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On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis
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
##### References:
 [1] R. S. Cantrell and C. Cosner, Permanence in ecological systems with spatial heterogeneity,, Proc. Roy. Soc. Edinburgh, 123A (1993), 533. [2] E. N. Dancer, The existence and uniqueness of positive solutions of competing species equations with diffusion,, Trans. Amer. Math. Soc., 326 (1991), 829. [3] Wei Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model,, J. Math. Anal. Appl., 179 (1993), 592. [4] Wei Feng, Permanence effect in a three-species food chain model,, Applicable Analysis, 54 (1994), 195. [5] W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects,, J. Math. Anal. Appl., 206 (1997), 547. [6] W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model,, Diff. Integ. Eq.s, 8 (1995), 617. [7] W. Feng and X. Lu, Asymptotic periodicity in diffusive logistic equations with discrete delays,, Nonlinear Analysis, 26 (1996), 171. [8] A. Leung, "Systems of Nonlinear Partial Differential Equations,'', Kluwer Publ., (1989). [9] X. Lu, Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays,, Numer. Meth. Part. Diff. Eqn.s, 11 (1995), 591. [10] X. Lu, Numerical solutions of coupled parabolic systems with time delays,, Differential Equa tions and Nonlinear Mechanics (Orlando, 528 (2001), 201. [11] X. Lu and W. Feng, Periodic solution and oscillation in a competition model with diffusion and distributed delay effects,, Nonlinear Analysis, 27 (1996), 699. [12] C. V. Pao, "On Nonlinear Parabolic and Elliptic Equations,'', Plenum Press, (1992). [13] C. V. Pao, Numerical methods for semilinear parabolic equations,, SIAM J. Numer. Anal., 24 (1987), 24. [14] C. V. Pao, Coupled nonlinear parabolic system with time delays,, J. Math. Anal. Appl., 196 (1995), 237. [15] C. V. Pao, Dynamics of nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 198 (1996), 751. [16] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays,, Nonlinear Analysis T.M.A., 48 (2002), 349. [17] C. V. Pao, Numerical analysis of coupled system of nonlinear parabolic equations,, SIAM J. Numer Anal., 36 (1999), 394. [18] C. V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays,, J. Math. Anal. Appl., 240 (1999), 249. [19] M. H. Posey and A. H. Hines, Complex predator-prey interactions within an estuarine benthic community,, Ecology, 72 (1991), 2155. [20] J. T. Rowell and Wei Feng, Coexistence and permanence in a four-species food chain model,, Nonlinear Times and Digest, 2 (1995), 191.

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##### References:
 [1] R. S. Cantrell and C. Cosner, Permanence in ecological systems with spatial heterogeneity,, Proc. Roy. Soc. Edinburgh, 123A (1993), 533. [2] E. N. Dancer, The existence and uniqueness of positive solutions of competing species equations with diffusion,, Trans. Amer. Math. Soc., 326 (1991), 829. [3] Wei Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model,, J. Math. Anal. Appl., 179 (1993), 592. [4] Wei Feng, Permanence effect in a three-species food chain model,, Applicable Analysis, 54 (1994), 195. [5] W. Feng and X. Lu, Harmless delays for permanence in a class of population models with diffusion effects,, J. Math. Anal. Appl., 206 (1997), 547. [6] W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model,, Diff. Integ. Eq.s, 8 (1995), 617. [7] W. Feng and X. Lu, Asymptotic periodicity in diffusive logistic equations with discrete delays,, Nonlinear Analysis, 26 (1996), 171. [8] A. Leung, "Systems of Nonlinear Partial Differential Equations,'', Kluwer Publ., (1989). [9] X. Lu, Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays,, Numer. Meth. Part. Diff. Eqn.s, 11 (1995), 591. [10] X. Lu, Numerical solutions of coupled parabolic systems with time delays,, Differential Equa tions and Nonlinear Mechanics (Orlando, 528 (2001), 201. [11] X. Lu and W. Feng, Periodic solution and oscillation in a competition model with diffusion and distributed delay effects,, Nonlinear Analysis, 27 (1996), 699. [12] C. V. Pao, "On Nonlinear Parabolic and Elliptic Equations,'', Plenum Press, (1992). [13] C. V. Pao, Numerical methods for semilinear parabolic equations,, SIAM J. Numer. Anal., 24 (1987), 24. [14] C. V. Pao, Coupled nonlinear parabolic system with time delays,, J. Math. Anal. Appl., 196 (1995), 237. [15] C. V. Pao, Dynamics of nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 198 (1996), 751. [16] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays,, Nonlinear Analysis T.M.A., 48 (2002), 349. [17] C. V. Pao, Numerical analysis of coupled system of nonlinear parabolic equations,, SIAM J. Numer Anal., 36 (1999), 394. [18] C. V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays,, J. Math. Anal. Appl., 240 (1999), 249. [19] M. H. Posey and A. H. Hines, Complex predator-prey interactions within an estuarine benthic community,, Ecology, 72 (1991), 2155. [20] J. T. Rowell and Wei Feng, Coexistence and permanence in a four-species food chain model,, Nonlinear Times and Digest, 2 (1995), 191.
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