Stochastic bifurcations and tipping phenomena of insect outbreak systems driven by α-stable Lévy processes

In this work, we mainly characterize stochastic bifurcations and tipping phenomena of insect outbreak dynamical systems driven by $\alpha$-stable L\'evy processes. In one-dimensional insect outbreak model, we find the fixed points representing refuge and outbreak from the bifurcation curves, and carry out a sensitivity analysis with respect to small changes in the model parameters, the stability index and the noise intensity. We perform the numerical simulations of dynamical trajectories using Monte Carlo methods, which contribute to looking at stochastic hysteresis phenomenon. As for two-dimensional insect outbreak system, we identify the global stability properties of fixed points and  express the probability density function by the stationary solution of the nonlocal Fokker-Planck equation. Through numerical modelling,  the phase portrait makes it possible to detect critical transitions among the stable states. It is then worth describing stochastic bifurcation associated with tipping points induced by noise through a careful examination of the dynamical paths of the insect outbreak system with external forcing. The results give new insight into the sensitivity of ecosystems to realistic environmental changes represented by stochastic perturbations.