Coexistence of chaotic and non-chaotic attractors in a three-species slow-fast system

The dynamical complexity of conceptual few-species systems has long been attracting considerable attention. In particular, significant attention has been paid to the three trophic level systems to demonstrate that even a baseline prey/predator/top-predator system can exhibit complex dynamics. However, previous studies left many open questions. In this paper, we consider a generalization of the Hastings-Powell model that include intraspecific competition among the predators and top predators. To enhance the ecological relevance of the model, we incorporate different timescales for different species. Specifically, we aim to study how the dynamics of the system is affected in the presence of multiple timescales along the trophic levels. We employ the geometric singular perturbation approach to analyze the change in species abundance over three different timescales, slow, fast, and intermediate. We analytically determine the entry and exit points from the vicinity of the critical manifold; the corresponding values act as critical thresholds for a sudden, fast transition in population density. We show that the properties of this generalized form of the Hastings-Powell model are dynamically more rich than it was observed in previous studies; particularly, the system can exhibit bi-stability or tri-stability in certain parameter intervals. The existence of a homoclinic orbit in a limiting subsystem (prey- predator) indicates period-doubling cascades to chaos in the full system (prey-predator-top predator). We further investigate the impact of the intraspecific competition of predators and top predators on the system's dynamics. Strong intraspecific competition among the predators and top predators shows periodic coexistence of all species. In contrast, weak competition can drive the system to exhibit tri-stability, which includes the coexistence of chaotic and periodic (of different periods) attractors.