A Controlled Picard Semi-Analytical Framework for Nonlinear Fractional Stochastic Differential Equations with Delay in Biological and Population Models

Abstract

In this paper, a controlled Picard semi-analytical technique is improved for a branch of nonlinear fractional stochastic delay differential equations since the nonlinear terms always prevent the establishment of closed-form solutions. The proposed approach extends the known Picard iteration by embedding a convergence-control parameter that improves the flexibility and stability of the iterative scheme while keeping the original mathematical formulation. A careful theoretical analysis is presented to establish the existence of the iterative sequence, its mean-square boundedness, convergence, and an explicit error estimate under standard Lipschitz continuity and linear growth assumptions. Moreover, a Numerical Picard implementation is updated to rebuild stochastic sample trajectories and to give an independent illustration through comparison with a predictor–corrector scheme. The presented methodology is applied to fractional stochastic models of human postural sway and logistic population models. The numerical results illustrate that the semi-analytical framework evaluates the expectation of and variance in the stochastic response for various fractional orders accurately. Although the semi-analytical controlled Picard method is incapable of performing a lot of iterations, it generates statistical moments that agree with those obtained using both the Numerical Picard and predictor–corrector methods, while explicit semi-analytical representations of the solution. These results show that the presented technique gives an accurate and effective approach for examining nonlinear fractional stochastic delay systems from biological, ecological, and engineering applications.