Analysis of a Fractional-Order Leslie–Gower Prey–Predator–Parasite System with Dual Delays and Reaction–Diffusion Dynamics: A Statistical Approach

Thisarticle develops and analyzes a fractional-order Leslie–Gower prey–predator–parasite system incorporating two discrete delays and nonlocal spatial diffusion. The model’s central novelty lies in the simultaneous integration of three biologically realistic features that have not previously been combined: (i) fractional-order memory effects via a Caputo derivative of order α∈(0,1], (ii) two distinct biological delays—an infection transmission delay τ1 and a predator handling delay τ2—and (iii) nonlocal spatial dispersal modeled through fractional Laplacian operators (−Δ)γ/2. This triple integration enables the model to capture long-range temporal memory, delayed biological responses, and nonlocal spatial interactions simultaneously, offering insights into dynamics that are challenging to capture with classical integer-order or single-delay formulations. The fractional Laplacian generalizes classical diffusion by allowing long-range dispersal events (Lévy flights), where individuals can occasionally move over large distances with heavy-tailed step-size distributions—a phenomenon observed in many animal movement patterns but absent from standard diffusion models. We provide rigorous proofs of solution existence, uniqueness, non-negativity, and boundedness in both temporal and spatiotemporal settings. Local asymptotic stability conditions are derived for all feasible equilibrium states via characteristic equation analysis. The coexistence equilibrium undergoes a Hopf bifurcation when either delay crosses a critical threshold, with fractional order α modulating the bifurcation point and post-bifurcation oscillation frequency. A Lyapunov functional demonstrates global asymptotic stability of the infection-free equilibrium under biologically interpretable conditions. Turing instability analysis reveals conditions for spontaneous pattern formation, with the fractional exponent γ controlling pattern wavelength and correlation length. Numerical simulations validate theoretical predictions, including spatial patterns, traveling waves, and chaos. To bridge theory with potential applications, we outline a statistical framework for parameter estimation and uncertainty quantification, suggesting that β, α, and τ1 may be priority targets for parameter estimation.