Controlling of periodicity and chaos in a three dimensional prey predator model introducing the memory effect

Abstract

In an ecological system, omnivores are often significantly important as they feed upon more than one trophic level in a food chain model. Intraguild predation is a special kind of omnivory that is ubiquitous in many ecological communities. A lot of field experiments suggest that the experience over a time duration in the recent past affects the growth rate of all species at the present time. Here, we have considered the Caputo type fractional ordered three-species food chain model. The memory effect is explained in terms of the order of the fractional derivative. Intermediate, top predators feed upon prey by Holling type I functional response and the top predator consumes intermediate predator by Holling type II functional response. Here, we established the existence and uniqueness of the solution of the considered system. Also, we prove the positivity and boundedness of the system solution. All possible feasible equilibrium points with required parametric conditions are discussed in the text. The local stability of all feasible equilibrium points and the parametric conditions of global stability of all non-trivial equilibrium points are discussed. The memory effect can stabilize the system from the periodic oscillatory situation which has been proved through the Hopf bifurcation analysis. Finally, some numerical simulation has been carried out to make significant comments on the dynamics of the considered system about the memory effect and other system parameters. It is clear from the numerical simulations that the order of fractional derivative can be taken as the indicator of chaos control. The biological interpretations make the article most significant from the ecological point of view.