### Mixed-mode and relaxation oscillations in slow-fast predator-prey problems

- Series
- CDSNS Colloquium
- Time
- Monday, March 30, 2020 - 11:15 for 1 hour (actually 50 minutes)
- Location
- Skyles 006
- Speaker
- Susmita Sadhu – Georgia College & State University Milledgeville – susmita.sadhu@gcsu.edu

The entry-exit function for the phenomenon of delay of stability loss (Pontryagin’s delay) arising in certain classes of slow-fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations. In the first part of my talk, I will present an elementary approach to study the entry-exit function for a general class of slow-fast systems, and apply this function to a broad class of slow-fast planar systems to obtain existence, global uniqueness and asymptotic orbital stability of periodic solutions that exhibit relaxation oscillations. The obtained results will then be applied to some predator-prey models. This research was conducted in collaboration with Dr. Shangbing Ai. In the second part of my talk, I will present a slow-fast system comprising of three species where further complex oscillatory patterns such as mixed mode oscillations (MMOs) are observed. MMOs are concatenations of small amplitude oscillations and large amplitude oscillations which are of relaxation types. In a neighborhood of singular Hopf bifurcation, these types of oscillations occur as long lasting chaotic transients as the system approaches a periodic attractor. The transients could persist for thousands of generations, reflecting that dynamics on an ecological timescale can be completely different than asymptotic dynamics. The goal of the talk is to find conditions that will determine whether a trajectory exhibits another cycle of MMO dynamics before reaching its asymptotic state.