Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension. (English) Zbl 1484.35047
Summary: Resorting to M.G. Crandall and P.H. Rabinowitz’s well-known bifurcation theory we first obtain the local structure of steady states concerning the ratio-dependent predator-prey system with prey-taxis in spatial one dimension, which bifurcate from the homogeneous coexistence steady states when treating the prey-tactic coefficient as a bifurcation parameter. Based on this, then the global structure of positive solution is established. Moreover, through asymptotic analysis and eigenvalue perturbation we find the stability criterion of such bifurcating steady states. Finally, several numerical simulations are performed to show the pattern formation.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B09 | Positive solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 92C15 | Developmental biology, pattern formation |
| 92C17 | Cell movement (chemotaxis, etc.) |
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