Dynamical Behaviors of Fractional Order Prey Predator Interactions.

Abstract

This paper investigates the dynamical behavior of a Fractional order Prey -Predator interaction model. A discretization process is applied to obtain its discrete version. The fixed points are obtained and the stability properties are discussed. …………………………………………………………………………………………………….... Introduction:-Recently, population models have received increasing attention by scientists due to their importance in ecology. Indeed, there are different approaches to study population models, e.g. ordinary differential equations, difference equations, partial differential equations and fractional order differential equations. Fractional -order differential equations (FOD) are used since they are naturally related to systems with memory which exists in most biological systems [2,4]. Many phenomena in population dynamics can be described successfully by the models using fractional order differential equations. In this paper, we study the dynamical behaviors of fractional-order LotkaVolterra predator prey system. It is shown that the discretized fractional-order system produces a much richer set of patterns than those observed in the systems counterpart. Fractional Order Prey -Predator Model and its Discretization:-The Lotka Volterra equations, also known as the predator prey equations, are a pair of first-order, nonlinear, differential equations frequently used to describe the dynamics in which two species interact, one as a predator and the other as prey. The populations change through time according to the following pair of equations:

Dynamical Behaviours of Stage-Structured Fractional-Order Prey-Predator Model with Crowley–Martin Functional Response

“…A stage-structured predator-prey model is considered and elucidated by the authors in Reference [9]. Some authors [10][11][12][13][14][15][16] have studied two species models with stages structured in the predators. ey discussed and investigated the stability of the all equilibrium points as well as the dynamic behaviour of their systems.…”

Dynamic Analysis of a Fractional Order Three Species Interaction Model with Holling Type II Response

“…Especially, lots of works are carried out on fractional order systems and its applications in various fields and recent developments on the theory on existence and uniqueness, stability conditions of solutions of non-linear FDEs. [ [3], [5]]…”