Abstract:

This research paper delves into the intricate realm of mathematical modeling through the lens of nonlinear Volterra and Fredholm integro-differential equations (IDEs). These equations are pivotal in capturing dynamic behaviors and interactions in various real-world systems. We begin by providing an overview of the theoretical foundations of Volterra and Fredholm equations, elucidating their significance in modeling complex dynamical systems. Subsequently, we explore the nonlinear extensions of these equations and discuss their applications across diverse domains, including population dynamics, epidemiology, neuroscience, and ecology. Through numerical simulations and analytical techniques, we analyze the behavior of solutions to these equations, uncovering rich dynamical phenomena such as stability, bifurcations, and chaos. Furthermore, we investigate the implications of nonlinearities on system behavior, highlighting the emergence of non-trivial dynamics that defy simple intuition. Our findings underscore the profound insights gained from studying nonlinear Volterra and Fredholm IDEs, emphasizing their utility in understanding and predicting the behavior of complex systems.

Description:

Population, Ecological, Mathematical, Complex, Nonlinear