How predator harvesting affects prey-predator dynamics in deterministic and stochastic environments?

This study investigates the dynamics of predator-prey interactions in both deterministic and stochastic environments, with a focus on the ecological implications of predator harvesting. Theoretical and numerical analyses explore local stability, bifurcations, and bionomic equilibria to identify sustainable harvesting strategies. Our findings reveal that increasing predator harvesting rates can induce up to four interior equilibrium points via saddle-node bifurcations, including catastrophic transitions that destabilize the system. At high harvesting rates, the predator-free equilibrium becomes globally stable, while low and intermediate rates result in bistability or tristability, allowing coexistence of prey and predator populations. For the stochastic model, we derive conditions for species persistence and extinction, using the confidence ellipse method to quantify threshold noise intensities that trigger critical transitions between stable states. At low noise levels, predator and prey populations fluctuate around stable equilibria, but higher noise intensities can drive shifts to alternate states or predator extinction. The key factors influencing system dynamics include the predator's intrinsic growth rate, alternative food sources, and harvesting intensity. Our analysis underscores the vulnerability of ecological systems to stochastic disturbances and emphasizes the importance of carefully managed harvesting practices. These findings contribute to the development of strategies that balance ecological stability with economic objectives, ensuring the long-term sustainability of predator-prey populations.