Using first passage times to manage eco-system regime shifts

Regime shifts refer to sudden changes in the structure or function of an eco-system due to external forces on the system. Such shifts arise because these systems have multiple equilibria so external disturbances may drive the state between different regions of attraction. Examples of such shifts include the shift in eutrophic state of shallow lakes in response to nutrient loading as well as the collapse of fisheries in response to the introduction of invasive species. A key measure of an ecosystem´s resilience to such shifts is measured by first passage times (FPT) between the basins of attraction for different equilibria. Prior work in eco-system management has assumed low-dimensional linearized models driven by Brownian motions. This paper uses sum-of-square (SOS) programs to bound FPT probabilities for more complex nonlinear population processes in which the primary disturbance is a Poisson jump process. The paper uses this approach to design management policies controlling the level of invasive species in lake systems.