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

• Published in: Mathematical methods in the applied sciences 2023-05, Vol.46 (8), p.9652-9675
• Main Authors: Han, Renji, Röst, Gergely
• Format: Article
• Language: English
• Subjects: Diffusion; Food chains; Hopf bifurcation; Mathematical models; non‐constant positive steady state; oscillatory pattern; Predators; predator‐taxis; reaction‐diffusion food chain model; Smooth boundaries; Stability analysis; stationary turing pattern
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9079

In this paper, we investigate pattern dynamics in a reaction‐diffusion‐chemotaxis 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⊂ℝn(n≥1)$$ \Omega \subset {\mathbb{R}}^n\kern0.1em \left(n\ge 1\right) $$ 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.