Combining Perturbations and Parameter Variation to Influence Mean First Passage Times

Perturbations are relatively large shocks to state variables that can drive transitions between stable states, while drift in parameter values gradually alters equilibrium magnitudes. This latter effect can lead to equilibrium bifurcation, the generation, or annihilation of equilibria. Equilibrium annihilations reduce the number of equilibria and so are associated with catastrophic population collapse. We study the combination of perturbations and parameter drift, using a two-species intraguild predation (IGP) model. For example, we use bifurcation analysis to understand how parameter drift affects equilibrium number, showing that both competition and predation rates in this model are bifurcating parameters. We then introduce a stochastic process to model the effects of population perturbations. We demonstrate how to evaluate the joint effects of perturbations and drift using the common currency of mean first passage time to transitions between stable states. Our methods and results are quite general, and for example, can relate to issues in both pest control and sustainable harvest. Our results show that parameter drift (1) does not importantly change the expected time to reach target points within a basin of attraction, but (2) can dramatically change the expected time to shift between basins of attraction, through its effects on equilibrium resilience.