scholarly journals Persistence and Nonpersistence of a Food Chain Model with Stochastic Perturbation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Haihong Li ◽  
Fuzhong Cong ◽  
Daqing Jiang ◽  
Hongtu Hua
Keyword(s):  
Numerical Simulations ◽  
Positive Solution ◽  
Food Chain ◽  
Stochastic Perturbation ◽  
Chain Model ◽  
Unique Positive Solution ◽  
Predator Prey ◽  
Time Average ◽  
Food Chain Model

We analyze a three species predator-prey chain model with stochastic perturbation. First, we show that this system has a unique positive solution and itspth moment is bounded. Then, we deduce conditions that the system is persistent in time average. After that, conditions for the system going to be extinction in probability are established. At last, numerical simulations are carried out to support our results.

scholarly journals Persistence and Nonpersistence of a Predator Prey System with Stochastic Perturbation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Haihong Li ◽  
Daqing Jiang ◽  
Fuzhong Cong ◽  
Haixia Li
Keyword(s):  
Numerical Simulations ◽  
Positive Solution ◽  
Stationary Distribution ◽  
Stochastic Perturbation ◽  
Unique Positive Solution ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Time Average

We analyze a predator prey model with stochastic perturbation. First, we show that this system has a unique positive solution. Then, we deduce conditions that the system is persistent in time average. Furthermore, we show the conditions that there is a stationary distribution of the system which implies that the system is permanent. After that, conditions for the system going extinct in probability are established. At last, numerical simulations are carried out to support our results.

Spatial patterns created by cross-diffusion for a three-species food chain model

2014 ◽  
Vol 07 (02) ◽  
pp. 1450013 ◽  
Author(s):  
Canrong Tian ◽  
Zhi Ling ◽  
Zhigui Lin
Keyword(s):  
Stability Analysis ◽  
Numerical Simulations ◽  
Spatial Patterns ◽  
Food Chain ◽  
Turing Instability ◽  
Steady States ◽  
Chain Model ◽  
Cross Diffusion ◽  
The Stability ◽  
Food Chain Model

This paper deals with the stability analysis to a three-species food chain model with cross-diffusion, the results of which show that there is no Turing instability but cross-diffusion makes the model instability possible. We then show that the spatial patterns are spotted patterns by using numerical simulations. In order to understand why the spatial patterns happen, the existence of the nonhomogeneous steady states is investigated. Finally, using the Leray–Schauder theory, we demonstrate that cross-diffusion creates nonhomogeneous stationary patterns.

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

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 229 ◽  
Author(s):  
Nitu Kumari ◽  
Nishith Mohan
Keyword(s):  
Pattern Formation ◽  
Positive Solution ◽  
Functional Response ◽  
Food Chain ◽  
Turing Patterns ◽  
Chain Model ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Wide Range ◽  
Food Chain Model

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.

scholarly journals Chaos in Beddington–DeAngelis food chain model with fear

2020 ◽  
Vol 1591 ◽  
pp. 012082
Author(s):  
Hiba Abdullah Ibrahim ◽  
Raid Kamel Naji
Keyword(s):  
Food Chain ◽  
Chain Model ◽  
Food Chain Model
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