Dynamics of a Leslie-Gower type predation model with non-monotonic functional response and weak Allee effect on prey

Authors

  • Francisco J. Reyes Bahamón Escuela de Ciencias Básicas Tecnología e Ingeniería, Universidad Nacional Abierta y a Distancia, Colombia.
  • Simeon Casanova Trujillo Departamento de Matemática y Estadística, Universidad Nacional de Colombia, Colombia.
  • Eduardo Gonzalez-Olivares Pontificia Universidad Católica de Valparaíso, Chile.

DOI:

https://doi.org/10.17268/sel.mat.2023.02.07

Keywords:

Predator-prey model, bifurcation, Allee effect, holling functional response, Local stability, Hopf bifurcation, limit cycles

Abstract

This research concerns with analysis of a class of modified predator- prey type Leslie-Gower models. The model is described by an autonomous nonlinear ordinary differential equation system. The functional response of predators is Holling IV type or non-monotone, and the growth of prey is affected by the Allee effect. An important aspect is the study of the point (0, 0) since it has a strong influence on the behavior of the system being essential for the existence and extinction of both species, although the proposed system is not define there.

Author Biography

Francisco J. Reyes Bahamón, Escuela de Ciencias Básicas Tecnología e Ingeniería, Universidad Nacional Abierta y a Distancia, Colombia.

Programa de Licenciatura en Matemáticas

References

Courchamp F, Clutton-Brock T, Grenfell B. Inverse density dependence and the Allee effect. Trends in Ecology and Evolution. 1999; 14(6):405-410.

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

González-Olivares E, Mena-Lorca J, Rojas-Palma A, Flores JD. Dynamical complexities in the Leslie–Gower predator–prey model as consequences of the Allee effect on prey. Applied Mathematical Modelling. 2011; 35(1):366-381.

Lamontagne Y, Coutu C, Rousseau C. Bifurcation Analysis of a Predator-Prey System with Generalised Holling Type III Functional Response. J. of Dynamics and Differential Equations. 2008; 20(3):535-571.

Broer H, Saleh K, Naudot V, Roussarie R. A Dynamics of a predator-prey model with nonmonotonic response function. Discrete and Continuous Dynamical Systems. 2007; 18(2):221-251.

Gallego-Berrío L, González-Olivares E. The Holling-Tanner predation model with a special weak Allee effect on prey. Proceedings of the 2014 International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE. 2014; 2:585-5961.

Li Y, Xiao D. Bifurcations of a predator-prey system of Holling and Leslie types. Chaos, Solitons and Fractals. 2007; 34(2):606-620.

González-Olivares E, Tintinago-Ruiz P, Rojas-Palma A. A Leslie–Gower-type predator–prey model with sigmoid functional response. Int. J. of Computer Mathematics. 2015; 92(9):1895-1909.

Saez E, González-Olivares E. Dynamics of a Predator-Prey Model. SIAM Journal on Applied Mathematics. 1999; 59:1867-1878.

Leslie P. Some Further Notes on the Use of Matrices in Population Mathematics. Biometrika. 1948; 35:213-245.

Korobeinikov A. A Lyapunov function for Leslie-Gower predator-prey models. Applied Mathematics Letters. 2001; 14(6):697-699.

Stephens P, Sutherland W, Freckleton R. What is the Allee effect?. Source: Oikos. 1999; 87:185-190.

Liermann M, Hilborn R. Depensation: evidence, models and implications. Fish Fisheries. 2001; 2:33-58.

Courchamp F, Berec L, Gascoigne J. Allee Effects in Ecology and Conservation. Oxford University Press. 2008.

Stephen P, Sutherland W. Consequences of the Allee effect for behaviour, ecology and conservation. Trends Ecologics. 1999 14:401-405.

González-Olivares E, González B, Mena-Lorca J, Ramos-Jiliberto R. Modelling the Allee effect: are the different mathematical forms proposed equivalents. In Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006. 2007; 53-71.

Perko L. Differential Equations and Dynamical Systems, Springer. 7 (2013).

Kuznetsov Y. Elements of applied bifurcation theory. Springer Applied Mathematics Science. 112, 2013.

Downloads

Published

2023-12-27

How to Cite

Reyes Bahamón, F. J., Casanova Trujillo, S. ., & Gonzalez-Olivares, E. (2023). Dynamics of a Leslie-Gower type predation model with non-monotonic functional response and weak Allee effect on prey. Selecciones Matemáticas, 10(02), 310 - 323. https://doi.org/10.17268/sel.mat.2023.02.07

Issue

Vol. 10 No. 02 (2023): August - December

Section

Articles

License

Copyright (c) 2023 Selecciones Matemáticas

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.

Most read articles by the same author(s)