Abstract
Spatial predator-prey models have been studied by researchers for many years, because the exact distributions of the population can be well illustrated via pattern formation. In this paper, amplitude equations of a spatial Holling–Tanner predator-prey model are studied via multiple scale analysis. First, by amplitude equations, we obtain the corresponding intervals in which different kinds of patterns will be onset. Additionally, we get the conclusion that pattern transitions of the predator are induced by the increasing rate of conversion into predator biomass. Specifically, pattern transitions of the predator between distinct Turing pattern structures vary in an orderly manner: from spotted patterns to stripe patterns, and finally to black-eye patterns. Moreover, it is discovered that pattern transitions of prey can be induced by cross-diffusion; that is, patterns of prey transmit from spotted patterns to stripe patterns and finally to a mixture of spot and stripe patterns. Meanwhile, it is found that both effects of cross-diffusion and interaction between the prey and predator can lead to the complicated phenomenon of dynamics in the system of biology.
Highlights
Analysis of the dynamics of the predator-prey model is one of the most interesting topics in mathematics as well as in ecology
Pattern formation has already been analyzed in ecosystems [8,9,10,11] and epidemics for many years [12, 13]
Pattern selection and formation of the model were analyzed via weakly multiple scale analysis
Summary
Analysis of the dynamics of the predator-prey model is one of the most interesting topics in mathematics as well as in ecology. When the prey grows logistically with growth rate r and carrying capacity K in the absence of predator, i.e., φ(N) r(1 − N/K), we obtain the following Holling–Tanner model with Holling type III functional response: rN1. Taking the prey–predator model as an example, cross-diffusion terms explain the following biological meaning: predator species will move towards various directions and affect the density of different prey species at distinct places, and vice versa [33]. An attempt is made in this paper to understand the effect of cross-diffusion on the prey-predator model as well. Ouyang’s work [45], multiple scale analysis is used to study a ratio-dependent predator-prey model with spatial motion in the present paper. E results reveal the complicated mutual effects of cross-diffusion and the intrinsic mechanism, e.g., biomass on the predator-prey system Pattern transitions are obtained both by the biomass from prey to predator and by the crossdiffusion of the prey. e results reveal the complicated mutual effects of cross-diffusion and the intrinsic mechanism, e.g., biomass on the predator-prey system
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