A predation model considering a generalist predator and the Rosenzweig functional response
DOI: https://doi.org/10.3846/mma.2025.22892Abstract
This work deals with the dynamics of an ordinary differential equation system describing a Leslie-Gower predator-prey model with a generalist predator and a non-differentiable functional response proposed by M. L. Rosenzweig, given by h(x) = qxα with 0 < α < 1. Two aspects have a significant impact on the model: (1) the predator’s carrying capacity depends on both the favorite prey population and an alternative food source, and (2) consumers have access to an alternative food source. Among the main results, a separatrix curve Σ arises dividing the phase plane into regions with different dynamic behaviors. Trajectories above the separatrix curve Σ reach the vertical axis in finite time, while those below Σ may converge to positive equilibrium points, limit cycles, or homoclinic connections. Furthermore, the system is non-Lipschitz, implying non-uniqueness of solutions at points of the vertical axis. Several bifurcations, including saddle-node, homoclinic, Hopf, generalized Hopf, and Bogdanov-Takens bifurcations, are identified through the use of computational techniques. The dynamics of the system are visualized by presenting a bifurcation diagram in a convenient parameter space.
Keywords:
predator-prey model, Leslie-Gower model, separatrix, bifurcations, limit cycles, homoclinic connectionHow to Cite
Share
License
Copyright (c) 2025 The Author(s). Published by Vilnius Gediminas Technical University.
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
V. Ajraldi, M. Pittavino and E. Venturino. Modeling herd behavior in population systems. Nonlinear Analysis: Real World Applications, 12(4):2319–2338, 2011. https://doi.org/10.1016/j.nonrwa.2011.02.002
A.D. Bazykin. Nonlinear dynamics of interacting populations, volume 11 of World Scientific Series on Nonlinear Science, Series A. World Scientific, Singapore; River Edge, NJ, 1998. https://doi.org/10.1142/2284
A.A. Berryman, A.P. Gutierrez and R. Arditi. Credible, parsimonious and useful predator-prey models: a reply to Abrams, Gleeson, and Sarnelle. Ecology, 76(6):1980–1985, 1995. https://doi.org/10.2307/1940728
G. Birkhoff and R. Gian-Carlo. Ordinary differential equations. John Wiley & Sons, New York, NY, 1989.
J.L. Bravo, M. Fernández, M. Gámez, B. Granados and A. Tineo. Existence of a polycycle in non-Lipschitz Gause-type predator–prey models. Journal of Mathematical Analysis and Applications, 373(2):512–520, 2011. https://doi.org/10.1016/j.jmaa.2010.08.001.
C. Chicone. Ordinary differential equations with applications, volume 34 of Texts in Applied Mathematics. Springer-Verlag, New York, 2 edition, 2006.
Colin W Clark. The worldwide crisis in fisheries: economic models and human behavior. Cambridge University Press, 2006.
C.W. Clark. Mathematical Bioeconomics. In P. van den Driessche(Ed.), Mathematical Problems in Biology, volume 2, pp. 29–45, Berlin, Heidelberg, 1974. Springer, Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-454554_3
A. Dhooge, W. Govaerts and Yu.A. Kuznetsov. MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software (TOMS), 29(2):141–164, 2003. https://doi.org/10.1145/779359.779362
J. Díaz-Avalos and E. González-Olivares. Dynamics of a predator-prey model with a non-differentiable functional response. In J. Vigo-Aguiar (Ed.), Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2017), volume 3, pp. 765–776, Costa Ballena, Rota, Cádiz, Spain, 2017. Arizona State University.
F. Dumortier, J. Llibre and J.C. Artés. Qualitative theory of planar differential systems, volume 2 of Universitext. Springer-Verlag, Berlin, Heidelberg, 1 edition, 2006. https://doi.org/10.1007/978-3-540-32902-2
H.I. Freedman. Deterministic mathematical models in population ecology, volume 57 of Pure and Applied Mathematics. Marcel Dekker Incorporated, New York, 1980.
E. González-Olivares and V. Rivera-Estay and A. Rojas-Palma and K. VilchesPonce. A Leslie–Gower type predator-prey model considering herd behavior. RicerchediMatematica, 73(4):1683–1706,2024. https://doi.org/10.1007/s11587022-00694-5
E. González-Olivares, E. Sáez, E. Stange and I. Szantó. Topological description of a non-differentiable bioeconomics model. The Rocky Mountain Journal of Mathematics, 35(4):1133–1155, 2005. https://doi.org/10.1216/rmjm/1181069680
J. Hadamard. Lectures on Cauchy’s problem in linear partial differential equations, volume 15 of Yale Studies in Mathematics. Yale university press, New Haven, CT, 1923.
C.S. Holling. The components of predation as revealed by a study of smallmammal predation of the European pine sawfly. The Canadian Entomologist, 91(5):293–320, 1959. https://doi.org/10.4039/Ent91293-5
A. Korobeinikov. A Lyapunov function for Leslie-Gower predatorprey models. Applied Mathematics Letters, 14(6):697–699, 2001. https://doi.org/10.1016/S0893-9659(01)80029-X
Y. Kuang and H.I. Freedman. Uniqueness of limit cycles in Gause-type models of predator–prey systems. Mathematical Biosciences, 88(1):67–84, 1988. https://doi.org/10.1016/0025-5564(88)90049-1
Y.A. Kuznetsov. Elements of applied bifurcation theory, volume 112 of Applied Mathematical Sciences. Springer-Verlag, New York, 2 edition, 1998. https://doi.org/10.1007/978-1-4612-0745-9
P.H. Leslie. Some further notes on the use of matrices in population mathematics. Biometrika, 35(3-4):213–245, 1948. https://doi.org/10.1093/biomet/35.3-4.213
W.-M. Liu, S.A. Levin and Y. Iwasa. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. Journal of Mathematical Biology, 23(2):187–204, 1986. https://doi.org/10.1007/BF00276956
R.M. May. Stability and Complexity in Model Ecosystems. Princeton Landmarks in Biology. Princeton University Press, Princeton, NJ, 2 edition, 2001. https://doi.org/10.1515/9780691206912. Reprint of the 1973 work with a new introduction
M.R. Myerscough, M.J. Darwen and W.L. Hogarth. Stability, persistence and structural stability in a classical predator-prey model. Ecological Modelling, 89(1-3):31–42, 1996. https://doi.org/10.1016/0304-3800(95)00117-4
L. Perko. Differential equations and dynamical systems, volume 7 of Texts in Applied Mathematics. Springer-Verlag, New York, 3 edition, 2001. https://doi.org/10.1007/978-1-4613-0003-8
V. Rivera-Estay, E. González-Olivares, A. Rojas-Palma and K. Vilches-Ponce. Dynamics of a class of Leslie-Gower predation models with a non-differentiable functional response. In H. Dutta and J.F. Peters (Eds.), Applied Mathematical Analysis: Theory, Methods, and Applications, volume 177 of Studies in Systems, Decision and Control, pp. 433–457. Springer, Cham, 2019. https://doi.org/10.1007/978-3-319-99918-0_14
M.L. Rosenzweig. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science, 171(3969):385–387, 1971. https://doi.org/10.1126/science.171.3969.385
E. Sáez and I. Szántó. A polycycle and limit cycles in a non-differentiable predator-prey model. Proceedings – Mathematical Sciences, 117(2):219–231, 2007. https://doi.org/10.1007/s12044-007-0018-9
P. Turchin. Complex Population Dynamics: A Theoretical/Empirical Synthesis, volume 35 of Monographs in Population Biology. Princeton University Press, Princeton, 2003.
E. Venturino and S. Petrovskii. Spatiotemporal behavior of a prey–predator system with a group defense for prey. Ecological Complexity, 14:37–47, 2013. https://doi.org/10.1016/j.ecocom.2013.01.004
K. Vilches, E. González-Olivares and A. Rojas-Palma. Prey herd behavior modeled by a generic non-differentiable functional response Mathematical Modelling of Natural Phenomena, 13(3):26, 2018. https://doi.org/10.1051/mmnp/2018038
View article in other formats
CrossMark check
Published
Issue
Section
Copyright
Copyright (c) 2025 The Author(s). Published by Vilnius Gediminas Technical University.
License
This work is licensed under a Creative Commons Attribution 4.0 International License.