This thesis aims to achieve the oscillatory and asymptotic behavior of solutions to logistic differential equations (LDEs) with multiple delays. Following the foundation laid by Verhulst (1838) and Hutchinson (1948), we extend the classical logistic model to include variable and constant delays, addressing the gaps in stability analysis and oscillation criteria for higher-order delay systems. The study establishes the following: theoretical conditions for oscillation and stability in first-, second-, and third-order logistic differential equations (LDEs) with periodic coefficients, deriving sufficient criteria for oscillatory solutions. Numerical verification of delay-induced bifurcations, showing the transitions from stable equilibria to periodic oscillations. Comparative analysis of retarded, neutral, and advanced types of delay, with applications in population dynamics, control theory, and economics

Several delays were used to develop the logistic equation and the difference equation, using a hybrid methodology that combines analytical techniques (method of steps, Laplace transforms) and numerical simulations. We generalize existing oscillation theorems and provide explicit conditions for non-oscillation. The main results include the dependence of solution behavior on delay configurations and periodic coefficients, illustrated through case studies. This study contributes to the broader understanding of complex systems with memory effects, providing tools for modeling real-world phenomena in ecology, epidemiology, and engineering.

very good.