Controlling transient of species in a food chain

Abstract

Extinction of species has been one of the biggest mysteries in nature. A dynamical model for the species extinction is very complex because each species interacts with many other species as well as physical factors such as weather. Therefore it looks difficult to determine the structure of food chain mathematically. However, in certain situations the problem of species extinction can be modelled by a coupled ordinary differential equation. In this work, we demonstrate that if species extinction is caused by transient chaos, the population size of some species behaves chaotically for a period time and then decreases to zero in a short period of time. it can be prevented from being extinct by controlling transient chaos. The proposed scheme applies feedback perturbations to populations of the species, which are based on the return map. We illustrate the efficiency of the scheme by applying to a food chain system. Considering the perturbation, we have to think the quantity we will add most efficiently. Therefore, we focused on the reasonable perturbation ways by using various factors. Especially, we will show the phase drawn by behaviour depending on initial values respectively. We think its the key for finding the way giving reasonable perturbation. Not only that, we will suggest the new definition of the feedback point so called crisis point. Finally, we consider the fractional system of food chain and their transient chaos.