| Abstract: |
| In this talk, I will discuss methods of analyzing transient dynamics to detect early warning signals of population extinction or population outbreaks in a three-dimensional predator-prey model featuring two-timescales. The model under consideration studies the interaction between two species of predators competing for their common prey with explicit interference competition. We will consider two different scenarios in a parameter regime near {\emph{singular Hopf bifurcation}} of the coexistence equilibrium. In one case, the system exhibits bistability between a periodic attractor and a boundary equilibrium state with long transients characterized by rapid oscillations and slow variation in amplitudes, while in the other case, the system exhibits chaotic {\emph{mixed-mode oscillations}}, featuring concatenation of small and large-amplitude oscillations as long transients before approaching a stable limit cycle. To analyze the transients, the system is reduce to a suitable normal form. Exploiting the timescales separation and the underlying geometry of the normal form, the transient dynamics are analyzed. The analysis is then used to devise methods for identifying early warning signals of a large population transition leading to an outbreak or extinction of one of the species. |
|