Abstract:

This paper proposes a stochastic single-species model with Allee effect and fear effect. Based on the theory of stochastic dynamical systems, the survivability of the model is analyzed, and it is found that the survival characteristics are determined by a critical parameter $\sigma$. Specifically, when $\sigma \leq 0$, the population goes extinct; when $\sigma > 0$, the system admits a unique non-trivial stationary distribution with an explicit density function, indicating the persistence of the population. Moreover, the parameter $\sigma$ governs both the D-bifurcation and P-bifurcation of the system. In particular, variations in the noise intensity can induce three significant changes in the shape of the density function, leading to triple P-bifurcations. Ecologically, high levels of noise increase the risk of population extinction, while moderate noise levels contribute to maintaining ecological balance. This study reveals the combined regulatory effects of noise and ecological effects on population dynamics, providing a theoretical basis for understanding the persistence mechanisms of populations in stochastic environments.

Key words: Allee effect, fear effect, extinction and persistence, D-bifurcation, P-bifurcation