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

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. The domain where dynamics live is characterised, as well as the so-called escaping regions, which involve species extinctions. 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 via a supercritical Neimark-Sacker bifurcation. Then, period-doubling bifurcations of invariant curves take place. 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 chaotic attractors.The research leading to these results has received funding from “la Caixa” Foundation, from a MINECO grant awarded to the Barcelona Graduate School of Mathematics (BGSMath) under the “María de Maeztu” Program (grant MDM-2014-0445), and from the CERCA Programme of the Generalitat de Catalunya. LlA has been supported by the Spain’s ”Agencial Estatal de Investigación” (AEI) grant MTM2017-86795-C3-1-P. JTL has been partially supported by the Catalan grant 2017SGR1049, by the MINECO grant MTM2015-65715-P, and by grant PGC2018-098676-B-100 (AEI/FEDER/UE). JS has been also funded by a “Ramón y Cajal” Fellowship (RYC-2017-22243) and by a MINECO grant MTM-2015-71509-C2-1-R and the AEI grant RTI2018-098322-B-I00. RS and BV have been partially funded by the Botin Foundation, by Banco Santander through its Santander Universities Global Division and by the PR01018-EC-H2020-FET-Open MADONNA project. RS also acknowledges support from the Santa Fe Institute.Peer reviewe