Fractional-order prey--predator dynamics: Existence, uniqueness, and parameter estimation via fractional physics-informed neural networks

Abstract

This paper investigates a three-compartment fractional-order predator--prey system modelling the interaction between the Snowshoe Hare (prey) and two competing predators, the Canadian Lynx and the Coyote, in North American ecosystems. The dynamics are governed by a Caputo fractional-order system of order $\rho\in(0,1)$, which naturally captures memory and hereditary effects inherent in ecological processes. We establish the non-negativity and boundedness of solutions, and prove the existence and uniqueness of a global solution via the Banach fixed-point theorem applied to the equivalent Volterra integral formulation. To solve both the forward problem (trajectory reconstruction for $\rho\in\{0.2,0.4,0.6,0.8\}$) and the inverse problem (simultaneous estimation of the seven biological parameters $\theta=(r,a_1,a_2,b_1,b_2,d_1,d_2)$ at $\rho=0.8$), we develop a fractional physics-informed neural network (fPINN) framework that approximates the Caputo derivative with the L1 scheme and trains a fully connected network $[1\!\to\!20\!\to\!3]$ by minimising a composite loss comprising equation-residual, initial-condition, and data-fidelity terms.
Five representative functional responses are examined: basic Lotka--Volterra (Holling~I), prey carrying capacity, predator carrying capacities, Holling~type~II, and Holling~type~III. Across all examples the fPINN recovers the biological parameters with relative errors below $3\%$ when full population data are available, and below $10\%$ when only prey observations are provided, demonstrating that the governing equations act as a strong inductive bias that compensates for missing observational data.