Dynamics analysis of a generalized stochastic predator-prey model based on DD-RBFNN algorithm

The Fokker Planck Kolmogorov (FPK) equation stands as an indispensable instrument for the stochastic analysis of predator-prey systems, facilitating the conversion of probabilistic challenges into deterministic formulations. We propose the Data-Driven Radial Basis Function Neural Network (DD-RBFNN) algorithm to solve the FPK equations, which can be obtained via a generalized two-dimensional predator-prey system with multiplicative Gaussian white noise excitations, for scrutinizing the corresponding stochastic characteristics. The key innovation of this study lies in addressing the inherent non-negativity constraint of predator-prey models that complicates target domain determination, achieved through the modification made to the conventional RBFNN algorithm. To circumvent the trial-and-error costs, optimize computational costs and enhance overall computational efficiency, we introduce a novel method for delineating the target domain for the training dataset and use the number theoretical method (NTM) for the selection of training points. Precision and effectiveness are corroborated through a series of numerical examples that vary in functional nutritional functions. Moreover, the semi-analytical solutions derived from the DD-RBFNN algorithm exhibit a superior degree of smoothness when contrasted with the numerical solutions yielded by the Monte-Carlo Simulation (MCS) or the fourth-order Runge-Kutta algorithm.