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Mathematics 2019, 7(3), 229;

Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response

School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, India
Author to whom correspondence should be addressed.
Received: 20 January 2019 / Revised: 22 February 2019 / Accepted: 23 February 2019 / Published: 1 March 2019
(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)
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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. View Full-Text
Keywords: cross diffusion; Turing patterns; non-constant positive solution cross diffusion; Turing patterns; non-constant positive solution

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Kumari, N.; Mohan, N. Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response. Mathematics 2019, 7, 229.

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