Spatio-temporal steady-state analysis in a prey-predator model with saturated hunting cooperation and chemotaxis. (English) Zbl 07873861
Summary: In this paper, we propose a diffusive prey-predator model with saturated hunting cooperation and predator-taxis. We first establish the global classical solvability and boundedness, and provide some sufficient conditions to assure the existence of a unique positive homogeneous steady state and the global uniform asymptotic stability of the predator-free homogeneous steady state. Secondly, we study the pattern formation mechanism and reveal that pattern formation is driven by the joint effect of predator-taxis, hunting cooperation, and slow diffusivity of predators. Moreover, we find that a strong predator-taxis can annihilate the spatiotemporal patterns, but a weak predator-taxis supports the pattern formation when diffusion-driven instability is present in the model without predator-taxis. However, if diffusion-driven instability is absent, predator-taxis cannot destabilize the unique positive spatially homogeneous steady state. Additionally, we highlight that spatially heterogeneous steady states do not exist when the diffusion coefficient ratio of predators to prey is sufficiently large under specific parametric conditions. To explore the various types of spatially heterogeneous steady states, we derive amplitude equations based on the weakly nonlinear analysis theory. Finally, numerical simulations, including the hexagonal pattern, stripe pattern, a mixed pattern combining hexagons and stripes, and the square pattern, are presented to illustrate the theoretical results.
MSC:
| 35B36 | Pattern formations in context of PDEs |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34D23 | Global stability of solutions to ordinary differential equations |
| 35B32 | Bifurcations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 92D25 | Population dynamics (general) |
| 92D40 | Ecology |
Keywords:
hunting cooperation; predator-taxis; global stability; pattern selection; amplitude equation; non-existence of non-homogeneous steady statesReferences:
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