A Radial Basis Function Neural Network Surrogate for a Four-Dimensional Ecological Population Model with Refuge and Migration

This paper presents a Radial Basis Function Neural Network (RBFNN) that approximates the vector field of a four-dimensional prey–migration–predator–superpredator ecological model with prey refuge, considered in deterministic and stochastic formulations. The network is trained directly on the governing equations without the need for any simulated trajectories for training. The network learns the deterministic vector field, and hence analytical verification is carried out for the deterministic model, with a proof of global existence of the solution, a proof of the boundedness of the solution, a classification of five equilibria via the Jacobian matrix, and a proof of local exponential stability of the coexistence equilibrium via a Lyapunov function. The RBFNN is validated by comparing its trajectories to those of the RK45 solver, checking its long-term behavior, performing phase-space analysis, and computing the eigenvalues of the equilibrium and Jacobian of the learned field. For the stochastic formulation, the learned drift is added with explicit noise terms and is compared with the Euler–Maruyama and Milstein methods using a finer-step Milstein method as a reference. The numerical results presented here establish that the RBFNN can be used as an alternative to a deterministic vector field in standard numerical solvers, and that the framework can be used to compare the computational efficiency of the vector field evaluation in more complex dynamical systems where the field evaluation is more costly.