Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain

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December 2014, 7(6): 1215-1230. doi: 10.3934/dcdss.2014.7.1215

Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain

Wei Feng ^1, , Nicole Rocco ^1, , Michael Freeze ^1, and Xin Lu ^1,

1.	Mathematics and Statistics Department, University of North Carolina Wilmington, Wilmington, NC 28403-5970, United States, United States

Received February 2013 Revised August 2013 Published June 2014

Abstract
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In this paper, we study a new model as an extension of the Rosenzweig-MacArthur tritrophic food chain model in which the super-predator consumes both the predator and the prey. We first obtain the ultimate bounds and conditions for exponential convergence for these populations. We also find all possible equilibria and investigate their stability or instability in relation with all the ecological parameters. Our main focus is on the conditions for the existence, uniqueness and stability of a coexistence equilibrium. The complexity of the dynamics in this model is theoretically discussed and graphically demonstrated through various examples and numerical simulations.

Keywords: stability and asymptotic behavior, extinction or coexistence, numerical simulations., limiting nutrient response, Tri-trophic food-chain models.

Mathematics Subject Classification: Primary: 34A34, 34C11, 34D23; Secondary: 92D2.

Citation: Wei Feng, Nicole Rocco, Michael Freeze, Xin Lu. Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1215-1230. doi: 10.3934/dcdss.2014.7.1215

References:

[1]	M. Candaten and S. Rinaldi, Peak-to-peak dynamics in food chain models, Theoretical Population Biology, 63 (2003), 257-267.
[2]	L. Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill, Inc., 1977. doi: 10.1137/1.9780898719147.
[3]	W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209. doi: 10.1080/00036819408840277.
[4]	W. Feng, C. V. Pao and X. Lu, Global attractors of reaction-diffusion systems modeling food chain populations with delays, Commun. Pure Appl. Anal., 10 (2011), 1463-1478. doi: 10.3934/cpaa.2011.10.1463.
[5]	O. De Feo and S. Rinaldi, Yield and dynamics of tritrophic food chains, The American Naturalist, 150 (1997), 328-345.
[6]	M. Haque, Ratio-dependent predator-prey models of interacting populations, Bulletin of Mathematical Biology, 71 (2009), 430-452. doi: 10.1007/s11538-008-9368-4.
[7]	S.-B. Hsu, T.-W. Hwang and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Mathematical Biosciences, 181 (2003), 55-83. doi: 10.1016/S0025-5564(02)00127-X.
[8]	Y. A. Kuznetsov, O. De Feo and S. Rinaldi, Belyakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math., 62 (2001), 462-487. doi: 10.1137/S0036139900378542.
[9]	Y. A. Kuznetsov and S. Rinaldi, Remarks on food chain dynamics, Mathematical Biosciences, 134 (1996), 1-33. doi: 10.1016/0025-5564(95)00104-2.
[10]	Y. Kuang, Some mechanistically derived population models, Mathematical Biosciences and Engineering, 4 (2007), 1-11.
[11]	C. Lu, W. Feng and X. Lu, Long-term survival in a 3 -species ecological system, Dynam. Contin. Discrete Impuls. Systems, 3 (1997), 199-213.
[12]	S. Rinaldi, A. Gragnani and S. DeMonte, Remarks on antipredator behavior and food chain dynamics, Theoretical Population Biology, 66 (2004), 277-286. doi: 10.1016/j.tpb.2004.07.002.
[13]	M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223. doi: 10.1086/282272.
[14]	D. M. Wrzosek, Limit cycles in predator-prey models, Mathematical Biosciences, 98 (1990), 1-12. doi: 10.1016/0025-5564(90)90009-N.

show all references

References:

[1]	M. Candaten and S. Rinaldi, Peak-to-peak dynamics in food chain models, Theoretical Population Biology, 63 (2003), 257-267.
[2]	L. Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill, Inc., 1977. doi: 10.1137/1.9780898719147.
[3]	W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209. doi: 10.1080/00036819408840277.
[4]	W. Feng, C. V. Pao and X. Lu, Global attractors of reaction-diffusion systems modeling food chain populations with delays, Commun. Pure Appl. Anal., 10 (2011), 1463-1478. doi: 10.3934/cpaa.2011.10.1463.
[5]	O. De Feo and S. Rinaldi, Yield and dynamics of tritrophic food chains, The American Naturalist, 150 (1997), 328-345.
[6]	M. Haque, Ratio-dependent predator-prey models of interacting populations, Bulletin of Mathematical Biology, 71 (2009), 430-452. doi: 10.1007/s11538-008-9368-4.
[7]	S.-B. Hsu, T.-W. Hwang and Y. Kuang, A ratio-dependent food chain model and its applications to biological control, Mathematical Biosciences, 181 (2003), 55-83. doi: 10.1016/S0025-5564(02)00127-X.
[8]	Y. A. Kuznetsov, O. De Feo and S. Rinaldi, Belyakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math., 62 (2001), 462-487. doi: 10.1137/S0036139900378542.
[9]	Y. A. Kuznetsov and S. Rinaldi, Remarks on food chain dynamics, Mathematical Biosciences, 134 (1996), 1-33. doi: 10.1016/0025-5564(95)00104-2.
[10]	Y. Kuang, Some mechanistically derived population models, Mathematical Biosciences and Engineering, 4 (2007), 1-11.
[11]	C. Lu, W. Feng and X. Lu, Long-term survival in a 3 -species ecological system, Dynam. Contin. Discrete Impuls. Systems, 3 (1997), 199-213.
[12]	S. Rinaldi, A. Gragnani and S. DeMonte, Remarks on antipredator behavior and food chain dynamics, Theoretical Population Biology, 66 (2004), 277-286. doi: 10.1016/j.tpb.2004.07.002.
[13]	M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223. doi: 10.1086/282272.
[14]	D. M. Wrzosek, Limit cycles in predator-prey models, Mathematical Biosciences, 98 (1990), 1-12. doi: 10.1016/0025-5564(90)90009-N.

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