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The pursuit problem is a historical issue of the application of mathematicsin physics, which has been discussed for centuries since the time of LeonardoDa Vinci, and its applications are wide ranging from military and industrial torecreational, but its place of interest is nowhere but nature and inspirationfrom the way of migration of birds and hunting of archer fish. The pursuitproblem involves one or more pursuers trying to catch a target that is movingin a certain direction. In this article, we delve into two modes of movement:movement on a straight line and movement on a curve. Our primary focus is onthe latter. Within the context of movement on a straight line, we explore twomethods and compare their respective results. Furthermore, we investigate themovement of two particles chasing each other and extend these findings to Nparticles that are chasing each other in pairs. By leveraging these two modesof movement, we present a novel relationship for two-particle and N-particlesystems in pursuit. Lastly, we analyze the movement of moths around a lamp andevaluate their motion in relation to two-particle and N-particle systems inpursuit. The results of this analysis are carefully examined.

Investigating the classical problem of pursuit, in two modes