Orbital stability and chaotic dynamics in a diffusive prey–predator ecological model

This study investigates the orbital stability and chaotic dynamics of a diffusive prey–predator ecological model. The stability properties of the system are analyzed through Jacobian-based linear stability analysis at the equilibrium points by evaluating the corresponding eigenvalues. This analysis reveals different dynamical regimes, including centers, saddles, and spiral behaviors, each exhibiting distinct stability properties. Phase portraits are employed to illustrate the qualitative structure of the system trajectories and to visualize the orbital stability of the ecological interactions. To assess the sensitivity of the system to initial conditions, numerical simulations are performed using the fourth-order Runge–Kutta method. The results show that small perturbations in initial conditions can produce significantly divergent trajectories, indicating the presence of chaotic dynamics. Furthermore, specific parameter values responsible for the onset of chaos are identified through bifurcation diagrams, Lyapunov exponent analysis, two-dimensional phase portraits, and time-series representations. The occurrence of positive Lyapunov exponents provides strong evidence of chaotic behavior and confirms the effectiveness of this approach in characterizing complex ecological dynamics. The bifurcation and Lyapunov analyses further demonstrate that the system undergoes transitions among stable, periodic, and chaotic regimes as key parameters vary. The emergence of irregular time series and disordered phase portraits further supports the existence of chaos in the system. This study enhances the understanding of diffusive prey–predator interactions and demonstrates the effectiveness of various tools in uncovering chaotic ecological dynamics.