Global dynamics of Bazykin-type cross-diffusive model with indirect predator-taxis. (English) Zbl 07903727
Summary: This paper investigates the Bazykin-type cross-diffusive system which incorporates the prey-taxis with “volume-filling” effect and the evasive defense with signaling mechanism in two-dimensional case. To compare with the fully cross-diffusive prey-predator model, this system exhibits two new features: The prey directly move towards to the opposite concentration gradient of the chemical signal which released by the predators, on the other hand, the predator engage competition interactions. Given suitably regular initial value, it is shown that this system admits a globally bounded classical solution without any restrictiveness on the parameters of the system. The current results supplement the conditions [P. Mishra and D. Wrzosek, J. Differ. Equations 361, 391–416 (2023; Zbl 1512.35086)] which are used to prevent any blow-up of classical solution to this system. Moreover, we also prove that the prey-only steady state and coexistence steady state is globally convergent.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 92C17 | Cell movement (chemotaxis, etc.) |
Citations:
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