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Permanence and coexistence in a diffusive complex ratio-dependent food chain

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Abstract

In this article, we study a complex ratio-dependent food chain model with diffusion in bounded habitat and various boundary conditions. As an extension of simple food chain models studied in earlier literature (Ko and Ahn, Math Biosci Eng 4:1–11, 2007; Kuang and Beretta, J Math Biol 36:389–406, 1998), this model has a more complicated (and general) dynamical structure where the super-predator consumes both the prey and predator. For Dirichlet, Robin, and Neumann boundary conditions, we obtain straightforward criterion for the existence of a positive global attractor which ensures the permanence effect in the ecological system and the presence of a positive steady-state solution. Sufficient conditions on the interaction rates are further given for the uniqueness and global stability of the coexistence state. Several examples give specific parameter sets which satisfy the permanence criteria, or violate the criteria with multiple coexistence states. Numerical simulations of the diffusive food-chain models are also provided to demonstrate and compare the asymptotic behavior of the time-dependent density functions.

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References

  1. Feng W (1994) Permanence effect in a three-species food chain model. Appl Anal 54:195–209

    Article  MathSciNet  Google Scholar 

  2. Feng W, Lu X (1995) Some coexistence and extinction results in a three species ecological model. Differ Integral Equ 8:617–626

    MathSciNet  Google Scholar 

  3. Feng W, Pao CV, Lu X (2011) Global attractors of reaction-diffusion systems modeling food chain populations with delays. Commun Pure Appl Anal 10:1463–1478

    Article  MathSciNet  MATH  Google Scholar 

  4. Anthony W, Leung AW (2009) Nonlinear systems of partial differential equations, applications to life and physical sciences. World Scientific Publishing, Singapore

    MATH  Google Scholar 

  5. Lu C, Feng W, Lu X (1997) Long-term survival in a 3-species ecological system. Dyn Contin Discret Impuls Syst 3: 199–213

  6. Lu X, Feng W (1994) Dynamics and numerical simulations of food chain populations. Appl Math Comput 65:335–344

    Article  MathSciNet  Google Scholar 

  7. Pao CV (1992) Nonlinear parabolic and elliptic equations. Plenum Press, New York

    Google Scholar 

  8. Arditi R, Ginzburg LR (1989) Coupling in predator-prey dynamics: ratio-dependence. J Theor Biol 139:311–326

    Article  Google Scholar 

  9. Candaten M, Rinaldi S (2003) Peak-to-peak dynamics in food chain models. Theor Popul Biol 63:257–267

    Article  Google Scholar 

  10. Gutierrez AP (1992) The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson’s blowflies as an example. Ecology 73:1552–1563

    Article  Google Scholar 

  11. Haque M (2009) Ratio-dependent predator-prey models of interacting populations. Bull Math Biol 71:430–452

    Article  MathSciNet  Google Scholar 

  12. Kuang Y, Beretta E (1998) Global qualitative analysis of a ratio-dependent predator-prey system. J Math Biol 36:389–406

    Article  MathSciNet  Google Scholar 

  13. Feng W, Rocco N, Freeze M, Lu X (2014) Mathematical analysis on an extended Rosenzwig-MaCarthur model of Tri-trophic food chain. Discret Contin Dyn Syst Ser S 4:69–87

    MathSciNet  Google Scholar 

  14. De Feo O, Rinaldi S (1997) Yield and dynamics of tritrophic food chains. Am Nat 150:328–345

    Article  MATH  Google Scholar 

  15. Kuznetsov YA, De Feo O, Rinaldi S (2001) Belyakov homoclinic bifurcations in a tritrophic food chain model. SIAM J Appl Math 62:462–487

    Article  MathSciNet  MATH  Google Scholar 

  16. Kuznetsov YA, Rinaldi S (1996) Remarks on food chain dynamics. Math Biosci 134:1–33

    Article  MathSciNet  MATH  Google Scholar 

  17. Rinaldi S, Gragnani A, DeMonte S (2004) Remarks on antipredator behavior and food chain dynamics. Theor Popul Biol 66: 277–286

  18. Hsu SB, Hwang TW, Kuang Y (2003) A ratio-dependent food chain model and its applications to biological control. Math Biosci 181:55–83

    Article  MathSciNet  MATH  Google Scholar 

  19. Ko W, Ahn I (2007) Dynamics of a simple food chain with a ratio-dependent functional response. Math Biosci Eng 4:1–11

  20. Lu X (1995) Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays. Numer Methods Partial Differ Equ 11:591–602

  21. Pao CV (1987) Numerical methods for semilinear parabolic equations. SIAM J Numer Anal 24:24–35

    Article  MathSciNet  MATH  Google Scholar 

  22. Pao CV, Lu X (2010) Block monotone iterative methods for numerical solutions of nonlinear parabolic equations. SIAM J Numer Anal 47:4581–4606

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics and Statistics Department, University of North Carolina Wilmington, Wilmington, NC, USA

    Yaw Chang, Wei Feng, Michael Freeze & Xin Lu

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  1. Yaw Chang
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  2. Wei Feng
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  3. Michael Freeze
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  4. Xin Lu
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Correspondence to Wei Feng.

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Chang, Y., Feng, W., Freeze, M. et al. Permanence and coexistence in a diffusive complex ratio-dependent food chain. Int. J. Dynam. Control 3, 262–274 (2015). https://doi.org/10.1007/s40435-014-0131-4

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  • DOI: https://doi.org/10.1007/s40435-014-0131-4

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