_i(x,1)=0$ , it is straightforward to check that ${\partial }_s{f}_i(x,1)<0$ for $x\in \mathbb{R}$ , and hence, ${\lambda }_{\ast ,i}^{+}<0$ . Denote by ${\psi }_i^{\pm }$ the positive periodic eigenfunctions with respect to ${\lambda }_{\ast ,i}^{\pm }$ , respectively. Since ${\psi }_i^{\pm }$ are unique up to multiplication by a positive constant, for definiteness and later use, we normalize them in a way such that

2.3 $$\begin{array}{c}\hfill \Vert {\psi }_1^{+}\Vert =1,\Vert {\psi }_2^{+}\Vert =1,∥\frac{k_1{\psi }_2^{-}}{{\psi }_1^{+}}∥=-\frac{{\lambda }_{\ast ,1}^{+}}{4},\text{and}∥\frac{k_2{\psi }_1^{-}}{{\psi }_2^{+}}∥=-\frac{{\lambda }_{\ast ,2}^{+}}{4},\\ \hfill \end{array}$$

Graph

where $\Vert ·\Vert$ denotes the $L^{\infty }$ -norm in $C(\mathbb{R})$ . Let $\rho \in C^2(\mathbb{R})$ be a nonnegative function satisfying

2.4 $$\begin{array}{c}\hfill \rho (\xi )=0\text{in}[2,\infty ),\rho (\xi )=1\text{in}(-\infty ,0],-1\le {\rho }^{\prime }(\xi )\le 0\text{and}|{\rho }^{\prime \prime }(\xi )|\le 1\text{in}\mathbb{R}.\\ \hfill \end{array}$$

Graph

In our discussion below, the direction $e\in \{\pm 1\}$ is fixed, and thus, by a pulsating wave of (2.1), we always mean one which connects ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ , and moves in the direction e. For convenience, we denote $c:=c(e)$ .

Lemma 2.3

Let (U, V)(t, x) be a pulsating wave of (2.1) with speed $c\ne 0$ . There exist ${\epsilon }_0>0$ , $\mu >0$ and $K\in \mathbb{R}$ (which has the sign of c) such that for any $\epsilon \in (0,{\epsilon }_0]$ and $m_0\in \mathbb{Z}$ , the pair of functions $(u^{+},v^{+})(t,x)$ (resp. $(u^{-},v^{-})(t,x)$ ) defined by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\pm }(t,x)=U(t\pm \epsilon K(1-{\mathrm{e}}^{-\mu t}),x+m_{0}L)\pm \epsilon {\mathrm{e}}^{-\mu t}\left(\rho ,(xe-ct),{\psi }_1^{+},(x),+,(1-\rho (xe-ct)),{\psi }_1^{-},(x)\right)\hfill \\ v^{\pm }(t,x)=V(t\pm \epsilon K(1-{\mathrm{e}}^{-\mu t}),x+m_{0}L)\pm \epsilon {\mathrm{e}}^{-\mu t}\left(\rho ,(xe-ct),{\psi }_2^{-},(x),+,(1-\rho (xe-ct)),{\psi }_2^{+},(x)\right)\hfill \end{array}\right)\end{array}$$

Graph

is a supersolution (resp. subsolution) of (2.1).

We point out that, similar super and subsolutions were constructed to show the globally asymptotic stability of traveling waves of the bistable Lotka–Volterra competition systems in various situations (see e.g., [[3], [14], [20], [38]]). In the spatially periodic case, an additional assumption was imposed in [[14]] to ensure that the linearized operators of the elliptic part of system (2.1) at ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ have a principal eigenvalue with positive eigenfunction in $C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ , respectively. Such an assumption is not needed in our proof below.

Proof

We only prove that $(u^{-},v^{-})$ is a subsolution of (2.1), since the verification of the supersolution part is analogous. Without loss of generality, we assume that $c>0$ (as we will sketch at the end of the proof, the case where $c<0$ can be treated similarly). It follows from Theorem 1.1 that U(t, x) and V(t, x) are increasing in $t\in \mathbb{R}$ .

Let us first set some notations. Since for each $i=1,2$ , the function $s\mapsto f_i(x,s)$ is of class $C^{1,1}$ uniformly in $x\in \mathbb{R}$ , there exists a small constant ${\delta }_0\in (0,1/4)$ such that

2.5 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle 0<{\delta }_0\le min\left\{\frac{-{\lambda }_{\ast ,1}^{-}}{8\Vert k_1\Vert },,,,,\frac{-{\lambda }_{\ast ,2}^{-}}{8\Vert k_2\Vert },,,,,\frac{-{\lambda }_{\ast ,1}^{-}}{8},{∥\frac{k_1{\psi }_2^{+}}{{\psi }_1^{-}}∥}^{-1},,,,,\frac{-{\lambda }_{\ast ,2}^{-}}{8},{∥\frac{k_2{\psi }_1^{+}}{{\psi }_2^{-}}∥}^{-1}\right\},}\hfill \\ {\displaystyle {sup}_{x\in \mathbb{R},s\in (-2{\delta }_0,2{\delta }_0)}\left|{\partial }_s,f_i,(x,0),-,{\partial }_s,f_i,(x,s)\right|\le -\frac{{\lambda }_{\ast ,i}^{-}}{4}\text{for}i=1,2,}\hfill \\ {\displaystyle {sup}_{x\in \mathbb{R},s\in (1-2{\delta }_0,1+2{\delta }_0)}\left|{\partial }_s,f_i,(x,1),-,{\partial }_s,f_i,(x,s)\right|\le -\frac{{\lambda }_{\ast ,i}^{+}}{4}\text{for}i=1,2.}\hfill \end{array}\right)\end{array}$$

Graph

Let $m_0\in \mathbb{Z}$ be arbitrarily fixed. By the definition of pulsating waves, there exists $M>3+|m_0|L$ sufficiently large such that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}0<U(t,x),V(t,x)<{\delta }_0,\hfill & \text{for}\text{all}xe-ct\ge M,\hfill \\ 1-{\delta }_0<U(t,x),V(t,x)<1,\hfill & \text{for}\text{all}xe-ct\le -M,\hfill \\ {\delta }_0/2<U(t,x),V(t,x)<1-{\delta }_0/2\hfill & \text{for}\text{all}-M<xe-ct<-M.\hfill \end{array}\right)\end{array}$$

Graph

Set

$$\begin{array}{c}\hfill \mu =\underset{i=1,2}{min}\left\{-,\frac{{\lambda }_{\ast ,i}^{+}}{2},,,,-,\frac{{\lambda }_{\ast ,i}^{-}}{2}\right\},\end{array}$$

Graph

and

$$\begin{array}{c}\hfill K=\frac{A+B+(\mu +\Vert k_1+k_2\Vert )\left(\Vert ,{\psi }_1^{+},+,{\psi }_1^{-},\Vert +\Vert ,{\psi }_2^{+},+,{\psi }_2^{-},\Vert \right)}{\beta \mu },\end{array}$$

Graph

where A and B and $\beta$ are positive constants given by

$$\begin{array}{cc}& \beta =min\left\{\underset{{\delta }_0/2\le U(t,x)\le 1-{\delta }_0/2}{min},{\partial }_t,U,(t,x),,,,,\underset{{\delta }_0/2\le V(t,x)\le 1-{\delta }_0/2}{min},{\partial }_t,V,(t,x)\right\},\hfill \\ \hfill & A=\underset{i=1,2}{max}\left\{(|c|,+,1)\Vert ,{\psi }_i^{+},-,{\psi }_i^{-},\Vert +2\Vert ,{({\psi }_i^{+})}^{\prime },-,{({\psi }_i^{-})}^{\prime },\Vert +\Vert ,{({\psi }_i^{+})}^{\prime \prime },+,{({\psi }_i^{-})}^{\prime \prime },\Vert \right\},\hfill \end{array}$$

Graph

and

$$\begin{array}{c}\hfill B=\underset{i=1,2,x\in \mathbb{R},s\in [-{\delta }_0,1+{\delta }_0]}{max}|{\partial }_s{f}_i(x,s)|\Vert {\psi }_i^{+}+{\psi }_i^{-}\Vert .\end{array}$$

Graph

Furthermore, we choose a small constant ${\epsilon }_0\in (0,{\delta }_0)$ such that

2.6 $$\begin{array}{c}\hfill 0<{\epsilon }_0\le min\left\{\frac{{\delta }_0}{\Vert {\psi }_1^{-}+{\psi }_1^{+}\Vert },,,,,\frac{{\delta }_0}{\Vert {\psi }_2^{-}+{\psi }_2^{+}\Vert },,,,,\frac{1}{\mathit{cK}}\right\}.\end{array}$$

Graph

Next, for $(t,x)\in (0,\infty )\times \mathbb{R}$ , we define

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{L}}_1{u}^{-}:={\partial }_t{u}^{-}-{\partial }_{\mathit{xx}}{u}^{-}-f_1(x,u^{-})+k_1(x)u^{-}(1-v^{-}),\hfill \\ {\mathcal{L}}_2{v}^{-}:={\partial }_t{v}^{-}-{\partial }_{\mathit{xx}}{v}^{-}+f_2(x,1-v^{-})-k_2(x)u^{-}(1-v^{-}).\hfill \end{array}\right)\end{array}$$

Graph

For the above $\mu >0$ , $K>0$ and ${\epsilon }_0>0$ , we will show that for any $0<\epsilon \le {\epsilon }_0$ , ${\mathcal{L}}_1{u}^{-}\le 0$ and ${\mathcal{L}}_2{v}^{-}\le 0$ in $(0,\infty )\times \mathbb{R}$ . According to Definition 2.1, this immediately implies that $(u^{-},v^{-})$ is a subsolution of (2.1). We will only show that ${\mathcal{L}}_1{u}^{-}\le 0$ , as the proof of ${\mathcal{L}}_2{v}^{-}\le 0$ is similar.

For any $0<\epsilon \le {\epsilon }_0$ , denote

2.7 $$\begin{array}{c}\hfill q(t)=\epsilon {\mathrm{e}}^{-\mu t}\text{and}\eta (t)=\epsilon K(1-{\mathrm{e}}^{-\mu t})\text{for}t>0.\end{array}$$

Graph

Since (U, V)(t, x) is an entire solution of (2.1) and since the functions $k_i$ and $f_i$ are t-independent and L-periodic in x, direct computations give that, for $(t,x)\in (0,\infty )\times \mathbb{R}$ ,

2.8 $$\begin{array}{c}\hfill \left(\begin{array}{cc}& {\mathcal{L}}_1{u}^{-}=-{\eta }^{\prime }{\partial }_{t}U-q^{\prime }(\rho {\psi }_1^{+}+(1-\rho ){\psi }_1^{-})+cq{\rho }^{\prime }({\psi }_1^{+}-{\psi }_1^{-})\hfill \\ \hfill & +q\left[{\rho }^{\prime \prime },({\psi }_1^{+}-{\psi }_1^{-}),+,2,{\rho }^{\prime },({({\psi }_1^{+})}^{\prime }-{({\psi }_1^{-})}^{\prime }),+,(\rho {({\psi }_1^{+})}^{\prime \prime }+(1-\rho ){({\psi }_1^{-})}^{\prime \prime })\right]\hfill \\ \hfill & -\left[f_1,(x,u^{-}),-,f_1,(x,U)\right]+\left[k_1,(x),u^{-},(1-v^{-}),-,k_1,(x),U,(1-V)\right],\hfill \end{array}\right)\\ \hfill \end{array}$$

Graph

where $\rho$ , ${\rho }^{\prime }$ and ${\rho }^{\prime \prime }$ are evaluated at $xe-ct$ , q, $q^{\prime }$ , $\eta$ and ${\eta }^{\prime }$ are evaluated at t, and ${\psi }_1^{\pm }$ , ${({\psi }_1^{\pm })}^{\prime }$ and ${({\psi }_1^{\pm })}^{\prime \prime }$ are evaluated at x. We will complete the proof of ${\mathcal{L}}_1{u}^{-}\le 0$ by considering three cases: (a) $xe+m_{0}Le-c(t-\eta (t))\le -M$ ; (b) $xe+m_{0}Le-c(t-\eta (t))\ge M$ ; (c) $-M<xe+m_{0}Le-c(t-\eta (t))<M$ .

In the first case, since $0<\epsilon cK\le 1$ (see (2.6)), we have $0\le c\eta (t)\le 1$ , whence $xe-ct\le -M+|m_{0}L|<0$ . Then, by our choice of $\rho$ , we have $\rho =1$ , ${\rho }^{\prime }={\rho }^{\prime \prime }=0$ . Thus,

$$\begin{array}{cc}& {\mathcal{L}}_1{u}^{-}=-{\eta }^{\prime }{\partial }_{t}U-q^{\prime }{\psi }_1^{+}+q{({\psi }_1^{+})}^{\prime \prime }\hfill \\ \hfill & +{\partial }_s{f}_1(x,U-{\theta }_{1}q{\psi }_1^{+})q{\psi }_1^{+}+k_{1}q\left[{\psi }_2^{-},U,-,{\psi }_1^{+},(1-V),-,q,{\psi }_2^{-},{\psi }_1^{+}\right],\hfill \end{array}$$

Graph

for some ${\theta }_1={\theta }_1(t,x)\in [0,1]$ . Noticing that ${\eta }^{\prime }(t)>0$ , that U is increasing in its first variable, and that ${\lambda }_{\ast ,1}^{+}$ is the principal eigenvalue of (2.2) with $i=1$ , we have

$$\begin{array}{cc}& {\mathcal{L}}_1{u}^{-}\le -q^{\prime }{\psi }_1^{+}+{\lambda }_{\ast ,1}^{+}q{\psi }_1^{+}-q{\psi }_1^{+}\hfill \\ \hfill & \left[{\partial }_s,f_1,(x,1),-,{\partial }_s,f_1,(x,U-{\theta }_{1}q{\psi }_1^{+})\right]+q(U-q{\psi }_1^{+})\frac{k_1{\psi }_2^{-}}{{\psi }_1^{+}}{\psi }_1^{+}.\hfill \end{array}$$

Graph

Since $U-{\theta }_{1}q{\psi }_1^{+}\in (1-2{\delta }_0,1)$ (remember that $\Vert {\psi }_1^{+}\Vert =1$ and $0\le q\le \epsilon <{\delta }_0$ ), by the third inequality of (2.5) and the normalization condition (2.3), it follows that

$$\begin{array}{c}\hfill {\mathcal{L}}_1{u}^{-}\le -q^{\prime }{\psi }_1^{+}+{\lambda }_{\ast ,1}^{+}q{\psi }_1^{+}-\frac{{\lambda }_{\ast ,1}^{+}}{4}q{\psi }_1^{+}-\frac{{\lambda }_{\ast ,1}^{+}}{4}q{\psi }_1^{+}=-q^{\prime }{\psi }_1^{+}+\frac{{\lambda }_{\ast ,1}^{+}}{2}q{\psi }_1^{+}.\end{array}$$

Graph

Furthermore, since $q(t)=\epsilon {\mathrm{e}}^{-\mu t}$ and since $0<\mu \le -{\lambda }_{\ast ,1}^{+}/2$ , we obtain that ${\mathcal{L}}_1{u}^{-}\le 0$ for all $(t,x)\in (0,\infty )\times \mathbb{R}$ such that $xe+m_{0}Le-c(t-\eta (t))\le -M$ .

Next, we consider the case where $xe+m_{0}Le-c(t-\eta (t))\ge M$ . Similarly as above, we have $0\le c\eta (t)\le 1$ , $xe-ct\ge M-c\eta (t)-|m_{0}L|>2$ , and hence, $\rho ={\rho }^{\prime }={\rho }^{\prime \prime }=0$ . It follows that

$$\begin{array}{c}\hfill \left(\begin{array}{cc}{\mathcal{L}}_1{u}^{-}\hfill & =-{\eta }^{\prime }{\partial }_{t}U-q^{\prime }{\psi }_1^{-}+q{({\psi }_1^{-})}^{\prime \prime }+{\partial }_s{f}_1(x,U-{\theta }_{2}q{\psi }_1^{-})q{\psi }_1^{-}\hfill \\ \hfill & +k_{1}q\left[{\psi }_2^{+},U,-,{\psi }_1^{-},(1-V),-,q,{\psi }_1^{-},{\psi }_2^{+}\right]\hfill \\ \hfill & \le -q^{\prime }{\psi }_1^{-}+{\lambda }_{\ast ,1}^{-}q{\psi }_1^{-}-q{\psi }_1^{-}\left[{\partial }_s,f_1,(x,0),-,{\partial }_s,f_1,(x,U-{\theta }_{2}q{\psi }_1^{-})\right]\hfill \\ \hfill & {\displaystyle +k_{1}q{\psi }_1^{-}V+q\left(U,-,q,{\psi }_1^{-}\right)\frac{k_1{\psi }_2^{+}}{{\psi }_1^{-}}{\psi }_1^{-}}\hfill \end{array}\right)\end{array}$$

Graph

for some ${\theta }_2={\theta }_2(t,x)\in [0,1]$ , where ${\lambda }_{\ast ,1}^{-}$ is the principal eigenvalue of (1.3) with $i=1$ . Notice that $U-{\theta }_{2}q{\psi }_1^{-}\in (-{\delta }_0,{\delta }_0)$ (since $0<q{\psi }_1^{-}\le \epsilon \Vert {\psi }_1^{-}\Vert \le {\delta }_0$ , due to (2.6)), and that $V\in (0,{\delta }_0)$ . It then follows from the first two inequalities of (2.5) that

$$\begin{array}{cc}& {\mathcal{L}}_1{u}^{-}\le -q^{\prime }{\psi }_1^{-}+{\lambda }_{\ast ,1}^{-}q{\psi }_1^{-}\hfill \\ \hfill & -\frac{{\lambda }_{\ast ,1}^{-}}{4}q{\psi }_1^{-}-\frac{{\lambda }_{\ast ,1}^{-}}{8}q{\psi }_1^{-}-\frac{{\lambda }_{\ast ,1}^{-}}{8}q{\psi }_1^{-}=-q^{\prime }{\psi }_1^{-}+\frac{{\lambda }_{\ast ,1}^{-}}{2}q{\psi }_1^{-}.\hfill \end{array}$$

Graph

Therefore, thanks to $0<\mu \le -{\lambda }_{\ast ,1}^{-}/2$ , we complete the proof of ${\mathcal{L}}_1{u}^{-}\le 0$ when $xe+m_{0}Le-c(t-\eta (t))\ge M$ .

It remains to consider the values where $-M<xe+m_{0}Le-c(t-\eta (t))<M$ . In this case, we have $U(t-\eta (t),x+m_{0}L),V(t-\eta (t),x+m_{0}L)\in [{\delta }_0/2,1-{\delta }_0/2]$ , whence by (2.6), $u^{-}(t,x),v^{-}(t,x)\in [-{\delta }_0/2,1-{\delta }_0/2]$ . By the choice of the constants A, B and the function $\rho \in C^2(\mathbb{R};[0,1])$ satisfying (2.4), we have

$$\begin{array}{cc}& cq{\rho }^{\prime }({\psi }_1^{+}-{\psi }_1^{-})+q\hfill \\ \hfill & \left[{\rho }^{\prime \prime },({\psi }_1^{+}-{\psi }_1^{-}),+,2,{\rho }^{\prime },({({\psi }_1^{+})}^{\prime }-{({\psi }_1^{-})}^{\prime }),+,(\rho {({\psi }_1^{+})}^{\prime \prime }+(1-\rho ){({\psi }_1^{-})}^{\prime \prime })\right]\le Aq,\hfill \end{array}$$

Graph

and

$$\begin{array}{c}\hfill f_1(x,U)-f_1(x,u^{-})={\partial }_s{f}_1\left(x,,,U,-,{\theta }_3,q,(\rho {\psi }_1^{+}+(1-\rho ){\psi }_1^{-})\right)q(\rho {\psi }_1^{+}+(1-\rho ){\psi }_1^{-})\le Bq,\end{array}$$

Graph

for some ${\theta }_3={\theta }_3(t,x)\in [0,1]$ . Moreover, it is straightforward to check that

$$\begin{array}{c}\hfill \left(k_1,u^{-},(1-v^{-}),-,k_1,U,(1-V)\right)\le k_{1}q{u}^{-}\left(\rho ,{\psi }_2^{-},+,(1-\rho ),{\psi }_2^{+}\right)\le k_{1}q\Vert {\psi }_2^{+}+{\psi }_2^{-}\Vert .\end{array}$$

Graph

Then, with the choice of the positive constants $\beta$ and K, it follows from (2.8) that

$$\begin{array}{c}\hfill \left(\begin{array}{cc}{\mathcal{L}}_1{u}^{-}\hfill & \le -\beta {\eta }^{\prime }-\Vert {\psi }_1^{+}+{\psi }_1^{-}\Vert q^{\prime }+Aq+Bq+k_{1}q\Vert {\psi }_2^{+}+{\psi }_2^{-}\Vert \hfill \\ \hfill & \le -\mu K\beta q+\left(\mu \Vert ,{\psi }_1^{+},+,{\psi }_1^{-},\Vert +\Vert ,k_1,\Vert \Vert ,{\psi }_2^{+},+,{\psi }_2^{-},\Vert \right)q+Aq+Bq\le 0,\hfill \end{array}\right)\end{array}$$

Graph

for all $(t,x)\in (0,\infty )\times \mathbb{R}$ such that $-M<xe+m_{0}Le-c(t-\eta (t))<M$ .

Combining the above, we immediately obtain that ${\mathcal{L}}_1{u}^{-}\le 0$ for all $(t,x)\in (0,\infty )\times \mathbb{R}$ . With similar arguments, we see that ${\mathcal{L}}_2{v}^{-}\le 0$ for all $(t,x)\in (0,\infty )\times \mathbb{R}$ . Therefore, $(u^{-},v^{-})$ is a subsolution of (2.1).

Finally, we consider the case where $c<0$ . Let ${\delta }_0$ , M, A and B be the positive constants defined above. Since $c<0$ , U(t, x) and V(t, x) are decreasing in $t\in \mathbb{R}$ , and hence,

$$\begin{array}{c}\hfill \tilde{\beta }:=min\left\{\underset{{\delta }_0/2\le U(t,x)\le 1-{\delta }_0/2}{min},-,{\partial }_t,U,(t,x),,,,,\underset{{\delta }_0/2\le V(t,x)\le 1-{\delta }_0/2}{min},-,{\partial }_t,V,(t,x)\right\}>0.\end{array}$$

Graph

Now, take a negative constant

$$\begin{array}{c}\hfill \tilde{K}=-\frac{A+B+(\mu +\Vert k_1+k_2\Vert )\left(\Vert ,{\psi }_1^{+},+,{\psi }_1^{-},\Vert +\Vert ,{\psi }_2^{+},+,{\psi }_2^{-},\Vert \right)}{\tilde{\beta }\mu },\end{array}$$

Graph

and let ${\epsilon }_0\in (0,{\delta }_0)$ be a small constant satisfying (2.6) with K replaced by $\tilde{K}$ . For any $0<\epsilon \le {\epsilon }_0$ , let q(t) and $\eta (t)$ be the functions given in (2.7) with K replaced by $\tilde{K}$ . Since $0<\epsilon c\tilde{K}\le 1$ , with the same reasoning as before, one can conclude that ${\mathcal{L}}_1{u}^{-}\le 0$ for $xe+m_{0}Le-c(t-\eta (t))\ge M$ or $xe+m_{0}Le-c(t-\eta (t))\le -M$ . Now, in the case where $-M<xe+m_{0}Le-c(t-\eta (t))<M$ , since ${\eta }^{\prime }(t)<0$ , it follows that $-{\eta }^{\prime }{\partial }_{t}U\le {\eta }^{\prime }\tilde{\beta }\le \mu \tilde{K}\tilde{\beta }q$ . Proceeding similarly as above, we have

$$\begin{array}{c}\hfill \left(\begin{array}{cc}{\mathcal{L}}_1{u}^{-}\hfill & \le -{\eta }^{\prime }{\partial }_{t}U-\Vert {\psi }_1^{+}+{\psi }_1^{-}\Vert q^{\prime }+Aq+Bq+k_{1}q\Vert {\psi }_2^{+}+{\psi }_2^{-}\Vert \hfill \\ \hfill & \le \tilde{\beta }\mu \tilde{K}q+\left(\mu \Vert ,{\psi }_1^{+},+,{\psi }_1^{-},\Vert +\Vert ,k_1,\Vert \Vert ,{\psi }_2^{+},+,{\psi }_2^{-},\Vert \right)q+Aq+Bq\le 0.\hfill \end{array}\right)\end{array}$$

Graph

As a consequence, we obtain that ${\mathcal{L}}_1{u}^{-}\le 0$ for all $(t,x)\in (0,\infty )\times \mathbb{R}$ . The verification of ${\mathcal{L}}_1{v}^{-}\le 0$ is similar.

Combining the above, we can conclude that $(u^{-},v^{-})$ is a subsolution of (2.1) whenever $c\ne 0$ , and that the constant $K\in \mathbb{R}$ has the sign of c. As mentioned earlier, the supersolution can be verified similarly. This completes the proof of Lemma 2.3. $\square$

As a consequence of Lemma 2.3, we have the following observation.

Lemma 2.4

Assume that (H1)–(H2) hold for the following system

2.9 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}u={\partial }_{\mathit{xx}}u+{\tilde{f}}_1(x,u)-{\tilde{k}}_1(x)u(1-v),\hfill \\ {\partial }_{t}v={\partial }_{\mathit{xx}}v-{\tilde{f}}_2(x,1-v)+{\tilde{k}}_2(x)u(1-v).\hfill \end{array}\right)\end{array}$$

Graph

Let (U, V) (resp. $(\tilde{U},\tilde{V})$ ) be the pulsating wave of (2.1) (resp. (2.9)) with speed c (resp. $\tilde{c}$ ) in the same direction e. If $({\tilde{k}}_1,{\tilde{k}}_2)\le (k_1,k_2)$ and $(f_1(·,s),f_2(·,s))\le ({\tilde{f}}_1(·,s),{\tilde{f}}_2(·,s))$ for all $s\in [0,1]$ , then $c\le \tilde{c}$ .

Proof

Assume by contradiction that $c>\tilde{c}$ . Then, either $c\ne 0$ or $\tilde{c}\ne 0$ . Without loss of generality, we assume that $c\ne 0$ , that is, the wave (U, V)(t, x) is not stationary (the other case can be treated similarly).

Let ${\epsilon }_0>0$ , $\mu >0$ and $K\in \mathbb{R}$ be the constants obtained in Lemma 2.3. Since (U, V) and $(\tilde{U},\tilde{V})$ are, respectively, the pulsating waves of (2.1) and (2.9) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ in the same direction, one finds some $m_0\in \mathbb{Z}$ such that for any $x\in \mathbb{R}$ ,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\tilde{U}(0,x)\ge U(0,x+m_{0}L)-{\epsilon }_0\left(\rho ,(xe),{\psi }_1^{+},(x),+,(1-\rho (xe)),{\psi }_1^{-},(x)\right),\hfill \\ \tilde{V}(0,x)\ge V(0,x+m_{0}L)-{\epsilon }_0\left(\rho ,(xe),{\psi }_2^{-},(x),+,(1-\rho (xe)),{\psi }_2^{+},(x)\right),\hfill \end{array}\right)\end{array}$$

Graph

where $\rho (·)\in C^2(\mathbb{R};[0,1])$ satisfies (2.4). Since $({\tilde{k}}_1,{\tilde{k}}_2)\le (k_1,k_2)$ and $(f_1(·,s),f_2(·,s))\le ({\tilde{f}}_1(·,s),{\tilde{f}}_2(·,s))$ for all $s\in [0,1]$ , it is easily checked that $(\tilde{U},\tilde{V})(t,x)$ is a supersolution of (2.1). On the other hand, the pair of functions $(u^{-},v^{-})(t,x)$ provided by Lemma 2.3 is a subsolution of (2.1). Then, since ${\mathbf{e}}_1\gg (\tilde{U},\tilde{V})(t,·)\gg {\mathbf{e}}_0$ for any $t\in \mathbb{R}$ , Lemma 2.2 (i) immediately implies that

2.10 $$\begin{array}{c}\hfill (\tilde{U},\tilde{V})(t,·)\ge (u^{-},v^{-})(t,·)\text{for all}t\ge 0.\end{array}$$

Graph

We will derive a contradiction in the case where $c>0$ , as we will sketch below, similar arguments apply to the case where $c<0$ . For each $n\in \mathbb{N}$ , taking $x=neL$ and $t=nL/c$ in (2.10), we have $\tilde{U}(nL/c,neL)\ge u^{-}(nL/c,neL)$ . Since $c>0$ , U(t, x) is increasing in t, and the constant K obtained in Lemma 2.3 is positive. Thanks to the spatial periodicity and the property (1.4), we obtain that for each $n\in \mathbb{N}$ ,

2.11 $$\begin{array}{c}\hfill \tilde{U}(nL/c,neL)\ge U(-{\epsilon }_{0}K,m_{0}L)-{\epsilon }_0{\mathrm{e}}^{-\mu nL/c}\left(\rho ,(0),{\psi }_1^{+},(0),+,(1-\rho (0)),{\psi }_1^{-},(0)\right).\\ \hfill \end{array}$$

Graph

Notice that we have assumed $c>\tilde{c}$ and $c>0$ . It is clear that $nL-\tilde{c}/cnL\to \infty$ as $n\to \infty$ , and hence, by the characterization of pulsating wave, we have $\tilde{U}(nL/c,neL)\to 0$ as $n\to \infty$ . Passing to the limit as $n\to \infty$ in (2.11), we get $0\ge U(-{\epsilon }_{0}K,m_{0}L)$ , which is impossible.

If $c<0$ , then U(t, x) is decreasing in t, and the constant K is negative. In this case, for each $n\in \mathbb{N}$ , taking $x=-neL$ and $t=-nL/c$ in (2.10), one reaches a similar contradiction.

Finally, if $\tilde{c}\ne 0$ , by using $(\tilde{U},\tilde{V})$ to construct a supersolution of (2.9) similarly as in Lemma 2.3, and comparing this supersolution with the pulsating wave (U, V), one derives a similar contradiction. Therefore, the proof of $c\le \tilde{c}$ is complete. $\square$

We are now ready to complete the

Proof of Proposition 1.2

We only show statement (i), since the proof for statement (ii) is analogous. Recall that (U, V)(t, x) is the pulsating wave of (2.1) in the direction e. Define

$$\begin{array}{c}\hfill \tilde{U}(t,x)=1-V(t,-x)\text{and}\tilde{V}(t,x)=1-U(t,-x)\text{for}t\in \mathbb{R},x\in \mathbb{R}.\end{array}$$

Graph

It is easily checked that $(\tilde{U},\tilde{V})(t,x)$ is an entire solution of the following system

2.12 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t\tilde{u}={\partial }_{\mathit{xx}}\tilde{u}+f_2(-x,\tilde{u})-k_2(-x)\tilde{u}(1-\tilde{v})\hfill & \text{in}\mathbb{R}\times \mathbb{R},\hfill \\ {\partial }_t\tilde{v}={\partial }_{\mathit{xx}}\tilde{v}-f_1(-x,1-\tilde{v})+k_1(-x)\tilde{u}(1-\tilde{v})\hfill & \text{in}\mathbb{R}\times \mathbb{R},\hfill \end{array}\right)\end{array}$$

Graph

and that

$$\begin{array}{cc}& (\tilde{U},\tilde{V})(t,x)\to {\mathbf{e}}_1\text{as}xe+ct\to -\infty \hfill \\ \hfill & \text{and}(\tilde{U},\tilde{V})(t,x)\to {\mathbf{e}}_0\text{as}xe+ct\to \infty .\hfill \end{array}$$

Graph

Namely, $(\tilde{U},\tilde{V})(t,x)$ is a pulsating wave of system (2.12) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ in the direction e, and $-c$ is the wave speed. Remember that (U, V)(t, x) is the pulsating wave of (2.1) in the same direction with speed c. Since $k_1(x)\le k_2(-x)$ and $f_1(x,·)\ge f_2(-x,·)$ in [0, 1] for all $x\in \mathbb{R}$ , it follows directly from Lemma 2.4 that $-c\le c$ , and hence, $c\ge 0$ . This ends the proof of Proposition 1.2. $\square$

Proofs of Theorems 1.3 and 1.5

This section is devoted to the proof of Theorems 1.3 and 1.5. Both rely on some qualitative properties of the traveling wave of the following homogeneous system

3.1 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{t}u={\partial }_{\mathit{xx}}u+f_{\ast }(u)-k_{\ast }u(1-v)\hfill & \text{in}\mathbb{R}\times \mathbb{R},\hfill \\ {\partial }_{t}v={\partial }_{\mathit{xx}}v-f_{\ast }(1-v)+k_{\ast }u(1-v)\hfill & \text{in}\mathbb{R}\times \mathbb{R},\hfill \end{array}\right)\end{array}$$

Graph

where $k_{\ast }$ is a positive constant and $f_{\ast }\in C^{1,1}(\mathbb{R})$ is a function satisfying the assumptions of Theorem 1.3. It is well known (see e.g., [[8], [20], [26]]) that system (3.1) admits a traveling wave $(U_0,V_0)$ connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ in the direction $e=1$ , and this wave is unique up to shifts and decreasing in $x\in \mathbb{R}$ . Moreover, as an easy consequence of Proposition 1.2, this wave is stationary, that is $(U_0,V_0)=(U_0,V_0)(x)$ . Since system (3.1) is spatially homogeneous, it is clear that $(U_0(-x),V_0(-x))$ is a stationary wave in the opposite direction $e=-1$ .

Notice that, under the assumption (1.8), $(U_0,V_0)(xe)$ is a subsolution of (2.1), and it is strict if one of the conditions (a)–(c) in Theorem 1.3 holds. We will show that this strict subsolution forces the propagation of (2.1) with positive speeds in both directions $e=\pm 1$ . As for the proof of Theorem 1.5, we will perturb the homogeneous system (3.1) by a non-symmetric L-periodic function in the u-component, and show that the resulting competition system admits a bistable structure in the sense that (H2) and (H3) are satisfied. The specifically designed perturbation function will make the speeds of pulsating waves in two opposition directions have different signs. It should be pointed out that, this perturbation idea is adapted from [[9]] which is concerned with the dependency of the wave speeds on directions for the scalar bistable equation (1.7).

Sufficient conditions for non-zero wave speed

In this subsection, we give the proof of Theorem 1.3. Since $f_{\ast }(s)/s$ is decreasing in $s\ge 0$ and $f_{\ast }(1)=0$ , one checks that $f_{\ast }^{\prime }(1)<0$ . Let $\mu >0$ be a positive constant given by

3.2 $$\begin{array}{c}\hfill \mu =min\left\{-,\frac{f_{\ast }^{\prime }(1)}{2},,,,-,\frac{f_{\ast }^{\prime }(0)-k_{\ast }}{2}\right\}.\end{array}$$

Graph

Recall that $\rho \in C^2(\mathbb{R})$ is a nonnegative function satisfying (2.4). The following lemma is analogous to Lemma 2.3.

Lemma 3.1

There exist ${\epsilon }_0>0$ and $K>0$ such that for any $\epsilon \in (0,{\epsilon }_0]$ , ${\xi }_0\in \mathbb{R}$ and each $e\in \{\pm 1\}$ , the pair of functions $(u^{-},v^{-})(t,x)$ defined by

3.3 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle u^{-}(t,x;{\xi }_0)=U_0(xe+{\xi }_0+\epsilon K(1-{\mathrm{e}}^{-\mu t}))-\epsilon {\mathrm{e}}^{-\mu t}\left(\rho ,(xe),-,(1-\rho (xe)),\frac{f_{\ast }^{\prime }(1)}{4k_{\ast }}\right)}\hfill \\ {\displaystyle v^{-}(t,x;{\xi }_0)=V_0(xe+{\xi }_0+\epsilon K(1-{\mathrm{e}}^{-\mu t}))-\epsilon {\mathrm{e}}^{-\mu t}\left((1-\rho (xe)),-,\rho ,(xe),\frac{f_{\ast }^{\prime }(1)}{4k_{\ast }}\right)}\hfill \end{array}\right)\end{array}$$

Graph

is a subsolution of (2.1), where $\mu >0$ is given by (3.2).

Proof

We first claim that there exist ${\epsilon }_0>0$ and $K>0$ such that for any $\epsilon \in (0,{\epsilon }_0]$ , ${\xi }_0\in \mathbb{R}$ and each $e\in \{\pm 1\}$ , $(u^{-},v^{-})(t,x;{\xi }_0)$ is a subsolution of (3.1). Indeed, this claim follows from similar arguments to those used in the proof of Lemma 2.3, and the proof is even simpler here, since system (3.1) is spatially homogeneous. Hence, the periodic eigenfunctions of (1.3) and (2.2) are reduced to positive constants and the normalization condition (2.3) is reduced to ${\psi }_i^{+}=1$ and ${\psi }_i^{-}=-f_{\ast }^{\prime }(1)/(4k_{\ast })$ for $i=1,2$ . The only minor difference is that in Lemma 2.3, we assumed that the pulsating wave of (2.1) is not stationary and used the monotonicity of this wave with respect to time t, while in the spatially homogeneous case, the traveling wave of (3.1) is monotone in the spatial variable x even it is stationary (see [[20], [26]]). In order not to repeat the proof of Lemma 2.3, we omit the details of our claim.

Next, thanks to the assumption (1.8), according to Definition 2.1, it is straightforward to check that any subsolution of (3.1) is a subsolution of (2.1). In particular, $(u^{-},v^{-})(t,x;{\xi }_0)$ is a subsolution of (2.1). This ends the proof of Lemma 3.1. $\square$

We are now ready to complete the

Proof of Theorem 1.3

Let $e\in \{\pm 1\}$ be fixed. Due to the assumption (1.8), it follows directly from Proposition 1.2 that $c(e)\ge 0$ . Therefore, it suffices to show that c(e) is positive if one of the conditions (a)–(c) holds. Suppose to the contrary that $c(e)=0$ , that is, system (2.1) admits a stationary pulsating wave (U, V)(x) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ in the direction e.

Let ${\epsilon }_0>0$ and $K>0$ be the constants provided by Lemma 3.1. Since $(U_0(xe),V_0(xe))$ is a stationary wave of (3.1) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ in the direction e, and since $0\le \rho (·)\le 1$ , it is clear that there exists ${\xi }_0\in \mathbb{R}$ such that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle U(x)\ge U_0(xe+{\xi }_0)-{\epsilon }_0\left(\rho ,(xe),-,(1-\rho (xe)),\frac{f_{\ast }^{\prime }(1)}{4k_{\ast }}\right)}\hfill \\ {\displaystyle V(x)\ge V_0(xe+{\xi }_0)-{\epsilon }_0\left((1-\rho (xe)),-,\rho ,(xe),\frac{f_{\ast }^{\prime }(1)}{4k_{\ast }}\right)}\hfill \end{array}\right)\text{for all}x\in \mathbb{R}.\end{array}$$

Graph

Let $(u^{-},v^{-})(t,x;{\xi }_0)$ be the pair of functions defined as in (3.3) with $\epsilon ={\epsilon }_0$ . Since ${\mathbf{e}}_1\gg (U,V)\gg {\mathbf{e}}_0$ , it then follows directly from Lemmas 2.2 (i) and 3.1 that $(U,V)(·)\ge (u^{-},v^{-})(t,·;{\xi }_0)$ for any $t\ge 0$ . Passing to the limit as $t\to \infty$ , we obtain

$$\begin{array}{c}\hfill (U,V)(·)\ge (U_0,V_0)(·e+{\xi }_0+{\epsilon }_{0}K).\end{array}$$

Graph

Now, call

$$\begin{array}{c}\hfill {\xi }_{\ast }:=inf\{\xi \in \mathbb{R}:(U,V)(·)\ge (U_0,V_0)(·e+\xi )\}.\end{array}$$

Graph

Since $U_0(·)$ and $V_0(·)$ are decreasing functions, it is clear that ${\xi }_{\ast }$ is a real number and ${\xi }_{\ast }\le {\xi }_0+{\epsilon }_{0}K$ . It is also easily seen that $(U,V)(·)\ge (U_0,V_0)(·e+{\xi }_{\ast })$ .

Thanks to (1.8), one checks that $(U_0,V_0)(xe+{\xi }_{\ast })$ is a subsolution of (2.1). Then, letting (u, v)(t, x) be the solution of (2.1) with initial function $(U_0,V_0)(xe+{\xi }_{\ast })$ , we obtain from Lemma 2.2 (i) that $(u,v)(t,·)\ge (U_0,V_0)(·e+{\xi }_{\ast })$ for any $t\ge 0$ . Notice that, any of the conditions (a)–(c) implies that the subsolution $(U_0,V_0)(xe+{\xi }_{\ast })$ is strict, and hence, for any $t>0$ , $(u,v)(t,·)$ is not identically equal to $(U_0,V_0)(·e+{\xi }_{\ast })$ . Furthermore, since ${\mathbf{e}}_1\gg (U_0,V_0)(·e+{\xi }_{\ast })\gg {\mathbf{e}}_0$ , it follows from Lemma 2.2 (ii) that

$$\begin{array}{c}\hfill (u,v)(t,·)\gg (U_0,V_0)(·e+{\xi }_{\ast })\text{for any}t>0.\end{array}$$

Graph

Remember that (u, v)(t, x) and (U, V)(x) are two solutions of (2.1) with the initial data being ordered. By using Lemma 2.2 (i), we have $(U,V)(·)\ge (u,v)(t,·)$ for all $t\ge 0$ . Combining the above, we obtain that

3.4 $$\begin{array}{c}\hfill (U,V)(·)\gg (U_0,V_0)(·e+{\xi }_{\ast }).\end{array}$$

Graph

Denote ${\nu }_0=min\left\{1,,,,-,f_{\ast }^{\prime },(1),/,(4k_{\ast })\right\}$ , and take a large positive constant C such that

$$\begin{array}{c}\hfill \delta :=\underset{|x|\ge C-1}{sup}\{|U_0^{\prime }(xe+{\xi }_{\ast })|,|V_0^{\prime }(xe+{\xi }_{\ast })|\}\le \frac{{\nu }_0}{2K}.\end{array}$$

Graph

Thanks to (3.4) and the fact that $U_0(·)$ and $V_0(·)$ are decreasing, one finds a small constant $0<\widehat{\epsilon }\le min\{1,{\epsilon }_0{\nu }_0{\delta }^{-1}\}$ such that

$$\begin{array}{c}\hfill (U,V)(·)\ge (U_0,V_0)(·e+{\xi }_{\ast }-\widehat{\epsilon })\text{in}[-C,C].\end{array}$$

Graph

For $x\in (-\infty ,-C)\cup (C,+\infty )$ , by the definition of $\delta$ and the monotonicity of $U_0(·)$ and $V_0(·)$ , one can check that $(U,V)(·)\ge (V_0,U_0)(·e+{\xi }_{\ast }-\widehat{\epsilon })-\widehat{\epsilon }\delta$ . Therefore, since $0\le \rho \le 1$ , it follows from the choice of ${\nu }_0$ that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle U(x)\ge U_0(xe+{\xi }_{\ast }-\widehat{\epsilon })-\frac{\widehat{\epsilon }\delta }{{\nu }_0}\left(\rho ,(xe),-,(1-\rho (xe)),\frac{f_{\ast }^{\prime }(1)}{4k_{\ast }}\right)}\hfill \\ {\displaystyle V(x)=V_0(xe+{\xi }_{\ast }-\widehat{\epsilon })-\frac{\widehat{\epsilon }\delta }{{\nu }_0}\left((1-\rho (xe)),-,\rho ,(xe),\frac{f_{\ast }^{\prime }(1)}{4k_{\ast }}\right)}\hfill \end{array}\right)\text{for all}x\in \mathbb{R}.\end{array}$$

Graph

By using Lemmas 2.2 and 3.1 again, we obtain that

$$\begin{array}{c}\hfill (U,V)(·)\ge ({\tilde{u}}^{-},{\tilde{v}}^{-})(t,·;{\xi }_{\ast }-\widehat{\epsilon })\text{for all}t\ge 0,\end{array}$$

Graph

where $({\tilde{u}}^{-},{\tilde{v}}^{-})(t,x;{\xi }_{\ast }-\widehat{\epsilon })$ is the pair of functions defined as in (3.3) with $\epsilon$ replaced by $\widehat{\epsilon }\delta {\nu }_0^{-1}$ . Since $0<\delta \le {\nu }_0/(2K)$ , sending to the limit as $t\to \infty$ gives that

$$\begin{array}{c}\hfill (U,V)(·)\ge (U_0,V_0)(·e+{\xi }_{\ast }-\widehat{\epsilon }+\widehat{\epsilon }\delta {\nu }_0^{-1}K)\ge (U_0,V_0)(·e+{\xi }_{\ast }-1/2\widehat{\epsilon }),\end{array}$$

Graph

which is a contradiction with the definition of ${\xi }_{\ast }$ . Therefore, the pulsating wave (U, V) cannot be stationary, and consequently, $c(e)>0$ . This ends the proof of Theorem 1.3. $\square$

As for the Lotka–Volterra system (1.2), it is clear that for any $\xi \in \mathbb{R}$ , $(U_0,V_0)(·e+\xi )$ is a strict subsolution provided that (1.9) is satisfied and that either $k_1(x)\not\equiv k_2(x)$ or $a_1(x)\not\equiv a_2(x)$ . Therefore, the above arguments immediately imply Corollary 1.4.

Distinct wave speeds in two opposite directions

Let us now turn to the proof of Theorem 1.5 by using some perturbation arguments on the system (3.1). We begin with the introduction of a perturbation function. Notice that we have assumed $k_1=k_2$ is a positive constant. Denote $k_{\ast }:=k_1=k_2$ and let $f_{\ast }:\mathbb{R}\to \mathbb{R}$ be a specific function given by $f_{\ast }(s)=a_{\ast }s(1-s)$ for $s\in \mathbb{R}$ , where $a_{\ast }$ is a constant satisfying $0<a_{\ast }<k_{\ast }$ . Recall that for each $e\in \{\pm 1\}$ , $(U_0,V_0)(xe)$ is the stationary traveling wave of (3.1) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ in the direction e. Fix a small constant ${\delta }_0\in (0,1/4)$ and let

$$\begin{array}{c}\hfill \chi :{\mathbb{R}}^2\to [0,1],(x,s)\mapsto \chi (x,s)\end{array}$$

Graph

be a $C^1$ -function satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\chi (x,s)=0\text{for}x\in \mathbb{R},s\in (-\infty ,{\delta }_0]\cup [1-{\delta }_0,+\infty ),\hfill \\ \chi (x,U_0(x+mL))=0\text{for}x\in \mathbb{R},m\in \mathbb{Z},\hfill \\ \chi (x,s)=\chi (x+L,s)\text{for}x\in \mathbb{R},s\in \mathbb{R},\hfill \end{array}\right)\end{array}$$

Graph

and

3.5 $$\begin{array}{cc}& \chi (x,s)>0\text{in}\hfill \\ \hfill & \bigcup _{m\in \mathbb{Z}}\left\{(x,s),\in ,{\mathbb{R}}^2,:,U_0,(x+mL),<,s,<,U_0,(x+mL-L),,,,2,{\delta }_0,\le ,s,\le ,1,-,2,{\delta }_0\right\}.\hfill \\ \hfill \end{array}$$

Graph

For any small $\sigma >0$ , we consider the following system

3.6 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{t}u={\partial }_{\mathit{xx}}u+f_{\sigma }(x,u)-k_{\ast }u(1-v)\hfill & \text{in}\mathbb{R}\times \mathbb{R},\hfill \\ {\partial }_{t}v={\partial }_{\mathit{xx}}v-f_{\ast }(1-v)+k_{\ast }u(1-v)\hfill & \text{in}\mathbb{R}\times \mathbb{R},\hfill \end{array}\right)\end{array}$$

Graph

where

$$\begin{array}{c}\hfill f_{\sigma }(x,s)=f_{\ast }(s)+\sigma \chi (x,s).\end{array}$$

Graph

Clearly, $f_{\sigma }(x,s)$ is L-periodic in x, $f_{\sigma }(x,s)\ge f_{\ast }(s)$ for $x\in \mathbb{R}$ , $s\in \mathbb{R}$ , and $f_{\sigma }$ is not identically equal to $f_{\ast }$ . It is also easily checked that for all small $\sigma$ , say $0<\sigma \le {\sigma }_0$ , and each $x\in \mathbb{R}$ , $f_{\sigma }(x,s)/s$ is decreasing in $s>0$ and the kinetic problem

$$\begin{array}{c}\hfill \frac{\mathit{du}}{\mathit{dt}}=f_{\sigma }(x,u)-k_{\ast }uv,\frac{\mathit{dv}}{\mathit{dt}}=f_{\ast }(v)-k_{\ast }uv\end{array}$$

Graph

has a unique positive equilibrium. Namely, the assumption (H1) holds when $0<\sigma \le {\sigma }_0$ . Furthermore, since $\chi$ is null on a neighborhood near $s=0$ and $s=1$ , and since $f_{\ast }^{\prime }(1)<0$ and $0<f_{\ast }^{\prime }(0)=a_{\ast }<k_{\ast }$ , for any $\sigma >0$ , ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ are two linearly stable steady states of (3.6) in the space $C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ , that is, system (3.6) satisfies the assumption (H2).

Now, we prove the existence of pulsating waves of (3.6) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ when $\sigma$ is small enough. By Theorem 1.1 (i), it suffices to show that for all small $\sigma$ , system (3.6) admits a bistable structure, that is, any steady state $(\tilde{u},\tilde{v})\in C_{\mathit{per}}(\mathbb{R};{(0,1)}^2)$ is linearly unstable. Let ${\mathcal{L}}_{\sigma }$ be the linearized operator of (3.6) around $(\tilde{u},\tilde{v})$ defined on $C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ , that is,

$$\begin{array}{c}\hfill {\mathcal{L}}_{\sigma }({\varphi }_1,{\varphi }_2)=\left({\varphi }_1^{\prime \prime },,,{\varphi }_2^{\prime \prime }\right)+A_{\sigma }({\varphi }_1,{\varphi }_2)\text{for}({\varphi }_1,{\varphi }_2)\in C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2),\end{array}$$

Graph

where

3.7 $$\begin{array}{c}\hfill A_{\sigma }=\left(\begin{array}{cc}{\partial }_s{f}_{\sigma }(x,\tilde{u})-k_{\ast }(1-\tilde{v})& k_{\ast }\tilde{u}\\ k_{\ast }(1-\tilde{v})& f_{\ast }^{\prime }(1-\tilde{v})-k_{\ast }\tilde{u}\end{array}\right).\end{array}$$

Graph

Since the off-diagonal entries of $A_{\sigma }$ are positive, it is then known from the Krein–Rutman theory that the linear problem

$$\begin{array}{c}\hfill {\mathcal{L}}_{\sigma }({\varphi }_1,{\varphi }_2)=\lambda ({\varphi }_1,{\varphi }_2)\text{in}{C}_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)\end{array}$$

Graph

admits a principal eigenvalue which is a real number, and the corresponding eigenfunction is strongly positive and unique (up to multiplication); see e.g., [[34], Chapter 7, Theorem 6.1]. Denote by ${\lambda }_{1,\sigma }(\tilde{u},\tilde{v})$ this principal eigenvalue. By Mora's theorem [[31]], $(\tilde{u},\tilde{v})$ is linearly unstable if ${\lambda }_{1,\sigma }(\tilde{u},\tilde{v})>0$ .

Therefore, to obtain the pulsating waves, it suffices to show the following:

Lemma 3.2

There exists ${\sigma }^{\ast }\in (0,{\sigma }_0]$ such that for any $0<\sigma \le {\sigma }_{\ast }$ and any steady state $(\tilde{u},\tilde{v})\in C_{\mathit{per}}(\mathbb{R};{(0,1)}^2)$ , there holds ${\lambda }_{1,\sigma }(\tilde{u},\tilde{v})>0$ .

Proof

Suppose to the contrary that there exist some sequences ${({\sigma }_n)}_{n\in \mathbb{N}}\subset (0,{\sigma }_0]$ , ${({\tilde{u}}_n,{\tilde{v}}_n)}_{n\in \mathbb{N}}\subset C_{\mathit{per}}(\mathbb{R};{(0,1)}^2)$ and ${({\varphi }_{1,n},{\varphi }_{2,n})}_{n\in \mathbb{N}}\in C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ such that ${\sigma }_n\to 0$ as $n\to \infty$ , and that for each $n\in \mathbb{N}$ , the pairs of functions $({\tilde{u}}_n,{\tilde{v}}_n)$ and $({\varphi }_{1,n},{\varphi }_{2,n})$ satisfy

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\tilde{u}}_n^{\prime \prime }+f_{{\sigma }_n}(x,{\tilde{u}}_n)-k_{\ast }{\tilde{u}}_n(1-{\tilde{v}}_n)=0\hfill & \text{in}\mathbb{R},\hfill \\ {\tilde{v}}_n^{\prime \prime }-f_{\ast }(1-{\tilde{v}}_n)+k_{\ast }{\tilde{u}}_n(1-{\tilde{v}}_n)=0\hfill & \text{in}\mathbb{R},\hfill \end{array}\right)\end{array}$$

Graph

and

$$\begin{array}{c}\hfill {\mathcal{L}}_{{\sigma }_n}({\varphi }_{1,n},{\varphi }_{2,n})={\lambda }_{1,{\sigma }_n}({\tilde{u}}_n,{\tilde{v}}_n)({\varphi }_{1,n},{\varphi }_{2,n})\text{in}\mathbb{R},({\varphi }_{1,n},{\varphi }_{2,n})\gg {\mathbf{e}}_0\text{in}\mathbb{R},\end{array}$$

Graph

with ${\lambda }_{1,{\sigma }_n}({\tilde{u}}_n,{\tilde{v}}_n)\le 0$ . Since for each $n\in \mathbb{N}$ , the eigenfunction $({\varphi }_{1,n},{\varphi }_{2,n})$ is unique up to multiplication, we normalize it by requiring ${max}_{x\in \mathbb{R}}({\varphi }_{1,n}(x)+{\varphi }_{2,n}(x))=1$ .

Let $A_{{\sigma }_n}$ be the matrix defined as (3.7) with $\sigma$ and $(\tilde{u},\tilde{v})$ replaced by ${\sigma }_n$ and $({\tilde{u}}_n,{\tilde{v}}_n)$ , respectively. It is clear that each entry of $A_{{\sigma }_n}$ is bounded uniformly in $x\in \mathbb{R}$ and $n\in \mathbb{N}$ . Let $A_{\pm }$ be two $2\times 2$ constant matrices given by

$$\begin{array}{c}\hfill A_{\pm }=\left(\begin{array}{cc}\pm {max}_{s\in [0,1],x\in \mathbb{R},\sigma \in [0,{\sigma }_0]}|{\partial }_s{f}_{\sigma }(x,s)|\pm k_{\ast }& \pm k_{\ast }\\ \pm k_{\ast }& \pm {max}_{s\in [0,1]}|f_{\ast }^{\prime }(s)|\pm k_{\ast }\end{array}\right).\end{array}$$

Graph

Clearly, $A_{+}$ and $A_{-}$ are, respectively, positive and negative matrices. Denote by ${\lambda }_{\pm }$ the principal eigenvalues of $A_{\pm }$ in the space ${\mathbb{R}}^2$ and let ${\mathbf{w}}_{\pm }\in {\mathbb{R}}^2$ be the corresponding positive eigenvectors. Notice that for each $n\in \mathbb{N}$ , $A_{+}-A_{{\sigma }_n}$ and $A_{{\sigma }_n}-A_{-}$ are positive matrices. Then, we have ${\mathcal{L}}_{{\sigma }_n}({\mathbf{w}}_{+})\le {\lambda }_{+}{\mathbf{w}}_{+}$ and ${\mathcal{L}}_{{\sigma }_n}({\mathbf{w}}_{-})\ge {\lambda }_{-}{\mathbf{w}}_{-}$ . It follows from the Krein–Rutman theory that ${\lambda }_{-}\le {\lambda }_{1,{\sigma }_n}({\tilde{u}}_n,{\tilde{v}}_n)\le {\lambda }_{+}$ , that is, the sequence ${({\lambda }_{1,{\sigma }_n}({\tilde{u}}_n,{\tilde{v}}_n))}_{n\in \mathbb{N}}$ is bounded. Thus, up to extraction of some subsequence, one finds some $\tilde{\lambda }\in (-\infty ,0]$ such that

$$\begin{array}{c}\hfill {\lambda }_{1,{\sigma }_n}({\tilde{u}}_n,{\tilde{v}}_n)\to \tilde{\lambda }\text{as}n\to \infty .\end{array}$$

Graph

By standard elliptic estimates, there exists $(u_{\infty },v_{\infty })\in C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ such that, possibly up to a further subsequence, $({\tilde{u}}_n,{\tilde{v}}_n)\to (u_{\infty },v_{\infty })$ as $n\to \infty$ in $C^2(\mathbb{R})$ . By our choice of the function $f_{\sigma }$ , it is clear that $(u_{\infty },v_{\infty })$ is an L-periodic steady state of (3.1), that is, $(u_{\infty },v_{\infty })$ satisfies

3.8 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_{\infty }^{\prime \prime }+f_{\ast }(u_{\infty })-k_{\ast }{u}_{\infty }(1-v_{\infty })=0\text{in}\mathbb{R},\hfill \\ v_{\infty }^{\prime \prime }-f_{\ast }(1-v_{\infty })+k_{\ast }{u}_{\infty }(1-v_{\infty })=0\text{in}\mathbb{R},\hfill \end{array}\right)\end{array}$$

Graph

and ${\mathbf{e}}_1\ge (u_{\infty },v_{\infty })\ge {\mathbf{e}}_0$ . Proceeding similarly as above, we find a pair of nonnegative functions $({\varphi }_{1,\infty },{\varphi }_{2,\infty })\in C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ such that, up to extraction of a further subsequence, $({\varphi }_{1,n},{\varphi }_{2,n})\to ({\varphi }_{1,\infty },{\varphi }_{2,\infty })$ as $n\to \infty$ in $C^2(\mathbb{R})$ , and $({\varphi }_{1,\infty },{\varphi }_{2,\infty })$ satisfies ${max}_{x\in \mathbb{R}}({\varphi }_{1,\infty }(x)+{\varphi }_{2,\infty }(x))=1$ , and

3.9 $$\begin{array}{c}\hfill ({\varphi }_{1,\infty }^{\prime \prime },{\varphi }_{2,\infty }^{\prime \prime })+A({\varphi }_{1,\infty },{\varphi }_{2,\infty })=\tilde{\lambda }({\varphi }_{1,\infty },{\varphi }_{2,\infty })\text{in}\mathbb{R},\end{array}$$

Graph

where

$$\begin{array}{c}\hfill A=\left(\begin{array}{cc}{f}_{\ast }^{\prime }(u_{\infty })-k_{\ast }(1-v_{\infty })& k_{\ast }{u}_{\infty }\\ k_{\ast }(1-v_{\infty })& f_{\ast }^{\prime }(1-v_{\infty })-k_{\ast }{u}_{\infty }\end{array}\right).\end{array}$$

Graph

Now, we consider all possible L-periodic solutions of (3.8). We first observe that if $u_{\infty }$ attains the minimum 0, then $u_{\infty }\equiv 0$ . Indeed, since $u_{\infty }^{\prime \prime }-k_{\ast }{u}_{\infty }(1-v_{\infty })\le 0$ , this observation follows directly from the elliptic strong maximum principle. Next, we show that if $u_{\infty }(x_0)=1$ for some $x_0\in \mathbb{R}$ , then $u_{\infty }\equiv 1$ . As a matter of fact, by the equation of $u_{\infty }$ , we have $v_{\infty }(x_0)=1$ . This means that $v_{\infty }$ attains the maximum 1 at the point $x_0$ . Applying the strong maximum principle to the equation of $v_{\infty }$ , we have $v_{\infty }\equiv 1$ . Consequently, $u_{\infty }$ satisfies $u_{\infty }^{\prime \prime }+f_{\ast }(u_{\infty })=0$ in $\mathbb{R}$ and $u_{\infty }(x_0)=1$ . By the strong maximum principle again, there must hold $u_{\infty }\equiv 1$ . Therefore, either $0<u_{\infty }<1$ in $\mathbb{R}$ or $u_{\infty }$ is a constant equal to 0 or 1. With the same reasoning, it follows that $0<v_{\infty }<1$ in $\mathbb{R}$ or $v_{\infty }\equiv 0$ or $v_{\infty }\equiv 1$ . Moreover, it is easily seen that $u\equiv 1$ is the unique positive L-periodic solution of the equation $u_{\mathit{xx}}+f_{\ast }(u)=0$ in $\mathbb{R}$ . Then, we can conclude that there are only three possibilities of L-periodic solutions of (3.8):

• $(u_{\infty },v_{\infty })=(0,1)$ ;
• $(u_{\infty },v_{\infty })={\mathbf{e}}_0$ or $(u_{\infty },v_{\infty })={\mathbf{e}}_1$ ;
• ${\mathbf{e}}_1\gg (u_{\infty },v_{\infty })\gg {\mathbf{e}}_0$ and $(u_{\infty },v_{\infty })$ is L-periodic.

We will derive a contradiction in each of these cases.

If Case (a) happens, it is easily seen from (3.9) that $\tilde{\lambda }$ is an eigenvalue of the problem

3.10 $$\begin{array}{c}\hfill {\varphi }^{\prime \prime }+f_{\ast }^{\prime }(0)\varphi =\lambda \varphi \end{array}$$

Graph

in the space of L-periodic functions with nonnegative eigenfunctions ${\varphi }_{1,\infty }$ and ${\varphi }_{2,\infty }$ . Since ${max}_{x\in \mathbb{R}}({\varphi }_{1,\infty }(x)+{\varphi }_{2,\infty }(x))=1$ , by the strong maximum principle (applied to (3.10)), at least one of ${\varphi }_{1,\infty }$ and ${\varphi }_{2,\infty }$ is positive. This means that $\tilde{\lambda }$ is an eigenvalue of (3.10) with a positive eigenfunction. Then, the Krein–Rutman theorem implies that $\tilde{\lambda }$ is the principal eigenvalue, and hence, $\tilde{\lambda }=f_{\ast }^{\prime }(0)$ . This is impossible, since $f_{\ast }^{\prime }(0)=a_{\ast }>0$ while $\tilde{\lambda }\le 0$ . Therefore, Case (a) is ruled out.

Suppose that Case (b) happens. Without loss of generality, we may assume that $(u_{\infty },v_{\infty })={\mathbf{e}}_0$ , as the other case can be treated identically. Since $0<a_{\ast }<k_{\ast }$ , ${\mathbf{e}}_0$ is an asymptotic stable steady state of the parabolic system (3.1). This in particular implies that there exists $\tilde{\delta }\in (0,{\delta }_0)$ such that the elliptic system (3.8) has no solutions in $B_{\tilde{\delta }}({\mathbf{e}}_0)\backslash \{{\mathbf{e}}_0\}$ , where ${\delta }_0>0$ is the constant given in definition of $\chi$ , and $B_{\tilde{\delta }}({\mathbf{e}}_0)$ denotes the ball in $C(\mathbb{R};{\mathbb{R}}^2)$ with radius $\tilde{\delta }$ and center ${\mathbf{e}}_0$ . On the other hand, since $({\tilde{u}}_n,{\tilde{v}}_n)\to {\mathbf{e}}_0$ as $n\to \infty$ in $C^2(\mathbb{R})$ , we have $({\tilde{u}}_n,{\tilde{v}}_n)\in B_{\tilde{\delta }}({\mathbf{e}}_0)$ for all large n. Remember that $\chi (x,s)\equiv 0$ for all $s\le {\delta }_0$ , and hence, for all large n, $({\tilde{u}}_n,{\tilde{v}}_n)$ is an L-periodic solution of (3.8). Combining the above, we obtain $({\tilde{u}}_n,{\tilde{v}}_n)={\mathbf{e}}_0$ for all large n, which is impossible, since for each $n\in \mathbb{N}$ , $({\tilde{u}}_n,{\tilde{v}}_n)\gg {\mathbf{e}}_0$ . Therefore, Case (b) is ruled out, too.

Finally, assume that Case (c) occurs. In this case, the off-diagonal entries of A are positive. Since the functions ${\varphi }_{1,\infty }$ and ${\varphi }_{2,\infty }$ are nonnegative and ${max}_{x\in \mathbb{R}}({\varphi }_{1,\infty }(x)+{\varphi }_{2,\infty }(x))=1$ , then the strong maximum principle for cooperation systems implies that $({\varphi }_{1,\infty },{\varphi }_{2,\infty })\gg {\mathbf{e}}_0$ . This means that $\tilde{\lambda }$ is an eigenvalue of the linear problem

3.11 $$\begin{array}{c}\hfill ({\varphi }_1^{\prime \prime },{\varphi }_2^{\prime \prime })+A({\varphi }_1,{\varphi }_2)=\lambda ({\varphi }_1,{\varphi }_2)\text{for}({\varphi }_1,{\varphi }_2)\in C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)\end{array}$$

Graph

with a strongly positive eigenfunction. By the Krein–Rutman theorem, $\tilde{\lambda }$ is the principal eigenvalue which is simple and has the greatest real part among all eigenvalues. If $(u_{\infty },v_{\infty })$ is a constant solution of (3.8), that is, $(u_{\infty },v_{\infty })=(a_{\ast }/(a_{\ast }+k_{\ast }),k_{\ast }/(a_{\ast }+k_{\ast }))$ , then A is a constant matrix with positive off-diagonal entries. Since $0<a_{\ast }<k_{\ast }$ , it is straightforward to check that the matrix A has a positive eigenvalue $a_{\ast }(k_{\ast }-a_{\ast })/(a_{\ast }+k_{\ast })$ with a positive eigenvector in ${\mathbb{R}}^2$ . Clearly, by the uniqueness of principal eigenvalue, $\tilde{\lambda }$ is equal to this positive eigenvalue, which is a contradiction with $\tilde{\lambda }\le 0$ . On the other hand, if $(u_{\infty },v_{\infty })$ is a non-constant L-periodic solution of (3.8), then we see from (3.8) that $(u_{\infty }^{\prime },v_{\infty }^{\prime })$ is a pair of sign-changing functions satisfying

$$\begin{array}{c}\hfill (u_{\infty }^{\prime \prime \prime },v_{\infty }^{\prime \prime \prime })+A(u_{\infty }^{\prime },v_{\infty }^{\prime })=0\text{in}\mathbb{R},\end{array}$$

Graph

that is, 0 is an eigenvalue of (3.11). Consequently, we have $\tilde{\lambda }>0$ , which is a contradiction with $\tilde{\lambda }\le 0$ again. Therefore, Case (c) is ruled out, and the proof of Lemma 3.2 is complete. $\square$

Now, we have prepared to complete the

Proof of Theorem 1.5

Let ${\sigma }_{\ast }>0$ be the constant obtained in Lemma 3.2. It follows from Theorem 1.1 (i) that for any $0<\sigma \le {\sigma }_{\ast }$ and any direction $e\in \{\pm 1\}$ , system (3.6) admits a pulsating wave connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ with speed c(e). Moreover, c(e) is the unique wave speed in the direction e. In what follows, let $0<\sigma \le {\sigma }_{\ast }$ be fixed.

Remember that $(U_0,V_0)(·)$ is a stationary traveling wave of (3.1) in the direction $e=1$ . Since $\chi (x,U_0(x))\equiv 0$ , we have $f_{\sigma }(x,U_0(x))\equiv f_{\ast }(U_0(x))$ . It then follows that $(U_0,V_0)(·)$ is also a stationary pulsating wave of (3.6) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ in the same direction. By the uniqueness of wave speeds, we immediately obtain $c(1)=0$ .

On the other hand, in the direction $e=-1$ , since the function $\chi$ is nonnegative, applying Proposition 1.2 to system (3.6), we have $c(-1)\ge 0$ . It remains to show that $c(-1)>0$ . Because of (3.5), it is easily checked that for any $\xi \in \mathbb{R}$ , $\chi (x,U_0(-x+\xi ))$ is a nonnegative and nonzero function, whence there holds $f_{\sigma }(x,U_0(-x+\xi ))\ge ,\not\equiv f_{\ast }(U_0(-x+\xi ))$ . Notice that $(U_0,V_0)(-x+\xi )$ is an entire solution of (3.1). This implies that $(U_0,V_0)(-x+\xi )$ is a strict subsolution of (3.6). It further follows from Lemma 2.2 that

$$\begin{array}{c}\hfill (u,v)(t,·)\gg (U_0,V_0)(-x+\xi )\text{for any}t>0,\end{array}$$

Graph

where (u, v)(t, x) is the solution of the Cauchy problem of (3.6) with initial data $(U_0,V_0)(-x+\xi )$ . The remaining proof is almost identical to that of Theorem 1.3; therefore, we omit the details. In conclusion, we obtain $c(-1)>0=c(1)$ , and the proof of Theorem 1.5 is thus complete. $\square$

Proof of Theorem 1.6

Throughout this section, we fix $e=1$ and consider the stationary pulsating waves of (1.11) in this direction, since the case where $e=-1$ can be treated identically. Recall that we assumed that $k_1=k_2$ , and ${\partial }_s{f}_i(·,0)$ , ${\partial }_s{f}_i(·,1)$ are constants. Due to the assumption (1.10), it is clear that ${\partial }_s{f}_1(·,0)={\partial }_s{f}_2(·,0)$ and ${\partial }_s{f}_1(·,1)={\partial }_s{f}_2(·,1)$ . For convenience, we denote

$$\begin{array}{c}\hfill k_{\ast }:=k_1=k_2,{\mu }_{+}:={\partial }_s{f}_1(·,0)={\partial }_s{f}_2(·,0),\text{and}{\mu }_{-}:={\partial }_s{f}_1(·,1)={\partial }_s{f}_2(·,1).\end{array}$$

Graph

Since $L>0$ is fixed, even if it means rescaling the variables, we may assume that $L=1$ for simplicity. Therefore, $f_i(x,·)$ is 1-periodic in x for $i=1,2$ . Moreover, without loss of generality, we assume that $\overline{x}\in (0,1]$ and $\underline{x}\in [-1,0)$ such that (1.10) holds.

Similarly as in Sect. 2, by a change of variables, the competition system (1.11) is transformed into the following cooperation system

4.1 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}u={\partial }_{\mathit{xx}}u+f_1(x/l,u)-k_{\ast }u(1-v),\hfill \\ {\partial }_{t}v={\partial }_{\mathit{xx}}v-f_2(x/l,1-v)+k_{\ast }u(1-v).\hfill \end{array}\right)\end{array}$$

Graph

To prove Theorem 1.6, it amounts to show the existence and non-uniqueness of stationary pulsating waves connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ for (4.1). The strategy of our proof is adapted from that of [[10], Theorem 1.7] which is concerned with the existence of a new type of wave solutions connecting two ordered stationary pulsating waves (in time) for the scalar bistable equation (1.7). Here, for simplicity, we only consider the existence of ordered stationary pulsating waves that are not identically equal up to any l-periodic shift. Based on our arguments below, one may further achieve that [[10], Theorem 1.7] remains valid for system (4.1).

Existence of stationary pulsating waves

In this subsection, we construct super and subsolutions of (4.1) by using some homogeneous traveling waves when l is large, and then show the existence of stationary pulsating waves. Let us first consider the following homogeneous system

4.2 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}u={\partial }_{\mathit{xx}}u+f_1(\overline{x},u)-k_{\ast }u(1-v),\hfill \\ {\partial }_{t}v={\partial }_{\mathit{xx}}v-f_2(\overline{x},1-v)+k_{\ast }u(1-v).\hfill \end{array}\right)\end{array}$$

Graph

Since $0<{\mu }_{+}<k_{\ast }$ and since (H1) holds, it follows from [[8], [20]] that system (4.2) admits a traveling wave $(u,v)(t,x)=({\overline{\mathrm{\Phi }}}_1,{\overline{\mathrm{\Phi }}}_2)(x-\overline{c}t)$ such that $0<{\overline{\mathrm{\Phi }}}_i<1$ in $\mathbb{R}$ for $i=1,2$ , and $({\overline{\mathrm{\Phi }}}_1,{\overline{\mathrm{\Phi }}}_2)(-\infty )={\mathbf{e}}_1$ , $({\overline{\mathrm{\Phi }}}_1,{\overline{\mathrm{\Phi }}}_2)(+\infty )={\mathbf{e}}_0$ . The front $({\overline{\mathrm{\Phi }}}_1,{\overline{\mathrm{\Phi }}}_2)(x)$ is decreasing in x and unique up to shift in x. Furthermore, $({\overline{\mathrm{\Phi }}}_1,{\overline{\mathrm{\Phi }}}_2)$ approaches the limiting states ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ exponentially fast (see e.g, [[26], [32]]). More precisely, denoting

4.3 $$\begin{array}{c}\hfill {\overline{\lambda }}_{1,\pm }=\frac{-\overline{c}\mp \sqrt{{\overline{c}}^2-4({\mu }_{+}-k_{\ast })}}{2},\text{and}{\overline{\lambda }}_{2,\pm }=\frac{-\overline{c}\mp \sqrt{{\overline{c}}^2-4{\mu }_{-}}}{2},\end{array}$$

Graph

we have

4.4 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{{\overline{\mathrm{\Phi }}}_1^{\prime }(x)}{{\overline{\mathrm{\Phi }}}_1(x)}\to {\overline{\lambda }}_{1,+},\text{and}\frac{{\overline{\mathrm{\Phi }}}_2^{\prime }(x)}{{\overline{\mathrm{\Phi }}}_2(x)}\to {\overline{\sigma }}_{+}\text{as}x\to \infty ,}\hfill \\ {\displaystyle \frac{{\overline{\mathrm{\Phi }}}_1^{\prime }(x)}{1-{\overline{\mathrm{\Phi }}}_1(x)}\to -{\overline{\sigma }}_{-},\text{and}\frac{{\overline{\mathrm{\Phi }}}_2^{\prime }(x)}{1-{\overline{\mathrm{\Phi }}}_2(x)}\to -{\overline{\lambda }}_{1,-}\text{as}x\to -\infty ,}\hfill \end{array}\right)\end{array}$$

Graph

where ${\overline{\sigma }}_{+}=max\{{\overline{\lambda }}_{1,+},{\overline{\lambda }}_{2,+}\}$ , and ${\overline{\sigma }}_{-}=min\{{\overline{\lambda }}_{1,-},{\overline{\lambda }}_{2,-}\}$ . Since $0<{\mu }_{+}<k_{\ast }$ and ${\mu }_{-}<0$ , it is clear that ${\overline{\lambda }}_{i,+}<0$ , ${\overline{\lambda }}_{i,-}>0$ , ${\overline{\sigma }}_{+}<0$ , and ${\overline{\sigma }}_{-}>0$ . It is also known from [[20]] that the wave speed $\overline{c}$ is unique. Regarding the sign of $\overline{c}$ , we have the following lemma.

Lemma 4.1

The wave speed $\overline{c}$ is negative.

Proof

Notice that if one assumes that $f_1(\overline{x},·)<f_2(\overline{x},·)$ in (0, 1), then the negativity of $\overline{c}$ follows directly from Theorem 1.3. Yet, for the homogeneous equation (4.2), the weaker assumption $f_1(\overline{x},·)\le ,\not\equiv f_2(\overline{x},·)$ in [0, 1] is sufficient to ensure the same result. Indeed, applying Proposition 1.2 to the homogeneous system (4.2), we immediately obtain $\overline{c}\le 0$ . Therefore, it suffices to show $\overline{c}\ne 0$ . Assume by contradiction that $\overline{c}=0$ . Define $\tilde{U}(x)=1-{\overline{\mathrm{\Phi }}}_2(x)$ , and $\tilde{V}(t,x)=1-{\overline{\mathrm{\Phi }}}_1(x)$ for $x\in \mathbb{R}$ . It is easily checked that for any ${\xi }_0\in \mathbb{R}$ , $(\tilde{U},\tilde{V})(·+{\xi }_0)$ is a strict supersolution of (4.2). By the strong maximum principle (see Lemma 2.2), this strict supersolution forces the wave $({\overline{\mathrm{\Phi }}}_1,{\overline{\mathrm{\Phi }}}_2)$ to propagate with a negative speed, which is a contradiction with $\overline{c}=0$ . The rigorous details are similar to the proof of Theorem 1.3; therefore, we do not repeat them here. $\square$

Our supersolution of (4.1) is stated as follows:

Lemma 4.2

There exists $l_1>0$ such that for any $l\ge l_1$ , the pair of functions ${\overline{\mathbf{u}}}_l:=({\overline{u}}_l,{\overline{v}}_l)(x)$ defined by

$$\begin{array}{c}\hfill ({\overline{u}}_l,{\overline{v}}_l)(x)=({\overline{\mathrm{\Phi }}}_1,{\overline{\mathrm{\Phi }}}_2)(x-l\overline{x})\text{for}x\in \mathbb{R}\end{array}$$

Graph

is a strict supersolution of (4.1).

Proof

According to Definition 2.1, to prove Lemma 4.2, we need to check that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{L}}_1{\overline{u}}_l:=-{\overline{u}}_l^{\prime \prime }-f_1(x/l,{\overline{u}}_l)+k_{\ast }{\overline{u}}_l(1-{\overline{v}}_l)\ge 0\text{in}\mathbb{R},\hfill \\ {\mathcal{L}}_2{\overline{v}}_l:=-{\overline{v}}_l^{\prime \prime }+f_2(x/l,1-{\overline{v}}_l)-k_{\ast }{\overline{u}}_l(1-{\overline{v}}_l)\ge 0\text{in}\mathbb{R},\hfill \end{array}\right)\end{array}$$

Graph

for all large $l>0$ . Since $({\overline{\mathrm{\Phi }}}_1,{\overline{\mathrm{\Phi }}}_2)(x-\overline{c}t)$ is an entire solution of (4.2), it follows that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{L}}_1{\overline{u}}_l=\overline{c}{\overline{\mathrm{\Phi }}}_1^{\prime }(x-l\overline{x})+f_1(\overline{x},{\overline{\mathrm{\Phi }}}_1(x-l\overline{x}))-f_1(x/l,{\overline{\mathrm{\Phi }}}_1(x-l\overline{x})),\hfill \\ {\mathcal{L}}_2{\overline{v}}_l=\overline{c}{\overline{\mathrm{\Phi }}}_2^{\prime }(x-l\overline{x})-f_2(\overline{x},1-{\overline{\mathrm{\Phi }}}_2(x-l\overline{x}))+f_2(x/l,1-{\overline{\mathrm{\Phi }}}_2(x-l\overline{x})).\hfill \end{array}\right)\end{array}$$

Graph

Notice that the coupled term ${\overline{u}}_l(1-{\overline{v}}_l)$ disappears in the above two equalities. Then, thanks to (4.4) and the negativity of $\overline{c}$ , proceeding similarly as in the proof of [[10], Lemma 4.1] for scalar equations, one can prove that ${\mathcal{L}}_1{\overline{u}}_l\ge 0$ and ${\mathcal{L}}_2{\overline{v}}_l\ge 0$ in $\mathbb{R}$ for all large l. Here, we omit the details. $\square$

Next, we construct a subsolution of (4.1) in a similar way. Let $(u,v)(t,x)=({\underline{\mathrm{\Phi }}}_1,{\underline{\mathrm{\Phi }}}_2)(x-\underline{c}t)$ be the unique (up to shifts) traveling wave of the homogeneous system

4.5 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}u={\partial }_{\mathit{xx}}u+f_1(\underline{x},u)-k_{\ast }u(1-v),\hfill \\ {\partial }_{t}v={\partial }_{\mathit{xx}}v-f_2(\underline{x},1-v)+k_{\ast }u(1-v),\hfill \end{array}\right)\end{array}$$

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such that $0<\underline{{\mathrm{\Phi }}_i}<1$ in $\mathbb{R}$ for $i=1,2$ , and $({\underline{\mathrm{\Phi }}}_1,{\underline{\mathrm{\Phi }}}_2)(-\infty )={\mathbf{e}}_1$ , $({\underline{\mathrm{\Phi }}}_1,{\underline{\mathrm{\Phi }}}_2)(+\infty )={\mathbf{e}}_0$ . The front $({\underline{\mathrm{\Phi }}}_1,{\underline{\mathrm{\Phi }}}_2)(x)$ is decreasing in x, and approaches the limiting states ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ exponentially fast. Defining ${\underline{\lambda }}_{1,\pm }$ and ${\underline{\lambda }}_{2,\pm }$ similarly to ${\overline{\lambda }}_{1,\pm }$ and ${\overline{\lambda }}_{2,\pm }$ in (4.3) with $\overline{c}$ replaced by $\underline{c}$ , respectively, we have

4.6 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{{\underline{\mathrm{\Phi }}}_1^{\prime }(x)}{{\underline{\mathrm{\Phi }}}_1(x)}\to {\underline{\lambda }}_{1,+},\text{and}\frac{{\underline{\mathrm{\Phi }}}_2^{\prime }(x)}{{\underline{\mathrm{\Phi }}}_2(x)}\to {\underline{\sigma }}_{+}\text{as}x\to \infty ,}\hfill \\ {\displaystyle \frac{{\underline{\mathrm{\Phi }}}_1^{\prime }(x)}{1-{\underline{\mathrm{\Phi }}}_1(x)}\to -{\underline{\sigma }}_{-},\text{and}\frac{{\underline{\mathrm{\Phi }}}_2^{\prime }(x)}{1-{\underline{\mathrm{\Phi }}}_2(x)}\to -{\underline{\lambda }}_{1,-}\text{as}x\to -\infty ,}\hfill \end{array}\right)\end{array}$$

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where ${\underline{\sigma }}_{+}=max\{{\underline{\lambda }}_{1,+},{\underline{\lambda }}_{2,+}\}<0$ , and ${\underline{\sigma }}_{-}=min\{{\underline{\lambda }}_{1,-},{\underline{\lambda }}_{2,-}\}>0$ . Moreover, thanks to the assumption $f_1(\underline{x},·)\ge ,\not\equiv f_2(\underline{x},·)$ in [0, 1], by the proof of Lemma 4.1, we can conclude that the speed $\underline{c}$ is positive.

The following lemma is analogous to Lemma 4.2.

Lemma 4.3

There exists $l_2>0$ such that for any $l\ge l_2$ , the pair of functions ${\underline{\mathbf{u}}}_l:=({\underline{u}}_l,{\underline{v}}_l)(x)$ defined by

$$\begin{array}{c}\hfill ({\underline{u}}_l,{\underline{v}}_l)(x)=({\underline{\mathrm{\Phi }}}_1,{\underline{\mathrm{\Phi }}}_2)(x-l\underline{x})\text{for}x\in \mathbb{R}\end{array}$$

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is a strict subsolution of (4.1).

Now, we have prepared to show the existence of stationary pulsating waves when l is large. In the sequel, for any ${\mathbf{u}}_0\in C(\mathbb{R};{\mathbb{R}}^2)$ , denote by $\mathbf{u}(t,x;{\mathbf{u}}_0)$ the solution of the Cauchy problem of (4.1) with initial function ${\mathbf{u}}_0$ .

Lemma 4.4

There exists $l_{\ast }\ge max\{l_1,l_2\}$ such that for all $l\ge l_{\ast }$ , system (4.1) admits two stationary waves ${\mathbf{u}}_1:=(u_1,v_1)$ and ${\mathbf{u}}_2:=(u_2,v_2)$ such that

4.7 $$\begin{array}{c}\hfill {\mathbf{e}}_1\gg {\overline{\mathbf{u}}}_l\gg {\mathbf{u}}_2\ge {\mathbf{u}}_1\gg {\underline{\mathbf{u}}}_l\gg {\mathbf{e}}_0\text{in}\mathbb{R}.\end{array}$$

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Furthermore, for $i=1,2$ , ${\mathbf{u}}_i$ is strictly l-decreasing in the sense that ${\mathbf{u}}_i(·)\gg {\mathbf{u}}_i(·+l)$ in $\mathbb{R}$ .

Proof

We first show that ${\mathbf{e}}_1\gg {\overline{\mathbf{u}}}_l\gg {\underline{\mathbf{u}}}_l\gg {\mathbf{e}}_0$ in $\mathbb{R}$ when l is large. Remember that $\overline{c}<0<\underline{c}$ . This implies that ${\underline{\lambda }}_{i,+}<{\overline{\lambda }}_{i,+}<0$ , $0<{\underline{\lambda }}_{i,-}<{\overline{\lambda }}_{i,-}$ for $i=1,2$ , and ${\underline{\sigma }}_{+}<{\overline{\sigma }}_{+}<0$ , $0<{\underline{\sigma }}_{-}<{\overline{\sigma }}_{-}$ . It then follows from (4.4) and (4.6) that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{{\overline{\mathrm{\Phi }}}_1(x)}{{\underline{\mathrm{\Phi }}}_1(x)}\to +\infty ,\text{and}\frac{{\overline{\mathrm{\Phi }}}_2(x)}{{\underline{\mathrm{\Phi }}}_2(x)}\to +\infty \text{as}x\to \infty ,}\hfill \\ {\displaystyle \frac{1-{\overline{\mathrm{\Phi }}}_1(x)}{1-{\underline{\mathrm{\Phi }}}_1(x)}\to 0,\text{and}\frac{1-{\overline{\mathrm{\Phi }}}_2(x)}{1-{\underline{\mathrm{\Phi }}}_2(x)}\to 0\text{as}x\to -\infty .}\hfill \end{array}\right)\end{array}$$

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This in particular implies that there exists $C>0$ such that ${\overline{\mathrm{\Phi }}}_i(x)>{\underline{\mathrm{\Phi }}}_i(x)$ for $|x|\ge C$ and $i=1,2$ . Furthermore, since ${\overline{\mathrm{\Phi }}}_i(x)$ and ${\underline{\mathrm{\Phi }}}_i(x)$ are both decreasing in $x\in \mathbb{R}$ , and since $-1\le \underline{x}<0<\overline{x}\le 1$ , one finds some $l_{\ast }\ge max\{l_1,l_2\}$ such that for all $l\ge l_{\ast }$ , ${\overline{\mathrm{\Phi }}}_i(·-l\overline{x})>{\underline{\mathrm{\Phi }}}_i(·-l\underline{x})$ in $\mathbb{R}$ for $i=1,2$ . This immediately gives ${\mathbf{e}}_1\gg {\overline{\mathbf{u}}}_l\gg {\underline{\mathbf{u}}}_l\gg {\mathbf{e}}_0$ in $\mathbb{R}$ if $l\ge l_{\ast }$ .

Next, we show the existence of ordered stationary pulsating waves. Let $l\ge l_{\ast }$ be fixed. Since ${\overline{\mathbf{u}}}_l$ (resp. ${\underline{\mathbf{u}}}_l$ ) is a strict supersolution (resp. subsolution) of (4.1) and since ${\mathbf{e}}_1\gg {\overline{\mathbf{u}}}_l\gg {\underline{\mathbf{u}}}_l\gg {\mathbf{e}}_0$ in $\mathbb{R}$ , it follows from Lemma 2.2 that

$$\begin{array}{c}\hfill {\mathbf{e}}_1\gg {\overline{\mathbf{u}}}_l(·)\gg \mathbf{u}(t,·;{\overline{\mathbf{u}}}_l)\gg \mathbf{u}(t,·;{\underline{\mathbf{u}}}_l)\gg {\underline{\mathbf{u}}}_l(·)\gg {\mathbf{e}}_0\text{in}\mathbb{R}\end{array}$$

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for all $t>0$ , and that $\mathbf{u}(t,·;{\overline{\mathbf{u}}}_l)$ (resp. $\mathbf{u}(t,·;{\underline{\mathbf{u}}}_l)$ ) is strictly decreasing (resp. increasing) in $t\ge 0$ , that is, $\mathbf{u}(t_1,·;{\overline{\mathbf{u}}}_l)\gg \mathbf{u}(t_2,·;{\overline{\mathbf{u}}}_l)$ (resp. $\mathbf{u}(t_2,·;{\underline{\mathbf{u}}}_l)\gg \mathbf{u}(t_1,·;{\underline{\mathbf{u}}}_l)$ ) whenever $0\le t_1<t_2$ . Then, by standard parabolic estimates, the functions

$$\begin{array}{c}\hfill {\mathbf{u}}_1(·):=\underset{t\to \infty }{lim}\mathbf{u}(t,·;{\underline{\mathbf{u}}}_l)\text{and}{\mathbf{u}}_2(·):=\underset{t\to \infty }{lim}\mathbf{u}(t,·;{\overline{\mathbf{u}}}_l)\end{array}$$

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are two stationary solutions of (4.1), and they satisfy (4.7). Remember that ${\overline{\mathbf{u}}}_l(-\infty )={\underline{\mathbf{u}}}_l(-\infty )={\mathbf{e}}_1$ and ${\overline{\mathbf{u}}}_l(\infty )={\underline{\mathbf{u}}}_l(\infty )={\mathbf{e}}_0$ . This implies that ${\mathbf{u}}_i(-\infty )={\mathbf{e}}_1$ and ${\mathbf{u}}_i(\infty )={\mathbf{e}}_0$ for $i=1,2$ . And hence, ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ are stationary pulsating waves of (4.1) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ .

Finally, since ${\overline{\mathbf{u}}}_l(x)$ is strictly decreasing in $x\in \mathbb{R}$ , it follows that ${\overline{\mathbf{u}}}_l(·)\gg {\overline{\mathbf{u}}}_l(·+l)$ . Then, by Lemma 2.2 and the periodicity, we have

$$\begin{array}{c}\hfill \mathbf{u}(t,·;{\overline{\mathbf{u}}}_l)\gg \mathbf{u}(t,·;{\overline{\mathbf{u}}}_l(·+l))\equiv \mathbf{u}(t,·+l;{\overline{\mathbf{u}}}_l)\text{in}\mathbb{R}\end{array}$$

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for all $t>0$ . Passing to the limit as $t\to \infty$ gives ${\mathbf{u}}_1(·)\ge {\mathbf{u}}_1(·+l)$ in $\mathbb{R}$ . Furthermore, since ${\mathbf{u}}_1$ is a pulsating wave connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ , ${\mathbf{u}}_1$ cannot be identically equal to ${\mathbf{u}}_1(·+l)$ . It then follows from Lemma 2.2 (ii) that ${\mathbf{u}}_1(·)\gg {\mathbf{u}}_1(·+l)$ in $\mathbb{R}$ . With the same arguments applied to $\mathbf{u}(t,·;{\underline{\mathbf{u}}}_l)$ , we can conclude that ${\mathbf{u}}_2(·)\gg {\mathbf{u}}_2(·+l)$ in $\mathbb{R}$ . This ends the proof of Lemma 4.4. $\square$

Exponential decay of stationary pulsating waves

Notice that whether the stationary pulsating waves obtained in Lemma 4.4 are identically equal up to an l-periodic shift is unclear. To further show the non-uniqueness result, we need to establish some exponential bounds for the stationary pulsating waves when they approach the limiting states ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ , as stated below:

Lemma 4.5

Let $l>0$ be arbitrary and let $({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)\in C^2(\mathbb{R};{(0,1)}^2)$ be a stationary pulsating wave of (4.1) connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ . Then for any small $\epsilon >0$ , there exists some constants $0<C_2<C_1$ such that

4.8 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{C}_2{\mathrm{e}}^{({\lambda }_{1,+}-\epsilon )x}\le {\mathrm{\Phi }}_1(x)\le C_1{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}\hfill & \text{for}x\ge 0,\hfill \\ C_2{\mathrm{e}}^{({\sigma }_{+}-\epsilon )x}\le {\mathrm{\Phi }}_2(x)\le C_1{\mathrm{e}}^{({\sigma }_{+}+\epsilon )x}\hfill & \text{for}x\ge 0,\hfill \\ C_2{\mathrm{e}}^{({\sigma }_{-}+\epsilon )x}\le 1-{\mathrm{\Phi }}_1(x)\le C_1{\mathrm{e}}^{({\sigma }_{-}-\epsilon )x}\hfill & \text{for}x\le 0,\hfill \\ C_2{\mathrm{e}}^{({\lambda }_{1,-}+\epsilon )x}\le 1-{\mathrm{\Phi }}_2(x)\le C_1{\mathrm{e}}^{({\lambda }_{1,-}-\epsilon )x}\hfill & \text{for}x\le 0,\hfill \end{array}\right)\end{array}$$

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where

$$\begin{array}{cc}& {\lambda }_{1,\pm }=\mp \sqrt{-({\mu }_{+}-k_{\ast })},\hfill \\ \hfill & {\lambda }_{2,\pm }=\mp \sqrt{-{\mu }_{-}},{\sigma }_{+}=\underset{i=1,2}{max}{\lambda }_{i,+},\text{and}{\sigma }_{-}=\underset{i=1,2}{min}{\lambda }_{i,-}.\hfill \end{array}$$

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We remark that, based on the above lemma, one may further establish more accurate exponential decay rates (see e.g., [[13], [26], [32]] for spatially homogeneous competition systems). Since this is not needed in showing Theorem 1.6, we do not pursue it here.

Since $({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)$ is a stationary wave of (4.1), it satisfies

4.9 $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{\Phi }}_1^{\prime \prime }+f_1(x/l,{\mathrm{\Phi }}_1)-k_{\ast }{\mathrm{\Phi }}_1(1-{\mathrm{\Phi }}_2)=0\text{in}\mathbb{R},\hfill \\ {\mathrm{\Phi }}_2^{\prime \prime }-f_2(x/l,1-{\mathrm{\Phi }}_2)+k_{\ast }{\mathrm{\Phi }}_1(1-{\mathrm{\Phi }}_2)=0\text{in}\mathbb{R},\hfill \\ ({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)(-\infty )={\mathbf{e}}_1,({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)(+\infty )={\mathbf{e}}_0.\hfill \end{array}\right)\end{array}$$

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Notice that here we cannot obtain the exponential decay rate of $({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)$ by linearizing system (4.9) around ${\mathbf{e}}_0$ or ${\mathbf{e}}_1$ . In fact, the linearized operator only admits the weak maximum principle but not the strong one. This means that the dominant eigenvalue may not have a strictly positive eigenfunction in $C_{\mathit{per}}(\mathbb{R};{\mathbb{R}}^2)$ , and hence it seems difficult to use such an eigenvalue to estimate the exponential decay rates of ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ simultaneously. Here, we prove this lemma by constructing sub and supersolutions directly rather than using the complicated classification analysis based on the Floquet theory of periodic ordinary differential equations. We first give the estimate of ${\mathrm{\Phi }}_1$ by some linearization arguments on the equation of ${\mathrm{\Phi }}_1$ , and then estimate ${\mathrm{\Phi }}_2$ by constructing suitable sub and supersolutions for the equation of ${\mathrm{\Phi }}_2$ .

Proof

We only prove the first two estimates of (4.8), as we will sketch below, the proof of the other two is similar. Let $\epsilon >0$ be a fixed small constant such that $\epsilon \le {min}_{i=1,2}\{-{\lambda }_{i,+}/2,{\lambda }_{i,-}/2\}$ . For clarity, we divide the proof into several steps. The proof of the first step follows the main lines of that of [[10], Lemma 4.4], and we outline it for completeness.

Step 1: There exist $0<A_2<A_1$ such that $A_2{\mathrm{e}}^{({\lambda }_{1,+}-\epsilon )x}\le {\mathrm{\Phi }}_1(x)\le A_1{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}$ for $x\ge 0$ .

Multiplying the equation of ${\mathrm{\Phi }}_1$ by ${\mathrm{\Phi }}_1^{-1}$ , we obtain

$$\begin{array}{c}\hfill \frac{{\mathrm{\Phi }}_1^{\prime \prime }}{{\mathrm{\Phi }}_1}+\frac{f_1(x/l,{\mathrm{\Phi }}_1)}{{\mathrm{\Phi }}_1}-k_{\ast }(1-{\mathrm{\Phi }}_2)=0\text{in}\mathbb{R}.\end{array}$$

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Remember that $({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)(+\infty )={\mathbf{e}}_0$ and ${\mu }_{+}={\partial }_s{f}_1(x,0)<k_{\ast }$ . It follows that ${\mathrm{\Phi }}_1^{\prime \prime }(x)>0$ for all large x, whence ${\mathrm{\Phi }}_1^{\prime }(+\infty )=0$ and ${\mathrm{\Phi }}_1^{\prime }(x)<0$ for all large x. Moreover, by standard interior estimates and Harnack inequality for cooperative elliptic systems (see e.g., [[1], [7]]), the function ${\mathrm{\Phi }}_1^{\prime }(x)/{\mathrm{\Phi }}_1(x)$ is bounded in $\mathbb{R}$ . Therefore, we have $\lambda :={lim\; sup}_{x\to \infty }{\mathrm{\Phi }}_1^{\prime }(x)/{\mathrm{\Phi }}_1(x)\in (-\infty ,0]$ .

Let ${(x_n)}_{n\in \mathbb{N}}\subset \mathbb{R}$ be a sequence such that $x_n\to \infty$ and ${\mathrm{\Phi }}_1^{\prime }(x_n)/{\mathrm{\Phi }}_1(x_n)\to \lambda$ as $n\to \infty$ . Write $x_n=x_n^{\prime }+x_n^{\prime \prime }$ with $x_n^{\prime }\in l\mathbb{Z}$ , $x_n^{\prime \prime }\in (0,l]$ , and introduce

$$\begin{array}{c}\hfill w_n(x):=\frac{{\mathrm{\Phi }}_1(x+x_n^{\prime })}{{\mathrm{\Phi }}_1(x_n^{\prime })}\text{for}x\in \mathbb{R}.\end{array}$$

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Since the function ${\mathrm{\Phi }}_1^{\prime }(x)/{\mathrm{\Phi }}_1(x)$ is bounded in $\mathbb{R}$ , $w_n(x)$ is bounded uniformly in x and locally uniformly in $x\in \mathbb{R}$ , and by the periodicity, it satisfies

$$\begin{array}{c}\hfill w_n^{\prime \prime }+\frac{f_1(x/l,{\mathrm{\Phi }}_1(x_n^{\prime })w_n)}{{\mathrm{\Phi }}_1(x_n^{\prime })}-k_{\ast }{w}_n(1-{\mathrm{\Phi }}_2(x+x_n^{\prime }))=0\text{in}\mathbb{R}.\end{array}$$

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Then, since $({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2)(+\infty )={\mathbf{e}}_0$ , by standard elliptic estimates, up to extraction of some subsequence, there is a nonnegative function $w_{\infty }\in C^2(\mathbb{R})$ such that $w_n\to w_{\infty }$ in $C_{\mathit{loc}}^2(\mathbb{R})$ . Clearly, $w_{\infty }$ satisfies $w_{\infty }^{\prime \prime }+({\mu }_{+}-k_{\ast })w_{\infty }=0$ in $\mathbb{R}$ . Since $w_{\infty }(0)>0$ , the strong maximum principle implies $w_{\infty }>0$ in $\mathbb{R}$ . Furthermore, since $w_{\infty }(x)$ is nonincreasing when x is large and since ${\lambda }_{1,+}=-\sqrt{-({\mu }_{+}-k_{\ast })}<0$ , it follows that $w_{\infty }(x)={\mathrm{e}}^{{\lambda }_{1,+}x}$ for $x\in \mathbb{R}$ . Finally, noticing that $w_n^{\prime }(x_n^{\prime \prime })/w_n(x_n^{\prime \prime })\to \lambda$ as $n\to \infty$ , we obtain $\lambda ={\lambda }_{1,+}$ .

Similarly as above, we can prove that ${lim\; inf}_{x\to \infty }{\mathrm{\Phi }}_1^{\prime }(x)/{\mathrm{\Phi }}_1(x)={\lambda }_{1,+}$ , and hence, we have ${lim}_{x\to \infty }{\mathrm{\Phi }}_1^{\prime }(x)/{\mathrm{\Phi }}_1(x)={\lambda }_{1,+}$ . As a consequence, there exists $x_1>0$ sufficiently large such that

$$\begin{array}{c}\hfill ({\lambda }_{1,+}-\epsilon )(x-x_1)\le ln{\mathrm{\Phi }}_1(x)-ln{\mathrm{\Phi }}_1(x_1)\le ({\lambda }_{1,+}+\epsilon )(x-x_1)\text{for all}x\ge x_1.\end{array}$$

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Since $0<{\mathrm{\Phi }}_1<1$ in $\mathbb{R}$ , this clearly implies the conclusion of Step 1.

Remember that $f_2(x,1)\equiv 0$ and ${\mu }_{-}={\partial }_s{f}_2(x,1)$ is a constant. For the above $\epsilon >0$ , we find a large constant $M>0$ such that

4.10 $$\begin{array}{c}\hfill {\mu }_{-}-{\epsilon }^2\le -\frac{f_2(x/l,1-{\mathrm{\Phi }}_2(x))}{{\mathrm{\Phi }}_2(x)}\le {\mu }_{-}+{\epsilon }^2\text{for all}x\ge M,\end{array}$$

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and that

4.11 $$\begin{array}{c}\hfill M\ge \frac{ln(-2\epsilon {\sigma }_{+})-ln(k_{\ast }{A}_1)}{{\lambda }_{1,+}+\epsilon },\end{array}$$

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where $A_1>0$ is the constant obtained in Step 1.

Step 2: There exists $B_1>0$ such that ${\mathrm{\Phi }}_2(x)\le B_1{\mathrm{e}}^{({\sigma }_{+}+\epsilon )x}$ for $x\ge M$ .

Since ${\mathrm{\Phi }}_2^{\prime \prime }-f_2(x/l,1-{\mathrm{\Phi }}_2)+k_{\ast }{\mathrm{\Phi }}_1(1-{\mathrm{\Phi }}_2)=0$ in $\mathbb{R}$ , it follows from the second inequality of (4.10) that

$$\begin{array}{c}\hfill 0<{\mathrm{\Phi }}_2^{\prime \prime }-\frac{f_2(x/l,1-{\mathrm{\Phi }}_2)}{{\mathrm{\Phi }}_2}{\mathrm{\Phi }}_2+k_{\ast }{\mathrm{\Phi }}_1\le {\mathrm{\Phi }}_2^{\prime \prime }+({\mu }_{-}+{\epsilon }^2){\mathrm{\Phi }}_2+k_{\ast }{\mathrm{\Phi }}_1\end{array}$$

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for all $x\ge M$ , whence by Step 1, ${\mathrm{\Phi }}_2^{\prime \prime }+({\mu }_{-}+{\epsilon }^2){\mathrm{\Phi }}_2+k_{\ast }{A}_1{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}>0$ . This means that ${\mathrm{\Phi }}_2(x)$ is a strict subsolution of

4.12 $$\begin{array}{c}\hfill w^{\prime \prime }+({\mu }_{-}+{\epsilon }^2)w+k_{\ast }{A}_1{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}=0\text{in}[M,\infty ).\end{array}$$

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On the other hand, define $w_{+}(x)=B_1{\mathrm{e}}^{({\sigma }_{+}+\epsilon )x}$ for $x\ge M$ , where $B_1$ is a large constant satisfying $w_{+}(M)\ge {\mathrm{\Phi }}_2(M)$ and

$$\begin{array}{c}\hfill B_1\ge -\frac{k_{\ast }{A}_1}{2\epsilon ({\sigma }_{+}+\epsilon )}.\end{array}$$

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If ${\lambda }_{1,+}\le {\lambda }_{2,+}$ , then ${\sigma }_{+}={\lambda }_{2,+}$ . Since ${\lambda }_{2,+}+\epsilon <0$ , it is straightforward to check that for any $x\ge M$ ,

$$\begin{array}{c}\hfill \left(\begin{array}{cc}& {\displaystyle w_{+}^{\prime \prime }+({\mu }_{-}+{\epsilon }^2)w_{+}+k_{\ast }{A}_1{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}}\hfill \\ \hfill & {\displaystyle \le B_1{\mathrm{e}}^{({\lambda }_{2,+}+\epsilon )x}\left({({\lambda }_{2,+}+\epsilon )}^2,+,{\mu }_{-},+,{\epsilon }^2\right)+k_{\ast }{A}_1{\mathrm{e}}^{({\lambda }_{2,+}+\epsilon )x}}\hfill \\ \hfill & {\displaystyle =2\epsilon ({\lambda }_{2,+}+\epsilon ){\mathrm{e}}^{({\lambda }_{2,+}+\epsilon )x}\left(B_1,+,\frac{k_{\ast }{A}_1}{2\epsilon ({\lambda }_{2,+}+\epsilon )}\right)\le 0.}\hfill \end{array}\right)\end{array}$$

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If ${\lambda }_{1,+}>{\lambda }_{2,+}$ , then ${\sigma }_{+}={\lambda }_{1,+}$ and ${\lambda }_{1,+}^2<-{\mu }_{-}$ . Remember that ${\lambda }_{1,+}+\epsilon <0$ . One easily checks that

$$\begin{array}{cc}& w_{+}^{\prime \prime }+({\mu }_{-}+{\epsilon }^2)w_{+}+k_{\ast }{A}_1{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}\hfill \\ \hfill & \le 2\epsilon ({\lambda }_{1,+}+\epsilon ){\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}\left(B_1,+,\frac{k_{\ast }{A}_1}{2\epsilon ({\lambda }_{1,+}+\epsilon )}\right)\le 0\hfill \end{array}$$

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for all $x>M$ . Namely, in both cases, $w_{+}$ is a supersolution of (4.12). Since $w_{+}(M)\ge {\mathrm{\Phi }}_2(M)$ and $w_{+}(+\infty )={\mathrm{\Phi }}_2(+\infty )=0$ , by the elliptic maximum principle, we obtain that ${\mathrm{\Phi }}_2(x)\le w_{+}(x)$ for all $x\ge M$ . This completes the proof of Step 2.

Step 3: There exists $B_2>0$ such that ${\mathrm{\Phi }}_2(x)\ge B_2{\mathrm{e}}^{({\sigma }_{+}-\epsilon )x}$ for $x\ge M$ .

By the first inequality of (4.10) and Step 1, we have

$$\begin{array}{c}\hfill \left(\begin{array}{cc}{\displaystyle 0}\hfill & {\displaystyle ={\mathrm{\Phi }}_2^{\prime \prime }-\frac{f_2(x/l,1-{\mathrm{\Phi }}_2)}{{\mathrm{\Phi }}_2}{\mathrm{\Phi }}_2+k_{\ast }{\mathrm{\Phi }}_1(1-{\mathrm{\Phi }}_2)}\hfill \\ \hfill & \ge {\mathrm{\Phi }}_2^{\prime \prime }+({\mu }_{-}-{\epsilon }^2){\mathrm{\Phi }}_2+k_{\ast }{A}_2{\mathrm{e}}^{({\lambda }_{1,+}-\epsilon )x}-k_{\ast }{A}_1{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}{\mathrm{\Phi }}_2\hfill \end{array}\right)\end{array}$$

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for all $x\ge M$ , that is, ${\mathrm{\Phi }}_2(x)$ is a supersolution of

4.13 $$\begin{array}{c}\hfill v^{\prime \prime }+\left({\mu }_{-},-,{\epsilon }^2,-,k_{\ast },A_1,{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}\right)v+k_{\ast }{A}_2{\mathrm{e}}^{({\lambda }_{1,+}-\epsilon )x}=0\text{in}[M,\infty ).\end{array}$$

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Let $v_{-}(x)=B_2{\mathrm{e}}^{({\sigma }_{+}-\epsilon )x}$ for $x\ge M$ , where $B_2>0$ is a small constant satisfying $v_{-}(M)\le {\mathrm{\Phi }}_2(M)$ and

$$\begin{array}{c}\hfill 0<B_2\le -\frac{{\lambda }_{1,+}^2+{\mu }_{-}}{k_{\ast }{A}_2}\text{if}{\lambda }_{1,+}>{\lambda }_{2,+}.\end{array}$$

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If ${\lambda }_{1,+}\le {\lambda }_{2,+}$ , then ${\sigma }_{+}={\lambda }_{2,+}$ . It follows from (4.11) that

$$\begin{array}{c}\hfill \left(\begin{array}{cc}& {\displaystyle v_{-}^{\prime \prime }+\left({\mu }_{-},-,{\epsilon }^2,-,k_{\ast },A_1,{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}\right)v_{-}+k_{\ast }{A}_2{\mathrm{e}}^{({\lambda }_{1,+}-\epsilon )x}}\hfill \\ \hfill & {\displaystyle >\left({({\lambda }_{2,+}-\epsilon )}^2,+,{\mu }_{-},-,{\epsilon }^2,-,k_{\ast },A_1,{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}\right)v_{-}}\hfill \\ \hfill & {\displaystyle \ge \left(-,2,\epsilon ,{\lambda }_{2,+},-,k_{\ast },A_1,{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )M}\right)v_{-}\ge 0}\hfill \end{array}\right)\end{array}$$

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for all $x\ge M$ . On the other hand, if ${\lambda }_{1,+}>{\lambda }_{2,+}$ , then ${\sigma }_{+}={\lambda }_{1,+}$ and ${\lambda }_{1,+}^2<-{\mu }_{-}$ . By (4.11) again, we have

$$\begin{array}{c}\hfill \left(\begin{array}{cc}& {\displaystyle v_{-}^{\prime \prime }+\left({\mu }_{-},-,{\epsilon }^2,-,k_{\ast },A_1,{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )x}\right)v_{-}+k_{\ast }{A}_2{\mathrm{e}}^{({\lambda }_{1,+}-\epsilon )x}}\hfill \\ \hfill & {\displaystyle \ge \left({({\lambda }_{1,+}-\epsilon )}^2,+,{\mu }_{-},-,{\epsilon }^2,-,k_{\ast },A_1,{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )M}\right)v_{-}+k_{\ast }{A}_2{\mathrm{e}}^{({\lambda }_{1,+}-\epsilon )x}}\hfill \\ \hfill & {\displaystyle =\left({\lambda }_{1,+}^2,+,{\mu }_{-},-,2,\epsilon ,{\lambda }_{1,+},-,k_{\ast },A_1,{\mathrm{e}}^{({\lambda }_{1,+}+\epsilon )M}\right)v_{-}+k_{\ast }{A}_2{\mathrm{e}}^{({\lambda }_{1,+}-\epsilon )x}}\hfill \\ \hfill & {\displaystyle \ge \left({\lambda }_{1,+}^2,+,{\mu }_{-},+,\frac{k_{\ast }{A}_2}{B_2}\right)B_2{\mathrm{e}}^{({\lambda }_{1,+}-\epsilon )x}\ge 0}\hfill \end{array}\right)\end{array}$$

Graph

for all $x\ge M$ . Therefore, $v_{-}(x)$ is a subsolution of (4.13). It then follows from the elliptic maximum principle that ${\mathrm{\Phi }}_2(x)\ge v_{-}(x)$ for all $x\ge M$ . This ends the proof of Step 3.

Finally, since $0<{\mathrm{\Phi }}_i<1$ for $i=1,2$ , one finds some constants $C_1\ge max\{A_1,B_1\}$ and $0<C_2\le min\{A_2,B_2\}$ such that the first two estimates of (4.8) hold true. Notice that the pair of functions $({\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2):=(1-{\mathrm{\Phi }}_2,1-{\mathrm{\Phi }}_1)$ satisfies

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathrm{\Psi }}_1^{\prime \prime }+f_2(x/l,{\mathrm{\Psi }}_1)-k_{\ast }{\mathrm{\Psi }}_1(1-{\mathrm{\Psi }}_2)=0\text{in}\mathbb{R},\hfill \\ {\mathrm{\Psi }}_2^{\prime \prime }-f_1(x/l,1-{\mathrm{\Psi }}_2)+k_{\ast }{\mathrm{\Psi }}_1(1-{\mathrm{\Psi }}_2)=0\text{in}\mathbb{R},\hfill \\ ({\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2)(-\infty )={\mathbf{e}}_0,({\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2)(+\infty )={\mathbf{e}}_1.\hfill \end{array}\right)\end{array}$$

Graph

Proceeding similarly as above, and making some adjustment to the constants $C_1$ and $C_2$ if necessary, we have $C_2{\mathrm{e}}^{({\lambda }_{1,-}+\epsilon )x}\le {\mathrm{\Psi }}_1(x)\le C_1{\mathrm{e}}^{({\lambda }_{1,-}-\epsilon )x}$ and $C_2{\mathrm{e}}^{({\sigma }_{-}+\epsilon )x}\le {\mathrm{\Psi }}_2(x)\le C_1{\mathrm{e}}^{({\sigma }_{-}-\epsilon )x}$ for $x\le 0$ . This immediately gives the last two estimates of (4.8). The proof of Lemma 4.5 is thus complete. $\square$

Non-uniqueness of stationary pulsating waves

Now, to prove Theorem 1.6, it suffices to show the following

Theorem 4.6

Let $l_{\ast }>0$ be the constant provided by Lemma 4.4. For any $l\ge l_{\ast }$ , system (4.1) admits two ordered stationary pulsating waves connecting ${\mathbf{e}}_1$ and ${\mathbf{e}}_0$ that are not identically equal up to any l-periodic shift.

Proof

Let $l\ge l_{\ast }$ be fixed. Recall that for any ${\mathbf{u}}_0\in C(\mathbb{R};{\mathbb{R}}^2)$ , $\mathbf{u}(t,x;{\mathbf{u}}_0)$ denotes the solution of the Cauchy problem of (4.1) with initial function ${\mathbf{u}}_0$ . In Lemma 4.4, we have obtained two stationary pulsating waves ${\mathbf{u}}_1$ , ${\mathbf{u}}_2$ satisfying (4.7) and

4.14 $$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\Vert \mathbf{u}(t,·;{\overline{\mathbf{u}}}_l)-{\mathbf{u}}_2(·)\Vert =0,\underset{t\to \infty }{lim}\Vert \mathbf{u}(t,·;{\underline{\mathbf{u}}}_l)-{\mathbf{u}}_1(·)\Vert =0,\end{array}$$

Graph

where $\Vert ·\Vert$ denotes the usual $L^{\infty }$ -norm in $C(\mathbb{R};{\mathbb{R}}^2)$ .

If there is no $m\in \mathbb{Z}$ such that ${\mathbf{u}}_1(·-ml)\ge {\mathbf{u}}_2(·)$ in $\mathbb{R}$ , then it is clear that ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ are the desired stationary pulsating waves, that is, ${\mathbf{u}}_1$ is not identically equal to ${\mathbf{u}}_2$ up to any l-periodic shift. Hence, nothing is left to prove in this case.

Suppose, on the other hand, that there is $m_0\in \mathbb{Z}$ such that ${\mathbf{u}}_1(·-m_{0}l)\ge {\mathbf{u}}_2(·)$ in $\mathbb{R}$ . Then, since ${\mathbf{u}}_1(·)\gg {\mathbf{u}}_1(·+l)$ in $\mathbb{R}$ by Lemma 4.4, the integer

$$\begin{array}{c}\hfill m_{\ast }:=inf\left\{m,\in ,\mathbb{Z},:,,{\mathbf{u}}_1,(·-ml),\gg ,{\mathbf{u}}_2,(·),,,\text{in},,,\mathbb{R}\right\}\end{array}$$

Graph

is well-defined and $0<m_{\ast }\le m_0+1$ . Denote ${\mathbf{u}}_3=(u_3,v_3):={\mathbf{u}}_1(·-m_{\ast }l)$ and

$$\begin{array}{c}\hfill I:=\left\{\mathbf{u},\in ,C,(\mathbb{R};{\mathbb{R}}^2),:,,{\mathbf{u}}_3,\ge ,\mathbf{u},\ge ,{\mathbf{u}}_2\right\}.\end{array}$$

Graph

Now, we show that ${\mathbf{u}}_2$ is a stable (from above) stationary solution of (4.1) in I in the sense there exists $\delta >0$ such that

4.15 $$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\Vert \mathbf{u}(t,·;{\mathbf{u}}_0)-{\mathbf{u}}_2(·)\Vert =0\text{for all}{\mathbf{u}}_0\in I\cap B_{\delta }({\mathbf{u}}_2),\end{array}$$

Graph

where $B_{\delta }({\mathbf{u}}_2)$ denotes the ball in $C(\mathbb{R};{\mathbb{R}}^2)$ with center ${\mathbf{u}}_2$ and radius $\delta$ , that is, $B_{\delta }({\mathbf{u}}_2)=\{\mathbf{u}\in C(\mathbb{R};{\mathbb{R}}^2):\Vert \mathbf{u}-{\mathbf{u}}_2\Vert \le \delta \}$ . Indeed, since $\overline{c}<0$ , we have ${\lambda }_{i,+}<{\overline{\lambda }}_{i,+}<0$ , $0<{\lambda }_{i,-}<{\overline{\lambda }}_{i,-}$ for $i=1,2$ , and ${\sigma }_{+}<{\overline{\sigma }}_{+}<0$ , $0<{\sigma }_{-}<{\overline{\sigma }}_{-}$ . Remember that ${\mathbf{u}}_3$ is a stationary pulsating wave of (4.1). It then follows from (4.4) and Lemma 4.5 that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{{\overline{u}}_l(x)}{u_3(x)}\to +\infty ,\text{and}\frac{{\overline{v}}_l(x)}{v_3(x)}\to +\infty \text{as}x\to \infty ,}\hfill \\ {\displaystyle \frac{1-{\overline{u}}_l(x)}{1-u_3(x)}\to 0,\text{and}\frac{1-{\overline{v}}_l(x)}{1-v_3(x)}\to 0\text{as}x\to -\infty .}\hfill \end{array}\right)\end{array}$$

Graph

Therefore, there exists $C>0$ such that ${\overline{u}}_l(x)>u_3(x)$ and ${\overline{v}}_l(x)>v_3(x)$ for $|x|\ge C$ . On the other hand, by continuity and the inequalities ${\overline{\mathbf{u}}}_l\gg {\mathbf{u}}_2$ and ${\mathbf{u}}_3\gg {\mathbf{u}}_2$ , one finds some small $\delta >0$ such that $min\{{\overline{u}}_l(x),u_3(x)\}>u_2(x)+\delta$ and $min\{{\overline{v}}_l(x),v_3(x)\}>v_2(x)+\delta$ for $|x|\le C$ . Combining the above, we obtain that for any ${\mathbf{u}}_0\in I\cap B_{\delta }({\mathbf{u}}_2)$ , there holds ${\overline{\mathbf{u}}}_l\gg {\mathbf{u}}_0\gg {\mathbf{u}}_2$ in $\mathbb{R}$ . It then follows from Lemma 2.2 that $\mathbf{u}(t,·;{\overline{\mathbf{u}}}_l)\gg \mathbf{u}(t,·;{\mathbf{u}}_0)\gg {\mathbf{u}}_2(·)$ in $\mathbb{R}$ for any $t>0$ . This together with the first convergence of (4.14) yields the stability of ${\mathbf{u}}_2$ .

Similarly, since $\underline{c}>0$ , we have $0>{\lambda }_{i,+}>{\underline{\lambda }}_{i,+}$ , ${\lambda }_{i,-}>{\underline{\lambda }}_{i,-}>0$ for $i=1,2$ , and $0>{\sigma }_{+}>{\underline{\sigma }}_{+}$ , ${\sigma }_{-}>{\underline{\sigma }}_{-}>0$ ; and hence, by the arguments used above, we find some ${\delta }^{\prime }>0$ such that ${\mathbf{u}}_3(·)\gg {\mathbf{u}}_0(·)\gg {\underline{\mathbf{u}}}_l(·-m_{\ast }l)$ in $\mathbb{R}$ for all ${\mathbf{u}}_0\in I\cap B_{{\delta }^{\prime }}({\mathbf{u}}_3)$ . Then, by Lemma 2.2 and the periodicity, we get ${\mathbf{u}}_3(·)\gg \mathbf{u}(t,·;{\mathbf{u}}_0)(·)\gg \mathbf{u}(t,·-m_{\ast }l;{\underline{\mathbf{u}}}_l)$ in $\mathbb{R}$ for any $t>0$ . Therefore, by the second convergence of (4.14), we obtain

4.16 $$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\Vert \mathbf{u}(t,·;{\mathbf{u}}_0)-{\mathbf{u}}_3(·)\Vert =0\text{for all}{\mathbf{u}}_0\in I\cap B_{{\delta }^{\prime }}({\mathbf{u}}_3),\end{array}$$

Graph

namely, ${\mathbf{u}}_3$ is a stable (from below) stationary solution of (4.1) in I.

We are now ready to complete the proof by applying the Dancer-Hess connecting orbit theorem (see e.g., [[25]]). For any ${\mathbf{u}}_0\in I$ , define $S_t[{\mathbf{u}}_0]=\mathbf{u}(t,·;{\mathbf{u}}_0)$ for $t\ge 0$ . Clearly, ${(S_t)}_{t\ge 0}$ is a continuous semiflow on I, and thanks to Lemma 2.2, it is strongly order-preserving in the sense that $S_t[{\mathbf{u}}_0]\gg S_t[{\tilde{\mathbf{u}}}_0]$ in $\mathbb{R}$ for every $t>0$ whenever ${\mathbf{u}}_0>{\tilde{\mathbf{u}}}_0$ in $\mathbb{R}$ . Furthermore, by standard estimates of parabolic systems and the fact that ${\mathbf{u}}_2(-\infty )={\mathbf{u}}_3(-\infty )={\mathbf{e}}_1$ and ${\mathbf{u}}_2(+\infty )={\mathbf{u}}_3(+\infty )={\mathbf{e}}_0$ , one easily checks that for any $t>0$ , the set $S_t[I]$ is relatively compact in $C(\mathbb{R};{\mathbb{R}}^2)$ with respect to the $L^{\infty }$ -norm. Moreover, in view of (4.15) and (4.16), ${\mathbf{u}}_2$ (resp. ${\mathbf{u}}_3$ ) is stable from above (resp. from below) in I for ${(S_t)}_{t\ge 0}$ . As a consequence of the connecting orbit theorem, the semiflow ${(S_t)}_{t\ge 0}$ admits an equilibrium ${\mathbf{u}}_{\ast }\in I$ such that ${\mathbf{u}}_3\gg {\mathbf{u}}_{\ast }\gg {\mathbf{u}}_2$ . Clearly, ${\mathbf{u}}_{\ast }$ is a stationary pulsating wave of (4.1) connecting ${\mathbf{e}}_0$ and ${\mathbf{e}}_1$ .

Finally, we observe that ${\mathbf{u}}_{\ast }$ is not identically equal to ${\mathbf{u}}_2$ up to any l-periodic shift. Otherwise, since ${\mathbf{u}}_{\ast }\gg {\mathbf{u}}_2$ , one would find some integer $\tilde{m}\ge 1$ such that ${\mathbf{u}}_{\ast }(·)\equiv {\mathbf{u}}_2(·-\tilde{m}l)$ in $\mathbb{R}$ , whence due to ${\mathbf{u}}_3\gg {\mathbf{u}}_{\ast }$ , there would hold ${\mathbf{u}}_1(·-(m_{\ast }-\tilde{m})l)\equiv {\mathbf{u}}_3(·+\tilde{m}l)\gg {\mathbf{u}}_2(·)$ in $\mathbb{R}$ ; this last property would contradict the definition of $m_{\ast }$ . The proof of Theorem 4.6 is thus complete. $\square$

Acknowledgements

The authors would like to thank the anonymous referees for careful reading and valuable comments. Weiwei Ding was partly supported by the National Natural Science Foundation of China (12001206) and the Basic and Applied Basic Research Foundation of Guangdong Province (2019A1515110506). Xing Liang was partially supported by the National Natural Science Foundation of China (11971454).

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