Approximating Solution of Fabrizio-Caputo Volterra’s Model for Population Growth in a Closed System by Homotopy Analysis Method

Abstract

Volterra’s model for population growth in a closed system consists in an integral term to indicate accumulated toxicity besides the usual terms of the logistic equation. Scudo in 1971 suggested the Volterra model for a population u(t) of identical individuals to show crowding and sensitivity to “total metabolism”: du/dt=au(t)-bu2(t)-cu(t)∫0tu(s)ds. In this paper our target is studying the existence and uniqueness as well as approximating the following Caputo-Fabrizio Volterra’s model for population growth in a closed system: CFDαu(t)=au(t)-bu2(t)-cu(t)∫0tu(s)ds, α∈[0,1], subject to the initial condition u(0)=0. The mechanism for approximating the solution is Homotopy Analysis Method which is a semianalytical technique to solve nonlinear ordinary and partial differential equations. Furthermore, we use the same method to analyze a similar closed system by considering classical Caputo’s fractional derivative. Comparison between the results for these two factional derivatives is also included.