A modified Leslie-Gower-type predation model considering collaboration between predators


  • Eduardo González Olivares Pontificia Universidad Católica de Valparaíso, Chile. http://orcid.org/0000-0003-3907-0076
  • Alejandro Rojas-Palma Departamento de Matemática, Física y Estadística, Facultad de Ciencias Básicas, Universidad Católica del Maule, Talca, Chile.




Predator-prey model, functional response, stability, bifurcations, limit cycles


The interactions between predators and their prey are one of the most important aspects in the dynamics of food chains or trophic webs.

Usually, this relationship in the real world is influenced by various behaviors, both from prey and  predators. Collaboration or cooperation between predators is one of those behaviors, which has received less attention than other behaviors of predators, such as competition between them.

In this work, we will model the cooperation between predators to capture (or consume) their favorite

prey using a recent proposition that modifies the linear functional response of the Leslie-Gower model.

We show that this modified model has richer dynamics than the original, obtaining varied results.

Among the main ones, populations can oscillate around a point where population sizes are fixed.

Author Biography

Eduardo González Olivares, Pontificia Universidad Católica de Valparaíso, Chile.

Profesor titular Jubilado


Antonelli PL, Kazarinoff ND. Starfish predation of a growing coral reef community. J. Theor. Biol. 1984;107:667-684.

Antonelli PL, Kazarinoff ND. Modelling density-dependent aggregation and reproduction in certain terrestrial and marine ecosystems: A comparative study, Ecological Modelling 1988;41:219-227.

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

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

Birkhoff G, Rota GS. Ordinary Differential Equations (4th ed.). John Wiley & Sons, New York 1989.

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

Courchamp F, Berec L, Gascoigne J. Allee effects in Ecology and Conservation, Oxford University Press 2007.

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

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

González-Olivares E, González-Yañez B, Becerra-Klix R. Prey refuge use as a function of predator-prey encounters, Private communication, submitted to International Journal of Biomathematics (2012).

González-Olivares E, Cabrera-Villegas J, Córdova-Lepe F, Rojas Palma A. Competition among predators and Allee effect on prey: their influence on a Gause-type predation model, Mathematical Problems in Engineering, vol. 2019, Article ID 3967408, 19 pages, 2019.

González-Olivares E, Valenzuela-Figueroa S. and Rojas-Palma A. A simple Gause type predator-prey model considering social predation, Mathematical Methods in the Applied Sciences 42 (2019) 5668-5686.

González-Olivares E, Rojas Palma A. lnfluencia del efecto Allee fuerte en las presas y de la competición entre los depredadores en modelos de depredación del tipo Leslie-Gower, Selecciones Matemáticas 7(2) (2020) 302-313.

Hilker FM, Paliaga M, Venturino E. Diseased social predators, Bulletin of Mathematical Biology 2017;79:2175-2196.

Leslie PH. Some further notes on the use of matrices in population mathematics, Biometrica 35 (1948) 213-245.

Leslie PH, Gower JC. The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika 47 (1960) 219-234.

MacNulty DR, Tallian A, Stahler DR, Smith DW. Influence of group size on the success of wolves hunting bison, Plos One 2014;9(11):e112884 1-8.

Maynard Smith J. Models in Ecology, University Press 1974.

Molla H, Rahman MS and Sarwardi S. Dynamical study of a prey-predator model incorporating nonlinear prey refuge and additive Allee effect acting on prey species. Model. Earth Syst. Environ. 7, 749-765 (2021).

Taylor RJ. Predation, Chapman and Hall, 1984.

Teixeira Alves M, Hilker FM. Hunting cooperation and Allee effects in predators, Journal of Theoretical Biology 419 (2017) 13-22.

Turchin P. Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, Princeton, New Jersey, 2003.

Ye P, Wu D. Impacts of strong Allee effect and hunting cooperation for a Leslie-Gower predator-prey system, Chinese Journal of Physics 68 (2020) 49-64.



How to Cite

González Olivares, E., & Rojas-Palma, A. (2021). A modified Leslie-Gower-type predation model considering collaboration between predators. Selecciones Matemáticas, 8(02), 379-385. https://doi.org/10.17268/sel.mat.2021.02.14