Existence of a Limit Cycle in an Intraguild Food Web Model with Holling Type II and Logistic Growth for the Common Prey

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DOI: 10.4236/am.2017.83030    523 Downloads   677 Views  


In this paper, we prove the existence of a limit cycle for a given system of differential equations corresponding to an asymmetrical intraguild food web model with functional responses Holling type II for the middle and top predators and logistic grow for the (common) prey. The existence of such limit cycle is guaranteed, via the first Lyapunov coefficient and the Andronov-Hopf bifurcation theorem, under certain conditions for the parameters involved in the system.

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Castillo-Santos, F. , Rosa, M. and Loreto-Hernández, I. (2017) Existence of a Limit Cycle in an Intraguild Food Web Model with Holling Type II and Logistic Growth for the Common Prey. Applied Mathematics, 8, 358-376. doi: 10.4236/am.2017.83030.


[1] Polis, G.A. and Myers, C.A. (1989) The Ecology and Evolution of Intraguild Predation: Potential Competitors That Eat Each Other. Annual Review of Ecology, Evolution, and Systematics, 20, 297-330.
[2] Root, R. (1967) The Niche Exploitation Pattern of the Blue-Gray Gnat Catcher. Ecological Monographs, 37, 317-350.
[3] Holt, R.D. and Polis, G.A. (1996) A Theoretical Framework for Intraguild Predation. The American Naturalist, 149, 745-764.
[4] Blé, G., Castellanos, V. and Llibre, J. (2016) Existence of Limit Cycles in a Tritrophic Food Chain Model with Holling Functional Responses of Type II and III. Mathematical Methods in the Applied Sciences, 39, 3996-4006.
[5] Castellanos, V., Llibre, J. and Quilantan, I. (2013) Simultaneous Periodic Orbits Bifurcating from Two Zero-Hopf Equilibria in a Tritrophic Food Chain Model. Journal of Applied Mathematics and Physics, 1, 31-38.
[6] Castellanos, V. and Chan-López, R.E. (2017) Existence of Limit Cycles in a Three Level Trophic Chain with Lotka-Volterra and Holling Type II Functional Responses. Chaos, Solitons & Fractals, 95, 157-167.
[7] Francoise, J.P. and Llibre, J. (2011) Analytical Study of a Triple Hopf Bifurcation in a Tritrophic Food Chain Model. Applied Mathematics and Computation, 217, 7146-7154.
[8] Kuznetsov, Y.A. (2004) Elements of Applied Bifurcation Theory. 3rd Edition, Springer-Verlag.
[9] Perko, L. (2001) Differential Equations and Dynamical Systems. 3rd Edition, Springer-Verlag.
[10] Andronov, A.A., Leontovich, E.A., Gordon, I.I. and Maier, A.G. (1971) Theory of Bifurcations of Dynamic Systems on a Plane. Israel Program for Scientific Translations, Jerusalem.
[11] Marsden, J.E. and McCracken, M. (1976) The Hopf Bifurcation and Its Applications. Springer-Verlag, New York.

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