Global attractors of reaction-diffusionsystems modeling food chain populations with delays

Wei Feng; C. V. Pao; Xin Lu

doi:10.3934/cpaa.2011.10.1463

Article Contents

2011, Volume 10, Issue 5: 1463-1478. Doi: 10.3934/cpaa.2011.10.1463

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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
3.
Department of Math and Stat. UNCW, 601 S. College Road, Wilmington NC 28403

Received: June 30, 2009

Revised: June 30, 2010

Published: August 2011

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Abstract

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.

Keywords:

Population dynamics,
reaction diffusion systems,
4-species food chain,
coexistence and permanence,
positive steady state and global attractor,
numerical simulations.

Mathematics Subject Classification: Primary: 35K57, 35B40. Secondary: 935B41, 92D25.

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