Global bifurcation analysis of the Leslie – Gower system with additive Allee effect and Holling functional response

Gaiko V.A.
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In this paper, we consider predator – prey models and carry out a global bifurcation analysis of the Leslie –Gower system with an additive Allee effect and a simplified Holling type III functional response, which models the dynamics of predator and prey populations in a given ecological or biomedical system. This system uses the most common mathematical form of expressing the Allee effect (or law) through the prey growth function. Allee’s law states that there is a very specific relationship between individual fitness to living conditions and the number or density of individuals of a given species, namely: with an increase in the population size, the ability to survive and reproductive ability also increases. After algebraic transformations, the rational Leslie –Gower system with additive Allee effect and simplified Holling type III functional response can be written as a quantic-sextic dynamical system, i. e., as a system with polynomials of the fifth and sixth degrees. Using information about its singular points and applying our bifurcation-geometric approach to qualitative analysis, we study global bifurcations of limit cycles of the quintic-sextic system. To control all limit cycle bifurcations, especially bifurcations of multiple limit cycles, it is necessary to know the properties and combine the actions of all parameters rotating the vector field of the system. This can be done using the Wintner – Perko termination principle, according to which a maximal one-parameter family of multiple limit cycles terminates either at a singular point, which typically has the same multiplicity (cyclicity), or at a separatrix cycle, which also typically has the same multiplicity (cyclicity). This principle is a consequence of the principle of natural termination which was stated for higher-dimensional dynamical systems by Wintner who studied one-parameter families of periodic orbits of the restricted three-body problem and proved that in the analytic case any oneparameter family of periodic orbits can be uniquely continued through any bifurcation except a period-doubling bifurcation. Applying the planar Wintner – Perko principle, we prove that if the cyclicity of the focus in the system under consideration is three, then the system can have at most three limit cycles surrounding one singular point.

Keywords: predator – prey model, Leslie –Gower system, Allee effect, simplified Holling type III functional response, field rotation parameter, bifurcation, singular point, limit cycle, Wintner – Perko termination principle
Citation in English: Gaiko V.A. Global bifurcation analysis of the Leslie – Gower system with additive Allee effect and Holling functional response // Computer Research and Modeling, 2025, vol. 17, no. 1, pp. 125-138
Citation in English: Gaiko V.A. Global bifurcation analysis of the Leslie – Gower system with additive Allee effect and Holling functional response // Computer Research and Modeling, 2025, vol. 17, no. 1, pp. 125-138
DOI: 10.20537/2076-7633-2025-17-1-125-138
URL: http://crm-en.ics.org.ru/journal/article/3580/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

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