Dynamics of Complexity of a Discrete – Time Prey-Predator system

Abstract

Investigations were carried out on a discrete-time prey-predator system. The characteristic pattern within periodic windows appearing in the
chaotic region of bifurcation of this proposed prey-predator system indicates that the system is of a complex nature. This implies that the system
is internally composed of a multicomponent structure. Such components evolve independently, and to understand the evolution of these
components, one has to follow the rule of probability. Topological entropy is a measure of complexity; "a higher topological entropy indicates
a more complex system." During the process of study, fixed points are calculated and their stability criteria discussed in detail. Bifurcation
diagrams of the system obtained by varying the parameter and, also, a typical region of the bifurcation displaying a periodic window magnified.
A complex pattern of scenarios observed within a periodic window is discussed with proper justification. Numerical work is performed to get
attractors of regular and chaotic motion, followed by calculations of Lyapunov exponents. Numerical calculations were again extended to find
the topological entropies, a measure of complexity, of the system, which were displayed graphically. Finally, correlation dimensions for some
chaotic attractors were also calculated. The results obtained through this investigation are interesting and very significant.