Analysis of long transient dynamics and detecting early warning signals of regime shifts in a class of predator-prey models featuring two timescales
The leading role of long transient dynamics in ecological timescales can be very important
in explaining regime shifts. However, analytical techniques for studying long transients in relevant timescales in three or higher-dimensional ecological models is still at its infancy. In this talk, I will consider a three-dimensional predator-prey model featuring two-timescales that governs the interaction between two species of predators competing for their common prey with explicit interference competition. I will consider two different scenarios in a parameter regime near singular Hopf bifurcation of the coexistence equilibrium point. In one case, the system exhibits bistability between a periodic attractor and a boundary equilibrium state, with long transients characterized by rapid small-amplitude oscillations and slow variation in amplitudes, while in the other, the system exhibits chaotic 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 reduced to a suitable normal form near the singular Hopf point. Exploiting the separation of timescales and the underlying geometry of the normal form, the transient dynamics are analyzed. The analyses are then used to devise methods for identifying early warning signals of an abrupt transition resulting in an extinction of one of the species or a large population transition leading to an outbreak. I will finally end with some preliminary analysis on extending the results to stochastic settings.