Mathematical Sciences Colloquium Series presents Dr. Susmita Sadhu, from the Department of Mathematics at Georgia College & State University.

"Effect of Noise near Tipping Points in Predator-Prey Models with Time Scale Separation"

Many forest insects and agricultural pests exhibit sudden changes in their densities, where the densities can rise from undetectable to extremely high or vice-versa. These abrupt qualitative changes can be referred to as critical transitions in the state variables. Slow-fast systems can provide a natural framework to study such critical transitions. Furthermore, noise can play a crucial role near tipping points (bifurcation points) of such systems. In this talk, I will present two stochastic predator-prey models with time scale separation that can reproduce some of the features observed in population outbreaks. The interactions between the species are modeled by systems of slow-fast Itô stochastic differential equations. Using singular perturbation theory, we will study the effect of noise near tipping points. One of the interesting dynamics that we obtain are noise-induced mixed-mode oscillations, which are complex oscillatory patterns that feature concatenation of small and large amplitude oscillations. These oscillations capture the intermediate dynamics between two cycles of population outbreaks. We perform numerical simulations to study the distribution of a random number of small oscillations between two large oscillations, which can be related to the return time between the outbreak events. Our findings qualitatively resemble the return times of larch budmoth outbreaks in the subalpine larch forests in the European Alps recorded for 1200 years.