An Adaptive Time-Stepping Algorithm to Solve a Stochastic Lotka-Volterra Competition System with Time-Variable Delays

‎This paper introduces ‎a variable step size strategy for a stochastic time-delays Lotka-Volterra competition system‎. ‎This adaptive strategy utilizes the Milstein method for numerical solutions. It employs two local error estimates, corresponding to the diffusion and drift components of the model, to select and control the step sizes‎. ‎The algorithm is described in detail‎, ‎and numerical experiments are conducted to demonstrate the efficiency of the proposed method‎. ‎The primary objective of this research is to propose a dynamic strategy for generating and controlling the step sizes in the finite difference algorithm employed. ‎This adaptive approach accelerates the numerical procedure and improves efficiency compared to a constant-size scheme‎. ‎ As an analytical solution for the model is unavailable‎, ‎a numerical estimation with a small fixed step size is considered a reference solution‎. ‎The numerical results demonstrate the superior accuracy of the proposed strategy compared to a reference solution‎.