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 provide natural definition for such critical transitions and noise can play crucial role near tipping points (bifurcation points). 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 a system of slow-fast Ito stochastic differential equations. Using classical bifurcation 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 that capture the intermediate dynamics between two cycles of population outbreaks. We perform numerical simulations to study the distribution of random number of small oscillations between two large oscillations, which can be related to the return time between outbreaks. Our findings qualitatively resemble the return times of larch budmoth outbreak events in the subalpine larch forests in the European Alps. recorded for 1200 years.