The analysis of prey–predator interactions in deterministic and stochastic environments with parameter sensitivity

Population dynamics, mainly the predator–prey interactions, are fundamental to ecological systems. In this work, we focus on a predator–prey model with Allee effects in both populations. By selecting key parameters, we explore diverse bifurcations, including transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bautin bifurcation, cusp bifurcation, and Bogdanov–Takens bifurcation. To account for environmental variability, stochastic perturbations are introduced into prey growth rates and harvesting intensities, represented by stochastic differential equations with multiplicative white noise. At higher noise intensity, we observe a rare and intriguing phenomenon: simulations under identical parameters and noise intensity yield three distinct trajectory behaviors. These include all trajectories escaping to extinction, some trajectories dividing between extinction and persistence, and all trajectories converging to persistence. This variability, absent in the deterministic framework, underscores the critical role of noise in driving unpredictable ecological outcomes. We use Stochastic Sensitivity Function (SSF) technique and confidence ellipse method to find exact noise intensity of a transition. These findings provide new insights into the interplay of deterministic and stochastic forces in ecological systems, emphasizing the importance of noise in shaping population dynamics and informing conservation strategies under environmental uncertainty.