Pattern Formation in a Semi-Ratio-Dependent Predator-Prey System with Diffusion

Hunki Baek,Zhi-wen Wang,Dong Ick Jung

doi:10.1155/2013/657286

Abstract

We investigate spatiotemporal dynamics of a semi-ratio-dependent predator-prey system with reaction-diffusion and zero-flux boundary. We obtain the conditions for Hopf, Turing, and wave bifurcations of the system in a spatial domain by making use of the linear stability analysis and the bifurcation analysis. In addition, for an initial condition which is a small amplitude random perturbation around the steady state, we classify spatial pattern formations of the system by using numerical simulations. The results of numerical simulations unveil that there are various spatiotemporal patterns including typical Turing patterns such as spotted, spot-stripelike mixtures and stripelike patterns thanks to the Turing instability, that an oscillatory wave pattern can be emerged due to the Hopf and wave instability, and that cooperations of Turing and Hopf instabilities can cause occurrence of spiral patterns instead of typical Turing patterns. Finally, we discuss spatiotemporal dynamics of the system for several different asymmetric initial conditions via numerical simulations.

Highlights

In recent years, pattern formations in nonlinear complex systems have been one of the central problems of the natural, social, technological sciences and ecological systems [1,2,3,4,5,6,7]
In [8], Garvie and Trenchea presented the analysis of reaction-diffusion predator-prey systems with the Holling type II functional response and provided an L∞ a priori estimate
Different types of spatiotemporal dynamics are observed and we have found that the distributions of predator and prey are always of the same type

Summary

Introduction

Pattern formations in nonlinear complex systems have been one of the central problems of the natural, social, technological sciences and ecological systems [1,2,3,4,5,6,7]. Wang et al in [9] investigated the spatial pattern formation of a predatorprey system with prey-dependent functional response of Ivlev type and reaction-diffusion. Camara and Aziz-Alaoui, the authors of [11], considered a predator-prey system with a modified Leslie-Gower functional response modeled by a reaction-diffusion equation and derived the conditions for Hopf and Turing bifurcation in the spatial domain. In this context, in this paper, we will focus on the following a semi-ratio-dependent predator-prey system with reaction-diffusion:.

Bifurcation Analysis

Spatiotemporal Pattern Analysis via Numerical Simulations

Conclusion and Discussion