Stability and Coexistence in an Improved Three-Species Predator–Prey System with Logistic Growth and Nonlinear Interactions

Understanding the mechanisms that sustain coexistence among multiple interacting species remains a fundamental problem in mathematical ecology. In this study, we develop an enhanced three-trophic-level predator–prey model representing a food chain consisting of lions (top predators), leopards (intermediate predators/preys), and hares (preys). The model incorporates logistic growth functions to account for environmental carrying capacities and nonlinear predation terms with energetic conversion efficiencies to reflect realistic trophic interactions. Analytical results establish the positivity, boundedness, and biological feasibility of the model solutions. The equilibrium points and their local stability properties are examined through Jacobian linearization and Routh–Hurwitz conditions, while global stability is investigated via a suitable Lyapunov function. Numerical simulations illustrate diverse dynamic behaviors, including coexistence equilibria, extinction scenarios, and oscillatory coexistence depending on the predation and conversion parameters. The proposed framework provides a more biologically consistent and mathematically robust tool for studying multispecies interactions and can be extended to incorporate stochastic perturbations, time delays, or spatial heterogeneity.