Global solutions for a general predator-prey model with cross-diffusion effects

Lina Zhang,Shengmao Fu

doi:10.1186/1687-1847-2012-117

Abstract

This work investigates a general Lotka-Volterra type predator-prey model with nonlinear cross-diffusion effects representing the tendency of a predator to get closer to prey. Using the energy estimate method, Sobolev embedding theorems and the bootstrap arguments, the existence of global solutions is proved when the space dimension is less than 10. MSC: 35K57; 35B35

Highlights

Consider the following predator-prey model with cross-diffusion effects: ⎧⎪⎪⎪⎪⎪⎪⎪⎨vutt d d ( + δu)u = ug(u) – p(u, v)v, γ+ αv + + ul v = v –d – sv + cp(u, v),⎪⎪⎪⎪⎪⎪⎪⎩u∂∂(uνx,= )∂∂=νv =, u (x) ≥(≡), v(x, ) = v (x) ≥ (≡), x ∈, t >, x ∈, t >, x ∈ ∂, t >, x∈, ( )where ⊂ Rn (n ≥ ) is a bounded domain with smooth boundary ∂, ν is the outward unit normal vector of the boundary ∂, the given coefficients d, s, c, d, d, δ, α, γ and l are positive constants
The purpose of this paper is to establish a sufficient condition for the existence of global solutions for ( ) without any restrictions on the term p(u, v)v/u in the higher dimensional case
In order to establish Lp-estimates for solutions of ( ), we need the following result which can be found in [ ]

Summary

Introduction

The functions g ∈ C ([ , ∞)) and p ∈ C([ , ∞) × [ , ∞)) are assumed to satisfy the following two hypotheses throughout this paper: (H ) g( ) > and g (u) < for all u > , and there exists a positive constant K such that g(K) = . The purpose of this paper is to establish a sufficient condition for the existence of global solutions for ( ) without any restrictions on the term p(u, v)v/u in the higher dimensional case. In Section , we establish L∞-estimates for v and give a proof of Theorem .

Objectives

Results

Conclusion