Non-Constant Positive Steady States for a Predator-Prey Cross-Diffusion Model with Beddington-DeAngelis Functional Response

Shengmao Fu,Lina Zhang

doi:10.1155/2011/404696

Abstract

This paper deals with a predator-prey model with Beddington-DeAngelis functional response under homogeneous Neumann boundary conditions. We mainly discuss the following three problems: (1) stability of the nonnegative constant steady states for the reaction-diffusion system; (2) the existence of Turing patterns; (3) the existence of stationary patterns created by cross-diffusion.

Highlights

Introduction∂νu ∂νv 0, x ∈ ∂Ω, t > 0, u x, 0 u0 x > 0, v x, 0 v0 x ≥ 0, x ∈ Ω, where Ω ⊂ ÊN is a bounded domain with smooth boundary ∂Ω and ν is the outward unit normal vector of the boundary ∂Ω
Consider the following predator-prey system with diffusion: ut − d1Δu r1u − u K− fv, x ∈ Ω, t > 0, vt − d2Δv r2v v δu, x ∈ Ω, t > 0,∂νu ∂νv 0, x ∈ ∂Ω, t > 0, u x, 0 u0 x > 0, v x, 0 v0 x ≥ 0, x ∈ Ω, where Ω ⊂ ÊN is a bounded domain with smooth boundary ∂Ω and ν is the outward unit normal vector of the boundary ∂Ω
The homogeneous Neumann boundary conditions indicate that the system is self-contained with zero population flux across the boundary

Summary

Introduction

∂νu ∂νv 0, x ∈ ∂Ω, t > 0, u x, 0 u0 x > 0, v x, 0 v0 x ≥ 0, x ∈ Ω, where Ω ⊂ ÊN is a bounded domain with smooth boundary ∂Ω and ν is the outward unit normal vector of the boundary ∂Ω. In 7, 8 , Peng and Wang studied the long time behavior of time-dependent solutions and the global stability of the positive constant steady state for 1.1 with Holling-Tanner-type functional response i.e., f βu/ m u. They established some results for the existence and nonexistence of non-constant positive steady states with respect to diffusion and cross-diffusion rates. When the function f in the system 1.1 takes the form f βu/ u mv called ratio-dependent functional response, Peng, and Wang 10 studied the global stability of the unique positive constant steady state and gained several results for the non-existence of non-constant positive solutions.

The Long Time Behavior of Time-Dependent Solutions

Global Attractor and Permanence

Local Stability of Nonnegative Equilibria

B1 Ωu 4

B2 Ωu 4

Stationary Patterns for the PDE System without Cross-Diffusion

Non-Existence of Non-Constant Positive Steady States

Existence of Non-Constant Positive Steady States

Bifurcation

A Priori Upper and Lower Bounds

Conclusions