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

Tahereh Bashiri,S. Mansour Vaezpour,Juan J. Nieto

doi:10.1155/2018/3152502

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.

Highlights

Malthus was the first economist to propose a systematic theory of population [1] where he gathered experimental data to support his thesis. He proposed the principle that human populations grow exponentially
A nonlinear growth equation was introduced into population dynamics by Verhulst [2] to solve the unbounded growth in human population proposed by Malthus
Verhulst introduced the nonlinear term into the rate equation and reached what afterwards became nominated as the logistic equation: du (t) dt ku (t) [1

Summary

Introduction

Malthus was the first economist to propose a systematic theory of population [1] where he gathered experimental data to support his thesis. In the Caputo concept, the initial conditions are expressed in terms of integer-order derivatives, so it has physical meaning [13] These definitions have the disadvantage that their kernel has singularity; this kernel includes memory effects and both definitions cannot precisely describe the full effect of the memory [16]. By changing the kernel (t − s)−a by the function exp(−(α/(1 − α))(t − s)) and 1/Γ(1 − α) by M(α)/(1 − α), one obtains the new Caputo-Fabrizio fractional derivative of order 0 < α < 1, which has been recently introduced by Caputo and Fabrizio in [17]

Existence and Uniqueness

Homotopy Analysis Method

MATLAB Codes