Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response
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 cro...
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MDPI AG
2019-03-01
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| Series: | Mathematics |
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| Online Access: | http://www.mdpi.com/2227-7390/7/3/229 |
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| author | Nitu Kumari Nishith Mohan |
| author_facet | Nitu Kumari Nishith Mohan |
| author_sort | Nitu Kumari |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-23T23:55:57Z |
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| language | English |
| last_indexed | 2024-12-23T23:55:57Z |
| publishDate | 2019-03-01 |
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| spelling | doaj.art-4885683b19954523b33b79320da08bfc2022-12-21T17:25:15ZengMDPI AGMathematics2227-73902019-03-017322910.3390/math7030229math7030229Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional ResponseNitu Kumari0Nishith Mohan1School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, IndiaSchool of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, IndiaDiffusion 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.http://www.mdpi.com/2227-7390/7/3/229cross diffusionTuring patternsnon-constant positive solution |
| spellingShingle | Nitu Kumari Nishith Mohan Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response Mathematics cross diffusion Turing patterns non-constant positive solution |
| title | Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response |
| title_full | Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response |
| title_fullStr | Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response |
| title_full_unstemmed | Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response |
| title_short | Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response |
| title_sort | cross diffusion induced turing patterns in a tritrophic food chain model with crowley martin functional response |
| topic | cross diffusion Turing patterns non-constant positive solution |
| url | http://www.mdpi.com/2227-7390/7/3/229 |
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