Global Existence and Convergence of Solutions to a Cross-Diffusion Cubic Predator-Prey System with Stage Structure for the Prey

Huaihuo Cao,Shengmao Fu

doi:10.1155/2010/285961

Abstract

We study a cubic predator-prey system with stage structure for the prey. This system is a generalization of the two-species Lotka-Volterra predator-prey model. Firstly, we consider the asymptotical stability of equilibrium points to the system of ordinary differential equations type. Then, the global existence of solutions and the stability of equilibrium points to the system of weakly coupled reaction-diffusion type are discussed. Finally, the existence of nonnegative classical global solutions to the system of strongly coupled reaction-diffusion type is investigated when the space dimension is less than 6, and the global asymptotic stability of unique positive equilibrium point of the system is proved by constructing Lyapunov functions.

Highlights

−cv αu − βv v is said to be the general Lotka-Volterra predator-prey model in 1–3, and to be cubic predator-prey system in 4, where u, v are the population densities of prey and predator species at time t, respectively. b3, b4, c, α, β are positive constants, b1 is nonnegative as the intrinsic growth rate of prey population, and the sign of b2 is undetermined. c is the net Boundary Value Problems mortality rate of predator population, and the survival of predator species is dependent on the survival state of prey species, and b2u − b3u2, βv are the respective density restriction terms of prey and predator species. b4u is the predation rate of the predator, and αu is the conversion rate of the predator
Αx1 − βx[3] x3, where x1 and x2 are the population densities of the immature and mature prey species, respectively, and x3 denotes the density of the predator species
−b u1 − u3 u3, where a0 η1η2/r22, a1 r1 η2 /r2, a2 b2/α, a3 b3/r2, k b4/β, and b c/r2 are positive constants, and a2 b2/α is undetermined to the sign

Summary

Introduction and Mathematical Model

The predator-prey model as, which follows, the ordinary differential equation system du dt b1 b2u − b3u2 u − b4uv, 1.1 dv dt. C is the net Boundary Value Problems mortality rate of predator population, and the survival of predator species is dependent on the survival state of prey species, and b2u − b3u2, βv are the respective density restriction terms of prey and predator species.

Equilibrium Solution of the ODE System

Stability of the PDE System without Cross-Diffusion

Global Existence of Classical Solutions and Convergence