American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

S. R.-J. Jang, J. Baglama, P. Seshaiyer. Intratrophic predation in a simple food chain with fluctuating nutrient. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 335-352. doi: 10.3934/dcdsb.2005.5.335
[2]

Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823
[3]

Soliman A. A. Hamdallah, Sanyi Tang. Stability and bifurcation analysis of Filippov food chain system with food chain control strategy. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019244
[4]

Andrei Korobeinikov, William T. Lee. Global asymptotic properties for a Leslie-Gower food chain model. Mathematical Biosciences & Engineering, 2009, 6 (3) : 585-590. doi: 10.3934/mbe.2009.6.585
[5]

Guohong Zhang, Xiaoli Wang. Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3755-3786. doi: 10.3934/dcdsb.2018076
[6]

Yasuhisa Saito. A global stability result for an N-species Lotka-Volterra food chain system with distributed time delays. Conference Publications, 2003, 2003 (Special) : 771-777. doi: 10.3934/proc.2003.2003.771
[7]

Seung-Yeal Ha, Yongduck Kim, Zhuchun Li. Asymptotic synchronous behavior of Kuramoto type models with frustrations. Networks & Heterogeneous Media, 2014, 9 (1) : 33-64. doi: 10.3934/nhm.2014.9.33
[8]

Jian Yang. Asymptotic behavior of solutions for competitive models with a free boundary. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3253-3276. doi: 10.3934/dcds.2015.35.3253
[9]

Maria Paola Cassinari, Maria Groppi, Claudio Tebaldi. Effects of predation efficiencies on the dynamics of a tritrophic food chain. Mathematical Biosciences & Engineering, 2007, 4 (3) : 431-456. doi: 10.3934/mbe.2007.4.431
[10]

Jing Liu, Xiaodong Liu, Sining Zheng, Yanping Lin. Positive steady state of a food chain system with diffusion. Conference Publications, 2007, 2007 (Special) : 667-676. doi: 10.3934/proc.2007.2007.667
[11]

Florian De Vuyst, Francesco Salvarani. Numerical simulations of degenerate transport problems. Kinetic & Related Models, 2014, 7 (3) : 463-476. doi: 10.3934/krm.2014.7.463
[12]

Georg Hetzer, Tung Nguyen, Wenxian Shen. Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1699-1722. doi: 10.3934/cpaa.2012.11.1699
[13]

Cecilia Cavaterra, Maurizio Grasselli. Asymptotic behavior of population dynamics models with nonlocal distributed delays. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 861-883. doi: 10.3934/dcds.2008.22.861
[14]

Suzanne Lynch Hruska. Rigorous numerical models for the dynamics of complex Hénon mappings on their chain recurrent sets. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 529-558. doi: 10.3934/dcds.2006.15.529
[15]

Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551
[16]

Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049
[17]

Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343
[18]

Emmanuel Trélat. Optimal control of a space shuttle, and numerical simulations. Conference Publications, 2003, 2003 (Special) : 842-851. doi: 10.3934/proc.2003.2003.842
[19]

Tómas Chacón-Rebollo, Macarena Gómez-Mármol, Samuele Rubino. On the existence and asymptotic stability of solutions for unsteady mixing-layer models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 421-436. doi: 10.3934/dcds.2014.34.421
[20]

Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5673-5694. doi: 10.3934/dcdsb.2019101

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences