Eduardo Sáez

Dynamics of a predator-prey model

We describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator-prey model proposed by R. M. May [ Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1974]. In particular, we give a qualitative description of the bifurcation curve when two limit cycles collapse on a semistable limit cycle and disappear. Moreover, we show that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator-prey models.

Three Limit Cycles in a Leslie–Gower Predator-Prey Model with Additive Allee Effect

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.

Coexistence of algebraic and nonalgebraic limit cycles in Kukles systems

A class of Kukles differential systems of degree five having an invariant conic is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, under perturbations of the coefficients of the systems.

Small amplitude limit cycles for cubic systems

In this article we study the simultaneous generation of limit cycles out of singular points and infinity for the family of cubic planar systems x = y(ax + y 1 ) ley + sxy(—4 + y) y = ~x(x + by 1 ) + 2dx + exy(-4 + x). With a suitable choice of parameters, the origin and four other singularities are foci and infinity is a periodic orbit. We prove that it is possible to obtain the following configuration of limit cycles: two small amplitude limit cycles out of the origin, a small amplitude limit cycle out of each of the other four foci, and a large amplitude limit cycle out of infinity. We also obtain other configurations with fewer limit cycles.

Chaotic dynamics and coexistence in a three species interaction model

This work deals with a three-dimensional system, which describes a food web model consisting of a prey, a specialist predator and a top predator which is generalist as it consumes the other two species. Using tools of dynamical systems we prove that the trajectories of system are bounded and that open subsets of parameters exist, such that the system in the first octant has at most two singularities. For an open subset of the parameters space, the system is shown to have an invariant compact set and this is a topologically transitive attractor set. Finally, we find another open set in the parameters space, such that the system has two limit cycles each contained in different invariant planes. The work is completed with a numeric simulation showing the attractor is a strange attractor.

A predator-prey system involving five limit cycles

Abstract. In this paper we consider the multiparameter system introduced in [M. Scheffer et al. Can.J. Fish. Aquat. Sci. 57(6);1208-1219 (2000)] which corresponds to an extension of the classic minimal Daphnia-algae model. It is shown that there is a neighborhood in the parameter space where the system in the realistic quadrant has a unique equilibrium point which is a repelling weak focus of order four enclosed by a global attractor hyperbolic limit cycle. For a small enough change of the parameters in this neighborhood, bifurcate from the weak focus four infinitesimal Hopf limit cycles (alternating the type of stability) such that the last bifurcated limit cycle is an attractor. Moreover, for certain values of parameters we concluded that this applied model has five concentric limit cycles , three of them being stable hyperbolic limit cycles. This gives a positive answer to a question raised in [C.S. Coleman, Differential Equations Models, V 1. 279-297, Springer-Verlag (1978)] and [N.G. Lloyd et al. Appl. Math. Lett. Vol 9, No. 1, 15-18 (1996)].