Collaboration between predators and weak Allee effect in prey. Consequences on the dynamics of a predation model


  • Eduardo González-Olivares Pontificia Universidad Católica de Valparaíso, Chile.
  • 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 in the real world are usually affected by diverse ecological phenomena acting on both the prey and the predators.

Collaboration or cooperation between predators is one of those behaviors, which has received less attention from researchers than competition among consumers. These contacts are important aspects of the dynamics of food chains and trophic webs.

In this work, we will study the influence of collaboration or cooperation (hunting cooperation) between predators to consume (or capture) their favorite prey, which are affected by an effect Allee weak.

We extend the results obtained in a previously published model, considering only collaboration between predators and in which the Allee effect is absent.

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.

Antonelli PL, Lin X. Bifurcation analysis on a coral-starfish model. Math. and Computer Modelling. 1990; 13: 35-44.

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

Berec L, Angulo E, Courchamp F. Multiple Allee effects and population management. Trends in Ecology and Evolution. 2007; 22:185-191.

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

Boukal DS, Berec L. Single-species models and the Allee effect: Extinction boundaries, sex ratios and mate encounters. J. of Theoretical Biology. 2002; 218:375-394.

Boukal DS, SabelisMW, Berec L. How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses, Theoretical Population Biology. 2007; 72:136-147.

Boukal DS, Berec L. Modelling mate-finding Allee effects and populations dynamics, with applications in pest control, Population Ecology. 2009; 51: 445-458.

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

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

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

Dennis B. Allee effects: population growth, critical density, and the chance of extinction. Natural Resource Modeling. 1989; 3:481-538. Published by Princeton University Press 2002.

Dugatkin LA. (Editor) Model systems in Behavioral Ecology. Integrating conceptual, theoretical, and empirical approaches, Princeton University Press 2002.

Francomano E, Hilker FM, Paliaga M, Venturino E. Separatrix reconstruction to identify tipping points in an ecoepidemiological model, Appl Math Comput. 2018; 31(8):80-91.

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

Gaiko V. Global Bifurcation theory and Hilbert’s sixteenth problem, Springer, Boston, 2003.

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

González-Olivares E, González-Yañez B, Mena-Lorca J, Ramos-Jiliberto R. Modelling the Allee effect: Are the different mathematical forms proposed equivalents? In: R. Mondaini (Ed.) in Proceedings of the International Symposium on Mathematical and Computational Biology BIOMAT 2006, E-papers Servic¸os Editoriais Ltda., Rio de Janeiro. 2007; 53-71.

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; 35:366-381.

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

González-Olivares E, González-Yañez B, Becerra-Klix R, Ramos-Jiliberto R. Multiple stable states in a model based on predator-induced defenses, Ecological Complexity. 2017; 32:111-120.

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, Rojas-Palma A. A simple Gause type predator-prey model considering social predation, Mathematical Methods in the Applied Sciences. 2019; 42(1):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. 2020; 7(2):302-313.

González-Olivares E, Rojas Palma A. Un modelo de depredación del tipo Leslie-Gower modificado que considera la colaboración entre depredadores. (A modified Leslie-Gower-type predation model considering collaboration between predators), Selecciones Matemáticas. 2021; 8(2):379-385.

González-Olivares E, Rojas Palma A. Influencia del efecto Allee en las presas y de la colaboración entre los depredadores en un modelo de depredación del tipo Leslie-Gower, (Influence of the Allee effect on prey and collaboration between predators in a Leslie-Gower-type predation model). Modelamiento Matemático de Sistemas Biológicos (MMSB). 2021; 1(2):21-29.

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

Jang SR-J, Zhang W, Larriva V. Cooperative hunting in a predator-prey system with Allee effects in the prey, Natural Resource Modelling. 2018; 31:1-20.

Kot M. Elements of Mathematical Ecology, Cambridge University Press, Cambridge, New York, 2003.

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

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

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

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.

Monzón P. Almost global attraction in planar systems, System and Control Letter. 2005; 54:753-758.

Taylor RJ. Predation, Chapman and Hall, 1984.

Perko L. Differential equations and dynamical systems (3rd ed.) Springer, 2001.

Philip JR. Sociality and sparce populations, Ecology. 1957; 38:107-111.

Rantzer A. A dual to Lyapunov’s stability theorem, System and Control Letter. 2001; 42:161-168.

Stephens PA, Sutherland WJ. Consequences of the Allee effect for behaviour, ecology and conservation. Trends in Ecology and Evolution. 1999; 14:401-405.

Stephens PA, Sutherland WJ, Freckleton RP. What is the Allee effect?, Oikos. 1999; 87:185-190.

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

Thieme HR. Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2003.

Wang L, Qiu Z, Feng T, Kang Y. An eco-epidemiological model with social predation subject to a component Allee effect, Applied Mathematical Modelling. 2022; 101:111-131.

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



How to Cite

González-Olivares, E., & Rojas-Palma, A. (2022). Collaboration between predators and weak Allee effect in prey. Consequences on the dynamics of a predation model. Selecciones Matemáticas, 9(01), 173 - 183.

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