A Two‐Species Cooperative Lotka‐Volterra System of Degenerate Parabolic Equations

Jiebao Sun,Elena Braverman,Boying Wu,Dazhi Zhang

doi:10.1155/2011/714248

Abstract

We consider a cooperating two‐species Lotka‐Volterra model of degenerate parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time‐periodic solution for this system.

Highlights

IntroductionWe consider the following two-species cooperative system: ut Δum[1] uα a − bu cv , x, t ∈ Ω × Ê , 1.1 vt Δvm[2] vβ d eu − f v , x, t ∈ Ω × Ê , 1.2 u x, t 0, v x, t 0, x, t ∈ ∂Ω × Ê , 1.3 u x, 0 u0 x , v x, 0 v0 x , x ∈ Ω, 1.4 where m1, m2 > 1, 0 < α < m1, 0 < β < m2, 1 ≤ m1 − α m2 − β , a a x, t , b b x, t , c c x, t , d d x, t , e e x, t , f f x, t are strictly positive smooth functions and periodic in time with period T > 0 and u0 x and v0 x are nonnegative functions and satisfy um0 1 , v0m2 ∈ W01,2 Ω
Under the condition that blfl > cMeM, 1.8 where fM sup{f x, t | x, t ∈ Ω × Ê}, fl inf{f x, t | x, t ∈ Ω × Ê}, we show that the generalized solution is uniformly bounded
We show the existence and the attractivity of the maximal periodic solution

Summary

Introduction

We consider the following two-species cooperative system: ut Δum[1] uα a − bu cv , x, t ∈ Ω × Ê , 1.1 vt Δvm[2] vβ d eu − f v , x, t ∈ Ω × Ê , 1.2 u x, t 0, v x, t 0, x, t ∈ ∂Ω × Ê , 1.3 u x, 0 u0 x , v x, 0 v0 x , x ∈ Ω, 1.4 where m1, m2 > 1, 0 < α < m1, 0 < β < m2, 1 ≤ m1 − α m2 − β , a a x, t , b b x, t , c c x, t , d d x, t , e e x, t , f f x, t are strictly positive smooth functions and periodic in time with period T > 0 and u0 x and v0 x are nonnegative functions and satisfy um0 1 , v0m2 ∈ W01,2 Ω.

The Initial Boundary Value Problem

Periodic Solutions