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

Le résumé fourni par la source

In this paper, we investigate pattern dynamics in a reaction-diffusion-chemotaxis food chain model with predator-taxis, which enriches previous studies about diffusive food chain models. By virtue of diffusion semigroup theory, we first show the global classical solvability and uniform boundedness of the considered model over a bounded domain Ω ⊂ R N ( N ≥ 1 ) with smooth boundary. Then the linear stability analysis for the considered model shows that chemotaxis can induce the unique positive spatially homogeneous steady state loses its stability via Turing bifurcation and Turing-spatiotemporal Hopf bifurcation, which 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. In addition, the existence and stability of non-constant positive steady state that bifurcates from the positive constant steady state is investigated by the abstract bifurcation theory of Crandall-Rabinowitz and eigenvalue perturbation theory. Finally, numerical simulations are performed to verify our theoretical results, and some interesting non-Turing pattern are found in temporal Hopf parameter space by numerical simulation.

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