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September  2011, 10(5): 1463-1478. doi: 10.3934/cpaa.2011.10.1463

Global attractors of reaction-diffusion systems modeling food chain populations with delays

 1 Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403 2 Department of mathematics, North Carolina State University, Raleigh, NC27695, United States 3 Department of Math and Stat. UNCW, 601 S. College Road, Wilmington NC 28403

Received  July 2009 Revised  July 2010 Published  April 2011

In this paper, we study a reaction-diffusion system modeling the population dynamics of a four-species food chain with time delays. Under Dirichlet and Neumann boundary conditions, we discuss the existence of a positive global attractor which demonstrates the presence of a positive steady state and the permanence effect in the ecological system. Sufficient conditions on the interaction rates are given to ensure the persistence of all species in the food chain. For the case of Neumann boundary condition, we further obtain the uniqueness of a positive steady state, and in such case the density functions converge uniformly to a constant solution. Numerical simulations of the food-chain models are also given to demonstrate and compare the asymptotic behavior of the time-dependent density functions.
Citation: Wei Feng, C. V. Pao, Xin Lu. Global attractors of reaction-diffusion systems modeling food chain populations with delays. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1463-1478. doi: 10.3934/cpaa.2011.10.1463
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