Spatially Nonhomogeneous Periodic Solutions in a Delayed Predator‐Prey Model with Diffusion Effects

Jia-Fang Zhang

doi:10.1155/2012/856725

Abstract

This paper is concerned with a delayed predator‐prey diffusion model with Neumann boundary conditions. We study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. In particular, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that spatially nonhomogeneous periodic solutions bifurcate from the positive constant steady‐state solution when the system parameters are all spatially homogeneous. Meanwhile, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of partial functional differential equations (PFDEs).

Highlights

Functional differential equations have merited a great deal of attention due to its theoretical and practical significance; they are often used in population dynamics, epidemiology, and other important areas of science; see 1–6
H2 holds, 2.13 with k k0 has purely imaginary roots iωk[0] and system 1.3 has a family of spatially nonhomogeneous periodic solutions bifurcating from the spatially homogeneous steady state u∗, v∗, when τ crosses through the critical values τjk[0 ], where ωk[0] and τjk[0] are defined by 3.2 and 3.4 with k k0, k0 ∈ N, respectively
From Theorems 2.2 and 3.3, we can know that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that the system bifurcates spatially nonhomogeneous periodic solutions at the positive constant steady state under which the system parameters are all spatially homogeneous

Summary

Introduction

Functional differential equations have merited a great deal of attention due to its theoretical and practical significance; they are often used in population dynamics, epidemiology, and other important areas of science; see 1–6. Lu and Liu 7 proposed the following modified Holling-Tanner delayed predator-prey model: du t dt ru t. The first equation states that the prey grows logistically with carrying capacity K and Abstract and Applied Analysis intrinsic growth rate r in absence of predation. The second equation shows that predators grow logistically with intrinsic growth rate s and carrying capacity proportional to the prey populations size u t. Most population models are often formulated by ordinary differential equations with or without time delays 1, 2, 13–18. Time delays and spatial diffusion should be considered simultaneously in modeling biological interactions. The main purpose of this paper is to consider the effects of the delay and diffusion on the dynamics of system 1.3. We study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of PFDEs

Stability and Hopf Bifurcations

Effect of Small Diffusivity

Properties of Hopf Bifurcation

Conclusions