Limit cycles in a tritrophic food chain model with general functional responses
-
Gamaliel Blé
and
Iván Loreto-Hernández
Abstract
The conditions to have a stable limit cycle by an Andronov–Hopf bifurcation in a tritrophic model are given. A generalized logistic growth function for the prey is considered, and a general family of functional responses, including the Holling type, are taken for the predators. Some results obtained in previous works for tritrophic models, which consider logistic growth in the prey and Holling functional responses, are generalized.
Acknowledgment
We thank the referees for their valuable comments, which allowed us to improve this work.
-
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: None declared.
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
A The first Lyapunov coefficient
The first Lyapunov coefficient of the differential system (2) at the equilibrium point p is
where
References
[1] J. D. Murray, Mathematical Biology I: An Introduction, 3rd ed., New York, Springer, 2002.10.1007/b98868Search in Google Scholar
[2] H. I. Freedman and S. Ruan, “Hopf bifurcation in three-species food chain models with group defense,” Math. Biosci., vol. 111, pp. 73–87, 1992. https://doi.org/10.1016/0025-5564(92)90079-c.Search in Google Scholar
[3] J. P. Francoise and J. Llibre, “Analytical study of a triple Hopf bifurcation in a tritrophic food chain model,” Appl. Math. Comput., vol. 217, pp. 7146–7154, 2011. https://doi.org/10.1016/j.amc.2011.01.109.Search in Google Scholar
[4] G. Blé, V. Castellanos, and I. Loreto–Hernández, “Andronov–Hopf and Bautin Bifurcation in a tritrophic food chain model with Holling functional response types IV and II,” Electron. J. Qual. Theor. Differ. Equ., vol. 78, pp. 1–27, 2018. https://doi.org/10.14232/ejqtde.2018.1.78.Search in Google Scholar
[5] V. Castellanos, F. E. Castillo-Santos, M. A. Dela-Rosa, and I. Loreto–Hernández, “Hopf and Bautin bifurcation in a tritrophic food chain model with Holling functional response types III and IV,” Int. J. Bifurcat. Chaos., vol. 28, no. 3, p. 24, 2018. https://doi.org/10.1142/s0218127418500359.Search in Google Scholar
[6] H. Molla, M. S. Rahman, and S. Sarwardi, “Dynamics of a predator–prey model with Holling type II functional response incorporating a prey refuge depending on Both the species,” Int. J. Nonl. Sci. Num. Simul., vol. 20, no. 1, pp. 89–104, 2018. https://doi.org/10.1515/ijnsns-2017-0224.Search in Google Scholar
[7] R. Sigal, “Algorithms for the Routh–Hurwitz stability test,” Math. Comput. Model., vol. 13, no. 8, pp. 69–77, 1990. https://doi.org/10.1016/0895-7177(90)90072-u.Search in Google Scholar
[8] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., New York, Springer-Verlag, 2004.10.1007/978-1-4757-3978-7Search in Google Scholar
[9] Y. A. Kuznetsov, “Andronov–Hopf bifurcation,” Scholarpedia, vol. 1, no. 10, p. 1858, 2006. https://doi.org/10.4249/scholarpedia.1858.Search in Google Scholar
[10] J. E. Marsden and McCracken, The Hopf Bifurcation and its Applications, New York, Springer-Verlag, 1976.10.1007/978-1-4612-6374-6Search in Google Scholar
[11] L. Perko, Differential Equations and Dynamical Systems, 3rd ed., New York, Springer-Verlag, 2001.10.1007/978-1-4613-0003-8Search in Google Scholar
[12] J. L. Rocha and A. Taha, “Allee’s effect bifurcation in generalized logistic maps,” Int. J. Bifurcat. Chaos, vol. 29, no. 3, p. 19, 2019. https://doi.org/10.1142/S0218127419500391.Search in Google Scholar
[13] C. S. Holling, “Some characteristics of simple types of predation and parasitism,” Can. Entomol., vol. 91, pp. 385–398, 1959. https://doi.org/10.4039/ent91385-7.Search in Google Scholar
[14] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, New York, Marcel Dekker, 1980.Search in Google Scholar
[15] L. A. Real, “The kinetics of functional response,” Am. Nat., vol. 111, no. 978, pp. 289–300, 1977. https://doi.org/10.1086/283161.Search in Google Scholar
[16] J. H. P. Dawes and M. O. Souza, “A derivation of Holling’s type I, II and III functional responses in predator–prey systems,” J. Theor. Biol., vol. 327, pp. 11–22, 2013. https://doi.org/10.1016/j.jtbi.2013.02.017.Search in Google Scholar PubMed
[17] G. Blé, V. Castellanos, and M. A. Dela-Rosa, “Coexistence of species in a tritrophic food chain model with Holling functional response type IV,” Math. Methods Appl. Sci., vol. 16, pp. 6683–6701, 2018. https://doi.org/10.1002/mma.5184.Search in Google Scholar
[18] M. L. Rosenzweig, “Paradox of enrichment: destabilization of exploitation ecosystems in ecological time,” Science, vol. 171, no. 3969, pp. 385–387, 1971. https://doi.org/10.1126/science.171.3969.385.Search in Google Scholar PubMed
[19] W. T. Jamieson and J. Reis, “Global behavior for the classical Nicholson–Bailey model,” J. Math. Anal. Appl., vol. 461, pp. 492–499, 2018. https://doi.org/10.1016/j.jmaa.2017.12.071.Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Dynamics of synthetic drug transmission models
- Collision-induced amplitude dynamics of pulses in linear waveguides with the generic nonlinear loss
- Global dissipativity of non-autonomous BAM neural networks with mixed time-varying delays and discontinuous activations
- On the inverse problem for nonlinear strongly damped wave equations with discrete random noise
- A second-order nonlocal regularized variational model for multiframe image super-resolution
- Algebro-geometric integration of a modified shallow wave hierarchy
- Singularity analysis of a 7-DOF spatial hybrid manipulator for medical surgery
- Propagation of diffusing pollutant by kinetic flux-vector splitting method
- Limit cycles in a tritrophic food chain model with general functional responses
- Local and parallel stabilized finite element methods based on full domain decomposition for the stationary Stokes equations
- Stress concentration effect on deflection and stress fields of a master leaf spring through domain decomposition and geometry updation technique
- Electrostatically actuated double walled piezoelectric nanoshell subjected to nonlinear van der Waals effect: nonclassical vibrations and stability analysis
- Traveling wave solutions of the generalized Rosenau–Kawahara-RLW equation via the sine–cosine method and a generalized auxiliary equation method
- A predictor–corrector compact finite difference scheme for a nonlinear partial integro-differential equation
- Parameter inference with analytical propagators for stochastic models of autoregulated gene expression
- DCSK performance analysis of a chaos-based communication using a newly designed chaotic system
- On successive linearization method for differential equations with nonlinear conditions
- Comparison of different time discretization schemes for solving the Allen–Cahn equation
- The homoclinic breather wave solution, rational wave and n-soliton solution to a nonlinear differential equation
- Diversity of interaction phenomenon, cross-kink wave, and the bright-dark solitons for the (3 + 1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Dynamics of synthetic drug transmission models
- Collision-induced amplitude dynamics of pulses in linear waveguides with the generic nonlinear loss
- Global dissipativity of non-autonomous BAM neural networks with mixed time-varying delays and discontinuous activations
- On the inverse problem for nonlinear strongly damped wave equations with discrete random noise
- A second-order nonlocal regularized variational model for multiframe image super-resolution
- Algebro-geometric integration of a modified shallow wave hierarchy
- Singularity analysis of a 7-DOF spatial hybrid manipulator for medical surgery
- Propagation of diffusing pollutant by kinetic flux-vector splitting method
- Limit cycles in a tritrophic food chain model with general functional responses
- Local and parallel stabilized finite element methods based on full domain decomposition for the stationary Stokes equations
- Stress concentration effect on deflection and stress fields of a master leaf spring through domain decomposition and geometry updation technique
- Electrostatically actuated double walled piezoelectric nanoshell subjected to nonlinear van der Waals effect: nonclassical vibrations and stability analysis
- Traveling wave solutions of the generalized Rosenau–Kawahara-RLW equation via the sine–cosine method and a generalized auxiliary equation method
- A predictor–corrector compact finite difference scheme for a nonlinear partial integro-differential equation
- Parameter inference with analytical propagators for stochastic models of autoregulated gene expression
- DCSK performance analysis of a chaos-based communication using a newly designed chaotic system
- On successive linearization method for differential equations with nonlinear conditions
- Comparison of different time discretization schemes for solving the Allen–Cahn equation
- The homoclinic breather wave solution, rational wave and n-soliton solution to a nonlinear differential equation
- Diversity of interaction phenomenon, cross-kink wave, and the bright-dark solitons for the (3 + 1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation
