Effect of Differential Retardation Equations on Insect Life Cycle: Modeling and Analysis for Deeper Understanding

In this research, we explore the impact of using delay differential equations in analyzing and understanding actions within the framework of studying the life cycle of insects. It demonstrates how these equations can be used to predict insect population outbreaks and identify environmental conditions conducive to reproduction. In this paper, we rely on the growth function of the prey and predator equations to study the effect of delayed differential equations on the insect life cycle. The results and comparisons were obtained using MATLAB. Also, we give an applied example of the fruit fly and used numerical methods, namely Euler's method, to obtain accurate approximate values to study the phenomenon of the delay effect of the differential delay equations for the life cycle of the fruit fly. Finally, the new idea from this work is Include time delay effect where the Ordinary differential equations (ODEs) assume that changes occur instantaneously, which may not be accurate in many biological systems. While the delay differential equations (DDEs) allow the inclusion of time delays between different events, such as the time required for eggs to transform into larvae, larvae to pupae, and pupae to adults.