P-bifurcation and bistability arising from cross-correlated sine-Wiener bounded noises: A stochastic single-species model incorporating double Allee effects

The interplay between a noisy environment and system nonlinearity can produce counterintuitive phenomena unexplained by deterministic models. The detailed mechanisms of these phenomena, however, are not completely clear. In this paper, we propose a stochastic single-species model incorporating double Allee effects and cross-correlated sine-Wiener bounded noises to study how environmental fluctuations induce P-bifurcation and bistability. We begin by deriving an approximate Fokker–Planck equation and its stationary probability distribution. We then explore noise-induced phenomena under both strong and weak Allee effects, explaining these through potential well depth. Finally, using mean first passage time, we estimate the transition time from a persistent state to an extinction state. Our findings demonstrate that: (i) P-bifurcation occurs in both strong and weak Allee effect scenarios, signifying transitions between positive and trivial states that are not possible in the deterministic model; (ii) bistability can be induced only under a strong Allee effect; (iii) multiplicative noise, correlation degree, and correlation time increase the risk of population extinction, while additive noise and the strength of the Allee effect can delay extinction.