Andronov-Hopf and Bautin bifurcation in a tritrophic food chain model with Holling functional response types IV and II

The existence of an Andronov–Hopf and Bautin bifurcation of a given system of differential equations is shown. The system corresponds to a tritrophic food chain model with Holling functional responses type IV and II for the predator and superpredator, respectively. The linear and logistic growth is considered for the prey. In the linear case, the existence of an equilibrium point in the positive octant is shown and this equilibrium exhibits a limit cycle. For the logistic case, the existence of three equilibrium points in the positive octant is proved and two of them exhibit a simultaneous Hopf bifurcation. Moreover the Bautin bifurcation on these points are shown

Integrability and global dynamics of the May-Leonard model

We study when the celebrated May-Leonard model in R3, describing the competition between three species and depending on two positive parameters a and b, is completely integrable; i.e. when a+b = 2 or a = b. For these values of the parameters we shall describe its global dynamics in the compactification of the positive octant, i.e. adding its infinity. If a + b = 2 and a 6= 1 (otherwise the dynamics is very easy) the global dynamics was partially known, and roughly speaking there are invariant topological half-cones by the flow of the system. These half-cones have vertex at the origin of coordinates and surround the bisectrix x = y = z, and foliate the positive octant. The orbits of each half-cone are attracted to a unique periodic orbit of the half-cone, which lives on the plane x + y + z = 1. If b = a 6= 1 then we consider two cases. First, if 0 1 then there are three equilibria in the boundary of the positive octant, which attract almost all the orbits of the interior of the octant, we describe completely their bassins of attractions

On the global flow of a 3--dimensional Lotka--Volterra system

Agraïments: The first author is partially supported by the grant PROMEP/103.5/08/3189. The first three authors are partially supported by two CONACYT grants with numbers 58968 and 62613.In the study of the black holes with Higgs field appears in a natural way the Lotka-Volterra differential system x˙= x(y − 1), y˙= y(1 + y − 2x2 − z2), z˙= zy, in R3. Here we provide the qualitative analysis of the flow of this system describing the α-limit set and the ω-limit set of all orbits of this system in the whole Poincar'e ball, i.e. we identify R3 with the interior of the unit ball of R3 centered at the origin and we extend analytically this flow to its boundary, i.e. to the infinity

On the maximum number of limit cycles of a class of generalized Liénard differential systems

Agraïments: This work is partially supported by grant CONACYT-58968.Applying the averaging theory of first, second and third order to one class generalized polynomial Li'enard differential equations, we improve the known lower bounds for the maximum number of limit cycles that this class can exhibit