Abstract:  Predator-prey models form basic building blocks for studying population cycles and under- standing complex interactions in ecological communities. The temporal patterns can be broadly viewed as oscillatory dynamics that alternate between small-amplitude and large-amplitude oscillations, and thus can be represented by mixed-mode oscillations or bursting dynamics that typically occur in higher-dimensional slow-fast dynamical systems. In the first part of my talk, I will consider predator-prey models in the framework of singularly perturbed system of equations and discuss mechanisms underlying different kinds of oscillatory patterns. I will show the existence of periodic traveling waves in a class of reaction-diffusion system of equations that have oscillatory local kinetics. In the second part of my talk, I will address ecological "regime shifts" or the phenomenon of "tipping" and the leading role of long transient dynamics in ecological timescales in explaining such regime shifts. I will discuss some analytical techniques to analyze long transient dynamics preceding a regime shift in a two-timescale three-dimensional predator-prey model with explicit interference competition between the predators. The methods are then used for identifying early warning signals of a large population transition leading to an outbreak or resulting in an extinction of one of the species. Finally, I will end with some challenges and future directions.