Multiple stability and uniqueness of the limit cycle in a Gause-type predator–prey model considering the Allee effect on prey (original) (raw)

Two limit cycles in a Leslie–Gower predator–prey model with additive Allee effect

Nonlinear Analysis: Real World Applications, 2009

In this work, a bidimensional continuous-time differential equations system is analyzed which is derived of Leslie-type predator-prey schemes by considering a nonmonotonic functional response and Allee effect on 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 imply extinction of populations necessarily for our model.

Global dynamics of a predator–prey model

Journal of Mathematical Analysis and Applications, 2010

This paper deals with the dynamics of a predator-prey model with Hassell-Varley-Holling functional response. First, we show that the predator coexists with prey if and only if predator's growth ability is greater than its death rate. Second, using a blow-up technique, we prove that the origin equilibrium point is repelling and extinction of both predator and prey populations is impossible. Third, the local and global stability of the positive steady state coincide when the predator interference is large. Finally, for a typical biological case, we show instability of the positive equilibrium implies global stability of the limit cycle. Numerical simulations are carried out for a hypothetical set of parameter values to substantiate our analytical findings.

Heteroclinic orbits indicate overexploitation in predator–prey systems with a strong Allee effect

Mathematical Biosciences, 2007

Species establishment in a model system in a homogeneous environment can be dependent not only on the parameter setting, but also on the initial conditions of the system. For instance, predator invasion into an established prey population can fail and lead to system collapse, an event referred to as overexploitation. This phenomenon occurs in models with bistability properties, such as strong Allee effects. The Allee effect then prevents easy re-establishment of the prey species. In this paper, we deal with the bifurcation analyses of two previously published predator-prey models with strong Allee effects. We expand the analyses to include not only local, but also global bifurcations. We show the existence of a point-to-point heteroclinic cycle in these models, and discuss numerical techniques for continuation in parameter space. The continuation of such a cycle in two-parameter space forms the boundary of a region in parameter space where the system collapses after predator invasion, i.e. where overexploitation occurs. We argue that the detection and continuation of global bifurcations in these models are of vital importance for the understanding of the model dynamics.