How a responsive reproduction factor is determinant in a prey–predator dynamics: A numerical and analytical study in a discrete-time model

In this work, we investigate the dynamics of a discrete-time prey–predator model considering a prey reproductive response as a function of the predation risk, with the prey population growth factor governed by two parameters. The system can evolve toward scenarios of mutual or only of predators extinction, or species coexistence. We analytically show all different types of equilibrium points depending on the ranges of growth parameters. By numerical study, we find the occurrence of quasiperiodic, chaotic, and hyperchaotic behaviors. Our analytical results are corroborated by the numerical ones. We highlight Arnold tongue-like periodic structures organized according to the Farey sequence, as well as pairs of twin shrimps connected by two links. The mathematical model captures two possible prey responsive strategies, decreasing or increasing the reproduction rate under predatory threat. Our results support that both strategies are compatible with the populations coexistence and present rich dynamics.