We study when the celebrated May–Leonard model in ${\mathbb{R}}^3$, 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 $a\ne 1$ (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 a 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 $b=a\ne 1$ then we consider two cases. First, if $0<a<1$ then the unique positive equilibrium point attracts all the orbits of the interior of the positive octant. If $a>1$ 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.

In this paper we analyze the dynamics of a tritrophic food chain, with functional responses of Holling type III and II for the mesopredator and superpredator respectively and logistic growth rate for the prey. We show that there are parameter conditions for which the system has up to three equilibrium points. When three equilibrium points appear we show that each one of them exhibits a supercritical Hopf bifurcation. Moreover we can also show that there is a set of parameters for which the system exhibits a simultaneous Hopf bifurcation.

There is a correspondence between the boundary of the main hyperbolic component $W_0$ of the Mandelbrot set $M$ and $M \cap \mathbb R$ . It is induced by the map $T(\theta)=1/2+\theta/4$ defined on the set of external arguments of $W_0$. If $c$ is a point of the boundary of $W_0$ with internal argument $\gamma$ and external argument $\theta$ then $T(\theta)$ is an external argument of the real parameter $c'\in M.$ We give a characterization, for the parameter $c'$ corresponding to $\gamma$ rational, in terms of the Hubbard trees. If $\gamma$ is irrational, we prove that $P_{c'}$ does not satisfy the $CE$ condition. We obtain an asymmetrical diophantine condition implying the existence of an absolutely continuous invariant measure (a.c.i.m.) for $P_{c'}$. We also show an arithmetic condition on $\gamma$ preventing the existence of an a.c.i.m.