Dynamics in a time-discrete food-chain model with strong pressure on preys.

Ecological systems are complex dynamical systems. Modelling efforts on ecosystems' dynamical stability have revealed that population dynamics, being highly nonlinear, can be governed by complex fluctuations. Indeed, experimental and field research has provided mounting evidence of chaos in species' abundances, especially for discrete-time systems. Discrete-time dynamics, mainly arising in boreal and temperate ecosystems for species with non-overlapping generations, have been largely studied to understand the dynamical outcomes due to changes in relevant ecological parameters. The local and global dynamical behaviour of many of these models is difficult to investigate analytically in the parameter space and, typically, numerical approaches are employed when the dimension of the phase space is large. In this article we provide topological and dynamical results for a map modelling a discrete-time, three-species food chain with two predator species interacting on the same prey population. The domain where dynamics live is characterized, as well as the so-called escaping regions, for which the species go rapidly to extinction after surpassing the carrying capacity. We also provide a full description of the local stability of equilibria within a volume of the parameter space given by the prey's growth rate and the predation rates. We have found that the increase of the pressure of predators on the prey results in chaos. The entry into chaos is achieved via a supercritical Neimarck-Sacker bifurcation followed by period-doubling bifurcations of invariant curves. Interestingly, an increasing predation directly on preys can shift the extinction of top predators to their survival, allowing an unstable persistence of the three species by means of periodic and strange chaotic attractors.