Stability analysis of a three species food chain model with linear functional response via imprecise and parametric approach

Abstract

Currently, proper modeling of a biological system in an uncertain situation is a challenging task for researchers. Facing this challenge, in this article, a new concept of interval representation named as linear parametric representation is introduced by which a biological system in an uncertain situation can be formulated mathematically in a precise way. The aim of this work is to formulate a three species imprecise food chain model with one prey and two competing predators in an interval environment. Here, all the biological parameters except the conversion efficiency of the species are considered interval-valued. And interactions among the species are taken as Holling Type I functional response. Using the linear parametric representation of interval, a system in the interval form has been converted in the parametric form. Positivity and boundedness of the solutions of the proposed imprecise system are verified. Then, conditions of local stability, global stability and transcritical bifurcation of equilibrium points of the proposed system are established. Thereafter, all the theoretical results of this work are validated by a numerical example with interval-valued parametric data and results are presented pictorially. Finally, some biological implications of the obtained results are discussed and a fruitful conclusion has been made.