Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. (English) Zbl 1184.92046

Aguirre, Pablo; González-Olivares, Eduardo; Sáez, Eduardo

SIAM J. Appl. Math. 69, No. 5, 1244-1262 (2009).

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:

92D40	Ecology
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations
34C60	Qualitative investigation and simulation of ordinary differential equation models

Keywords:

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

Software:

Mathematica