A neural network framework for simulating drought impacts on predator– prey dynamics

This study examines the influence of drought on predator–prey systems under the variable-order (VO) fractional derivative. It is applied to the wildebeest–lion system of the Serengeti. First, the well-posedness of the system is ensured by the existence, uniqueness, and Ulam–Hyers (UH) stability of the solution. A finite difference method is presented, coupled with a neural network (NN) approach for numerical validation. The numerical results show the effect of the VO fractional derivative and the intensity of the drought. The results demonstrate that a critical drought threshold exists for the drought impact parameter $ \gamma $, beyond which the healthy prey populations decline by over $ 90\% $ from 6643 when $ \gamma $ is 0.20 to 407 when $ \gamma $ is 0.40, and the risk of extinction is very high. As the fractional order decreases from 0.5, the ecological memory is increased, resulting in increased predator populations (from 4898 to 8974 when $ \gamma $ is 0.1) and the long-term effects of the drought. The VO framework produces qualitatively different dynamics than constant-order models, featuring time-dependent stability and attractor morphing, which makes it more suitable for modelling real-world ecological systems under climate stress. The NN approach also demonstrates excellent predictive capabilities, achieving $ R^2 = 1.0 $ and RMSE $ < 12 $ for all populations. These metrics validate our numerical scheme and provide a computationally efficient quick scenario analysis. The novelty of our analysis is the combination of a VO operator, finite difference method, and neural computing in a unified framework for analyzing nonlinear fractional ecological systems. This study provides a mathematically sound framework for understanding drought-induced population shifts and offers practical computational tools for ecological forecasting under climate change.