Stability transition of persistence and extinction in an avian influenza model with Allee effect and stochasticity

Abstract

Population persistence and extinction are the most important issues in ecosystems. In the past a few decades, various deterministic and stochastic mathematical models with Allee effect have been extensively studied. However, in both population and disease dynamics, the question of how structural transitions caused by internal or external environmental noise emerge has not been fully elucidated. In this paper, we introduce a semi-analytical method to explore the asymptotically convergent behavior of a stochastic avian influenza model with Allee effect. First, by introducing noise to the model, we observe numerically a significant transition from bistability to monostability. Next, a corresponding Fokker-Planck (FPK) equation is obtained to analytically describe the probability density distributions with long time evolution in order to reveal the transition characteristics. Ratio of the approximately convergent probabilities for the two key equilibria derived from the FPK equation confirms the stability transition observed by previous numerical simulations. Moreover, bifurcation analysis in two important parameters demonstrates that noise not only reduces the parametric zone of sustaining bistability but also drives the system to exhibit different monostabilities, which correspond numerically to population persistence and extinction at different parametric intervals, respectively. Furthermore, noise induces higher probabilities for the system to sustain persistence instead of extinction in this model. Our results could provide some suggestions to improve wildlife species survival in more realistic situations where noise exists.