Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey–predator system

Abstract

In this paper, we consider the positive steady state problem of a spatially heterogeneous cross-diffusion prey–predator model with modified Leslie–Gower and Holling-Type II schemes. The heterogeneity here is created by a protection zone for the prey. By the bifurcation method and a priori estimates, we discuss the existence and non-existence of positive steady states. Moreover, uniqueness and stability of positive steady states for small birth rate of the predator are shown as well as the asymptotic behavior of positive steady states when some coefficients tend to infinity. Our result reveals that large cross-diffusion in a heterogeneous environment has a profound effect on the positive steady state set, and the bifurcation continuum of positive steady states changes from a bounded one to an unbounded one as the cross-diffusion varies from 0 to a large number for suitable ranges of the parameter. Whereas it has little effect in the homogeneous environment. The impact of the protection zone is also quite important, and it deduces a critical number for the birth rate $\lambda$ of the prey which determines the bifurcation continuum of positive steady states to be bounded or unbounded. Furthermore, the modified Leslie–Gower term yields essentially different results of positive solutions from the Leslie–Gower term.

Introduction

In this paper, we shall consider the following prey–predator model with cross-diffusion and protection for the prey:

$$\{\begin{array}{c}{u}_t=\Delta \left[(1+\tau \rho \left(x\right)v)u\right]+u(\lambda -u-\frac{m\left(x\right)v}{k+u})\text{,}x\in \Omega ,t>0\text{,}\hfill \\ v_t=\Delta v+v(\mu -\frac{cv}{d+u})\text{,}x\in \Omega \setminus {\bar{\Omega }}_0,t>0\text{,}\hfill \\ {\partial }_{\nu }u=0\text{,}x\in \partial \Omega ,t>0\text{,}{\partial }_{\nu }v=0\text{,}x\in \partial \Omega \cup \partial {\Omega }_0,t>0\text{,}\hfill \\ u(x,0)=u_0\left(x\right)\ge 0\text{,}x\in \Omega \text{,}v(x,0)=v_0\left(x\right)\ge 0\text{,}x\in \Omega \setminus {\bar{\Omega }}_0\text{.}\hfill \end{array}$$

Here $\Omega$ is a bounded domain in ${\mathbb{R}}^n(n\ge 1)$, ${\Omega }_0$ is a smooth domain satisfying ${\bar{\Omega }}_0\subset \Omega ;\nu$ is the outward unit normal vector on the boundary and ${\partial }_{\nu }=\partial /\partial \nu$;$u(x,t)$ and $v(x,t)$ represent the population density of the prey and predator, respectively; $\rho \left(x\right)=1$ and $m\left(x\right)=m>0$ in $\bar{\Omega }\setminus {\Omega }_0$, whereas $\rho \left(x\right)=m\left(x\right)=0$ in ${\Omega }_0;{\Omega }_0$ is a protection for the prey $u$, the predator $v$ cannot enter ${\Omega }_0$, and so ${\partial }_{\nu }v=0$ on $\partial {\Omega }_0$, while the prey can enter and leave ${\Omega }_0$ freely; $\tau \ge 0$ is the cross-diffusion coefficient and represents the sensitivity of the prey species to the population pressure from the predator species; $\lambda >0$ and $\mu >0$ denote the birth rate of the respective species; $k>0$ (resp. $d>0$) measures the extent to which the environment provides protection to $u$ (resp. $v$); $m>0$ in $\bar{\Omega }\setminus {\Omega }_0$ is the maximum value which per capita reduction rate of $u$ can attain; $c>0$ has a similar meaning to $m$. The system is self-contained, and there is no flux on the boundary $\partial \Omega$.

There is a nonlinear diffusion term $\tau \Delta \left[\rho \left(x\right)vu\right]$ in the first equation of (1.1), which is usually referred to as the cross-diffusion term and reflects the fact that the movement of prey $u$ is affected by the population pressure from the predator $v$. This was first proposed by Shigesada et al. [1] to model the habitat segregation phenomena between two competing species, one can see [2] for more backgrounds for cross-diffusion. Many researchers have payed attention to studying the effect of cross-diffusion from various aspects since the pioneering work [1], including the global existence of solutions and positive steady state problem, one can refer to [3], [8], [10], [9], [4], [5], [7], [11], [14], [12], [13], [6], [15], [16], [17] and references therein for more details.

The reaction–diffusion system with spatially homogeneous coefficients have been widely and extensively studied since the 1970s, in particular, the ODE and diffusive versions of (1.1) have been studied in [18] and [19]. While some interesting papers investigating the heterogeneous effect of environment have appeared in recent years. Dancer and Du [20] and Du et al. [21], [23], [24], [25], [26], [27], [22] have studied the effects of the heterogeneous environment caused by the protection zone or the degeneracy of some intra-specific pressures. The effects of spatially heterogeneous birth rates have been shown by Dockery et al. [28] and Hutson et al. [29], [31], [30], [32] for some diffusive competition models. We note that cross-diffusion is not included in the above work. Taking cross-diffusion into account, there is little work. Oeda[33] studied a cross-diffusive prey–predator model with a protection zone, and obtained some interesting results. One can also refer to [34] and [35] for a discussion of the heterogeneous environment which is not caused by the above cases.

The main purpose of this paper is to consider the positive steady state problem of (1.1). That is to say, we shall study the following problem

$$\{\begin{array}{c}\Delta \left[(1+\tau \rho \left(x\right)v)u\right]+u(\lambda -u-\frac{m\left(x\right)v}{k+u})=0\text{,}x\in \Omega \text{,}\hfill \\ \Delta v+v(\mu -\frac{v}{1+u})=0\text{,}x\in {\Omega }_1=\Omega \setminus {\bar{\Omega }}_0\text{,}\hfill \\ {\partial }_{\nu }u=0\text{,}x\in \partial \Omega \text{,}{\partial }_{\nu }v=0\text{,}x\in \partial {\Omega }_1\text{,}\hfill \end{array}$$

where we take $c=d=1$ without loss of generality. From the biological viewpoint, we tend to obtain the bifurcation structure of the positive solution set of (1.2), where a positive solution corresponds to a coexistence steady state of the prey and predator.

One of our main goals in the paper is to investigate the effects of cross-diffusion and a heterogeneous environment on the positive solution set of (1.2). By Remark 3.7 in Section 3, we can see that cross-diffusion in the homogeneous environment (${\Omega }_0=0̸$) has little effect on (1.2). That is, when ${\Omega }_0=0̸$, the bifurcation curve of positive solutions of (1.2) remains bounded as the cross-diffusion varies from 0 to a large value. While if the environment is heterogeneous (${\Omega }_0\ne 0̸$), cross-diffusion has a profound impact on the structure of the positive solution set. In the heterogeneous case, there exists a critical value ${\lambda }^{\ast }(\tau ,{\Omega }_0)$ such that a bounded continuum $\mathcal{C}$ bifurcates from the semitrivial solution $(\lambda ,0)$ at $\mu =0$ and joins with $(0,\mu )$ at some point $\mu ={\mu }_1$ if $0<\lambda <{\lambda }^{\ast }(\tau ,{\Omega }_0)$; while if $\lambda \ge {\lambda }^{\ast }(\tau ,{\Omega }_0)$, the bounded continuum $\mathcal{C}$ bifurcating from $(\lambda ,0)$ at $\mu =0$ becomes unbounded and tends to $\infty$ as the bifurcation parameter $\mu$ tends to $\infty$. Furthermore, Lemma 2.1 deduces that ${\lambda }^{\ast }(\tau ,{\Omega }_0)\to 0$ as $\tau \to \infty$ and ${\lambda }^{\ast }(0,{\Omega }_0)={\lambda }_1^D\left({\Omega }_0\right)$, where ${\lambda }_1^D\left({\Omega }_0\right)$ denotes the principal eigenvalue of $-\Delta$ over ${\Omega }_0$ subject to Dirichlet boundary condition. Then as $\tau$ is sufficiently large such that $\lambda$ satisfies ${\lambda }^{\ast }(\tau ,{\Omega }_0)\le \lambda <{\lambda }_1^D\left({\Omega }_0\right)$, one sees that large cross-diffusion causes the original bounded continuum $\mathcal{C}$ to be an unbounded one. The bifurcation structure of the positive solution set totally changed as $\tau$ changes from 0 to a large number. The effect of a heterogeneous environment is the same as that discussed in [25] and [33]. Precisely, there exists a critical patch size of the protection zone for every model, and the critical size is determined by an equation of the form $\lambda ={\lambda }^{\ast }(\tau ,{\Omega }_0)$, which corresponds to the value ${\lambda }_1^D\left({\Omega }_0\right)$ in [25] as $\tau =0$. If the protection zone is below its critical size, namely $0<\lambda <{\lambda }^{\ast }(\tau ,{\Omega }_0)$, (1.2) has no positive solution for large $\mu >0$; while if the protection zone is above the critical size, namely $\lambda \ge {\lambda }^{\ast }(\tau ,{\Omega }_0)$, then there exits coexistence state of the prey and predator even though the predator $v$ has a large birth rate $\mu$.

The other goal of this paper is to study the asymptotic behavior of positive solutions of (1.2). Our result shows that if $0<\lambda \le {\lambda }_1^D\left({\Omega }_0\right)$, then all positive solutions $(u_{\tau },v_{\tau })$ of (1.2) tend to a uniform steady state, that is, $(0,\mu )$; while if $\lambda >{\lambda }_1^D\left({\Omega }_0\right)$, the prey and predator species become spatially segregated. The asymptotic behavior of positive solution $u$ as $\mu \to \infty$ is similar to that in case $\tau \to \infty$, while $v\to \infty$ uniformly in ${\bar{\Omega }}_1$ as $\mu \to \infty$. Furthermore, Theorem 4.3 shows that the positive non-constant solution of (1.2) is unique and linearly stable for small $\mu >0$.

We next compare the results when $\tau =0$ with those of [24], where the same kind of protection zone is introduced. In [24], the authors considered the following system

$$\{\begin{array}{c}-\Delta u=u(\lambda -u-b\left(x\right)v)\text{,}x\in \Omega \text{,}\hfill \\ -\Delta v=v(\mu -\frac{v}{u})\text{,}x\in {\Omega }_1\text{,}\hfill \\ {\partial }_{\nu }u=0\text{,}x\in \partial \Omega \text{,}{\partial }_{\nu }v=0\text{,}x\in \partial {\Omega }_1\text{,}\hfill \end{array}$$

where $\lambda$ and $\mu$ are positive constants, $b\left(x\right)$ satisfies

$$b\left(x\right)=\beta {\chi }_{\bar{\Omega }\setminus {\bar{\Omega }}_0}=\{\begin{array}{cc}\beta \text{,}& \text{when }x\in \bar{\Omega }\setminus {\bar{\Omega }}_0\text{,}\\ 0\text{,}& \text{when }x\in {\bar{\Omega }}_0\text{.}\end{array}$$

As pointed out in their paper, the techniques developed there can be easily modified to obtain similar results for the following Leslie prey–predator model with the Holling-Type II interaction term for the prey:

$$\{\begin{array}{c}-\Delta u=u(\lambda -u-\frac{m\left(x\right)v}{k+u})\text{,}x\in \Omega \text{,}\hfill \\ -\Delta v=v(\mu -\frac{v}{u})\text{,}x\in {\Omega }_1\text{,}\hfill \\ {\partial }_{\nu }u=0\text{,}x\in \partial \Omega \text{,}{\partial }_{\nu }v=0\text{,}x\in \partial {\Omega }_1\text{,}\hfill \end{array}$$

where we have adopted the same notations as those used in our paper for convenience. There is a fundamentally different feature between (1.2), (1.4). As shown in [18], the Leslie–Gower term $v/u$ measures the loss in the predator population due to the rarity of its favorite food. In the case of severe scarcity, $v$ can switch over to other populations but its growth will be limited by the fact that its favorite food $u$ is not available in abundance, $v/(1+u)$ is suitable to capture such feature in this case. Thus the predator $v$ is a little stronger than the one in (1.4).

Due to the difference, the structure of the positive solution set of (1.2), (1.4) is quite different. There does not exist a critical value for the birth rate $\lambda$ of $u$ in (1.4), (1.4) has a positive solution for any $\lambda ,\mu >0$. This implies that even though $0<\lambda <{\lambda }_1^D\left({\Omega }_0\right)$, there exists coexistence state of the prey and predator for large $\mu$. While (1.2) with $\tau =0$ and $0<\lambda <{\lambda }_1^D\left({\Omega }_0\right)$ has no positive solution for large $\mu >0$ due to the stronger predator $v$ in (1.2). However, in the limiting case $\tau \to \infty$, ${\lambda }^{\ast }(\tau ,{\Omega }_0)\to 0$, we see that the prey species of (1.2) can coexist with the predator species for any $\lambda ,\mu >0$. Besides the difference between the bifurcation structure, we can also see that the asymptotic behavior of positive solutions as $\mu \to \infty$ is essentially different. From Theorems 1.2 and 1.3 in [24], the positive solution $(u,v)$ of (1.4) remains bounded as $\mu \to \infty$ when either $0<\lambda \le {\lambda }_1^D\left({\Omega }_0\right)$ or $\lambda >{\lambda }_1^D\left({\Omega }_0\right)$; in particular, for $\lambda >{\lambda }_1^D\left({\Omega }_0\right)$, $v\to V_{\lambda }$ uniformly on compact subset of $\bar{\Omega }\setminus {\bar{\Omega }}_0$, where $V_{\lambda }$ is the unique positive solution of the following boundary blow-up problem

$$\{\begin{array}{c}-\Delta v=\lambda v-m{v}^2\text{,}x\in {\Omega }_1\text{,}\hfill \\ {\partial }_{\nu }v=0\text{,}x\in \partial \Omega \text{,}\hfill \\ v=\infty \text{,}x\in \partial {\Omega }_0\text{.}\hfill \end{array}$$

However, Theorem 4.1 in this paper shows that for any positive solution $(u,v)$ of (1.2) with $\tau =0$, it holds that $v\to \infty$ uniformly in ${\bar{\Omega }}_1$, that is, $v/\mu \to 1$ uniformly in ${\bar{\Omega }}_1$. The asymptotic behavior of $u$ is similar between (1.2), (1.4). To be more precise, $u\to {\tilde{u}}_{\lambda }$ in $\bar{\Omega }$ as $\mu \to \infty$ for both (1.2), (1.4) if $\lambda >{\lambda }_1^D\left({\Omega }_0\right)$, where ${\tilde{u}}_{\lambda }$ is defined in Theorem 4.1.

If the interaction term is the classical Lotka–Volterra type, which has been discussed in [33], the structure of the positive solution set of two models are also slightly different. The predator in such a situation can have a death rate and there exists a coexistence state of the two species for negative $\mu$; whereas, the predator with the modified Leslie–Gower type must have a positive birth rate to assure the coexistence of the two species.

Throughout the paper, for any bounded domain $O\subset {\mathbb{R}}^n$ with smooth boundary, let us define the norms of spaces $L^p\left(O\right)$($p\ge 1$) and $C\left(\bar{O}\right)$ by

$${\Vert u\Vert }_{p,O}={\left({\int }_O{\left|u\right|}^p\right)}^{\frac{1}{p}}\text{and}{\Vert u\Vert }_{\infty ,O}=\underset{\bar{O}}{max}\left|u\right|\text{,}$$

respectively. Furthermore, for $q\in L^{\infty }\left(O\right)$, we denote ${\lambda }_1^D(q,O)$ and ${\lambda }_1^N(q,O)$ by the principal eigenvalue of $-\Delta +q$ over $O$ subject to Dirichlet and Neumann boundary condition, respectively.

The rest of the paper is organized as follows. In Section 2, we give some preliminary results and a priori estimates of positive solutions of (1.2). Some important nonexistence results will also be shown. In Section 3, we obtain the bifurcation structure of positive solutions of (1.2) by the bifurcation theory. In Section 4, asymptotic behavior of positive solutions of (1.2) will be given. We will also show the uniqueness and linear stability of (nonconstant) positive solutions for suitable ranges of $\mu$.