Abstract
Spatiotemporal patterns driven by cross-diffusion of a uni-directional consumer-resource (C-R) system with Holling-II type functional response are investigated in this paper. The existence of a unique positive steady state of the considered system is studied first. The linear stability analysis shows that the cross-diffusion is the key mechanism for the formation of spatiotemporal patterns through Turing bifurcation. We choose the cross-diffusion coefficient as bifurcation parameter and discuss three different types of Turing bifurcations, corresponding to simple, double non-resonant and double resonant cases. Based on weakly nonlinear analysis with the multiple scale method and the adjoint system theory, we derive the amplitude equations of the Turing patterns near the Turing bifurcation point and obtain the analytical approximation solutions of the patterns for each case. Specially, some qualitative results of amplitude equations of the resonant case are given in detail. Finally, numerical simulations are performed to illustrate the weakly nonlinear theoretical predictions and through these simulations some patterns (single mode pattern, mixed-mode pattern, super-squares pattern, roll pattern, hexagonal pattern) are found. Simultaneously, numerical simulations show that the resource supplying rate has an important impact on the direction of Turing bifurcation.
Highlights
The interaction of populations is one of the basic interspecies relations in biology and ecology [ ]
Many works have proposed to investigate the Turing instability driven by cross-diffusion and prove the existence of inhomogeneous steady states that induce the emergence of spatial patterns ([ – ] and [ – ])
Based on the statements above, in this paper, we introduce the spatial diffusion with zero-flux boundary conditions into system ( . ), and further assume that species u is subject to self-diffusion, and that species v is subject to nonlinear positive cross-diffusion
Summary
The interaction of populations is one of the basic interspecies relations in biology and ecology [ ]. A resource is considered to be a biotic or abiotic species that increases the population growth of its consumers, whereas a consumer exploits a resource and reduces its growth rate In this way, species interactions include bi-directional, uni-directional, and indirect C-R interactions. There exists a motional state of interaction as well, one that recognizes the possible bias, say, of the motion of one species toward or away from another species [ , ] Many works have proposed to investigate the Turing instability driven by cross-diffusion and prove the existence of inhomogeneous steady states that induce the emergence of spatial patterns ([ – ] and [ – ]). In our current work we first investigate the effect of cross-diffusion on the spatial inhomogeneous distribution of the two species in a uni-directional C-R system.
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