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Complex patterns in a predator-prey model with self and cross-diffusion. (English) Zbl 1221.35423
Summary: We present a theoretical analysis of processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self as well as cross-diffusion in a Beddington-DeAngelis-type predator-prey model. The instability of the uniform equilibrium of the model is discussed, and the sufficient conditions for the instability with zero-flux boundary conditions are obtained. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to stripes-spots, but also to hot/cold spots, stripes and wave pattern replication. This may enrich the pattern formation in cross-diffusive predator-prey model.

MSC:
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)
35K51 Initial-boundary value problems for second-order parabolic systems
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