BIFURCATIONS AND HYDRA EFFECTS IN A REACTION-DIFFUSION PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE

Abstract

In this paper, through bifurcation analysis and numerical simulations, we consider a reaction-diffusion predator-prey model with Holling Ⅱ functional response to analyze the existence of hydra effect and the relationship between mortality independent of predator density and different steady-state solutions of the system. The hydra effect, which is a paradoxical result in both theoretical and applied ecology, refers to the phenomenon in which an increase in population mortality enhances its own population size. We investigate the existence of the hydra effect when the positive equilibrium point is locally asymptotically stable and Turing unstable. Meanwhile, numerical simulations verify the existence of the hydra effect when the one-dimensional reaction-diffusion system has a spatially inhomogeneous steady-state solution. In addition, we introduce the existence of the Turing bifurcation, the Hopf bifurcation, and the Turing-Hopf bifurcation with the parameters $ d_{2} $ and $ m_{C} $, respectively, as well as the normal form for the Turing-Hopf bifurcation. Based on the obtained normal form, we analyze the complex spatio-temporal dynamics near the Turing-Hopf bifurcation point. Finally, the numerical simulations are carried out to corroborate the obtained theoretical results.