Hopf bifurcation and Turing instability in the reactiondiffusion HollingTanner predatorprey model

Author: Li Xin   Jiang Weihua   Shi Junping  

Publisher: Oxford University Press

ISSN: 1464-3634

Source: IMA Journal of Applied Mathematics, Vol.78, Iss.2, 2013-04, pp. : 287-306

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Abstract

The reactiondiffusion HollingTanner predatorprey model with Neumann boundary condition is considered. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for determining the direction of bifurcation and the stability of the bifurcating periodic solution. For partial differential equation (PDE), we consider the Turing instability of the equilibrium solutions and the bifurcating periodic solutions. Through both theoretical analysis and numerical simulations, we show the bistability of a stable equilibrium solution and a stable periodic solution for ordinary differential equation and the phenomenon that a periodic solution becomes Turing unstable for PDE.