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Bifurcations and global dynamics in a predator-prey model with a strong Allee effect on the prey, and a ratio-dependent functional response. (English) Zbl 1298.34078
Summary: We extend a previous study of a predator-prey model with strong Allee effect on the prey in which the functional response is a function of the ratio of prey to predator. We prove that the solutions are always bounded and non-negative, and that the species can always tend to long-term extinction. By means of bifurcation analysis and advanced numerical techniques for the computation of invariant manifolds of equilibria, we explain the consequences of the (dis)appearance of limit cycles, homoclinic orbits, and heteroclinic connections in the global arrangement of the phase plane near a Bogdanov-Takens bifurcation. In particular, we find that the Allee threshold in the two-dimensional system is given as the boundary of the basin of attraction of an attracting positive equilibrium, and determine conditions for the mutual extinction or survival of the populations.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
Software:
AUTO; AUTO-07P; MATCONT
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References:
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