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Pattern formation in a cross-diffusive Holling-Tanner model. (English) Zbl 1257.92012
Summary: We present a theoretical analysis of the processes of pattern formation that involves organisms distributions and their interactions of spatially distributed populations with self- as well as cross-diffusion in a Holling-Tanner predator-prey model; sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained; and Hopf and Turing bifurcations in a spatial domain are presented, too. 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 exhibit a cross-diffusion controlled formation growth not only to spots, but also to strips, holes, and stripes-spots replications. The methods and results in the present paper may be useful for the research of pattern formations in the cross-diffusive models.
MSC:
92C15Developmental biology, pattern formation
92D40Ecology
34C23Bifurcation (ODE)