Stationary and oscillatory patterns of a food chain model with diffusion and predator‐taxis

Abstract

In this paper, we investigate pattern dynamics in a reaction-diffusionchemotaxis food chain model with predator-taxis, which extends previous studies of reaction-diffusion food chain model. By virtue of diffusion semigroup theory, we first prove global classical solvability and boundedness for the considered model over a bounded domain Ω ⊂ R n (n ≥ 1) with smooth boundary for arbitrary predator-taxis sensitivity coefficient. Then the linear stability analysis for the considered model shows that chemotaxis can induce the losing of stability of the unique positive spatially homogeneous steady state via Turing bifurcation and Turing-spatiotemporal Hopf bifurcation. These bifurcations results in the formation of two kinds of important spatiotemporal patterns: stationary Turing pattern and oscillatory pattern. Simultaneously, the threshold values for Turing bifurcation and Turing-spatiotemporal Hopf bifurcation are given explicitly. Finally, numerical simulations are performed to illustrate and support our theoretical findings, and some interesting non-Turing patterns are found in temporal Hopf parameter space by numerical simulation.

Taxis-Driven Pattern Formation in Tri-Trophic Food Chain Model with Omnivory

“…where Z k = (U k , V k , W k ) are the Fourier coefficients. Substituting these series into (24) gives rise to…”

“…A number of works [13,23,24] studying the effect of taxis on the model dynamics state the stabilizing role of this factor in the case of positive values and the possibility of pattern formation in the case of negative ones. The purpose of this study is to determine the effects of positive prey-taxis and negative predator-taxis on the spatiotemporal dynamics of the system and pattern formation.…”