Abstract
Diffusion has long been known to induce pattern formation in predator prey systems. For certain prey-predator interaction systems, self diffusion conditions ceases to induce patterns, i.e., a non-constant positive solution does not exist, as seen from the literature. We investigate the effect of cross diffusion on the pattern formation in a tritrophic food chain model. In the formulated model, the prey interacts with the mid level predator in accordance with Holling Type II functional response and the mid and top level predator interact via Crowley-Martin functional response. We prove that the stationary uniform solution of the system is stable in the presence of diffusion when cross diffusion is absent. However, this solution is unstable in the presence of both self diffusion and cross diffusion. Using a priori analysis, we show the existence of a inhomogeneous steady state. We prove that no non-constant positive solution exists in the presence of diffusion under certain conditions, i.e., no pattern formation occurs. However, pattern formation is induced by cross diffusion because of the existence of non-constant positive solution, which is proven analytically as well as numerically. We performed extensive numerical simulations to understand Turing pattern formation for different values of self and cross diffusivity coefficients of the top level predator to validate our results. We obtained a wide range of Turing patterns induced by cross diffusion in the top population, including floral, labyrinth, hot spots, pentagonal and hexagonal Turing patterns.
Highlights
The two primary objectives of studying systems ecology are to get an understanding of the dynamics of the ecological systems and the nature of the forces, which determine the community structure
We concluded a three-dimensional continuous time ecological system, modeling a tritrophic food chain based on a hybrid type of Holling Type II and Crowley-Martin functional response
The phenomenon of pattern formation in spatially extended models including cross diffusion as a destabilizing mechanism was studied with varying the self diffusion and cross diffusion coefficients over a wide range, which serves as an extension to Turing pattern formations exhibited by reaction diffusion system
Summary
The two primary objectives of studying systems ecology are to get an understanding of the dynamics of the ecological systems and the nature of the forces, which determine the community structure. Three species continuous time models have observed more complex dynamical behavior [4,5,6,7] All these studies depend upon the classical types of functional responses, which include Holling. In the case of a three species prey-predator model, such a migratory behavior depends on the concentration of both predators (i.e., the mid level predator and the top level predator) This leads to cross diffusive system, in addition to each species natural tendency to diffuse i.e., self diffusion; such models are studied in [14,15,16,17,18]. We consider a tritrophic food chain model in which the species interact via Crowley-Martin and Holling type II functional response at different levels.
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