Existence and stability of traveling wavefronts for discrete three species competitive-cooperative systems

1.   Introduction

This paper is concerned with the existence and stability of traveling wavefronts for the following discrete three species competitive-cooperative systems of Lotka-Volterra type:

Here $D[u](t, x)$ and $\triangle[u_i](t)$ mean the discrete diffusive operators given by

Systems (1.1) and (1.2) can be considered as discrete versions of the following continuous system:

In system (1.3), $u(\cdot), v(\cdot)$ and $w(\cdot)$ represent the population density of the species. Each $d_i > 0 (i = 1, 2, 3)$ stands for the diffusion rate of each species, and $r_i > 0 (i = 1, 2, 3)$ is the growth rate of species. The parameter $b_{ii} > 0 (i = 1, 2, 3)$ means the intraspecific competition rates of a species, and $b_{12}, b_{21}, b_{23}, b_{32 } > 0$ describe the interspecific competition rates between species. Noting that $b_{13}$ and $b_{31}$ maybe positive or negative constants. If $b_{13}$ and $b_{31} < 0$ then (1.3) is a competitive system among three species and any two of the three species $u$, $v$ and $w$ are in a competitive manner. On the other hand, if $b_{13}$ and $b_{31} > 0$, then (1.3) becomes the competitive-cooperative system of three species. That is, $u$ and $v$ compete and $w$ and $v$ also compete with each other, while $u$ and $w$ are in a cooperative way to help each other.

Due to different signs of the parameters, the interacting behavior between the species of (1.3) are quite complicated and different. In biology, one of the important issue is to investigate the invasion phenomenon for system (1.3). Thus it is very nature to study the propagation of traveling wave solutions. The concept of traveling wave solutions was introduced by Fisher ^[1] in 1937 in reaction diffusion equations, which represents a segregated spatial pattern propagating through the spatial domain at a constant speed. In addition, such solutions are natural phenomena ubiquitously for many reaction-diffusion systems, e.g., biophysics, population genetics, mathematical ecology, chemistry, chemical physics, and so on. In past years, there have many progresses on this topic in various fields. Here we only illustrate some literature for system (1.3) in the sequel.

For the competitive case, Chen et al. ^[2,3] and Mimura and Tohma ^[4] used numerical approaches or the construction of exact traveling wave solutions to establish many kinds of pattern formulations. In addition, when the diffusion coefficients are small, Ikeda ^[5,6] considered traveling wave solutions and dynamics of weakly interacting front and back waves. Other related works, we refer Kan-on and Mimura ^[7], Miller ^[8] and Mimura and Fife ^[9]. On the other hand, for the competitive-cooperative case, one can see that system (1.3) is a monotone system which has some ordering structures. Based on the monotone structure, Guo et al. ^[10] proved the existence of traveling wave solutions under the assumption $b_{13} = b_{31} = 0$. Hung ^[11] further considered the existence of traveling wave solutions in the case $b_{13}, \, b_{31} > 0$, $d_1 = d_2 = d_3$ and $r_1 = r_2 = r_3$. Recently, Chang ^[12] improved the results of ^[10,11] to more general parameters. Motivated by the above mentioned literature, it is natural and important to study the same problems for the discrete systems (1.1) and (1.2). In this paper, we first establish the existence of traveling wavefronts for discrete systems (1.1) and (1.2). However, when these solutions are disturbed under small perturbations, only stable such solutions can be visualized in the real world. Therefore, it is quite important to study the stability problem of the traveling wavefronts. We also focus on the stability problem in this work.

Since there are many parameters appearing in the above systems, we first rescale the systems (1.1)–(1.2) into the following simpler forms:

Furthermore, replacing $(u, v, w)$ and $(u_i, v_i, w_i)$ by $(u, r_2-v, w)$ and $(u_i, r_2-v_i, w_i)$ respectively, we can transform systems (1.4)–(1.5) into the following systems

Since systems (1.6)–(1.7) are monotone systems, for simplicity, hereinafter we will consider our subject on the systems (1.6)–(1.7). By elementary computations, systems (1.6) or (1.7) have the following eight equilibria:

A traveling wave solution $(u(t, x), v(t, x), w(t, x))$ for (1.6) means that

for some smooth functions $\phi_i(\cdot)$, $i = 1, 2, 3$ with wave speed $c\in \mathbb{R}$. If $\Phi(\cdot) = (\phi_1(\cdot), \phi_2(\cdot), \phi_3(\cdot))$ is monotone, then it is called a traveling wavefront. Then, taking the moving coordinate $\xi: = x+ct$, we see the profile function $(\phi_1(\xi), \phi_2(\xi), \phi_3(\xi))$ for system (1.6) satisfy the system

where

Different to system (1.6), a traveling wave solution $(u_i(t), v_i(t), w_i(t))$ for (1.7) means that

for some smooth functions $\phi_i(\cdot)$, $i = 1, 2, 3$ with wave speed $c\in \mathbb{R}$. Then, taking the moving coordinate $\xi: = i+ct$, we see the profile function $(\phi_1(\xi), \phi_2(\xi), \phi_3(\xi))$ for system (1.7) is the same as system (1.8). From the viewpoint of biology, we are interested in the existence and stability of solutions for system (1.8) connecting the trivial equilibria ${\bf{E}}_1$ and positive co-exist equilibrium ${\bf{E}}_2$, that is satisfy the following conditions:

It is easy to see that ${\bf{E}}_2\gg {\bf 0}$ when $c_{31}c_{13} < 1$. Here and in the sequel, we always use the usual notations for the standard ordering in $\mathbb{R}^3$.

In this article we first consider the existence problem of traveling wavefronts for systems (1.6) and (1.7), i.e., looking for solutions of (1.8) satisfying the condition (1.9). Since (1.8) is a monotone system, the existence problem could be reduced to find a pair of supersolution and subsolution of system (1.8). To this end, throughout this article, we assume the following assumption:

${\rm(H1)}$ $d_1\ge d_3\ge d_2$, $c_{32} < 1 < c_{21}+c_{23}$ and $(c_{21}+c_{23})r_2\leq r_3-c_{32}r_2\leq r_1-c_{12}r_2.$

${\rm(H2)}$ $(c_{21}+c_{31}c_{23})r_1+(c_{23}+c_{21}c_{13})r_3 > r_2(1-c_{13}c_{31}) { > 0}.$

Note that (H2) holds when $r_2$ is small enough. The assumption (H1) will be used in proving the existence of traveling wavefronts. In addition, one can verify that

If $\Theta > 0$ then (H2) implies that $e_2 < u_2^*$. On the other hand, if $\Theta < 0$ then (H2) implies that $e_2 > u_2^*$. Hence, under the assumptions (H1)–(H2), we know that ${\bf E}_8\notin [{\bf 0}, {\bf E}_2]$.

Based on the above assumptions, we can establish a pair of supersolution and subsolution of system (1.8). Then, applying the monotone iteration method, we show that (1.8) admits a strictly increasing solution satisfying (1.9) as long as the wave speed is greater than the minimum wave speed (see Theorem 3.1). That is the existence of monotonic traveling wave solutions connecting two equilibria for systems (1.1) and (1.2). In addition, we show that ${\rm(H1)}$ and ${\rm(H2)}$ are sufficient conditions which ensure the linear determinacy for the minimal speed is given, i.e., the minimal speed is determined by the linearization of the problem at some unstable equilibrium.

Next, we consider the stability of traveling wavefronts derived in Theorem 3.1. In past years, there have been extensive investigations on the stability of traveling wave solutions for reaction-diffusion systems, see e.g., ^[13,14,15], the monographs ^[16,17], the survey paper ^[18] and the references therein. For examples, Mei et al. ^[14] used the weighted energy method and the Green function technique to study the global stability of monostable traveling wave solutions for nonlocal time-delayed reaction-diffusion equations. Recently, by using the monotone scheme and spectral analysis, Chang ^[12] considered the existence and stability of traveling wave solutions for system (1.3). More precisely, the author showed that the traveling wave solutions of (1.3) are essentially unstable in the uniform continuous function space. On the other hand, if the initial perturbations of the traveling wave solutions belong to certain exponentially weighted Banach space, then the traveling wave solutions are asymptotically stable in the weighted Banach space. However, due to the discrete diffusion operator in (1.8), the method of spectral analysis used in Chang ^[12] no longer works in investigating the stability problems of the discrete systems (1.6) and (1.7). Motivated by the works ^[14,19], we will establish the comparison principle for systems (1.6) and (1.7). And then use the the weighted energy method (see ^{[19,20,21,22,14]}) to show that the traveling wave solutions of (1.6) and (1.7) with large wave speed are exponentially stable when the initial perturbation around them decay exponentially as the spatial variable tending to $-\infty$ (see Theorems 4.1 and 5.1). Moreover, using different weighted functions, we improve the stability results of Theorems 4.1 and 5.1 to any wave speed greater than the minimum wave speed (see Theorems 6.1 and 6.2).

2.   Construction of sub-super solutions for (1.8)

For convenience, we write ${\bf E}_2 = (u_1^*, u_2^*, u_3^*)$ in this section, and $F(u, v, w): = (f_1(u, v, w),$ $f_2(u, v, w), f_3(u, v, w))$ where

Then the profile system (1.8) can be written into the form:

To establish the existence of solutions for system (2.1) by using the technique of sub-super solutions, we first give the following definition.

Definition 2.1. A continuous function $(\phi_1(\xi), \phi_2(\xi), \phi_3(\xi))$ is called a subsolution or supersolution of (2.1), if each $\phi_i(\xi)$ is continuously differentiable in $\mathbb{R}$ except at finite points and satisfies (resp.)

Before constructing a pair of sub-super solutions for system (1.8), we first consider the characteristic polynomials of system (1.8) at ${\bf E}_1$ given by

where

It is clear that $\Gamma_2(\mu; c) = 0$ have a positive root for any $c > 0$. For $\Gamma_1(\mu; c)$ and $\Gamma_3(\mu; c)$, we have the following properties.

Lemma 2.1. There exist $c_1^*\geq c_3^* > 0$ such that (for $i = 1, 3$)

${\rm (1)}$ if $c > c_i^*$, there exist $0 < \mu_i^- < \mu_i^+$ such that

${\rm (2)}$ if $c = c_i^*$, there exists a unique $\mu_i^*\in (\mu_i^-, \mu_i^+)$ such that

${\rm (3)}$ if $0 < c < c_i^*$, then $\Gamma_i(\mu; c) > 0$ for all $\mu\in\mathbb{R}$.

In addition, we have $\mu_{3}^-\leq\mu_1^- < \mu_1^+\leq\mu_{3}^+$ when $c > c_1^*$.

By Lemma 2.1, we can construct a pair of sub-super solutions for (1.8) in the sequel.

Lemma 2.2. Assume $c > c^*_1.$ Let's set

for $i = 2, 3$, where $\hat{u}_i(\xi_i) = u^*_i\, (i = 1, 2, 3)$, $q$ and $\eta$ are positive constants with

Then $\hat{U}(\xi) = (\hat{u}_1(\xi), \hat{u}_2(\xi), \hat{u}_3(\xi))$ is a supersolution of (1.8) when $q$ is large enough.

Proof. Let us write $\xi_i = \xi_i(q)$ $(i = 1, 2, 3)$ as a function of $q$. Since $\hat{u}_i(\xi_i) = u^*_i$, one can easily verify that

Then, for $\xi\ge \xi_1(q),$ it is clear that

If $\xi < \xi_1(q),$ by (2.4) and (2.5) and elementary computations, we have

provided that $q$ is large enough.

Next, we set

Then, for all $\xi\in\mathbb{R}^-$, it is clear that $\max\{\hat{u}_1(\xi), \hat{u}_2(\xi), \hat{u}_3(\xi)\}\leq \hat{u}(\xi)$ and

Assuming that $q$ is large enough, we have $\xi_2(q) < 0$. For $\xi > \xi_2(q),$ it is clear that

If $\xi < \xi_2(q),$ by the fact $\hat{u}_2(\xi)\leq r_2$, (2.4), (2.8) and (H1), we can obtain

Finally, for $\xi > \xi_3(q),$ it is clear that

If $\xi < \xi_3(q),$ then (2.4) implies that

provided that $q$ is large enough. Hence, it follows from (2.6)–(2.12) that $\hat{U}(\xi)$ is a supersolution of system (1.8) when $q$ is large enough. The proof is complete.

Lemma 2.3. Assume $c > c^*_1.$ Let's set $\bar{u}_2(\xi):\equiv 0,$

where $\bar{u}_i(\bar\xi_i) = 0$ for $i = 1, 3$; $q$ and $\eta$ are positive constants with

Then $\bar{U}(\xi) = (\bar{u}_1(\xi), \bar{u}_2(\xi), \bar{u}_3(\xi))$ is a subsolution of (1.8) when $q$ is large enough.

Proof. Let us also write $\bar\xi_i = \bar\xi_i(q)$ as a function of $q$. Similarly, $\bar\xi_i(\infty) = -\infty$, for $i = 1, 3$. According to the definition of $\bar{u}_i(\xi)$, we only need to consider the cases $\xi < \bar\xi_1(q)$ and $\xi < \bar\xi_3(q)$ for $\bar{u}_1(\xi)$ and $\bar{u}_3(\xi)$, respectively.

If $\xi < \bar\xi_1(q)$, by (2.13), we have

provided that $q$ is large enough.

For $\xi < \xi_3(q),$ by (2.13) again, one can see that

if $q$ is large enough. Hence, it follows from (2.14) and (2.15) that $\bar{U}(\xi)$ is a subsolution of (1.8) when $q$ is large enough. The proof is complete.

3.   Existence of traveling wavefronts for (1.6) and (1.7)

Based on the supersolution and subsolution derived in previous section, we can apply the the monotone iteration method to obtain the following existence result.

Theorem 3.1. Given any $c\ge c^*_1$, system (1.8) admits a strictly increasing traveling wave solution $\Phi(\xi) = (\phi_1(\xi), \phi_2(\xi), \phi_3(\xi))$ satisfying (1.9) and with wave speed $c$.

Proof. Let $c > c^*_1$ and $\hat{U}(\xi)$ and $\bar{U}(\xi)$ be the supersolution and subsolution constructed in Lemmas 2 and 3 respectively. Since (1.8) is a monotone system on $[{\bf E}_1, {\bf E}_2]$, by the monotone iteration method, system (1.8) admits a non-decreasing solution $\Phi(\xi) = (\phi_1(\xi), \phi_2(\xi), \phi_3(\xi))$ satisfying

Since $\bar{U}(-\infty) = \hat{U}(-\infty) = {\bf E}_1,$ it follows that $\Phi(-\infty) = {\bf E}_1$. Moreover, we have $\Phi(\infty) = {\bf E}_* = ({\bf E}_*^1, {\bf E}_*^2, {\bf E}_*^3)$ for some equilibrium ${\bf E}_*\le {\bf E}_2$. By the non-decreasing property of $\Phi(\xi)$ and the fact $\bar{u}_1\not\equiv0$ and $\bar{u}_3\not\equiv0$, we see that ${\bf E}_*^1 > 0$ and ${\bf E}_*^2 > 0$, and hence ${\bf E}_*\in\{ {\bf E}_2, {\bf E}_8 \}$. Since ${\bf E}_8\notin [{\bf 0}, {\bf E}_2]$, we conclude that ${\bf E}_* = {\bf E}_2$. Hence $\Phi(\xi)$ satisfies the condition (1.9).

Next, we consider the case $c = c^*_1$. Let $\{\ell_n\}$ be a sequence with $\ell_n > c^*_1$ for all $n\in \mathbb{N}$, which converges to $c^*_1$. Denoting $\Phi_n(\xi)$ by the non-decreasing solution of (1.8) satisfying (1.9) with $c = \ell_n$. Then, by the limiting arguments (cf. ^[23]), $\{\Phi_n(\xi)\}$ has a convergent subsequence which converges to a function $\Phi_*(\xi)$ which satisfies (1.8) and (1.9) with $c = c^*_1$.

Finally, we show that $\Phi^\prime(\xi)\gg{\bf 0}$ for all $\xi\in\mathbb{R}$. We first claim that $\Phi (\xi)\gg{\bf 0}$ for all $\xi\in\mathbb{R}$. Note that $\phi_1(+\infty) = u_*$. If there exists $\xi_1\in\mathbb{R}$ such that $\phi_1(\xi_1) = 0$, we may assume that $\phi_1(\xi) > 0$ for all $\xi > \xi_1$. Since $\phi_1(\cdot)\geq0$, we have $\phi_1'(\xi_1) = 0$ and hence it follows the first equation of (1.8) that $\phi_1(\xi_1+1) = 0$, which contradicts to the definition of $\xi_1$. Thus, $\phi_1 (\xi) > 0$ for all $\xi\in\mathbb{R}$. Similarly, we can show that $\phi_3 (\xi) > 0$ for all $\xi\in\mathbb{R}$. Suppose that there exists $\xi_2\in\mathbb{R}$ such that $\phi_2(\xi_2) = 0$ and $\phi_2(\xi) > 0$ for all $\xi > \xi_2$. By the second equation of (1.8), we have

which implies that $\phi_2(\xi_2+1) = 0$. This contradiction shows that $\phi_2 (\xi) > 0$ for all $\xi\in\mathbb{R}$. Hence the claim holds.

According to (1.8), we know that

where $\ell$ is a positive constant and

Choosing $\ell$ large enough, we know that $H(\Psi)$ is monotone increasing for any $\Psi\in[{\bf E}_1, {\bf E}_2]$. Since $\Phi(\xi)$ is non-decreasing in $\xi$, by differentiating (3.1) with respect to $\xi$, we have

Suppose that $\phi_i^\prime(\xi_i) = 0$ for some $\xi_i\in\mathbb{R}$ ($i = 1, 2,$ or $3$), then (3.2) implies that $H_i(\Phi(s)) = H_i(\Phi(\xi_i))$ for all $s\le\xi_i$. Taking $s\to-\infty$, it follows that

which implies that $\phi_i(\xi_i) = 0$. This contradiction implies that $\Phi^\prime(\xi) > {\bf 0}$, $\forall\xi\in\mathbb{R}$. The proof is complete.

Next, we investigate the linear determinacy for the problem (1.8). The definition of linear determinacy was first introduced in ^[24], which means that the minimal speed is determined by the linearization of the problem at some unstable equilibrium. In the following theorem, we show that $c^*_1$ is the minimal speed of system (1.8).

Theorem 3.2. Assume $c < c^*_1$. System (1.8) has no strictly increasing solution $\Phi(\xi) = (\phi_1(\xi), \phi_2(\xi), \phi_3(\xi))\in {[{\bf E}_1, {\bf E}_2]}$ satisfying the condition (1.9).

Proof. Suppose that (1.8) admits a strictly increasing solution $\Phi(\xi) = (\phi_1(\xi), \phi_2(\xi), \phi_3(\xi))\in {[{\bf E}_1, {\bf E}_2]}$ satisfying (1.9) with $c < c^*_1$. Then we define ${\psi(\xi): = \phi_1^\prime(\xi)/\phi_1(\xi)}$. From (1.8), one can verify that $\psi(\xi)$ satisfies the equation

Since $\Phi(-\infty) = {\bf 0}$, we have

According to (3.3) and [10, Proposition 3], the limit $\psi(-\infty)$ exists and satisfies

Then it follows from the proof of Lemma 2.1 that $c\ge c^*_1$, which gives a contradiction. This completes the proof.

4.   Stability of traveling wavefronts for (1.6) with large $c$

In this section, we will apply the weighted energy method to study the stability of traveling wavefronts for (1.6). Inspired by ^[14,19], we first introduce the following definition.

Definition 4.1. Let $I$ be an interval and $\omega (x):\mathbb{R}\to\mathbb{R}$ be a differentiable function.

${\rm(1)}$ Let $L^2(I)$ be the space of square integrable functions defined on $I$. We denote $L_\omega ^2 (I)$ by the weighted $L^2$-space with the weight function $\omega (x)$, which endows with the norm

${\rm(2)}$ Let $H^k(I)$ ($k \ge 0$) be the Sobolev space of the $L^2$-functions $f(x)$ defined on $I$ whose $i$th-derivative also belong to $L^2(I)$ for $i = 1, \cdots, k$. We denote $H_\omega ^k (I)$ by the weighted Sobolev space with the weight function $\omega (x)$, which endows with the norm

${\rm(3)}$ Let $T >$ 0 and $\mathbb{B}$ be a Banach space. We denote $C([0, T]; \mathbb{B})$ by the space of the $\mathbb{B}$-valued continuous functions defined on $[0, T]$, and $L^2([0, T];\mathbb{B})$ is regarded as the space of $\mathbb{B}$-valued $L^2$-function on $[0, T]$. The corresponding spaces of the $\mathbb{B}$-valued function on $[0, \infty)$ can be defined similarly.

Note that we always assume (H1) and (H2) throughout this article. Moreover, we assume the parameters satisfying the following assumption:

First, we establish the following global existence and uniqueness of solutions, and the comparison theorem for system (1.6) with initial data

satisfying the following condition:

Here we assume that the weight function $\omega (\xi)$ in (S2) is given by

for some positive constants $\sigma$ and $\xi _0$ which will be determined later.

Lemma 4.1. (See also [19].) Assume (S1)–(S2). Then the following statements are valid.

${\rm(1)}$ There exists a unique solution $U (t, x) = (u(t, x), v(t, x), w(t, x))$ of (1.6) with initial data $U_0(x)$ such that ${\bf E}_1\le U (t, x)\le {\bf E}_2,$ $\forall t > 0, \ x\in \mathbb{R}.$ In addition,

${\rm(2)}$Let $U^\pm(t, x)$ be solutions of (1.6) with $U^\pm(0, x) = (u^\pm(x), v^\pm(x), w^\pm(x))$, respectively. If ${\bf E}_1\le U^- (0, x)\le U^+ (0, x)\le {\bf E}_2, \ \forall x\in \mathbb{R},$ then

Proof. (1) The assertion can be derived by the theory of abstract functional differential equation, see ^[25]. Also the standard energy method and continuity extension method, see ^[26]. Here we skip the details.

(2) The proof of this part is the same as that of [27,Lemma 3.2] and omitted.

Then, applying the technique of weighted energy estimate, we have the following stability result.

Theorem 4.1. Assume that (S1)–(S2) hold. Let $\Phi (x+ct)$ be a traveling wave front of (1.6) satisfying (1.9) and with speed $c > \max\{{c}_1^{*}, c_1, c_2, c_3\}$ (Note that $c_i$, $i = 1, 2, 3$ are defined in (4.23)–(4.25)). Let $U(t, x) = (u(t, x), v(t, x), w(t, x))$ be the unique solution of the initial value problem (1.6). In addition, there exist small $\sigma = \sigma_0 > 0$ and large $\xi_0 > 0$ such that

and

for some positive constants $C$ and $\mu$.

To prove the result of Theorem 4.1 by using the weighted energy method, we need to establish a priori estimate for the difference of solutions of systems (1.6) and (1.8). For convenience, we denote $U(t, x) = (u(t, x), v(t, x), w(t, x))$ by the solution of system (1.6) with initial data $U_{0}(x) = (u_0(x), v_0(x), w_0(x))$ satisfying (S2). Then, $\forall x\in \mathbb{R}$, we set

It is clear that $U^{\pm}_0 (x)$ satisfy

Let $U^\pm(t, x)$ be solutions of (1.6) with initial data $U_0^\pm(x)$, by Lemma 4.1, we have

Then it follows from (4.7) that

for $(t, x)\in \mathbb{R}^+\times \mathbb{R}^.$ Therefore, to derive a priori estimate of $U (t, x)-\Phi (x+ct)$, it suffices to estimate the functions $U^\pm (t, x)-\Phi (x+ct)$.

4.1.   Weighted energy estimate

For convenience, let's denote

Then it follows from (4.6) and (4.7) that

In the sequel, we only estimate $V^+(t, x)$, since $V^-(t, x)$ can also be discussed in the same way. For convenience, we drop the sign $"+"$ for $V^+(t, x)$, $U^+(t, x)$ and set

By systems (1.6) and (1.8), we can obtain

It is easy to see that $V_i^0(\xi) \in H_\omega ^1 (\mathbb{R})$, then we have $V_i (t, \xi)\in C([0, +\infty), H_\omega ^1 (\mathbb{R}))$, for $i = 1, 2, 3$. To employing the technique of energy estimate to the equations (4.8), (4.9) and (4.10), it is necessary to assure that the solutions $V_i (t, \xi)$ have sufficient regularity. To this end, we mollify the initial condition setting

where $J_\varepsilon (\xi)$ is the usual mollifier. Let $V^{\varepsilon } (t, \xi)$ be the solutions of (4.8), (4.9) and (4.10) with this mollified initial condition $V^{0\varepsilon } (\xi) = (V_{1}^{0\varepsilon } (\xi), {V_{2}^{0\varepsilon } (\xi)}, V_{3}^{0\varepsilon } (\xi))$. Then, we have

Letting $\varepsilon \to 0$, it follows that $V^\varepsilon (t, \xi)\to V (t, \xi)$ uniformly for all $(t, \xi)\in \mathbb{R}^+\times {\mathbb{R}}$. Therefore, without loss of generality, we may assume $V_{i} (t, \xi)\in C([0, +\infty), H_\omega ^2 (\mathbb{R}))$, for $i = 1, 2, 3$ in establishing the following energy estimates (cf. [14]).

First, let's multiply both sides of (4.8), (4.9) and (4.10) by $e^{2\mu t}\omega (\xi) V_i (\xi, t)$ with $i = 1, 2, 3,$ respectively, where $\mu > 0$ will be determined later. Direct computations give

where

Applying the Cauchy–Schwarz inequality $2xy\le x^2+y^2$, we can obtain

Since $V_i (t, \xi)\in H_\omega ^1$, we have $\left. {\left\{ e^{2\mu t}\omega V_i^2 \right\}} \right|_{\xi = -\infty }^{\xi = \infty } = 0, \ \mbox{for }i = 1, 2, 3.$ Therefore, integrating both sides of (4.11), (4.12) and (4.13) over $\mathbb{R}\times [0, t]$ with respect to $\xi$ and $t$ and using (4.14), we can obtain

Noting that $r_2-\phi_2 > 0$. Summing up the inequalities (4.16)–(4.18), we can derive

where

For convenience to estimate $R_1^\mu(t, \xi)$, we further set

Then we have the following properties.

Lemma 4.2. Assume that ${\rm(S1)}$ holds. There exist small $\sigma_0 > 0$ and large $\xi_0 > 0$ such that, for $i = 1, 2, 3$,

Proof. By (S1) and the fact

there exists a small $\sigma_0 > 0$ such that

Fixing this $\sigma_0$, then it follows from (1.9) and (4.20) that $\lim\limits_{\xi\to\infty}\Lambda_i(\xi) = \ell_i > 0\ \mbox{and }$

Hence, this assertion holds by the continuity argument.

Let's choose $\omega(\xi)$ as the form (4.1), where $\sigma = \sigma_0$ and $\xi_0$ are the positive constants derived in Lemma 4.2. It's easy to see that

Furthermore, let's fix three wave speeds $c_i > 0$ such that

Then we estimate $R_i^\mu(t, \xi)$, $i = 1, 2, 3$ in the following lemma.

Lemma 4.3. Assume that $\rm(S1) –{\rm(S2)}$ hold and $c > \max\{ c_1^*, c_1, c_2, c_3\}$. Then there exists a small $\mu > 0$ such that the following statements hold:

${\rm(1)}$There exists a positive constant $C_0$ such that

${\rm(2)}$ There exists a positive constant $C_1$ such that

Proof. (1) Noting that $(0, 0, 0) < (V_1+\phi_1, V_2+\phi_2, V_3+\phi_3) < (u_*, r_2, w_*)$. Let's prove the assertion by considering the following four cases.

Case 1: $\xi < \xi _0 -1$. By Lemma 4.2 and (4.21)–(4.25), we have

Case 2: $\xi _0 -1 < \xi\le \xi_0$. In this case, $d_ie^{-\sigma_0}+d_i(1-e^{\sigma_0(\xi-\xi_0)}) > 0$, for $i = 1, 2, 3$. By Lemma 4.2 and (4.21)–(4.25), we have

Case 3: $\xi _0 < \xi \le \xi _0 +1$. In this case, one can see that $d_1[e^{\sigma_0}-e^{-\sigma_0(\xi-\xi_0-1)}] \leq 0$. By Lemma 4.2, (4.21) and (4.22), we have

Case 4: $\xi > \xi _0 +1$. In this case, by Lemma 4.2, (4.21) and (4.22), we have

According to the above four cases, we may choose a small $\mu > 0$ such that (4.26) holds for some positive constant $C_0$.

(2) The inequality (4.27) is a direct consequence of (4.19) and (4.26).

4.2.   Derivative Estimates

Now we consider the derivative estimates of system (4.8). By differentiating (4.8), (4.9) and (4.10) with respect to $\xi$, it follows that

Multiplying (4.28)–(4.30) by $e^{2\mu t}\omega (\xi)V_{i\xi } (t, \xi)$ with $i = 1, 2, 3$, respectively, we can obtain

Then, applying the Cauchy–Schwarz inequality, it follows that

Since $V_i \in H_\omega ^2$, we know that $\{e^{2\mu t}\omega {V_{i\xi }^2 }\} \big|_{\xi = -\infty }^{\xi = \infty } = 0, \ \mbox{for }i = 1, 2, 3.$ Therefore, by (4.34), (4.35) and integrating both sides of (4.31)–(4.33) over $[0, t]\times\mathbb{R}$ with respect to $t$ and $\xi$, we have

Summing up the inequalities (4.36)–(4.39), we can derive

where

Similar to the discussion of Lemma 4.3, we have the following lemma.

Lemma 4.4. Assume $\rm(S1) –{\rm(S2)}$ and $c > \max\{{c_1^*}, c_1, c_2, c_3\}$. There exists a small $\mu > 0$ such that the following statements hold:

${\rm(1)}$ There exists a positive constant $\widehat{C}_0$ such that

${\rm(2)}$ There exists a positive constant $\widehat{C}_1$ such that

Proof. (1) Using the same definitions of $\Lambda_i(\xi)$ and $c_j$ ($i = 1, \cdots, 6$, $j = 1, 2, 3$), the proof of this assertion is similar to that of part (1) in Lemma 4.3 and omitted.

(2) According to (4.40), we first consider the following integral:

Based on the properties of the traveling wavefront $(\phi _1 (\xi), \phi _2 (\xi), \phi _3 (\xi))$, we can know that $({\phi }'_1 (\xi), {\phi }'_2 (\xi), {\phi }'_3 (\xi))$ is bounded for all $\xi \in \mathbb{R}$. Thus, by the Young-inequality $2xy\le \varepsilon^{-1} x^2+{\varepsilon }y^2$ with $\varepsilon > 0$, we have

for some constant $\bar{C}_2 > 0$. Then, by (4.27), one has

Choosing $\varepsilon$ small enough, it follows from (4.40) and (4.41) that the inequality (4.42) holds. The proof is complete.

4.3.   Proof of Theorem 4.1

Based on Lemmas 6 and 7, we know that there exist positive constant $C_3$ and small $\mu = \mu^+ > 0$ such that

Since $\omega (\xi)\ge 1$, we have $H_\omega ^1 (\mathbb{ R})\hookrightarrow H^1(\mathbb{ R})\hookrightarrow C(\mathbb{ R})$. Thus,

for some $C_4 > 0$. Hence, it follows from (4.44) that there exists a positive constant $C^+$ such that

Similar to the previous discussions, there exist positive constant $C^-$ and small $\mu = \mu^- > 0$ such that

Hence, we can conclude that

for some positive constants $C$ and $\mu$. The proof of Theorem 4.1 is complete.

5.   Stability of traveling wavefronts for (1.7) with large $c$

In this section, we will also apply the weighted energy method to study the stability of traveling wavefronts obtained in Theorem 3.1. However, due to the lattice structure of system (1.7), we should adopt different weighted spaces to derive the weighted energy estimates. Therefore, we first introduce the following notations.

Definition 5.1. Let $\omega(\cdot)\in C(\mathbb{R})$ be a given weighted function, for any fixed $t\ge 0$ and $c > {c}_1^*$, we denote the spaces $\ell^2$ and weighted spaces $\ell^2_\omega$ by

which are endowed with the following norms:

According to Definition 5.1, let us consider the initial value problem of (1.7) with initial data and $\{u_i(0)\}_{i\in\mathbb{Z}}, \ \{v_i(0)\}_{i\in\mathbb{Z}}, \ \{w_i(0)\}_{i\in\mathbb{Z}}$ satisfying the assumption

${\rm(L1)}$ $(u_i(0), v_i(0), w_i(0))\in[{\bf E}_1, {\bf E}_2]$ for all $i\in\mathbb{Z}$ and

Then we can obtain the following stability result.

Theorem 5.1. Assume that ${\rm(S1)}, {\rm(S2)}$ and ${\rm(L1)}$ hold. Let $\Phi (i+ct)$ be a traveling wavefront of (1.7) satisfying (1.9) and with speed $c > \max\{{c}_1^{*}, c_1, c_2, c_3\}$. Then the initial value problem of (1.7) admits a unique solution $\{u_i(t)\}_{i\in\mathbb{Z}}, \ \{v_i(t)\}_{i\in\mathbb{Z}}, \ \{w_i(t)\}_{i\in\mathbb{Z}}$ satisfying $(u_i(t), v_i(t), w_i(t))\in[{\bf E}_1, {\bf E}_2]$ for all $t > 0, \ i\in \mathbb{Z}$. In addition, for $t > 0$, we have

for some positive constants $C$ and $\mu$.

Proof. The proof is similar to that of Theorem 4.1 by replacing the weighted spaces $L^2$ and $L^2_{\omega}$ as $\ell^2$ and $\ell^2_{\omega}$ respectively, we sketch it in the sequel.

Step 1. Let $\{U_i(t)\}_{i\in\mathbb{Z}} = \{(u_i(t), v_i(t), w_i(t))\}_{i\in\mathbb{Z}}$ be the solution of system (1.7) with initial data $\{U_{i}(0)\}_{i\in\mathbb{Z}} = \{(u_i(0), v_i(0), w_i(0))\}_{i\in\mathbb{Z}}$ satisfying (L1). Then, $\forall i\in \mathbb{Z}$, we set

Based on assumption (A2), it is clear that $U^{\pm}_i (0)$ satisfy

Let $\{U_i^\pm(t)\}_{i\in\mathbb{Z}}$ be the solutions of (1.7) with initial data $\{U_i^\pm(0)\}_{i\in\mathbb{Z}}$, then we have

Then it follows from (4.7) that

for any $(t, i)\in \mathbb{R}^+\times \mathbb{Z}.$ Therefore, to derive a priori estimate of $U_i (t)-\Phi (i+ct)$, it suffices to estimate the functions $U_i^\pm (t)-\Phi (i+ct)$. For convenience, let's denote

Then it follows that

Hence, we only need to estimate $\{V_i^+(t)\}_{i\in\mathbb{Z}}$, since $\{V_i^-(t)\}_{i\in\mathbb{Z}}$ can also be discussed in the same way. For convenience, we drop the sign $"+"$ for $\{V_i^+(t)\}_{i\in\mathbb{Z}}$, $\{U_i^+(t)\}_{i\in\mathbb{Z}}$ and set

Step 2. Similar to (4.8)–(4.10), $V_i(t)$ satisfies

Step 3. Multiplying both sides of (5.4), (5.5) and (5.6) by $e^{2\mu t}\omega (\xi) X_i(t), \, e^{2\mu t}\omega (\xi) Y_i(t)$ and $e^{2\mu t}\omega (\xi) Z_i(t)$ respectively, we can obtain

where

Step 4. Let us set $X(t) = \{X_i(t)\}_{i\in\mathbb{Z}}, Y(t) = \{Y_i(t)\}_{i\in\mathbb{Z}}$ and $Z(t) = \{Z_i(t)\}_{i\in\mathbb{Z}}.$ Summing over all $i\in\mathbb{Z}$ for (5.7)–(5.9), integrating them over $[0, t]$ and applying the Cauchy-Schwarz inequality, we have

Summing up the inequalities (5.10)–(5.12), we can derive

where

Step 5. Similar to Lemma 4.3, there exists $\tilde{C}_0 > 0$ such that

Then, for $t\ge 0$, (5.13) implies that there exists a positive constant $\tilde{C}_1$ such that

Step 6. Since $\omega(\xi)\ge 1$, we have $\|\cdot\|_{\ell^2}\le\|\cdot\|_{\ell_\omega^2}.$ By the Sobolev's embedding inequality $\ell^2\hookrightarrow\ell^\infty$, we have

for some constant $C > 0$. Then it follows from (5.14) that

for some constant $C_1^+ > 0$. By (5.3) and similar arguments, we have

$\forall(t, i)\in \mathbb{R}^+\times \mathbb{Z},$ for some constant $C_2 > 0$. The proof is complete.

6.   Stability of traveling wavefronts for $c > c^*_1$

In this section, we will improve the stability results of Theorem 4.1 and Theorem 5.1 to any $c > c^*_1$. Different to (4.1), we consider the weighted function

Note that $\mu_1^* > 0$ is a constant given in Lemma 2.1 such that

Furthermore, we impose the following assumption:

${\rm (S3)}$ $r_1 > 2(c_{12}+c_{13}+c_{31})u_*+(3c_{12}+2c_{21}+4+2c_{23})r_2+2(c_{31}+c_{32})w_*.$

${\rm (S4)}$ $\hat{\mu}: = \min\{\frac{1}{2}[\min\{u_*-c_{12}r_2-c_{13}w_*, w_*-c_{31}u_*-c_{32}r_2\}-\max\{c_{13}u_*, c_{31}w_*\}], -2r_2+{2c_{21}u_*+2c_{23}w_*}\} > 0$.

Example 6.1. Assume that

Then the parameters satisfy the assumptions $\rm(H1), (H2), (S1), (S3)$ and $\rm(S4)$. In addition, we have

Similar to (4.19), we can obtain the following estimation:

where each ${\mathcal R}^{\mu}_i(t, \xi)$ has the same form as ${R}^{\mu}_i(t, \xi)$ but replacing $\omega(\cdot)$ as $\omega^*(\cdot)$. Similar and simpler than Lemma 4.3, we have the following result.

Lemma 6.1. Assume that $\rm(S3)$ holds and $c > c^*_1$. Then there exists a small $\mu > 0$ such that the following statements hold:

${\rm(1)}$There exists a positive constant $C_0$ such that

${\rm(2)}$ There exists a positive constant ${C}_1$ such that

Proof. (1) Noting that $(0, 0, 0) < (V_1+\phi_1, V_2+\phi_2, V_3+\phi_3) < (u_*, r_2, w_*)$. Since $d_1\ge d_2, d_3$, it follows from (6.2) that

By (6.6) and elementary computations, we have

Then it follows from (S3) that

Therefore, we may choose a small $\mu > 0$ such that (6.4) holds for some ${C}_0 > 0$.

(2) The proof of this part is the same as Lemma 4.3 and skipped. Similar to Lemmas 7 and, we have

Lemma 6.2. Assume $\rm(S3)$ and $c > c^*_1$. There exists a small $\mu > 0$ such that the following statements hold:

${\rm(1)}$ There exists a positive constant $\widehat{C}_0$ such that

${\rm(2)}$ There exists a positive constant $\widehat{C}_1$ such that

Note that each $\widehat{\mathcal R}^{\mu}_i(t, \xi)$ has the same form as $\widehat{R}^{\mu}_i(t, \xi)$ but replacing $\omega(\cdot)$ as $\omega^*(\cdot)$. as a consequence Lemmas 8 and 9, we know that there exist positive constant $\tilde{C}$ and small $\mu = \tilde{\mu} > 0$ such that

Since $\omega^*(\xi)\rightarrow 0$ as $\xi\rightarrow \infty$, it is not true that $H_{\omega^*} ^1(\mathbb{R})\hookrightarrow C(\mathbb{R})$. However, for any $I = (-\infty, \bar{\xi}]$ for some large $\bar{\xi}\gg 1$, we can obtain $H_{\omega^*} ^1(I)\hookrightarrow C(I)$. Thus, (6.9) implies the following lemma.

Lemma 6.3. For all $t > 0$, $i = 1, 2, 3$, it holds that

for some $\tilde{\mu} > 0$ and large $\bar{\xi}\gg 1$.

To extend the result of Lemma 6.3 to the whole space $(-\infty, \infty)$, we have to prove the convergence of $V_i(\xi, t)$ as $\xi\to\infty$.

Lemma 6.4. Assume that $\rm(S4)$ holds. There exists some constant $C > 0$ such that

Note that $\hat\mu$ is given in ${\rm(S4)}$.

Proof. It's easy to see that $V_{i\xi } (\infty, t) = 0$ and $d_i {D}[V_i ](+\infty) = 0$ for $i = 1, 2, 3$. Based on (4.8)–(4.10) and the boundedness of $\mathcal{V}_i(t): = V_i (\infty, t)$ for all $\xi \in (-\infty, \infty)$, letting $\xi \to \infty$, one immediately obtains

Let's set

By the assumption $\rm(S4)$, we see that $A_2 > 0$. Integrating (6.13) over $[0, t]$, we have

Then it follows from (6.12) and (6.14) that

where $\mathcal{A}: = \min\{A_1, A_3\}-\max\{c_{12}u_*, c_{32}w_*\}$. We claim that there exists some positive constant $\hat{ C}$ such that

Note that $\hat{\mu} = \min\{{\mathcal A}/{2}, A_2\}$. In fact, if ${\mathcal A}\ne A_2$, we then have

where

If ${\mathcal A} = A_2$, then we obtain

for some $\hat{ C}_2 > 0$. Thus, the claim holds. Therefore, we conclude that

for some positive constant $C$. This completes the proof.

Based on the above lemmas, we can also obtain the following stability result.

Theorem 6.1. Assume that ${\rm(S3)}–{\rm(S4)}$ hold. Let $\Phi (x+ct)$ be a traveling wavefront of (1.6) satisfying (1.9) and with speed $c > {c}^{*}_1$. Then the initial value problem (1.6) admits a unique solution $U(t, x) = (u(t, x), v(t, x), w(t, x))$ satisfying $U(t, x)\in[{\bf E}_1, {\bf E}_2]\ \mathit{\mbox{for all}}t > 0, \ x\in \mathbb{R}$. In addition, we have

and

for some positive constants $C$ and $\mu$.

By the same way, we also have the following stability result for (1.7).

Theorem 6.2. Assume that ${\rm(S3)}–{\rm(S4)}$ hold. Let $\Phi (i+ct)$ be a traveling wavefront of (1.7) satisfying (1.9) and with speed $c > {c}^{*}_1$. Then the initial value problem of (1.7) admits a unique solution $\{u_i(t)\}_{i\in\mathbb{Z}}, \ \{v_i(t)\}_{i\in\mathbb{Z}}, \ \{w_i(t)\}_{i\in\mathbb{Z}}$ satisfying $(u_i(t), v_i(t), w_i(t))\in[{\bf E}_1, {\bf E}_2]$ for all $t > 0, \ i\in \mathbb{Z}$. In addition, for $t > 0$, we have

for some positive constants $C$ and $\mu$.

7.   Discussion

In population dynamics, traveling wave solution can be used to describe the spatial spread or invasion of the species. In this article we consider the existence and stability of the traveling wavefronts of discrete diffusive systems which come from the competition and cooperations between three species.

In Theorem 3.1, we proved that both systems (1.6) and (1.7) admit traveling wavefronts connecting the extinct state ${\bf E_1}$ and co-existence state ${\bf E_2}$, provided the assumptions (H1)-(H2) hold and the propagation wave speed $c$ is greater than the minimum speed $c_1^*$. Roughly speaking, to guarantee the assumptions (H1)-(H2) hold, it is required that $d_2, r_2, c_{12}, c_{32}$ are small enough, and $d_1, \, r_1$ are large enough. Biologically, it means that the diffusion effect, growth rate for the species $v$ and the competition relation between $v$ and the other species are very weak. Since the species $u$ and $w$ cooperate with each other; the species $u$ has strong diffusion effect and growth rate; and their competition from the species $v$ are very weak, this gives us the reason why the minimal speed is determined by the linearization problem of the first $u$-equation of both systems. And also the existence of traveling wavefronts propagating from the extinct state to the co-existence state.

As mentioned in introduction, when the traveling wavefronts are disturbed under small perturbations, only stable such solutions can be visualized in the real world. However, since such solutions exist for all $c > c_1^*$, generically any one of them won't be globally asymptotic stable. Therefore, we introduce the weight functions to split the domain of attractions of traveling wavefronts with different speeds, and then obtain the stability results.

Acknowledgments.

The authors would like to thank the anonymous referees for their valuable comments and suggestions which have led to an improvement of the presentation. The first author was partially supported by the MOST of Taiwan (Grant No. MOST 107-2115-M-008-009-MY3) and NCTS of Taiwan, the second author was partially supported by the MOST of Taiwan (Grant No. MOST 107-2115-M-027-002), and the third author was partially supported by the NSF of China (Grant No. 11671315).

Conflict of interest.

The authors declare that they have no competing interests.