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<p>Predator–prey interactions are central to maintaining the stability and biodiversity of natural ecosystems. In this study, we develop and analyze a Leslie–Gower-type predator–prey model, incorporating hunting cooperation and nonlinear harvesting of predators to better capture the complex dynamics of ecological systems. Our theoretical analysis establishes the positivity of solutions as well as feasibility and stability of biologically feasible equilibria, revealing that while species-free equilibrium is inherently unstable prey-free equilibria may exhibit conditional stability. Through numerical simulations and sensitivity analysis, we show that both prey and predator populations are highly sensitive to prey growth rate, minimal reproductive capacity under fear, and intraspecific competition. One and two-parameter bifurcation analyses uncover rich dynamical patterns driven by variations in prey and predator reproduction rates. The system undergoes saddle-node, Hopf, and transcritical bifurcations, as well as codimension-two bifurcations such as cusp and Bogdanov–Takens. These bifurcations give rise to diverse dynamical regimes—mono-, bi- and tri-stability-indicating that small changes in the parameter can shift the system among coexistence, oscillatory, and extinction states. To further explore ecosystem resilience under uncertainty, a stochastic extension of the model is introduced by incorporating environmental noise. Using a framework based on transition probability densities and the stochastic sensitivity function technique, we determine the most probable transition pathway, tipping time, and critical noise intensity associated with population collapse. Our stochastic analysis demonstrates that strong environmental perturbations can induce regime shifts between the states of high and low population densities, destabilize coexistence equilibria, and lead to predator extinction. Overall, our findings highlight that both intrinsic biological interactions and extrinsic environmental fluctuations play decisive roles in shaping the long-term stability, resilience, and adaptability of predator–prey systems.</p>

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Coupled tipping dynamics: parameter-driven bifurcations and noise-induced transitions in a Leslie–Gower model

  • 期刊论文

  • 发表日期：

    2026年5月13日

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  • Bapin Mondal,
  • Purnendu Sardar,
  • Sayan Mandal,
  • Pankaj Kumar Tiwari

摘要

Predator–prey interactions are central to maintaining the stability and biodiversity of natural ecosystems. In this study, we develop and analyze a Leslie–Gower-type predator–prey model, incorporating hunting cooperation and nonlinear harvesting of predators to better capture the complex dynamics of ecological systems. Our theoretical analysis establishes the positivity of solutions as well as feasibility and stability of biologically feasible equilibria, revealing that while species-free equilibrium is inherently unstable prey-free equilibria may exhibit conditional stability. Through numerical simulations and sensitivity analysis, we show that both prey and predator populations are highly sensitive to prey growth rate, minimal reproductive capacity under fear, and intraspecific competition. One and two-parameter bifurcation analyses uncover rich dynamical patterns driven by variations in prey and predator reproduction rates. The system undergoes saddle-node, Hopf, and transcritical bifurcations, as well as codimension-two bifurcations such as cusp and Bogdanov–Takens. These bifurcations give rise to diverse dynamical regimes—mono-, bi- and tri-stability-indicating that small changes in the parameter can shift the system among coexistence, oscillatory, and extinction states. To further explore ecosystem resilience under uncertainty, a stochastic extension of the model is introduced by incorporating environmental noise. Using a framework based on transition probability densities and the stochastic sensitivity function technique, we determine the most probable transition pathway, tipping time, and critical noise intensity associated with population collapse. Our stochastic analysis demonstrates that strong environmental perturbations can induce regime shifts between the states of high and low population densities, destabilize coexistence equilibria, and lead to predator extinction. Overall, our findings highlight that both intrinsic biological interactions and extrinsic environmental fluctuations play decisive roles in shaping the long-term stability, resilience, and adaptability of predator–prey systems.