Open Journal Systems

Inhomogeneous spatial patterns in diffusive predator-prey system with spatial memory and predator-taxis

Yehu Lv

Article ID: 2289
Vol 1, Issue 1, 2023

DOI: https://doi.org/10.54517/mss.v1i1.2289
VIEWS - 1819 (Abstract)

Abstract

In this paper, by introducing predator-taxis into the diffusive predator-prey system with spatial memory, then we study the inhomogeneous spatial patterns of this system. Since in this system, the memory delay appears in the diffusion term, and the diffusion term is nonlinear, the classical normal form of Hopf bifurcation for the reaction-diffusion system with delay can’t be applied to this system. Thus, in this paper, we first derive an algorithm for calculating the normal form of Hopf bifurcation for this system. Then in order to illustrate the effectiveness of our newly developed algorithm, we consider the diffusive Holling-Tanner model with spatial memory and predator-taxis. The stability and Hopf bifurcation analysis of this model are investigated, and the direction and stability of Hopf bifurcation periodic solution are also studied by using our newly developed algorithm for calculating the normal form of Hopf bifurcation. At last, we carry out some numerical simulations to verify our theoretical analysis results, and two stable spatially inhomogeneous periodic solutions corresponding to the mode-1 and mode-2 Hopf bifurcations are found.

Keywords

predator-prey system; memory delay; predator-taxis; Hopf bifurcation; normal form; periodic solutions

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