Stochastic Bifurcation in Single-Species Model Induced by $\alpha$-Stable $L\'evy$ Noise

Bifurcation analysis has many applications in different scientific fields, such as electronics, biology, ecology, and economics. In population biology, deterministic methods of bifurcation are commonly used. In contrast, stochastic bifurcation techniques are infrequently employed. Here we establish stochastic P-bifurcation behavior of (i) a growth model with state-dependent birth rate and constant death rate, and (ii) a logistic growth model with state-dependent carrying capacity, both of which are driven by multiplicative non-Gaussian noise. Transcritical bifurcation occurs in the deterministic counterpart of the first model, while saddle-node bifurcation takes place in the logistic growth model. We focus on the impact of the deterministic parameters, the stability index, and the noise intensity on the stationary probability density functions (pdf) of the associated non-local Fokker-Planck equation. Implications of these bifurcations in population dynamics are discussed. In the first model, the bifurcation parameter is the ratio of the population birth rate to the population death rate. In the second model, the bifurcation parameter corresponds to the sensitivity of carrying capacity to change in the size of the population near equilibrium. In each case, we show that as the value of the bifurcation parameter increases, the shape of the steady-state pdf changes and that both stochastic models exhibit stochastic P-bifurcation. The unimodal pdf becomes more peaked around deterministic equilibrium points as the stability index increases. While an increase in any one of the other parameters has an effect on the stationary pdf. The peak appears in the middle of the domain, which means a transition occurs