Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model

Abstract

The reaction-diffusion Holling-Tanner predator-prey 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.

Travelling waves in the Holling–Tanner model with weak diffusion

“…For example, in [8,9], Turing patterns that bifurcate from A are observed in numerical simulations. In [10], conditions for Turing and Hopf bifurcations are derived, and the stability of the bifurcating periodic solutions is studied.…”

Supercritical Hopf bifurcation and Turing patterns for an activator and inhibitor model with different sources

“…For more results about reaction diffusion systems of activator-inhibitor type, see [19][20][21]. See [22][23][24] and the references therein for more real-world models on Turing instability. So far, most work on Turing instability of system (1) has been carried out by taking specific parameter values.…”

Delay-induced Hopf bifurcation in a diffusive Holling–Tanner predator–prey model with ratio-dependent response and Smith growth

“…In recent years, many authors investigate the diffusive predator-prey model with logistic prey growth rate [6][7][8][9][10] or strong Allee effect in prey [11]. It came out that this assumption is not realistic for a food-limited population under the effect of environmental toxicants.…”