Bounded environmental stochasticity generates secondary Allee thresholds

Abstract

A population exhibits an Allee effect when there is a critical density below which it goes extinct and above which it persists. Classical models with environmental stochasticity predict inevitable extinction, stemming from the assumption that environmental variation is normally distributed with rare but arbitrary large effect sizes. However, environmental fluctuations are bounded and often not normally distributed. To address this reality, I analyze piecewise deterministic Markov models (PDMPs) of populations experiencing Allee effects, where environmental dynamics are governed by a finite-state Markov chain. These models predict that populations can persist through the emergence of two threshold densities. Below the lower threshold, populations deterministically go extinct; above the higher threshold, they deterministically persist. At intermediate densities, populations experience stochastic bistability: with positive, complementary probabilities, they either go extinct or persist. Persistence becomes impossible when the carrying capacity in one environment falls below the Allee threshold in another. Such mismatch occurs only when the environmental state affects percapita growth rates non-monotonically, as when environments supporting higher carrying capacities also produce higher predation levels or greater mate limitation. This work demonstrates that incorporating realistic bounded environmental fluctuations substantially alters predictions about population persistence, with important implications for conservation and management.