Stability in a modified Leslie-Gower type predation model considering competence among predators

Authors

DOI:

https://doi.org/10.17268/sel.mat.2020.01.02

Keywords:

Predator-prey model, functional response, bifurcation, limit cycle, separatrix curve, stability

Abstract

In the ecological literature, the interference (self-interference) or competition among predators to effect the harvesting of their prey has been modeled by different mathematical formulations.

In this work, we will establish the dynamical properties of the Leslie-Gower type predation model, in which is incorporated one of these form, described by the function b (y) = yalfa
, with 0 < alfa < 1.

The main difficulty of the analysis is due to this function is not differentiable for y = 0, and the Jacobian matrix is not defined in the equilibrium points over the horizontal axis (x-axis).

Previously, we showed a summary with the fundamental properties of the Leslie-Gower type model, in order to carry out an adequate comparative analysis with models where self-interference between predators is incorporated.

To reinforce our results, some numerical simulations are shown.

References

Aguirre P., González-Olivares E., Sáez E. Two limit cycles in a Leslie.Gower predator-prey model with additive Allee effect. Nonlinear Analysis: Real World Applications. 2009; 10:1401-1416.

Aguirre P, González-Olivares E, Sáez E. Three limit cycles in a Leslie-Gower predator–prey model with additive Allee effect. SIAM Journal on Applied Mathematics. 2009; 69:1244-1262.

Arancibia-Ibarra C., and González-Olivares E. A modified Leslie-Gower predator-prey model with hyperbolic functional response and Allee effect on prey In R. Mondaini (Ed.) BIOMAT 2010 International Symposium on Mathematical and Computational Biology. World Scientific Co.Pte. Ltd., Singapore. 2011; 146-162.

Arrowsmith D.K, Place C.M. Dynamical System. Differential equations, maps and chaotic behaviour. Chapman and Hall; 1992.

Aziz-Alaoui M.A., Daher Okye M. Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Applied Mathematics Letters. 2003; 16:1069-1075.

Bacaër N. A short history of Mathematical Population Dynamics. Springer-Verlag. 2011.

Bazykin A. Nonlinear Dynamics of interacting populations. World Scientific Publishing Co. Pte. Ltd., 1998.

Berryman A. A., Gutierrez A.P., Arditi R. Credible, parsimonious and useful predator-prey models. A reply to Abrams, Gleeson and Sarnelle, Ecology. 1995; 76:1980-1985.

Birkhoff G.D, Rota G.C. Ordinary Differential Equations (Fourth Edition), John Wiley and Sons, New York; 1989.

Chicone C. Ordinary differential equations with applications. 2nd edition, Texts in Applied Mathematics 34, Springer, 2006.

Clark C.W. Mathematical Bioeconomic: The optimal management of renewable resources. (2nd edition). John Wiley and Sons,1990.

Clark C.W. The worldwide crisis in fisheries: Economic models and human behavior. Cambridge Univerity Press, 2006.

Coleman C.S. Hilbert’s 16th. Problem: How many cycles? In: Braun M., Coleman C.S., and Drew D. (Eds.). Differential Equations Model, Springer-Verlag. 1983; 279-297.

Dumortier F., Llibre J., Artïs J.C. Qualitative theory of planar differential systems, Springer; 2006.

Epstein J.M. Nonlinear Dynamics, Mathematical Biology, and Social Science, Addison-Wesley, 1997.

Freedman H.I. Deterministic Mathematical Model in Population Ecology, Marcel Dekker, 1980.

Freedman H.I. Stability analysis of a predator-prey system with mutual interference and density-dependent death rates. Bulletin of Mathematical Biology. 1979; 41:67-78.

Gallego-Berrío L.M. Consecuencias del efecto Allee en el modelo de depredación de May-Holling-Tanner. Tesis de Maestría, Maestría en Biomatemáticas Universidad del Quindío, Armenia, Colombia, 2004.

Gallegos-Zuñiga J. Modelo depredador-presa del tipo Leslie-Gower considerando interferencia entre los depredadores. Trabajo para optar al grado de Licenciado en Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Chile 2014.

Gause G.F. The Struggle for existence. Dover, 1934.

Geritz S., Gyllenberg M. A mechanistic derivation of the DeAngelis-Beddington functional response. Journal of Theoretical Biology. 2012; 314:106-108.

Goh B-S. Management and Analysis of Biological Populations. Elsevier Scientific Publishing Company, 1980.

González-Yañez B., González-Olivares E., Mena-Lorca J. Multistability on a Leslie-Gower Type predator-prey model with nonmonotonic functional response. In R. Mondaini and R. Dilao (eds.), BIOMAT 2006 - International Symposium on Mathematical and Computational Biology, World Scientific Co. Pte. Ltd., 2007; 359-384.

González-Olivares E., Mena-lorca J., Rojas-Palma A., Flores J.D. Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey. Applied Mathematical Modelling. 2011; 35:366-381.

González-Olivares E., Arancibia-Ibarra C., Rojas A, González-Yañez B. Dynamics of a Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response. Mathematical Biosciences and Engineering. 2019; 16(6):7995-8024.

Holling C.S. The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Canadian Entomologist. 1959; 91:293-320.

Korobeinikov A.A. Lyapunov function for Leslie-Gower predator-prey models. Applied Mathematical Letters. 2001; 14:697-699.

Leslie P.H. Some further notes on the use of matrices in population mathematics. Biometrika. 1948; 35:213-245.

Leslie P.H, and Gower J.C. The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrka. 1960; 47:219-234.

May R.M. Stability and complexity in model ecosystems (2nd edition), Princeton University Press. 2001.

Murray J.D. Mathematical Biology. Springer - Verlag New-York 1989.

Perko L. Differential Equations and Dynamical Systems. Springer-Verlag, 1991.

Sáez E., González-Olivares E. Dynamics on a predator-prey model. SIAM Journal of Applied Mathematics. 1999; 59:1867-1878.

Scudo F.M., Ziegler J.R. The golden age of Theoretical Ecology 1923-1940. Lecture Notes in Biomathematics 22. Springer-Verlag, Berlin 1978.

Tanner J.T. The stability and the intrinsic growth rates of prey and predator populations. Ecology. 1975; 56(4):855-867

Taylor R.J. Predation. London: Chapman and Hall, 1984.

Tintinago-Ruiz P.C., Gallego-Berrío L.M., González-Olivares E. Una clase de modelo de depredación del tipo Leslie-Gower con respuesta funcional racional no monotónica y alimento alternativo para los depredadores. Selecciones Matemáticas. 2019; 06(2):204-216.

Turchin P. Complex population dynamics. A theoretical/empirical synthesis. Monographs in Population Biology 35 Princeton University Press 2003.

Volterra V. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memorie della R. Accademia dei Lincei, S.VI, IT 1926; II:31-113.

Vera-Damián Y., Vidal C., González-Olivares E. Dynamics and bifurcations of a modified Leslie-Gower type model considering a Beddington-DeAngelis functional response. Mathematical Methods in the Applied Sciences. 2019; 42:3179-3210.

Published

2020-07-25

How to Cite

González-Olivares, E., & Gallegos-Zuñiga, J. (2020). Stability in a modified Leslie-Gower type predation model considering competence among predators. Selecciones Matemáticas, 7(01), 10-24. https://doi.org/10.17268/sel.mat.2020.01.02

Most read articles by the same author(s)