Analysis of nonlinear fractal-fractional differential equations with double discrete delays via Atangana-Baleanu-Caputo operator: theory and biological applications

This research article is mainly focussed on a general problem of fractal-fractional differential equations (FFDEs) with double discrete delays. For This article investigates a class of fractal-fractional differential equations (FFDEs) involving two discrete delays. To establish the existence and uniqueness of solutions, we employ tools from functional analysis, including Krasnoselskii’s and Banach’s fixed point theorems. The stability of the proposed model is analyzed within the framework of Hyers-Ulam (H-U) stability theory. Furthermore, we construct a numerical approximation using an Adams-Bashforth-type method combined with Lagrange interpolation to handle the delayed arguments. Finally, we apply the theoretical results to biologically relevant models, including a two-delay logistic equation and a housefly population model, verifying the existence and stability of solutions and interpreting the numerical outcomes graphically. We have presented comparison between the Adams-Bashforth method and Rungge-Kutta method of order for (RK4).