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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

Wei Feng 1, , C. V. Pao 2, and  Xin Lu 3,

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.
Keywords: coexistence and permanence, Population dynamics, numerical simulations., reaction diffusion systems, positive steady state and global attractor, 4-species food chain.
Mathematics Subject Classification: Primary: 35K57, 35B40. Secondary: 935B41, 92D2.
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), 533-559.  Google Scholar

[2]

E. N. Dancer, The existence and uniqueness of positive solutions of competing species equations with diffusion, Trans. Amer. Math. Soc., 326 (1991), 829-859.  Google Scholar

[3]

Wei Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179 (1993), 592-609.  Google Scholar

[4]

Wei Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209.  Google Scholar

[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), 547-566.  Google Scholar

[6]

W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model, Diff. Integ. Eq.s, 8 (1995), 617-626.  Google Scholar

[7]

W. Feng and X. Lu, Asymptotic periodicity in diffusive logistic equations with discrete delays, Nonlinear Analysis, T.M.A., 26 (1996), 171-178.  Google Scholar

[8]

A. Leung, "Systems of Nonlinear Partial Differential Equations,'' Kluwer Publ., Boston, 1989.  Google Scholar

[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), 591-602.  Google Scholar

[10]

X. Lu, Numerical solutions of coupled parabolic systems with time delays, Differential Equa tions and Nonlinear Mechanics (Orlando, FL, 1999), Math. Appl. 528 (2001), Kluwer Acad.Publ., Dordrecht, 201-212.  Google Scholar

[11]

X. Lu and W. Feng, Periodic solution and oscillation in a competition model with diffusion and distributed delay effects, Nonlinear Analysis, T.M.A., 27 (1996), 699-709.  Google Scholar

[12]

C. V. Pao, "On Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992.  Google Scholar

[13]

C. V. Pao, Numerical methods for semilinear parabolic equations, SIAM J. Numer. Anal., 24 (1987), 24-35.  Google Scholar

[14]

C. V. Pao, Coupled nonlinear parabolic system with time delays, J. Math. Anal. Appl., 196 (1995), 237-265.  Google Scholar

[15]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.  Google Scholar

[16]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Analysis T.M.A., 48 (2002), 349-362.  Google Scholar

[17]

C. V. Pao, Numerical analysis of coupled system of nonlinear parabolic equations, SIAM J. Numer Anal., 36 (1999), 394-416.  Google Scholar

[18]

C. V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays, J. Math. Anal. Appl., 240 (1999), 249-279.  Google Scholar

[19]

M. H. Posey and A. H. Hines, Complex predator-prey interactions within an estuarine benthic community, Ecology, 72 (1991), 2155-2169. Google Scholar

[20]

J. T. Rowell and Wei Feng, Coexistence and permanence in a four-species food chain model, Nonlinear Times and Digest, 2 (1995), 191-212.  Google Scholar

show all references

References:
[1]

R. S. Cantrell and C. Cosner, Permanence in ecological systems with spatial heterogeneity, Proc. Roy. Soc. Edinburgh, 123A (1993), 533-559.  Google Scholar

[2]

E. N. Dancer, The existence and uniqueness of positive solutions of competing species equations with diffusion, Trans. Amer. Math. Soc., 326 (1991), 829-859.  Google Scholar

[3]

Wei Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179 (1993), 592-609.  Google Scholar

[4]

Wei Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209.  Google Scholar

[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), 547-566.  Google Scholar

[6]

W. Feng and X. Lu, Some coexistence and extinction results in a three species ecological model, Diff. Integ. Eq.s, 8 (1995), 617-626.  Google Scholar

[7]

W. Feng and X. Lu, Asymptotic periodicity in diffusive logistic equations with discrete delays, Nonlinear Analysis, T.M.A., 26 (1996), 171-178.  Google Scholar

[8]

A. Leung, "Systems of Nonlinear Partial Differential Equations,'' Kluwer Publ., Boston, 1989.  Google Scholar

[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), 591-602.  Google Scholar

[10]

X. Lu, Numerical solutions of coupled parabolic systems with time delays, Differential Equa tions and Nonlinear Mechanics (Orlando, FL, 1999), Math. Appl. 528 (2001), Kluwer Acad.Publ., Dordrecht, 201-212.  Google Scholar

[11]

X. Lu and W. Feng, Periodic solution and oscillation in a competition model with diffusion and distributed delay effects, Nonlinear Analysis, T.M.A., 27 (1996), 699-709.  Google Scholar

[12]

C. V. Pao, "On Nonlinear Parabolic and Elliptic Equations,'' Plenum Press, New York, 1992.  Google Scholar

[13]

C. V. Pao, Numerical methods for semilinear parabolic equations, SIAM J. Numer. Anal., 24 (1987), 24-35.  Google Scholar

[14]

C. V. Pao, Coupled nonlinear parabolic system with time delays, J. Math. Anal. Appl., 196 (1995), 237-265.  Google Scholar

[15]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.  Google Scholar

[16]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Analysis T.M.A., 48 (2002), 349-362.  Google Scholar

[17]

C. V. Pao, Numerical analysis of coupled system of nonlinear parabolic equations, SIAM J. Numer Anal., 36 (1999), 394-416.  Google Scholar

[18]

C. V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays, J. Math. Anal. Appl., 240 (1999), 249-279.  Google Scholar

[19]

M. H. Posey and A. H. Hines, Complex predator-prey interactions within an estuarine benthic community, Ecology, 72 (1991), 2155-2169. Google Scholar

[20]

J. T. Rowell and Wei Feng, Coexistence and permanence in a four-species food chain model, Nonlinear Times and Digest, 2 (1995), 191-212.  Google Scholar

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