Positive solutions for a three-trophic food chain model with diffusion and Beddington–Deangelis functional response

Abstract

In this paper, we investigate the existence, stability, permanence, and global attractor of positive solutions for the following three species food chain model with diffusion and Beddington–Deangelis functional response

$$\{\begin{array}{cc}-\Delta u_1=u_1(r-u_1)-\frac{a_1{u}_1{u}_2}{e_1+u_1+u_2}\hfill & \mathrm{in}\Omega \text{,}\hfill \\ -\Delta u_2=\frac{m_1{u}_1{u}_2}{e_1+u_1+u_2}-b_1{u}_2-\frac{a_2{u}_2{u}_3}{e_2+u_2+u_3}\hfill & \mathrm{in}\Omega \text{,}\hfill \\ -\Delta u_3=\frac{m_2{u}_2{u}_3}{e_2+u_2+u_3}-b_2{u}_3\hfill & \mathrm{in}\Omega \text{,}\hfill \\ k_1\frac{\partial u_1}{\partial \nu }+u_1=k_2\frac{\partial u_2}{\partial \nu }+u_2=k_3\frac{\partial u_3}{\partial \nu }+u_3=0\hfill & \mathrm{on}\partial \Omega \text{,}\hfill \end{array}$$

where $\Omega$ is a bounded domain of ${\mathbb{R}}^N$, $N\ge 1$, with boundary $\partial \Omega$ of class ${\mathcal{C}}^{2+\alpha }$ for some $\alpha \in (0,1)$, $\nu$ is the outward unit vector on $\partial \Omega$, the parameters $r,a_i,b_i,e_i,m_i(i=1,2)$ are strictly positive, and $k_i\ge 0(i=1,2,3)$, $u_i(i=1,2,3)$ are the respective densities of prey, predator, and top predator.

Introduction

Understanding of spatial and temporal behaviors of interaction species in ecological systems is a center issue in population ecology. One aspect of great interest for a model with multispecies interactions is whether the involved species can persist or even stabilize at a coexistence steady state. In the case whether the species are homogeneously distributed, this would be indicated by a constant positive solution of a ordinary differential equation system. In the spatially inhomogeneous case, the existence of a nonconstant time-independent positive solution, also called stationary pattern, which is an indication of the richness of the corresponding partial differential equation dynamics. In recent years, stationary pattern induced by diffusion has been studied extensively, and many important phenomena have been observed.

The role of diffusion in the modeling of many physical, chemical and biological processes has been extensively studied. Starting with Turing’s seminal 1952 paper [1], diffusion and cross-diffusion have been observed as causes of the spontaneous emergence of ordered structures, called patterns, in a variety of non-equilibrium situations. These include the Gierer–Meinhardt model [2], [3], [4], [5], [6], the Sel’kov model [7], [8], the Noyes–Field model for Belousov–Zhabotinskii reaction [9], the chemotactic diffusion model [10], [11], the Brusselator model [12], the competition model [13], [14], [15], [16], [17], the predator–prey model [18], [19], [20], [21], [22], [23], [24], [25], [26], as well as models of semiconductors, plasmas, chemical waves, combustion systems, embryogenesis, etc., see e.g. [27], [28], [29] and references therein. Diffusion-driven instability, also called Turing instability, has also been verified empirically [30], [31].

This paper studies the stationary solution of the parabolic model

$$\{\begin{array}{cc}\frac{\partial u_1}{\partial t}-\Delta u_1=u_1(r-u_1)-\frac{a_1{u}_1{u}_2}{e_1+u_1+u_2}\hfill & \text{in }\Omega \times {\mathbb{R}}_{+}\text{,}\hfill \\ \frac{\partial u_2}{\partial t}-\Delta u_2=\frac{m_1{u}_1{u}_2}{e_1+u_1+u_2}-b_1{u}_2-\frac{a_2{u}_2{u}_3}{e_2+u_2+u_3}\hfill & \text{in }\Omega \times {\mathbb{R}}_{+}\text{,}\hfill \\ \frac{\partial u_3}{\partial t}-\Delta u_3=\frac{m_2{u}_2{u}_3}{e_2+u_2+u_3}-b_2{u}_3\hfill & \text{in }\Omega \times {\mathbb{R}}_{+}\text{,}\hfill \\ k_1\frac{\partial u_1}{\partial \nu }+u_1=k_2\frac{\partial u_2}{\partial \nu }+u_2=k_3\frac{\partial u_3}{\partial \nu }+u_3=0\hfill & \text{on }\partial \Omega \times {\mathbb{R}}_{+}\text{,}\hfill \end{array}$$

with initial values $u_i(x,0)=u_{i0}\left(x\right)\ge 0,\not\equiv 0,(i=1,2,3)$ in $\Omega$, where $\Omega$ is a bounded domain of ${\mathbb{R}}^N$, $N\ge 1$, with boundary $\partial \Omega$ of class ${\mathcal{C}}^{2+\alpha }$ for some $\alpha \in (0,1)$, $\nu$ is the outward unit vector on $\partial \Omega$, the parameters $r,a_i,b_i,e_i,m_i(i=1,2)$ are strictly positive, and $k_i\ge 0(i=1,2,3)$, and $u_{i0}(i=1,2,3)$ stand for the initial conditions. We denote ${\mathbb{R}}_{+}≔(0,\infty )$.

Problem (1.1) models three species food chain with diffusion and Beddington–Deangelis functional response, where $u_i(i=1,2,3)$ are the respective densities of prey, predator, and top predator (cf. [32], [33], [34], and the list of references therein).

In our work here, we are mainly characterizing the existence of steady-states of (1.1), which are the solution of

$$\{\begin{array}{cc}-\Delta u_1=u_1(r-u_1)-\frac{a_1{u}_1{u}_2}{e_1+u_1+u_2}\hfill & \text{in }\Omega \text{,}\hfill \\ -\Delta u_2=\frac{m_1{u}_1{u}_2}{e_1+u_1+u_2}-b_1{u}_2-\frac{a_2{u}_2{u}_3}{e_2+u_2+u_3}\hfill & \text{in }\Omega \text{,}\hfill \\ -\Delta u_3=\frac{m_2{u}_2{u}_3}{e_2+u_2+u_3}-b_2{u}_3\hfill & \text{in }\Omega \text{,}\hfill \\ k_1\frac{\partial u_1}{\partial \nu }+u_1=k_2\frac{\partial u_2}{\partial \nu }+u_2=k_3\frac{\partial u_3}{\partial \nu }+u_3=0\hfill & \text{on }\partial \Omega \text{.}\hfill \end{array}$$

Owing to the classical theory of parabolic equations, the solutions of (1.1) are globally defined in time and satisfy

$$u_i(\cdot ,t)\ge 0\text{,}(i=1,2,3),\text{for all }t\in {\mathbb{R}}_{+}\text{.}$$

So, in this paper, we will discuss the nonnegative solutions of (1.2). We have a trivial nonnegative solution $(u_1,u_2,u_3)=(0,0,0)$. As in [35], [36], the other nonnegative solutions of (1.2) can be classified by three types:

• nonnegative solutions with exactly two components identically zero

  $$({\bar{u}}_1,0,0)\text{,}(0,{\bar{u}}_2,0)\text{,}(0,0,{\bar{u}}_3)\text{;}$$

• nonnegative solutions with exactly one component identically zero

  $$(u_1^{\star },u_2^{\star },0)\text{,}({\bar{u}}_1,0,{\bar{u}}_3)\text{,}(0,{\hat{u}}_2,{\hat{u}}_3)\text{;}$$

• nonnegative solutions with no component identically zero.

We call solutions of the first two types semitrivial solutions, while those of the third type we call positive solutions.

In the present work, we attempt to further understand the influence of diffusion and functional response on pattern formation. As a consequence, the existence and nonexistence results for nonconstant positive steady state solution to (1.2) indicate the stationary pattern arises as the diffusion coefficient enter into certain regions. In other words, diffusion does help to create stationary pattern. On the other hand, our results also demonstrate that diffusion and functional response can become determining factors in the formation pattern.

The distribution of this paper is the following. In Section 2, we give some fundamental theorems. In Section 3, the necessary and sufficient conditions for positive solutions of (1.2) are investigated. To achieve this, some degree theorems are developed, which is playing an important role in the text. Finally, in Section 4, sufficient conditions for the extinction and permanence to the time-dependent system (1.1) are investigated.