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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

Malthus was the first economist to propose a systematic theory of population [

Volterra’s model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation. Scudo, in 1971, indicates that Volterra proposed his following model for a population

If the integral term on the right is removed the famous logistic equation with birth rate

The individual death rate is corresponding to this integral, so the population death rate by virtue of toxicity must include a factor

The tool for describing the behaviour of the equations such as logistic equation must be nonlocal differential equations in time. With this purpose, in the last decades, the Fractional Calculus (FC) allows investigating the nonlocal response of multiple phenomena [

Some fundamental definitions in the context of FC are Erdelyi-Kober, Riesz, Riemann-Liouville, Hadamard, Graunwald-Letnikov, Weyl, Jumarie, or Caputo (see, e.g., [

The main aims of this paper are studying the existence and uniqueness as well as approximating the Caputo-Fabrizio fractional population growth:

The auxiliary method used here for approximating the solution is a semianalytical technique known as Homotopy Analysis Method to solve nonlinear ordinary and partial differential equations.

Next, we introduce some basic definitions and notations that shall be used in next sections.

For at least

By changing the kernel

For

Let

In this section, we aim to show the existence and uniqueness of solution of the fractional Volterra population growth model (

If

First, we have

The Homotopy Analysis Method operates the theory of the homotopy from topology to generate a convergent series solution for nonlinear systems.

The HAM was first constructed in 1992 by Liao in his Ph.D. Thesis [

Now, by using HAM to approximate the solution of the model (

It is essential to know that we have a great freedom to choose auxiliary parameters in HAM. From base functions denoted by (

Let

Next, our target is to reach a recursive equation to be implemented, for example, in MATLAB, in order to provide an approximation of the solution.

From (

Then, we realize

Finally, we have the following equation:

As suggested by Liao [

The

The 8th order analytical approximations for

Xu [

The

The 8th order analytical approximations for

As a result, we can see that the solutions are similar with more oscillation in the case of Caputo-Fabrizio type of the population growth.

Here, we prepare codes written by MATLAB related to plotting the figures you can see as the results of approximation by HAM.

MATLAB code for Figure

clc; clear all; close all;

syms

for

if

else

end

end

for

for

end

for

end

end

for

end

syms

MATLAB code for Figure

clc; clear all; close all;

syms

for

if

else

end

end

for

for

end

for

end

end

for

end

syms

figure;

hold on;

set (

set (

set (

set (

legend (“

MATLAB code for Figure

clc; clear all; close all;

syms

for

if

else

end

end

for

for

end

for

end

for

end

end

for

end

syms

MATLAB code for Figure

clc; clear all; close all;

syms

for

if

else

end

end

for

for

end

for

end

for

end

end

for

end

syms

figure;

hold on;

set (

set (

set (

set (

legend (“

The authors declare that they have no conflicts of interest.

The work of J. J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under Grant MTM2016-75140-P, cofinanced by the European Community fund FEDER. The authors acknowledge the support of Xunta de Galicia, Grants GRC2015/004 and R2016/022. Tahereh Bashiri thanks the hospitality of Departamento de Análise Matemática of Universidade de Santiago de Compostela, where a significant part of this research was performed during her visit in 2016-2017.