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Zbl 1184.92046
Aguirre, Pablo; González-Olivares, Eduardo; Sáez, Eduardo
Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. (English)
[J] SIAM J. Appl. Math. 69, No. 5, 1244-1262 (2009). ISSN 0036-1399; ISSN 1095-712X/e

Summary: A bidimensional continuous-time differential equations system is analyzed which is derived from Leslie-type predator-prey schemes by considering a nonmonotonic functional response and an Allee effect on the population prey. For the system obtained we describe the bifurcation diagram of limit cycles that appears in the first quadrant, the only quadrant of interest for the sake of realism. We show that, under certain conditions over the parameters, the system allows the existence of three limit cycles: The first two cycles are infinitesimal ones generated by Hopf bifurcation; the third one arises from a homoclinic bifurcation. Furthermore, we give conditions over the parameters such that the model allows long-term extinction or survival of both populations. In particular, the presence of a weak Allee effect does not necessarily imply extinction of populations for our model.

MSC 2000:: *92D40 Ecology
34C05 Qualitative theory of some special solutions of ODE
34C23 Bifurcation (periodic solutions)
34C60 Applications of qualitative theory of ODE

Keywords: stability; limit cycles; homoclinic orbits; bifurcations; predator-prey models; Allee effect

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Zentralblatt MATH Berlin [Germany]