Dynamic Properties of the Feedback Control of a Distributed Delay Logistic Differential Equation

El-Sayed, Ahmed M. A.; Mousa, Sanna; Saad, Muhammed Ahmed

doi:10.21608/ajst.2025.361003.1066

Dynamic Properties of the Feedback Control of a Distributed Delay Logistic Differential Equation

Document Type : Original Article

Authors

¹ Mathematics and Computer Science Department, Faculty of Science, Alexandria University, Alexandria, Egypt

² Mathematics Department, Faculty of Education, Alexandria University, Alexandria, Egypt

10.21608/ajst.2025.361003.1066

Abstract

This paper investigates the stability characteristics of the logistic differential equation with distributed delay, exploring its dynamics in both discrete and continuous time frameworks. By examining the influence of varying system parameters, particularly the delay kernel, the study provides a comprehensive understanding of how these factors shape system stability. The analysis employs numerical simulations to delineate stability regions, focusing on the interplay between parameter variations and dynamic behavior.
Key findings highlight the critical role of the delay kernel in determining the transition between stable and unstable states. Numerical results are presented through visual tools such as phase portraits and plots of the maximal Lyapunov exponent, which capture the progression from stable equilibrium to oscillatory or chaotic dynamics. These illustrations offer valuable insights into the mechanisms underlying stability loss and the onset of complex behaviors.
The study emphasizes the sensitivity of the logistic model to distributed delay variations, showcasing the intricate dependency of system behavior on the kernel's properties. This sensitivity is pivotal in understanding the dynamics of real-world systems modeled by logistic equations with delay. Moreover, the results have broad implications for applications where distributed delay plays a significant role, such as population dynamics, biological systems, and control theory.
By elucidating the relationship between delay kernels, parameter changes, and system stability, this work contributes to a deeper understanding of delayed dynamical systems and their practical applications. The findings underscore the importance of delay structure in predicting and controlling system behavior.

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