Spatial Complexity of a Predator-Prey Model with Holling-Type Response

Lei Zhang,Zhibin Li

doi:10.1155/2014/675378

Abstract

We focus on a spatially extended Holling-type IV predator-prey model that contains some important factors, such as noise (random fluctuations), external periodic forcing, and diffusion processes. By a brief stability and bifurcation analysis, we arrive at the Hopf and Turing bifurcation surface and derive the symbolic conditions for Hopf and Turing bifurcation on the spatial domain. Based on the stability and bifurcation analysis, we obtain spiral pattern formation via numerical simulation. Additionally, we study the model with a color noise and external periodic forcing. From the numerical results, we know that noise or external periodic forcing can induce instability and enhance the oscillation of the species density, and the cooperation between noise and external periodic forces inherent to the deterministic dynamics of periodically driven models gives rise to the appearance of a rich transport phenomenology. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics.

Highlights

Predation, a complex natural phenomenon, exists widely in the world, for example, the sea, the plain, the forest, the desert, and so on [1]
Decades after Turing [28] demonstrated that spatial patterns could arise from the interaction of reactions or growth processes and diffusion; reaction-diffusion models have been studied in ecology to describe the population dynamics of predator-prey model for a long time since Segel and Jackson applied Turing’s idea [29]
We present a spatial Holling-type IV predatorprey model containing some important factors, such as noise, the external periodic forcing, and diffusion processes

Summary

Introduction

A complex natural phenomenon, exists widely in the world, for example, the sea, the plain, the forest, the desert, and so on [1]. Decades after Turing [28] demonstrated that spatial patterns could arise from the interaction of reactions or growth processes and diffusion; reaction-diffusion models have been studied in ecology to describe the population dynamics of predator-prey model for a long time since Segel and Jackson applied Turing’s idea [29]. It is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite naturally exposed [57] These periodic factors are regarded as external periodic forcing in the predator-prey systems. Si et al [61] studied the propagation of traveling waves in subexcitable systems driven; Liu et al [59] considered a spatially extended phytoplankton-zooplankton system with additive noise and periodic forcing Following these models they considered, the Holling-type IV predator-prey model with external periodic forcing and colored noise is as follows: rN In the last section, we give some discussions and remarks

Hopf and Turing Bifurcation

Spatiotemporal Dynamics of the Models

Conclusions and Remarks