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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

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 & 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.   Google Scholar

[2]

L. Edelstein-Keshet, Mathematical Models in Biology,, McGraw-Hill, (1977).  doi: 10.1137/1.9780898719147.  Google Scholar

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W. Feng, Permanence effect in a three-species food chain model,, Applicable Analysis, 54 (1994), 195.  doi: 10.1080/00036819408840277.  Google Scholar

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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.  doi: 10.3934/cpaa.2011.10.1463.  Google Scholar

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O. De Feo and S. Rinaldi, Yield and dynamics of tritrophic food chains,, The American Naturalist, 150 (1997), 328.   Google Scholar

[6]

M. Haque, Ratio-dependent predator-prey models of interacting populations,, Bulletin of Mathematical Biology, 71 (2009), 430.  doi: 10.1007/s11538-008-9368-4.  Google Scholar

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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.  doi: 10.1016/S0025-5564(02)00127-X.  Google Scholar

[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.  doi: 10.1137/S0036139900378542.  Google Scholar

[9]

Y. A. Kuznetsov and S. Rinaldi, Remarks on food chain dynamics,, Mathematical Biosciences, 134 (1996), 1.  doi: 10.1016/0025-5564(95)00104-2.  Google Scholar

[10]

Y. Kuang, Some mechanistically derived population models,, Mathematical Biosciences and Engineering, 4 (2007), 1.   Google Scholar

[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.   Google Scholar

[12]

S. Rinaldi, A. Gragnani and S. DeMonte, Remarks on antipredator behavior and food chain dynamics,, Theoretical Population Biology, 66 (2004), 277.  doi: 10.1016/j.tpb.2004.07.002.  Google Scholar

[13]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions,, American Naturalist, 97 (1963), 209.  doi: 10.1086/282272.  Google Scholar

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D. M. Wrzosek, Limit cycles in predator-prey models,, Mathematical Biosciences, 98 (1990), 1.  doi: 10.1016/0025-5564(90)90009-N.  Google Scholar

show all references

References:
[1]

M. Candaten and S. Rinaldi, Peak-to-peak dynamics in food chain models,, Theoretical Population Biology, 63 (2003), 257.   Google Scholar

[2]

L. Edelstein-Keshet, Mathematical Models in Biology,, McGraw-Hill, (1977).  doi: 10.1137/1.9780898719147.  Google Scholar

[3]

W. Feng, Permanence effect in a three-species food chain model,, Applicable Analysis, 54 (1994), 195.  doi: 10.1080/00036819408840277.  Google Scholar

[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.  doi: 10.3934/cpaa.2011.10.1463.  Google Scholar

[5]

O. De Feo and S. Rinaldi, Yield and dynamics of tritrophic food chains,, The American Naturalist, 150 (1997), 328.   Google Scholar

[6]

M. Haque, Ratio-dependent predator-prey models of interacting populations,, Bulletin of Mathematical Biology, 71 (2009), 430.  doi: 10.1007/s11538-008-9368-4.  Google Scholar

[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.  doi: 10.1016/S0025-5564(02)00127-X.  Google Scholar

[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.  doi: 10.1137/S0036139900378542.  Google Scholar

[9]

Y. A. Kuznetsov and S. Rinaldi, Remarks on food chain dynamics,, Mathematical Biosciences, 134 (1996), 1.  doi: 10.1016/0025-5564(95)00104-2.  Google Scholar

[10]

Y. Kuang, Some mechanistically derived population models,, Mathematical Biosciences and Engineering, 4 (2007), 1.   Google Scholar

[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.   Google Scholar

[12]

S. Rinaldi, A. Gragnani and S. DeMonte, Remarks on antipredator behavior and food chain dynamics,, Theoretical Population Biology, 66 (2004), 277.  doi: 10.1016/j.tpb.2004.07.002.  Google Scholar

[13]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions,, American Naturalist, 97 (1963), 209.  doi: 10.1086/282272.  Google Scholar

[14]

D. M. Wrzosek, Limit cycles in predator-prey models,, Mathematical Biosciences, 98 (1990), 1.  doi: 10.1016/0025-5564(90)90009-N.  Google Scholar

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