Stability Analysis of an Age Structured Population Model With Fractional Time

: In this paper, we analyse the large time behaviour in a fractional nonlinear model of population daynamics with age dependent. We show the existence and uniqueness of the solution by using the method of seperation of variables, and we studied the Ulam-Hyers stability of the model.


Introduction
The following model: Ṗ = δP, (1.1) where δ is the growth rate, is considered as the simplest model in the domain of dynamic of populations, and it was introduced by Malthus in [15].This model does not take into account the effect of overcrowding, for this reason Verhulst proposed the following model [19], Ṗ = (δ 0 − ω 0 P )P, ( where δ 0 and ω 0 are positive constants.In (1.1) the growth rate is given by (δ 0 − ω 0 P ) it then depends on the total population P , so it takes into account the effect of overcrowding.The models (1.1) and (1.2) have played a considerable role in the domain of dynamic of populations however they do not take into consideration the age of individuals, even though age is one of the most important parameters structuring a population.The first age-structured model was proposed by Lotka and Mckendrick, they assumed that the density of population u(t, a) satisfies the following problem [14], Where µ is the death rate and β is the birth rate.In this work we assumed that the parameters µ and β are given like in Michel Langlais's paper [13].We then propose the following model, Supplemented by the following initial condition, Where ∂ α ∂t α is the Caputo derivative of order α.For more details about the parameters used in this model see section 3. Using the method of separation of variables we proved the existence and uniqueness of the solution, and then we studied the Ulam Hyers stability of the model.

Preliminaries
In this section we introduce notations, definitions and preliminary facts which are used throughout this paper.We denote by X = C([0, T ], R + ) the Banach space of all continuous functions from [0, T ] into R + , with the norm P X = sup{|P (t)|, t ∈ [0, T ]}.We need some basic definitions and properties of the fractional calculus theory.For more details, see [11].Definition 2.1.[11] The fractional integral of the function h ∈ L 1 ([a, b]) of order α ∈ R + is defined by where Γ is the gamma function Definition 2.2.[11] For a function h given on the interval [a, b], the Caputo fractional order derivative of h, is given by (2.1) and the following inequalities Definition 2.6.[20] The equation (2.1) is Ulam-Hyers stable with respect to ϕ if there exists c f,ϕ > 0 such that for each ǫ > 0 and for each solution Definition 2.7.[20] The equation (2.1) is generalized Ulam-Hyers-Rassias stable with respect to ϕ if there exists c f,ϕ > 0 such that for each solution One can have similar remarks for the inequalities (2.3) and (2.4).

Formulation of the model
Let u(t, a) be the density of population having at time t > 0 the age a.We are going to study existence, uniqueness and Ulam-Hyers stability of this equation where, ∂ α ∂t α is the Caputo derivative, with 0 < α < 1, and (3.2) • P (t) is the total population at time t.
• µ n (a) is the probability of dying due to natural causes at age a, µ n (a) ≥ 0.
• µ e (P ) is the probability of death due to environmental factors.µ e is a function of the total population.
• The birth low is given by • β is called the birth-modulus which represents the fertility of the population One assumes that β has a compact support in [0, ∞[, so that where A = max{a, β(a) > 0} < ∞, is the minopause age.
We give the initial condition:

The characteristic equation
We can look for a solution of (3.1) in the form

Assumptions and notations (H1)
We assume that 0 where β 1 is a real constant, and where µ 1 is a real constant.

(H3)
We assume that where μ is a minimum mortality rate.

separable solution
We first define the notion of separable solution of the problem (3.1), (3.2), (3.3) and (3.4) which is a solution that can be written as: It remains to show the existence of a function P (t), such that the function u(t, a) = ψ(a)P (t) will be a separable solution.We will define this function in the following paragraph.In this equality the left term depends only on a, while the right term depends only on t.Then, these two terms are constant and equal to a constant r.This leads to the following two equations Hence, the solution of the equation (6.4) is where ψ 0 is given by (6.7).Study of equation (6.5).Let r = r * .The equation (6.5) can be written as where f : X −→ R + is a continuous function and satisfies the Lipschitz condition i.e ∃L > 0 such that Lemma 6.2.The function P ∈ X is a solution of the equation if and only if P satisfies the integral equation (6.9)

Proof.
Assume that P ∈ X is a solution of problem (6.9).Applying the Caputo fractional operator of the order α, we obtain the first equation in (6.8).Again, substituting t = 0 in (6.9) we have P (0) = P 0 .
Conversely, we have C D α P (t) = f (P (t)), so we get

Proof.
We define a subset U of X by, where, It is clear that U satisfies the hypothesis of Theorem 2.3.By application of Lemma 6.2, (6.8) is equivalent to the integral equation, Define the operator A : U → U , by Then our equation is transformed into the operator equation as, Let (P n ) n be a sequence in U converging to a point P ∈ U .Then, by the Lebesgue dominated convergence theorem, For all t ∈ [0, T ].This shows that A is a continuous operator on U .
• A is a compact operator on U .
First, we show that A(U ) is uniformly bounded set in X.
Let P ∈ U .Then for all t ∈ [0, T ], Thus, AP < ∞ for all P ∈ U .This shows that A is uniformly bounded on U .Next, we show that A(U ) is equicontinuous set on X.
Then, by the Arzela-Ascoli theorem, A is a continuous and compact operator on U .Thus by Schauder theorem, the operator A has a fixed point.Now, we must show that A has a unique fixed point.First, we prove that for all j ∈ N and t ∈ [0, T ], we have, for j = 0, the inequality is trivial.We suppose that the inequality holds for j − 1 and we prove that it holds for j.Let t ∈ [0, T ]. Thus, Hence, the operator A satisfies the assumptions of weissinger's fixed point theorem with α j = LT α Γ(α+1) j , we can therefore deduce the uniqueness of the solution of our equation.

Theorem 2 . 3 (
The Schauder's theorem).Let C be a nonempty closed convex subset of a Banach space X.Let T : C −→ C be a continuous mapping such that T (C) is relatively compact subset of X.Then T has at least one fixed point in C .Let ǫ be a positive number, f : [a, b] × B −→ B be a continuous function with B is a Banach space and ϕ : [a, b] −→ R + be a continuous function.We consider the following equation c