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

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) ∫t 0 u(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) ∫t 0 u(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.


Introduction
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.Consequently, 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: 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 () of identical individuals which exhibits crowding and sensitivity to "total metabolism": If the integral term on the right is removed the famous logistic equation with birth rate  and crowding coefficient  appears.The last term contains the integral that indicates the "total metabolism" or total amount of toxins produced since time zero.
The individual death rate is corresponding to this integral, so the population death rate by virtue of toxicity must include a factor .The existence of the toxic term as a result of the system being closed always causes the population level to fall to zero in the long run, as will be seen shortly.The relative size of the sensitivity to toxins, denoted by , determines the manner in which the population thrives before its decay.
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 [3][4][5][6][7][8][9].Fractional derivatives 2 Journal of Function Spaces are memory operators which usually represent dissipative effects or damage.Some fundamental definitions in the context of FC are Erdelyi-Kober, Riesz, Riemann-Liouville, Hadamard, Graunwald-Letnikov, Weyl, Jumarie, or Caputo (see, e.g., [10][11][12][13][14][15] and references therein).The Riemann-Liouville definition brings about physically unacceptable initial conditions (fractional order initial conditions) [12].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 therefore both definitions cannot precisely describe the full effect of the memory [16].Due to this problem, Caputo and Fabrizio in [17] introduced a new definition of fractional derivative without singular kernel, the Caputo-Fabrizio (CF) fractional derivative; this derivative possesses very interesting properties which are reviewed in detail in [18].Caputo and Fabrizio demonstrated that the new derivative involves further properties in comparison with the old version.They indicated, for instance, that it can interpret substance heterogeneities and configurations with different scales, which apparently cannot be investigated with the prominent local theories and also the old versions of fractional derivative.Another utilization is scrutinizing the macroscopic behaviours of some materials, identified with nonlocal communications between atoms.Other applications of the CF fractional derivative can be achieved, for example, in [19][20][21].
The main aims of this paper are studying the existence and uniqueness as well as approximating the Caputo-Fabrizio fractional population growth: subject to the initial condition Moreover, we shall compare the results with those obtained for the same problem by considering the classical Caputo fractional derivative in (3).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.

Existence and Uniqueness
In this section, we aim to show the existence and uniqueness of solution of the fractional Volterra population growth model (3).By Lemma 3 and using initial conditions we obtain the following result.Proof.First, we have Let We have to find a positive real number  such that We obtain Due to the assumption that  and V are bounded, and  ∈ [0, ], there exists a positive constant  > 0 such that ‖V‖ ≤  and ‖‖ ≤ .Thus, we get This shows the Lipschitz condition for Θ by taking  = 1 + 2 + 2.Regarding ( 9) we can get Now, we set   () =   () −  −1 ().So we have Taking the norm of the latter equation gives Then, by using the recursive principle it yields which proves that the solution exists and is continuous.We show that () defined as which is a solution for (3 We can take  as a solution of (3) that is continuous.Furthermore, applying the Lipschitz condition for Θ, we have Passing to the limit when  → ∞ and regarding the initial condition, we obtain For uniqueness we consider  and V to be two different solutions of (3); then, the Lipschitz condition for Θ gives the following result: rearranged to be Journal of Function Spaces Then, ‖ − V‖ = 0 if On the other hand, the constant function equal to zero is solution of the equation.Thus, the proof is completed.

Homotopy Analysis Method
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 [23] and later altered [24] in 1999 by proposing a nonzero auxiliary parameter, as the convergence-control parameter, ℎ, to construct a homotopy on a differential system in general form [25].The convergence-control parameter is a nonphysical variable that provides a simple way to certify convergence of a solution series.It is a series expansion method which is not directly dependent on small or large physical parameters.It also provides extreme flexibility to choose the basis functions of the desired solution and the corresponding auxiliary linear operator of the homotopy.Now, by using HAM to approximate the solution of the model (3), in view of properties of Caputo-Fabrizio fractional integrals, we assume that () can be expressed by the functions It is essential to know that we have a great freedom to choose auxiliary parameters in HAM.From base functions denoted by (27) and the initial condition given in (4), it is convenient to choose as the initial approximation of (), as well as as the auxiliary linear operator.Let ℎ denote a nonzero auxiliary parameter.We prepare the HAM deformation equation as subject to the initial condition where  ∈ [0, 1] is an embedding parameter and N[Ψ(, )] is a nonlinear operator, given by Evidently, when  = 0, the solution of (30) is When  = 1, (30) is the same as the original equation ( 3), provided Thus, Ψ(, ) alters from the initial approximation  0 () to the exact solution ().Expanding Taylor's series with respect to , we have where For briefness, one can define the vector Differentiating the HAM deformation equation ( 30)  times with respect to , then setting  = 0, and finally dividing them by !, we attain the th-order deformation equation subject to the initial condition where and () = 1.
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 (38) we have which implies that (42) then, Now, by Lemma 3 we have Hence, we can conclude By definition of the Caputo-Fabrizio fractional derivative, we get Now, using integration by parts as well as the initial condition we obtain So, Then, we realize

+ ( (ℎ + 𝜒
As suggested by Liao [25], the appropriate region for ℎ is a horizontal line segment.We can investigate the influence of ℎ on the convergence of , by plotting the curve of it versus ℎ, as shown in Figure 1.It seems that the best choice can be in the region [−1, 1].We can also see the results of the 8th order analytical approximations for () given by HMA for Fabrizio-Caputo differential equation in Figure 2.
Xu [26] has also studied the Caputo form of this equation for nonzero initial conditions.The result for this equation with zero initial conditions is as follows: As shown in Figure 3, it seems that the best choice for ℎ can be in the region [−1, 1].The 8th order analytical approximations for () given by HMA for the differential equation by considering the classical Caputo fractional derivative is shown  in Figure 4.This approximation is for  =  =  = 10 and  = 0.5.We also get here  0 () = .
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.

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

Figure 3 :
Figure 3: The ℎ-curves obtained from the 6th-order HAM approximate solution for Caputo logistic equation.

Figure 4 :
Figure 4: The 8th order analytical approximations for () given by HMA by considering the classical Caputo fractional derivative.