Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion. (English) Zbl 1505.35040
The paper under review discusses the global boundedness, asymptotic stability and pattern formation of predator-prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions. The authors establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. Furthermore, they use numerical simulations to demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. In particular, they also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different.
Reviewer: Wan-Tong Li (Lanzhou)
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 92D25 | Population dynamics (general) |
| 35B36 | Pattern formations in context of PDEs |
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