Non-linear Age-time Dependent Population Dynamics

Dissertation

zur Erlangung des Doktorgrades

der Mathematisch-Naturwissenschaftlichen Fakultäten der Georg-August-Universität zu Göttingen

vorgelegt von Md. Shahidul Islam

aus Noakhali, Bangladesh

Göttingen 2002 D 7 Referent: Prof. Dr. Manfred Denker Korreferent: Dr. Hartje Kriete Tag der mündlichen Prüfung: 26.04.2002

ACKNOWLEDGEMENTS

I take this opportunity to acknowledge my indebtness and gratitude to my advisor, Professor Dr. Manfred Denker, Institute for Mathematical Stochastic, under whose constant care and guidence this work has been accelerated and intensified greatly. I am grateful to my coadvisor Dr. Hartje Kriete, Institute for (Pure) Mathematics, for his so many valuable advices and kind hospitality in Göttingen.

I would like to thank Mario Roy, who spared his valuable time to go through most of the proofs.

I would further like to thank all my colleagues, in particular, Brice Franke, Robert Simon and Paul Besold with whom I shared some interesting discussions during this work.

I would like to thank my wife Nasrin for her support during this work. Finally, I would like to mention that this research was supported by the

grants from the Nidersächsischen Graduiertenstipendium.

Abstract

In this thesis, we introduce several non-linear population models, each of which is an extension of the celebrated Gurtin-MacCamy model [11]. We give constructive proofs of the existence and uniqueness of a global solution. More precisely, we show that our model has generally a unique solution in the future but not in the past. In fact, the model has a unique solution in the past under certain additional conditions on the birth rate of the population. Global stability in presence of noise is also considered.

Contents

1 Introduction 1 1.1 First Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Age-dependent Linear Models . . . . . . . . . . . . . . . . . . 3 1.3 Gurtin-MacCamy Model . . . . . . . . . . . . . . . . . . . . . 4 1.4 Extended Gurtin-MacCamy Model . . . . . . . . . . . . . . . 5

2 Preliminaries 7 2.1 Gronwall’s Inequality . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 The Extended Gurtin-MacCamy Model 13 3.1 Reduction of the Model Equations . . . . . . . . . . . . . . . . 13 3.2 Existence and Uniqueness Theorems . . . . . . . . . . . . . . 16 3.3 Birth Processes for a Given Population . . . . . . . . . . . . . 16 3.4 Extension of the Model to the Past . . . . . . . . . . . . . . . 19 3.5 Strong GMC Model . . . . . . . . . . . . . . . . . . . . . . . . 21 3.6 Proof of the Existence and Uniqueness Theorems . . . . . . . 29 3.7 Weak GMC Model . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Random Solutions 40 4.1 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 1

Introduction

One of the most interesting problem in science is the study of the dynamics of a biological population. Mathematical models describing the age-dependent populations have received increasing attention in recent years, both for the biological interest and for the mathematical one. In fact, age-dependent fertility and mortality rates are among the most basic parameters in the theory of population dynamics. There is an extensive literature for linear models of age-dependent populations, but the theory of nonlinear models are much more recent. The inclusion of nonlinearities in the equations of age-dependent population models increases considerably not only their mathematical difficulties, but also their reliability in the physical description of behaviours of a population. The purpose of this work is to describe a more general class of non-linear population models. We study the growth (or decay) in the number of individuals of a particular species in a given region. Immigration and emigration are assumed to play no significant role in the dynamics of the population, and our model does not take into account the sex of the individuals.

1.1 First Models

In 1798, T.R. Malthus made history by applying mathematical equations to the theory of populations. He came up with the now celebrated law of exponential change. In this law, the function P (t) represents the total number of individuals of a population at time t, and it satisfies the following differential equation:

\frac{dP (t)}{dt} = rP (t), t ≥ 0  (1.1)

where r ∈ IR is the relative growth (or decay when r < 0) modulus and is called Malthusian parameter. The solution of (1.1), the exponential function P (t) = P (0) e^{r t}, asserts that the population grows (or diminishes) exponentially.

However, the Malthusian model is clearly inapplicable to situations where crowding takes effect after some time, or when the population begins declining because of a birth rate decrease or mortality rate increase due to the inclination of one organism to destroy others or due to the limitations of resources (e.g. food or space). A more realistic model of population growth should therefore allow the Malthusian parameter to depend upon the size of the total population. In 1838, P.F. Verhulst assumed that the population respects the following differential equation:

\frac{dP (t)}{dt} = r P (t) − k P^{2}(t),  t ≥ 0  (1.2)

where r/k is called environmental carrying capacity (k > 0), that is, the maximum population the environment is capable of sustaining in the long run. The equation (1.2) has a stable equilibrium point, or limiting population, P (t) = r/k, and the population grows monotonically to r/k as t → ∞, whenever P (0) < r/k. If P (0) > r/k, then the population decreases monotonically to r/k. Finally, if P (0) = r/k, then the population remains constant forever. This model is also often called nowadays logistic growth model.

The models of Malthus and Verhulst are continuous, deterministic population models. The disadvantage of these models are that they do not take into account the age distribution of the population. In fact, the birth and death rates are independent of the age of the individuals in these models. However, it is obvious that, in many situations, the age distribution does influence population size and behaviour through the natality and mortality rates.

1.2 Age-dependent Linear Models

The theory of continuous, linear population dymamics has been developed by many mathematicans. The first age-dependent continuous, linear model was proposed by F.R. Sharpe and A.J. Lotka [1] in 1911, shortly followed by A.G. McKendrick in 1926. Their model is based on age-dependent natality and mortality moduli, but do not take account of crowding effects or environment limitations. In their theory, the age-structured population is specified by a density function x : IR_{+}^{2} → IR_{+}, which determines the number of individuals of age a at time t. The model is described by the following partial differential equation, the so-called Von Foerster equation

Dx(t, a) + μ(t, a)x(t, a) = 0, t > 0, a > 0  (1.3)

where μ is the (nonnegative) mortality modulus. Here, Dx denotes the directional derivative of x with respect to the direction (1, 1), that is,

Dx(t, a) = lim_{Δt→0} \frac{x(t + Δt, a + Δt) − x(t, a)}{Δt}.

The birth process is described by the birth law

x(t, 0) = ∫_{0}^{∞} ν(t, a)x(t, a)da, t > 0  (1.4)

where x(t, 0) is the number of newborn population at time t and ν is the (nonnegative) fertility modulus. The initial age distribution of the population is

x(0, a) = φ(a), a ≥ 0  (1.5)

where φ is a known nonnegative function.

Equations (1.3)-(1.5) constitute the classical linear model of age-dependent population dynamics, which is closely related to the theory of linear Volterra integral equations and the so-called renewal equation (see Feller [10]).

Some other models in population dynamics assume that the birth and death functions are time-independent. In this case, it was discovered that, ultimately, the total population always either decays or grows exponentially without limit (few real populations actually behave in either of these ways).

A detail treatment of these models can be found in Pollard [7], Keyfitz [8] and Langhaar [9].

All the models mentioned above are very simple from a mathematical point of view because of their linearity, but they are not applicable in many situations. For example, when diverse populations compete for resources or when densities influence fertility or mortality moduli.

A more realistic model of a stable population, crowding effects and the limitation of resources suggest that the fertility and mortality moduli vary with the population

P (t) = ∫_{0}^{∞} x(t, a)da, t ≥ 0  (1.6)

that is, the mortality μ should be an increasing function of P and the fertility ν a decreasing function of P.

1.3 Gurtin-MacCamy Model

In 1974, M.E. Gurtin and R.C. MacCamy [11], introduced a more realistic formulation of a nonlinear, continuous, age-dependent, deterministic population model (see also Sowunmi [18]). They studied the dynamics of a closed population, while taking account of its age-structure. Moreover, the effects of crowding appear in the model, for the fertility and mortality moduli are allowed to be nonlinear functionals of the population at any time. The problem is based on the following initial-boundary value problem:

Dx(t, a) + μ(a, P (t))x(t, a) = 0, t > 0, a > 0
x(t, 0) = ∫_{0}^{∞} ν(a, P (t))x(t, a)da, t > 0  (1.7)
x(0, a) = φ(a), a ≥ 0

where

P (t) = ∫_{0}^{∞} x(t, a)da, t ≥ 0

the parameters φ, μ and ν being the same as before.

The model consists of a pair of nonlinear Volterra integral equations. Existence, uniqueness and stability of the solution of the model was also investigated by Chipot [20] and Gopalsamy [21].

Moreover, Sinestrari [17] considered mortality and fertility moduli that further depended on the age density function x(t, a). Kostova and Milner [23] studied the model for populations of constant size P (t) = P_{0} for all t. Haimovici [22] discussed the model in which the mortality and fertility moduli depended on time t and age a, rather than on the population and age. He also took into account the amount of resources and pollution of the environment. Blasio [19] studied the age-specific fertility rate on the past history of the population. A more recent approach using semigroup theory was given by Webb [24].

1.4 Extended Gurtin-MacCamy Model

To overcome the deficiency of Sharpe-Lotka and Gurtin-MacCamy models, we study different models in which natality and mortality moduli are simultaneously age, time and population dependent. Nonlinear age-time dependent models of population provide a biologically more realistic description of the behaviour of the population when the past history of the population is considered. The problem can be stated as the following initial-boundary value problem:

Dx(t, a) + μ(t, a, p(t))x(t, a) = 0, a > 0, t ∈ IR
x(t, 0) = ∫_{0}^{∞} ν(t, a, p(t))x(t, a)da, t ∈ IR  (1.8)
x(0, a) = φ(a), a ≥ 0

the parameters φ, μ and ν being the same as before.

We will show that this problem can be reduced to a nonlinear integral equation which replaces the renewal equation that arises in linear models. Under certain conditions on φ, μ and ν, the solution of the equation exists and is unique in the future, though not generally in the past. We investigate the evolution of population

P (t) = ∫_{0}^{∞} x(t, a)da, t ∈ IR.  (1.9)

The models are studied in detail in Chapter 3. The set up is the following: the population is described by means of an age-time density function x(t, a). The basic hypotheses are that μ and ν are (non-negative) bounded and measurable functions. The initial birth rate is finite up to time T, and the initial age distribution φ(a) should belong to L_{1}^{+}(IR_{+}). The birth modulus is not specified for past but it has to satisfy the renewal equation. It turns out that the models are inconsistent when ν is integrable (the models have a trivial solution). If ν is not integrable, we show that the same model has a solution in the past (Theorem 3.2.2).

We also investigate a more general measure-theoretical approach in which the model has a solution in the past. If ν is uniformly bounded (not integrable), the renewal equation holds almost surely. We consider the space of Borel probability measures on the compact metric space [0, T], which is convex and weakly compact. We define an operator L acting on this space and apply Schauder-Tychonov fixed point theorem in order to prove that this operator admits a fixed point ρ. This fixed point is a measure which is absolutely continuous with respect to Lebesgue measure and the corresponding eigenvalue of L is λ = ∫ L 1 dρ. It turns out that three different dynamical behaviours can occur: The population grows exponentially when λ > 1, diminishes when λ < 1 and remains constant when λ = 1.

In Chapter 4, we introduce a non-deterministic Gurtin-MacCamy model, namely a model in which a random environment may affect our system. Since the environment changes are due to many factors, we can approximate by noise (e.g. a Gaussian white noise), although we do not know the exact behaviour of the noise term. By allowing some randomness in our system, we often obtain a more realistic mathematical model of the population. We investigate global stability in presence of noise, which characterizes the possible long term behaviour of the dynamical system.

Preliminaries

In this chapter, we recall generalizations of the well-known Gronwall’s inequality. We further state some useful fixed point theorems and integral equations.

2.1 Gronwall’s Inequality

Let a < b be real numbers. The following lemma is a generalization of Gronwall’s inequality. It follows from Gronwall’s inequality in [16] on [0, 1] by substitution. In fact, ρ(t) = (b − a)t + a maps homeomorphically the unit interval onto the interval [a, b] and the functions u ∘ ρ, f ∘ ρ, g ∘ ρ and h ∘ ρ satisfy the assumption of Gronwall’s inequality in [16].

Lemma 2.1.1 Let f, u : [a, b] → IR be continuous functions and g, h : [a, b] → IR_{+} be continuous positive functions. If

u(t) ≤ f(t) + g(t) ∫_{a}^{t} h(s) u(s) d s,  (2.1)

for each t ∈ [a, b], then

u(t) ≤ f(t) + g(t) ∫_{a}^{t} f(s)h(s) exp{∫_{s}^{t} g(v)h(v)dv}ds,

for each t ∈ [a, b].

Proof. Denote by

y(t) = ∫_{a}^{t} h(s) u(s) d s, t ∈ [a, b].

Using the equality

y′(t) = h(t)u(t), t ∈ [a, b]

and the hypotheses, it follows that

y′(t) ≤ h(t)f(t) + g(t)h(t)y(t), t ∈ [a, b].

Multiplying by exp{− ∫_{a}^{t} g(s)h(s) d s}, we obtain that

\frac{d}{dt}[y(t) exp\{− ∫_{a}^{t} g(s)h(s)ds\}] ≤ f(t) h(t) exp\{− ∫_{a}^{t} g(s)h(s)ds\}, t ∈ [a, b].

Integrating both sides with respect to Lebesgue measure on [a, b] and observing that y(a) = 0, we conclude that

y(t) ≤ ∫_{a}^{t} f(s) h(s) exp{∫_{s}^{t} g(v)h(v)dv}ds

for any t ∈ [a, b].

This completes the proof of the lemma. ■

Corollary 2.1.1 Under the same assumptions as in Lemma 2.1.1, it holds that

u(t) ≤ f(t) + ∫_{a}^{t} f(s)h(s) exp{∫_{s}^{t} h(v)dv}ds,  (2.2)

for each t ∈ [a, b].

Proof. Simply apply the previous lemma with g ≡ 1. ■

Corollary 2.1.2 Under the same assumptions as of Lemma 2.1.1, we have

u(t) ≤ c exp{∫_{a}^{t} h(s) ds},  (2.3)

for each t ∈ [a, b].

Proof. Simply apply the previous corollary with f ≡ c ≥ 0. It follows from condition (2.1) that

\frac{u(t)}{c + ∫_{a}^{t} h(s)u(s) ds} ≤ 1.

Multiplying by h(t) and integrating, we get the conclusion of the corollary.■

In the remaining part of this section, we study some properties related to the solution of the following linear integral equation with kernel K:

f(t) = ∫_{−∞}^{t} K(t, t − s)f(s)ds.  (2.4)

Lemma 2.1.2 Let K : IR^{2} → IR_{+} be continuous and bounded.

1. Every solution f of the equation (2.4) is continuous.

2. Let f̄ : (−∞, T] → IR_{+} be any function. Any function f satisfying equation (2.4) for t ≥ T and f = f̄ on (−∞, T] is unique.

Proof. [1.] The right-hand side of equation (2.4) is a continuous function of t, since

|f(t) − f(t_{1})| ≤ const · |t − t_{1}|.

[2.] To prove uniqueness of the solution f of equation (2.4), it suffices to show that two solutions which agree up to time T also agree up to time T_{1} for any T < T_{1} < ∞. Let f̃ be another solution on (−∞, T_{1}], which coincides with f on (−∞, T]. Then

|f(t) − f̃(t)| ≤ sup_{t≥T} |K(t, t − s)| · ∫_{T}^{t} |f(s) − f̃(s)|ds.  (2.5)

Using Lemma 2.1.1, it follows that f(t) = f̃(t). ■

Remark 2.1.1 1. If K depends on the second coordinate, then a solution of the equation (2.4) can be obtained by a trial solution f(t) ∼ const· e^{rt}, provided that r is a root of the characteristic equation

F (r) = ∫_{0}^{∞} K(x) e^{−rx}dx = 1.  (2.6)

For real values of r the function F (r) decreases monotonically as r increases. Hence equation (2.6) has only one real root r_{0} by Pollard [7, Theorem 3.2.1], and

r_{0} ⪌ 0 whenever ∫_{0}^{∞} K(x)dx ⪌ 1.

Langhaar [9, p. 220] and Keyfitz [8, §5.4] showed using Laplace transforms that equation (2.4) has a solution.

2. Lovitt [5, Theorem II.] proved that there is one and only one solution to equation (2.4) which is continuous on the interval 0 ≤ t < b, where b is an arbitrary bound.

2.2 Fixed Point Theorems

Fixed point theorems are the most important tools in analysis for proving the existence and/or uniqueness of solutions to some problems.

Theorem 2.2.1 (Contraction mapping principle)

Let (X, d) be a complete metric space and F : X → X be a contraction mapping, that is, there is 0 < k < 1 such that ∀x_{1}, x_{2} ∈ X we have

d(Fx_{1}, Fx_{2}) ≤ k · d(x_{1}, x_{2}).

Then there exists a unique fixed point of F in X and the orbit of any point is forward asymptotic to that fixed point.

The proof of this theorem can be found in any good introductory text on metric spaces and is based on an iterative method (see Smart [4, Theorem 1.2.2]).

Definition 2.2.1 Let X and Y be normed spaces. An operator T : X → Y is called compact if the image T (M) of every bounded subset M of X is a relatively compact subset of Y, that is, its closure \overline{T (M)} is a compact subset of Y.

Theorem 2.2.2 (Schauder-Tychonov theorem)

Let E be a non-empty closed, convex subset of a complete metric space X. Then every continuous map of E into itself has at least one fixed point.

The proof consists first in approximating the infinite-dimensional set E by a finite-dimensional set. Then one uses Brouwer’s theorem to deduce the existence of a fixed point of the finite-dimensional approximation. Finally, one takes the limit as the dimension of the approximating space tends to infinity to conclude.

There exists a vast literature on this topic. For details, we refer the reader to any good functional analysis book, for instance Dunford & Schwartz [3, p. 456] or Smart [4, Theorem 4.1.1].

Before using Schauder-Tychonov fixed point theorem, we would like to remind the reader about Arzelà-Ascoli theorem. For this, we first need to define the concept of equicontinuity of a family of maps under the supremum norm.

Definition 2.2.2 A family F of functions on a metric space (X, d) is equicontinuous if for every ϵ > 0, there exists δ > 0 such that if x, y ∈ X and d(x, y) < δ, then |f(x) − f(y)| < ϵ for all f ∈ F.

Theorem 2.2.3 (Arzelà-Ascoli theorem)

Let (X, d) be a compact metric space and let C(X) denote the set of all continuous functions on X. A closed subspace of C(X) is compact if and only if it is uniformly bounded and equicontinuous.

Every closed subspace of C(X) which is compact is clearly uniformly bounded. It remains to show that it is equicontinuous. Conversely, equicontinuity and uniform boundedness can be used to show that every sequence of functions of a closed subspace of C(X) has a convergent subsequence. The convergent subsequence is obtained via a Cantor diagonal argument.

For a proof of this theorem, we refer the reader to any real or functional analysis book, for example Bruckner, Bruckner & Thomson [6, Theorem 9.62].

2.3 Integral Equation

We now establish an important formula by integrating the following partial differential equation along characteristics.

We denote by D_{e} the derivative in the direction of the vector e = (1, 1) ∈ IR^{2}, that is,

D_{e}x(t, a) = lim_{h→0} \frac{x(t + h, a + h) − x(t, a)}{h}.

Lemma 2.3.1 Let p : IR ⟶ IR_{+} be a fixed measurable function and φ ∈ L_{1}^{+}(IR_{+}). Then there is a unique solution x : IR × IR_{+} ⟶ IR_{+} to the partial differential equation (PDE)

D_{e}x(t, a) = −μ(t, a, p(t))x(t, a), a > 0, t ∈ IR  (2.7)

with initial condition

x(0, a) = φ(a), a ≥ 0.  (2.8)

Remark 2.3.1 By a solution x of (2.7) and (2.8) on IR × IR_{+}, we mean a positive, measurable function x such that (2.7) and (2.8) hold, and such that ∫_{0}^{∞} x(t, a)da < ∞ for every t ∈ IR.

Proof. We first convert the system (2.7) and (2.8) into an equivalent system of an integral equation (see Corduneanu [12, §3.1]). Let x : IR × IR_{+} → IR_{+} be a function and choose a pair of arbitrary, but fixed, numbers (t_{0}, a_{0}) ∈ IR × IR_{+}, and denote

x̄(h) = x(t_{0} + h, a_{0} + h), μ̄(h) = μ(t_{0} + h, a_{0} + h, p(t_{0} + h)).

Then equation (2.7) can be rewritten as

\frac{dx̄}{dh} = −μ̄(h)x̄(h).

This equation has the following solution:

x̄(h) = x(t_{0}, a_{0}) exp{− ∫_{0}^{h} μ̄(τ)dτ}.  (2.9)

This relation gives the values of x at all points on the characteristic through (t_{0}, a_{0}).

In particular, if we take a_{0} = a − t, t_{0} = 0 and h = t, we obtain, using condition (2.8), that

x(t, a) = φ(a − t) exp{− ∫_{0}^{t} μ(τ, a + τ − t, p(τ))dτ}.  (2.10)

It is easy to show that equation (2.10) is the unique solution of the differential equation (2.7) with initial condition (2.8). ■

The Extended Gurtin-MacCamy Model

In this chapter, we extend Gurtin-MacCamy model. By allowing μ and ν to depend on a given measurable function p, the model assumes that migratory movements are negligible. The past history of the functions are also considered. The system describing our model consists of a pair of non-linear integral equations. The existence and uniqueness of solutions of the dynamics are studied at all time.

3.1 Reduction of the Model Equations

The general model we are interested in can be described by a map, defined on a certain function space, whose image is a set of integral operators. We begin with a general description of the model.

We consider the space IR × IR_{+}^{2} = {(t, a, p) : t ∈ IR, a, p ≥ 0} and refer to t, a and p as the time-coordinate, age-coordinate and population-coordinate, respectively. This space is equipped with the natural Euclidean topology. We let L_{p}(Ω) denote the space of p-integrable functions with respect to the Lebesgue measure on Ω, where Ω is a measurable subset of a Euclidean space of positive Lebesgue measure, whereas L_{p}^{+}(Ω) is the positive cone in L_{p}(Ω). We also let λ denote the Lebesgue measure on IR ×IR_{+}^{2}, and λ_{0} the Lebesgue measure on IR × IR_{+}.

Definition 3.1.1 The mortality space M is the set of positive functions μ such that log μ ∈ L_{∞}(λ) and

(t, a) ↦ sup_{p≠q∈IR_{+}} \frac{|μ(t, a, p) − μ(t, a, q)|}{|p − q|}

belongs to L_{∞}(λ_{0}).

Likewise, we define

Definition 3.1.2 The natality space N is the set of positive functions ν ∈ L_{∞}(λ), so that

(t, a) ↦ sup_{p≠q∈IR_{+}} \frac{|ν(t, a, p) − ν(t, a, q)|}{|p − q|}

belongs to L_{∞}(λ_{0}).

Remark 3.1.1 1. Note that M and N are cones, which are not closed in general.

2. The functions in M are called death moduli, whereas those in N are called birth moduli.

3. Every minorant of an integrable function μ ∈ M is itself integrable, that is, if μ_{0} ∈ L_{∞}(λ_{0}) satisfies

μ_{0}(t, a) ≤ μ(t, a, p)

for all p ∈ IR_{+}, given any arbitrary but fixed (t, a) ∈ IR × IR_{+}, then μ_{0} ∈ L_{1}(λ_{0}).

4. Since μ(t, a, p) ≥ \underline{μ} for some \underline{μ} > 0, the integral

∫_{−∞}^{t} exp{− ∫_{s}^{t} μ(τ, τ − s, p(τ))dτ}f(s)ds

exists for every integrable function f : (−∞, t] → IR_{+} and every measurable function p : IR → IR_{+}.

5. We denote by N (IR_{+}) the set of functions ν : IR_{+}^{3} → IR_{+} for which there exists ν_{0} ∈ N with ν = ν_{0}|_{IR_{+}^{3}}.

Let C denote the space of all non-negative bounded, continuous functions f : IR → IR_{+}, and C(T) be the space of all non-negative bounded, continuous functions f : [T, ∞) → IR_{+}, for some arbitrary but fixed T ∈ IR.

Our model comprises a non-negative function φ : IR_{+} → IR_{+}, whose value φ(a) symbolizes the population of age a at time 0. We assume that it is piecewise continuous and integrable on IR_{+}. We call it initial age distribution (function). The population at time 0 is therefore given by p(0) = ∫ φ(a)da.

Similarly, we define φ_{T} as the age distribution at time T. Given (B, P) ∈ C^{2} (resp. C(T)^{2}), we let φ(B,P)_{T} denote the age distribution of the Gurtin-MacCamy (GMC) model at time T, given the birth population B and the total population P (where ν and μ are fixed). Definition 3.1.3 1. A Gurtin-MacCamy model solution for (μ, ν) ∈

M × N is a pair (B, P) ∈ C^{2} satisfying the equations

B(t) = ∫_{−∞}^{t} ν(t, t−s, P (t)) exp{−∫_{s}^{t} μ(τ, τ − s, P (τ))dτ}B(s)ds (3.1)

and

P (t) = ∫_{−∞}^{t} exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ} B(s) ds  (3.2)

where t ∈ IR.

2. A pair as in 1. is called a weak Gurtin-MacCamy model solution if the equations (3.1) and (3.2) hold almost surely (abbreviated a.s. from now on) with respect to the Lebesgue measure on IR.

3. A pair as in 1. is called a strong Gurtin-MacCamy model solution if the equations (3.1) and (3.2) hold for t ≥ T for some T ∈ IR such that P (T) > 0, which implies that (B, P) ∈ C(T)^{2}.

The integral equation (3.1) is known as the Lotka-Sharpe renewal equation (see Feller [10]).

The general problems we are facing are the following:

1. Given (μ, ν) ∈ M × N, determine all GMC model solutions.

2. Given (μ, ν) ∈ M × N, determine all weak GMC model solutions.

3. Given (μ, ν) ∈ M × N with treshold time T and initial population φ_{T}, find all strong GMC model solutions.

3.2 Existence and Uniqueness Theorems

We want to prove the following generalization of the existence and “uniqueness after some time” of the population growth models.

Theorem 3.2.1 For any pair (μ, ν) ∈ M × N, the following possibilities occur:

1. If (B, P) ∈ C^{2}, with B|_{(−∞,T]} ∈ L_{1}((−∞, T]) is a solution satisfying the equations (3.1) and (3.2), then any other solution (B′, P′) which agrees with (B, P) on (−∞, T] also coincides with (B, P) on IR.

2. If (B, P) ∈ C(T)^{2} is a strong GMC model solution, then any other solution (B′, P′) ∈ C(T)^{2} with φ(B,P)_{T} = φ(B′,P′)_{T} coincides with (B, P).

3. Given φ_{T}, there exists a solution (B, P) ∈ C(T)^{2} with φ_{T} = φ(B,P)_{T}.

Let φ ∈ L_{1}^{+}(IR_{+}). We denote by C(φ) the set of population functions p ∈ C such that

p(0) = ∫_{0}^{∞} φ(a)da.

Given φ ∈ L_{1}^{+}(IR_{+}), p ∈ C(φ) and μ ∈ M, let N (φ, μ, p) be the set of birth moduli ν which satisfy

φ(a) = ∫_{a}^{∞} ν(−a, z−a, p(−a)) exp[∫_{−a}^{0}{μ(τ, z+τ, p(τ))−μ(τ, a+τ, p(τ))}dτ]φ(z)dz.

Theorem 3.2.2 For any positive function φ ∈ L_{1}(IR_{+}), μ ∈ M and any p ∈ C(φ), we have N (φ, μ, p) ≠ ∅. Moreover, if ν ∈ N (IR_{+}) there exists a solution (B, P) where P ∈ C(φ), satisfying the equations (3.1) and (3.2).

The proof of these theorems will be given in Section 3.6.

3.3 Birth Processes for a Given Population

For a fixed measurable function p : IR → IR_{+}, equation (2.10) reduces to

B̃(t) := x(t, 0) = φ(−t) exp{− ∫_{0}^{t} μ(τ, τ − t, p(τ))dτ},  (3.3)

when a = 0 and determines the population of age 0 at all times t ≤ 0. Equation (3.3) defines B̃(t), which may be taken as a new “initial condition”. The functions x, and thereby B̃, are non-negative by definition. It follows from our assumption on φ that B̃ is continuous on IR whenever φ is.

Combining equations (2.10) and (3.3), we can write

x(t, a) = B̃(t − a) exp{− ∫_{t−a}^{t} μ(τ, a + τ − t, p(τ))dτ}.  (3.4)

The birth process is described by the so-called birth law (boundary condition)

B(t) = ∫_{0}^{∞} ν(t, a, p(t))x(t, a)da, t ∈ IR  (3.5)

where B(t) is the number of newborn population at time t. Here ν(t, a, p(t)) x(t, a) gives the density of newborn descendants of parents of age a at time t.

Since we must have that B(t) = B̃(t) for t ≤ 0 in a solution of equation (3.1), we have

φ(−t) = \frac{∫_{0}^{∞} ν(t, a, p(t))x(t, a)da}{exp\{− ∫_{0}^{t} μ(τ, τ − t, p(τ))dτ\}}.  (3.6)

The following lemma shows that there is no nontrivial solution to the Lotka-Sharpe equation whenever the birth modulus ν is integrable.

Lemma 3.3.1 Given p : (−∞, 0] → IR_{+}, then for any pair (μ, ν) ∈ M × N with integrable ν ∈ L_{1}(IR × IR_{+}^{2}), there is no solution B : (−∞, 0] → IR_{+} of the Lotka-Sharpe renewal equation

B(t) = ∫_{0}^{∞} ν(t, a, p(t))B(t − a) exp{− ∫_{t−a}^{t} μ(τ, a + τ − t, p(τ))dτ}da (3.7)

other than B = 0.

Proof. Let

K(s, a) = 11_{[s,∞)}(a)ν(−s, a − s, p(−s)) exp{∫_{−s}^{−a} μ(τ, a + τ, p(τ))dτ} (3.8)

for s, a ≥ 0. Note that K is a uniformly bounded kernel on each set [0, M] × [0, ∞), and is integrable along the second coordinate. Formula (3.7) reads now

B(−s) = ∫_{0}^{∞} K(s, a)B(−a)da

and we therefore have to find a solution of

f(s) = ∫_{0}^{∞} K(s, a)f(a)da

where f : [0, ∞) → IR_{+}.

Choose T > 0 such that sup_{s≥T} ∫_{0}^{∞} K(s, a)da = q < 1. For f, g ∈ C(T), we have

∥ ∫_{0}^{∞} K(s, a)(f(a) − g(a))da∥ ≤ q∥f − g∥,

and hence K defines a contraction on C(T). By the contraction mapping principle, there exists a unique fixed point f_{0} ∈ C(T), that is, a function that satisfies

f_{0}(s) = ∫_{0}^{∞} K(s, a)f_{0}(a)da.

In order to extend this function, choose T_{1} < T with ∫_{T_{1}}^{T} K(s, a)da ≤ q_{1} for all T_{1} ≤ s ≤ T and some q_{1} < 1. Let C(T_{1}, f_{0}) denote the space of all bounded continuous functions on [T_{1}, ∞) which agree with f_{0} on [T, ∞). A function f ∈ C(T_{1}, f_{0}) satisfies equation (3.7) if and only if

f(s) = ∫_{T}^{∞} K(s, a)f_{0}(a)da + ∫_{s}^{T} K(s, a)f(a)da

holds for each T_{1} ≤ s < T. It follows as above that K defines a contraction on C(T_{1}, f_{0}), hence there exists a unique fixed point f_{1} ∈ C(T_{1}, f_{0}).

Continuing in this way, we can obtain a function f_{j} on the interval [T_{j}, ∞), which extends f_{j−1} on [T_{j−1}, ∞). Since μ and ν are bounded and integrable, the length of these intervals can be chosen to be bounded away from 0, whence, finally, the half line [0, ∞) is covered after finitely many steps and we hence get a unique f as the limit.

Since f ≡ 0 clearly satisfies the equation, we conclude that B = 0 is the only solution of equation (3.7). ■

The previous result shows that the trivial solution B = 0 prevails whenever ν is integrable. Therefore we will later on restrict our attention to the case where ν is not integrable. We shall require that the Lotka-Sharpe equation hold only almost surely with respect to the Lebesgue measure (explaining the introduction of the weak GMC model in Definition 3.1.3). We will come back to this concern in Section 3.7.

Corollary 3.3.1 Let T ∈ IR. If p and B_{T} on (−∞, T] are given so that they satisfy the model on (−∞, T], then there exists a unique B extending B_{T} .

Proof. Clearly, K (defined in (3.8)) is uniformly bounded kernel on [T, M]× [0, ∞) and decrease exponentially. It follows from Lemma 3.3.1 that there exists a unique fixed point in C(T). By using the same argument of the proof of Lemma 3.3.1, we can extend this function in C. ■

The corollary shows that for any p there exists B = ξp ∈ C such that equation (3.1) holds, reducing the proof of Theorem 3.2.1 to the solution of equation (3.2) with B = ξP.

3.4 Extension of the Model to the Past

We now turn our considerations to the preparation of Theorem 3.2.2. Let φ, μ and p be given so that p(0) = ∫_{0}^{∞} φ(a)da. According to Lemma 2.3.1, a solution for B satisfies equation (3.3)

B(t) = φ(−t) exp{∫_{t}^{0} μ(τ, τ − t, p(τ))dτ}

for t ≤ 0 and is uniquely determined by this equation. The birth modulus is not specified for t ≤ 0, but it must satisfy equation (3.1). Using (3.4), we obtain the Lotka-Sharpe renewal equation (3.1). Inserting (2.10) gives

φ(−t) exp{∫_{t}^{0} μ(τ, τ − t, p(τ))dτ}
= ∫_{−t}^{∞} ν(t, u + t, p(t))φ(u) exp{∫_{t}^{0} μ(τ, u + τ, p(τ))dτ}du. (3.9)

By definition, every ν ∈ N (φ, μ, p) satisfies this equation. It is immediate to see that such a ν exists for every absolutely continuous initial age distribution by setting

ν(t, u, p(t)) = \frac{−φ′}{φ} exp[∫_{t}^{0}\{μ(τ, τ − t, p(τ)) − μ(τ, u + τ, p(τ))\}dτ] (3.10)

for t ≤ 0 and u, p ≥ 0. Theorem 3.2.2 thereby follows if equation (3.1) holds for ν (as above) and B = ξp (as defined in Corollary 3.3.1). Hence Theorem 3.2.2 is a corollary to Theorem 3.2.1.

Remark 3.4.1 The solution of the model may not be always continuous in the past. However, if the system admits a positive solution up to time T, then it can be extended continuously on [T, ∞).

Our next result shows that solutions B and P of equations (3.1) and (3.2) can grow at most like e^{ν̄ (t−T)} for some T ∈ IR.

Lemma 3.4.1 Assume that (μ, ν) ∈ M × N. Let x ∈ C be a solution of (2.7) and (2.8) and (B, P) ∈ C^{2}, with B|_{(−∞,T]} ∈ L_{1}((−∞, T]) a solution of equations (3.1) and (3.2). Then for each t ≥ T, we have

B(t) ≤ ν̄f(T) e^{ν̄ (t−T)}, P (t) ≤ f(T)e^{ν̄ (t−T)},

and for t ≥ a,

x(t, a) ≤ f(T)e^{ν̄ (t−a−T)},

where f(T) = ∫_{−∞}^{T} B(s) ds (initial birth rate) and ν̄ = sup_{t,u,p} ν(t, u, p(t)).

Proof. Let Λ : IR^{2} × C → IR_{+} be defined for s ≤ t by

Λ(s, t, p) = exp\{− ∫_{s}^{t} μ(τ, τ − s, p(τ)) dτ\} ≤ exp\{−\underline{μ}(t − s)\} ≤ 1, (3.11)

where \underline{μ} = inf_{t,u,p} μ(t, u, p(t)) > 0 (by Definition 3.1.1).

We get from equation (3.1) for B ∈ C that

B(t) = ∫_{−∞}^{T} ν(t, t − s, p(t))Λ(s, t, p)B(s)ds + ∫_{T}^{t} ν(t, t − s, p(t))Λ(s, t, p)B(s)ds
≤ B(T, t) + ∫_{T}^{t} ν̄ B(s) ds,

where

B(T, t) = ∫_{−∞}^{T} ν(t, t − s, p(t))Λ(s, t, p)B(s) ds
≤ ∫_{−∞}^{T} ν(t, t − s, p(t))B(s) ds ≤ ν̄f(T)

is the inhomogeneous term of the integral equation for the birth rate.

Using Lemma 2.1.1 with f(t) = B(T, t), u(s) = B(s), h(s) = ν̄ and g(t) = 1, we obtain

B(t) ≤ B(T, t) + ∫_{T}^{t} ν̄ B(T, s)e^{ν̄ (t−s)}ds
≤ ν̄f(T)e^{ν̄ (t−T)}.

Substituting this result into (3.2), we get

P (t) ≤ f(T)e^{ν̄ (t−T)}.

Finally, for t ≥ a, we obtain from (3.4) that

x(t, a) ≤ B(t − a)
≤ f(T)e^{ν̄ (t−a−T)}

whenever t − a ≥ T. ■

Remark 3.4.2 To study the existence of solutions of equations (3.1) and (3.2), we first solve the linearized version of the problem, that is, we find functions that satisfy equation (3.7) for a fixed p ∈ C(T). In this case, the model is essentially equivalent to a linear Volterra integral equation for B (so-called renewal equation), and there exists a unique continuous solution B satisfying (3.7) for this choice of p. Using this solution, the right-hand side of (3.2) defines an operator A on C(T) which has a fixed point P. This will be established in the next section.

3.5 Strong GMC Model

In this section, we examine the existence of strong Gurtin-MacCamy model solutions (see Definition 3.1.3). We show, using a Schauder-Tychonov fixed point argument, that under certain regularity conditions on φ_{T} , μ and ν, the model has a solution (B, P) ∈ C(T)^{2} for some T ∈ IR. In [14], we investigated the model for t ≥ 0.

It follows from Lemma 2.3.1 that for any fixed measurable function

p : [T, ∞) → IR_{+} with initial condition x(T, a) = φ_{T} (a), a ≥ 0, equation (2.9) can be written by taking (t_{0} − T, a_{0}) = (T, a + T − t) and h = t − T with the aid of distribution function φ_{T} , as

x(t, a) = φ_{T} (a + T − t) exp{− ∫_{T}^{t} μ(τ, a + τ − t, p(τ))dτ} for a ≥ t − T.

On the other hand, if we substitute (t_{0} − T, a_{0}) = (t − a − T, 0) and h = a, then

x(t−T, a) = B(t−a−T) exp{− ∫_{t−a−T}^{t−T} μ(τ, a+τ −t+T, p(τ))dτ} for t−T > a

when

B(t) = x(t, 0).

We obtain from (3.5)

B(t) = ∫_{0}^{t−T} ν(t, a, p(t)) exp{− ∫_{t−a}^{t} μ(τ, a + τ − t, p(τ))dτ}B(t − a)da
+ ∫_{t−T}^{∞} ν(t, a, p(t)) exp{− ∫_{T}^{t} μ(τ, a + τ − t, p(τ))dτ}φ_{T} (a + T − t)da
= ∫_{T}^{t} ν(t, t − s, p(t))Λ(s, t, p)B(s)ds
+ ∫_{0}^{∞} ν(t, t + s′ − T, p(t))Γ_{T} (s′, t, p)φ_{T} (s′)ds′  (3.12)

and

P (t) = ∫_{0}^{t−T} exp{− ∫_{t−a}^{t} μ(τ, a + τ − t, p(τ))dτ}B(t − a)da
+ ∫_{t−T}^{∞} exp{− ∫_{T}^{t} μ(τ, a + τ − t, p(τ))dτ}φ_{T} (a + T − t)da
= ∫_{T}^{t} Λ(s, t, p)B(s)ds + ∫_{0}^{∞} Γ_{T} (s′, t, p)φ_{T} (s′)ds′,  (3.13)

where Λ : IR^{2} × C(T) → IR_{+} denotes

Λ(s, t, p) = exp{− ∫_{s}^{t} μ(τ, τ − s, p(τ))dτ}

and Γ_{T} : IR_{+} × IR × C(T) → IR_{+} denotes

Γ_{T} (s′, t, p) = exp{− ∫_{T}^{t} μ(τ, τ + s′ − T, p(τ))dτ}.

Note that for s ≤ t and p ≥ 0, we have

Λ(s, t, p) = exp{− ∫_{s}^{t} μ(τ, τ − s, p(τ))dτ}
≤ exp\{−\underline{μ}(t − s)\} ≤ 1  (3.14)

and, for T ≤ t and p ≥ 0,

Γ_{T} (s′, t, p) = exp{− ∫_{T}^{t} μ(τ, τ + s′ − T, p(τ))dτ}
≤ exp\{−\underline{μ}(t − T)\} ≤ 1.  (3.15)

We now state our existence theorem.

Theorem 3.5.1 Under the assumption (μ, ν) ∈ M × N , equations (3.12) and (3.13) have a solution (B, P) ∈ C(T)^{2}.

Proof. For fixed p ∈ C(T), according to Lovitt [5, Theorem II.], equation (3.12) is a linear Volterra integral equation for B having a unique continuous non-negative solution which we denote by B(t) = ξp(t).

We obtain from equation (3.12) that

ξp(t) ≤ ν̄ ϕ + ν̄ ∫_{T}^{t} ξp(s) ds

where ϕ := ∫_{0}^{∞} φ_{T} (s′)ds′.

Using Lemma 2.1.1, we conclude that

ξp(t) ≤ ν̄ϕ exp{ν̄(t − T)}.  (3.16)

Now, we define the operator A : C(T) → C(T) by

(Ap)(t) = ∫_{0}^{∞} Γ(s′, t, p)φ_{T} (s′)ds′ + ∫_{T}^{t} Λ(s, t, p) (ξp)(s)ds  (3.17)

where C(T) is endowed with the supremum norm ||.||.

Let k(M) = {f : [T, M] → IR_{+}, f is continuous, f(T) = ∫ φ_{T} (a)da}. Clearly, k(M) is a closed convex subset of C([T, M]).

We can show that, (B, P) ∈ C(T)^{2} is a solution of equations (3.12) and (3.13) if and only if the restrictions of B = ξ P and P on [T, M] is a fixed point of the operator A on k(M).

We obtain for h ∈ IR,

|(Ap)(t + h) − (Ap)(t)|

≤ ∫_{0}^{∞} |Γ(s′, t + h, p) − Γ(s′, t, p)|φ_{T} (s′)ds′ + ∫_{t}^{t+h} Λ(s, t + h, p) |ξp(s)|ds
+ ∫_{T}^{t} |Λ(s, t + h, p) − Λ(s, t, p)| |ξp(s)|ds
= I + II + III.  (3.18)

For s ≤ t and p ≥ 0, Λ satisfies the following inequality: |Λ(s, t + h, p) − Λ(s, t, p)| = | exp{− ∫_{s}^{t+h} μ(τ, τ − s, p(τ))dτ} − exp{− ∫_{s}^{t} μ(τ, τ − s, p(τ))dτ}| = exp{− ∫_{s}^{t} μ(τ, τ − s, p(τ))dτ} | exp{− ∫_{t}^{t+h} μ(τ, τ − s, p(τ))dτ} − 1|

≤ ∫_{t}^{t+|h|} μ(τ, τ − s, p(τ))dτ exp{∫_{t}^{t+|h|} μ(τ, τ − s, p(τ)) dτ}
≤ μ̄|h| exp{μ̄|h|},  (3.19)

and, when s′ ≥ 0 and p ≥ 0, Γ satisfies the following inequality: |Γ_{T} (s′, t + h, p) − Γ_{T} (s′, t, p)| = | exp{− ∫_{T}^{t+h} μ(τ, τ + s′ − T, p(τ))dτ} − exp{− ∫_{T}^{t} μ(τ, τ + s′ − T, p(τ)) dτ}| = exp{− ∫_{T}^{t} μ(τ, τ + s′ − T, p(τ)) dτ} | exp{− ∫_{t}^{t+h} μ(τ, τ + s′ − T, p(τ)) dτ} − 1| ≤ ∫_{t}^{t+|h|} μ(τ, τ + s′ − T, p(τ)) dτ exp{∫_{t}^{t+|h|} μ(τ, τ + s′ − T, p(τ)) dτ}

≤ μ̄|h| exp{μ̄|h|},  (3.20)

where

μ̄ = sup_{t,u,p} μ(t, u, p(t)).

Here we have used the fact that

| exp z − 1| ≤ |z| exp |z| for all z ∈ IR.

By (3.20), we obtain, for p ∈ k(M) and T ≤ t ≤ M,

I ≤ ∫_{0}^{∞} |Γ_{T} (s′, t + h, p) − Γ_{T} (s′, t, p)| φ_{T} (s′) ds′
≤ μ̄|h| exp{μ̄|h|}ϕ.  (3.21)

From (3.14) and (3.16), we get, for p ∈ k(M) and T ≤ t ≤ M,

II ≤ ∫_{t}^{t+h} Λ(s, t + h, p) |ξp(s)| ds
≤ exp\{−\underline{μ} h\}ϕ( exp\{ν̄(t + h − T)\} − exp\{ν̄(t − T)\}). (3.22)

Also, from (3.16) and (3.19), we obtain, for p ∈ k(M) and T ≤ t ≤ M,

III ≤ ∫_{T}^{t} |Λ(s, t + h, p) − Λ(s, t, p)| |ξp(s))|ds
≤ μ̄|h| exp{μ̄|h|}ϕ exp{ν̄ (t − T)},  (3.23)

where we have used the fact that

exp x − 1 ≤ exp x for all x ≥ 0.

Thus, from (3.18), (3.21), (3.22) and (3.23), we conclude that

|(Ap)(t + h) − (Ap)(t)| ≤ C_{M}h

where C_{M} is a constant only depending on M.

Hence A(k(M)) is equicontinuous for M ≥ T. Furthermore, Ap ≥ 0, as all functions involved in equations (3.12) and

(3.13) are nonnegative.

Clearly, Ap(t) is uniformly bounded for all T ≤ t ≤ M and p ∈ k(M). From above discussion, A(k(M)) is uniformly bounded and equicontin-

uous. Hence, by Arzelà-Ascoli theorem, A(k(M)) is a relatively compact (pre-compact) subset of C([T, M]).

It remains to show that A is continuous on k(M). Consider p, q ∈ k(M) and T ≤ t ≤ M, we obtain that

|(Ap)(t) − (Aq)(t)|

≤ ∫_{0}^{∞} |Γ_{T} (s′, t, p) − Γ_{T} (s′, t, q)|φ_{T} (s′)ds′

+∫_{T}^{t} |Λ(s, t, p) − Λ(s, t, q) | |ξp(s)|ds + ∫_{T}^{t} Λ(s, t, q) |ξp(s) − ξq(s)|ds

= IV + V + V I.  (3.24)

For p, q ∈ k(M), Γ satisfies the following inequality:

|Γ_{T} (s′, t, p) − Γ_{T} (s′, t, q)| = | exp{− ∫_{T}^{t} μ(τ, τ + s′ − T, p(τ)) dτ} − exp{− ∫_{T}^{t} μ(τ, τ + s′ − T, q(τ)) dτ}| ≤ ∫_{T}^{t} |μ(τ, τ + s′ − T, p(τ)) − μ(τ, τ + s′ − T, q(τ))|dτ,

where we have used the fact that |e^{x} − e^{y}| ≤ |x − y| for all x, y ≤ 0. Also, for p, q ∈ k(M) and T ≤ t ≤ M, we have

|Γ_{T} (s′, t, p) − Γ_{T} (s′, t, q)| ≤ ∫_{T}^{t} μ_{1}(τ, τ + s′ − T)|p(τ) − q(τ)| dτ
≤ μ̄_{1} ||p − q|| (t − T),  (3.25)

by setting μ_{1}(u, v) = sup_{y≠z} \frac{|μ(u,v,y)−μ(u,v,z)|}{|y−z|} and μ̄_{1} = sup μ_{1}(u, v).

Hence

IV ≤ ∫_{0}^{∞} |Γ_{T} (s′, t, p) − Γ_{T} (s′, t, q)| φ_{T} (s′)ds′
≤ μ̄_{1} ||p − q||(t − T) ϕ.  (3.26)

For s ≤ t and p, q ∈ k(M), Λ satisfies the following inequality: |Λ(s, t, p) − Λ(s, t, q)| = | exp{− ∫_{s}^{t} μ(τ, τ − s, p(τ)) dτ} − exp{− ∫_{s}^{t} μ(τ, τ − s, q(τ)) dτ}| ≤ ∫_{s}^{t} |μ(τ, τ − s, p(τ)) − μ(τ, τ − s, q(τ))|dτ

≤ μ̄_{1} ||p − q||(t − s).  (3.27)

From (3.16) and (3.27), we obtain

V ≤ ∫_{T}^{t} |Λ(s, t, p) − Λ(s, t, q)| |ξp(s)| ds
≤ ϕ μ̄_{1} ||p − q||(t − T) exp{ν̄ (t − T)}.  (3.28)

We further deduce from (3.12), for p, q ∈ k(M) and T ≤ t ≤ M, that |ξp(t) − ξq(t)| ≤ ∫_{0}^{∞} |ν(t, t + s′ − T, p(t))Γ_{T} (s′, t, p) − ν(t, t + s′ − T, q(t))Γ_{T} (s′, t, q)|φ_{T} (s′)ds′ + ∫_{T}^{t} |ν(t, t − s, p(t))Λ(s, t, p)ξp(s) − ν(t, t − s, q(t))Λ(s, t, q)ξq(s)|ds

= V II + V III.  (3.29)

Setting ν_{1}(u, v) = sup_{y≠z} \frac{|ν(u,v,y)−ν(u,v,z)|}{|y−z|} and ν̄_{1} = sup ν_{1}(u, v), we get

V II ≤ ∫_{0}^{∞} |ν(t, t + s′ − T, p(t)) − ν(t, t + s′ − T, q(t))| Γ(T, s′, t, p) φ_{T} (s′) ds′
+ ∫_{0}^{∞} ν(t, t + s′ − T, q(t)) |Γ(T, s′, t, p) − Γ(T, s′, t, q)| φ_{T} (s′) ds′
≤ ν̄_{1}||p − q||ϕ + ν̄μ̄_{1}||p − q||(t − T)ϕ  (3.30)

and

V III ≤ ∫_{T}^{t} |ν(t, t − s, p(t)) − ν(t, t − s, q(t))| Λ(s, t, p) |ξp(s)|ds
+ ∫_{T}^{t} ν(t, t − s, q(t)) |Λ(s, t, p) − Λ(s, t, q)| |ξp(s)|ds
+ ∫_{T}^{t} ν(t, t − s, q(t))Λ(s, t, q)|ξp(s) − ξq(s)|ds
≤ ϕ ν̄_{1}||p − q|| exp{ν̄ (t − T)} + ν̄ ϕ μ̄_{1} (t − T)||p − q|| exp{ν̄(t − T)}
+ ∫_{T}^{t} ν̄|ξp(s) − ξq(s)|ds.  (3.31)

Thus, from (3.29), (3.30) and (3.31), we obtain

|ξp(t) − ξq(t)|

≤ ν̄_{1}||p − q||ϕ + ν̄μ̄_{1}||p − q||(t − T)ϕ + ϕ ν̄_{1}||p − q|| exp{ν̄ (t − T)}

+ν̄ ϕ μ̄_{1}||p − q||(t − T) exp{ν̄ (t − T)} + ∫_{T}^{t} ν̄|ξp(s) − ξq(s)| ds
= α(t)||p − q|| + ν̄ ∫_{T}^{t} |ξp(s) − ξq(s)| ds,  (3.32)

where

α(t) = (ν̄_{1} + ν̄μ̄_{1}(t − T))ϕ + (ν̄_{1} + ν̄μ̄_{1}(t − T))ϕ exp{ν̄ (t − T)}.

It follows from Lemma 2.1.1 that

|ξp(t) − ξq(t)| ≤ α(t)||p − q|| + ν̄ ∫_{T}^{t} α(s) exp{ν̄(t − s)}||p − q||ds
= (α(t) + β(t))||p − q||,  (3.33)

where

β(t) = ν̄ exp{ν̄t} ∫_{T}^{t} α(s) exp{−ν̄s}ds.

Now, from (3.24), (3.26), (3.28) and (3.33), we conclude that

|Ap(t) − Aq(t)|
≤ μ̄_{1}||p − q||(t − T) ϕ + ϕ μ̄_{1}(t − T) exp{ν̄ (t − T)}||p − q||
+ ∫_{T}^{t}(α(s) + β(s))||p − q||ds
= [γ(t) + ψ(t)]||p − q||,

where

γ(t) = μ̄_{1} (t − T) ϕ + ϕ μ̄_{1}(t − T) exp{ν̄ (t − T)},

and

ψ(t) = ∫_{T}^{t}(α(s) + β(s))ds.

Consider t ∈ [T, M], then γ(t) + ψ(t) can be bounded by a positive constant K.

So we have, for p, q ∈ C([T, M])

||Ap − Aq|| ≤ K ||p − q||.

Hence, for all ϵ > 0 there exists 0 < δ ≤ \frac{ϵ}{K} such that

||Ap − Aq|| ≤ ϵ whenever ||p − q|| ≤ δ.

It follows that A is a continuous operator on k(M) ⊂ C([T, M]). By Schauder-Tychonov fixed point theorem, A has a fixed point P in k(M). This fixed point is, of course, a solution of equation (3.17) in C(T) when M → ∞, because it is unique by the argument in Lemma 2.1.2 or Corollary 3.3.1. The corresponding B = ξ P and this P then respect equations (3.12) and (3.13) for t ≥ T. ■

Observe also that it follows from (3.13) that P (T) = ∫ φ_{T} (a)da > 0, and from equation (3.12) that B(T) > 0. Hence P (t), B(t) > 0 for all t ≥ T.

Remark 3.5.1 Gurtin-MacCamy have studied this model when μ and ν depend on age and on the total population, such that equations (3.12) and (3.13) have a continuous solution (B, P) for all time t ≥ 0, but they did not consider the case where ν and μ are further directly time-dependent.

3.6 Proof of the Existence and Uniqueness Theorems

In this section, we prove Theorems 3.2.1 and 3.2.2. We investigate the Gurtin-MacCamy model for existence and unique solution for all time t ∈ IR. In [13], we established the model for t ≥ 0 by using contraction mapping principle.

Proof of Theorem 3.2.1.

1. Let \underline{μ} = inf_{t,a,P} μ(t, a, P) > 0 (by assumption that μ ∈ M). We obtain from equation (3.2) that

P (t) = ∫_{−∞}^{T} exp{− ∫_{s}^{T} μ(τ, τ − s, P (τ))dτ} exp{− ∫_{T}^{t} μ(τ, τ − s, P (τ))dτ}B(s)ds
+ ∫_{T}^{t} exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ} B(s) ds.  (3.34)

Let Λ : IR^{2} × C → IR_{+} be defined for s ≤ t and P ≥ 0 by

Λ(s, t, P) = exp\{− ∫_{s}^{t} μ(τ, τ − s, P (τ)) dτ\} ≤ exp\{−\underline{μ}(t − s)\} ≤ 1. Hence for s ≤ T and P ≥ 0

Λ(s, T, P) = exp\{− ∫_{s}^{T} μ(τ, τ − s, P (τ)) dτ\} ≤ exp\{−\underline{μ}(T − s)\} ≤ 1.

Define Λ_{T} : IR^{2} × C → IR_{+} for T ≤ t and P ≥ 0 by

Λ_{T} (s, t, P) = exp{− ∫_{T}^{t} μ(τ, τ − s, P (τ)) dτ}.

For s ≤ t and P ≥ 0, P′ ≥ 0, Λ satisfies the following inequality:

|Λ(s, t, P) − Λ(s, t, P′)| = | exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ} − exp{− ∫_{s}^{t} μ(τ, τ − s, P′(τ))dτ}|

≤ ∫_{s}^{t} |μ(τ, τ − s, P (τ)) − μ(τ, τ − s, P′(τ))|dτ  (3.35)

where we have used the fact that |e^{x} − e^{y}| ≤ |x − y| for all x, y ≤ 0. Also, for P, P′ ∈ C and s ≤ t, we have

|Λ(s, t, P) − Λ(s, t, P′)| ≤ ∫_{s}^{t} μ_{1}(τ, τ − s)|P (τ) − P′(τ)| dτ (3.36)

by setting μ_{1}(u, v) = sup_{y≠z} \frac{|μ(u,v,y)−μ(u,v,z)|}{|y−z|}. We also have,

|ν(t, t − s, P (t)) − ν(t, t − s, P′(t))| ≤ ν_{1}(t, t − s) |P (t) − P′(t)| (3.37)

where ν_{1}(u, v) = sup_{y≠z} \frac{|ν(u,v,y)−ν(u,v,z)|}{|y−z|}. By (3.34), we obtain for P, P′ ∈ C

|P (t) − P′(t)| ≤ ∫_{−∞}^{T} Λ(s, T, P)|Λ_{T} (s, t, P) − Λ_{T} (s, t, P′)|B(s) ds
+ ∫_{T}^{t} |Λ(s, t, P)B(s) − Λ(s, t, P′)B′(s)| ds
= I + II  (3.38)

where

I ≤ ∫_{−∞}^{T} Λ(s, T, P) · ∫_{T}^{t} |μ(τ, τ − s, P (τ)) − μ(τ, τ − s, P′(τ))|dτ B(s) ds
≤ ∫_{−∞}^{T} ∫_{T}^{t} μ_{1}(τ, τ − s)|P (τ) − P′(τ)|dτ B(s) ds

and

II ≤ ∫_{T}^{t} |Λ(s, t, P) − Λ(s, t, P′)| B(s) ds + ∫_{T}^{t} Λ(s, t, P′)|B(s) − B′(s)|ds
≤ ∫_{T}^{t} ∫_{s}^{t} μ_{1}(τ, τ − s)|P (τ) − P′(τ)|dτ B(s) ds + ∫_{T}^{t} |B(s) − B′(s)|ds.

We get from (3.38) that

|P (t) − P′(t)| ≤ ∫_{T}^{t} ∫_{−∞}^{T} μ_{1}(τ, τ − s)B(s) ds|P (τ) − P′(τ)|dτ
+ ∫_{T}^{t} ∫_{T}^{τ} μ_{1}(τ, τ − s)B(s) ds|P (τ) − P′(τ)|dτ
+ ∫_{T}^{t} |B(s) − B′(s)|ds.  (3.39)

From (3.1), we can write for P, P′ ∈ C |B(t) − B′(t)|

≤ ∫_{−∞}^{T} ν(t, t − s, P (t))Λ(s, T, P)|Λ_{T} (s, t, P) − Λ_{T} (s, t, P′)|B(s)ds

+ ∫_{T}^{t} |ν(t, t − s, P (t))Λ(s, t, P)B(s) − ν(t, t − s, P′(t))Λ(s, t, P′)B′(s)|ds

= III + IV  (3.40)

where

III ≤ ∫_{−∞}^{T} ν(t, t − s, P (t)) ∫_{T}^{t} |μ(τ, τ − s, P (τ)) − μ(τ, τ − s, P′(τ))|dτ B(s) ds
≤ ∫_{−∞}^{T} ∫_{T}^{t} ν̄μ_{1}(τ, τ − s)|P (τ) − P′(τ)|dτ B(s)ds

and

IV ≤ ∫_{T}^{t} ν(t, t − s, P′(t))|Λ(s, t, P) − Λ(s, t, P′)|B(s)ds
+ ∫_{T}^{t} |ν(t, t − s, P (t)) − ν(t, t − s, P′(t))|Λ(s, t, P)B(s)ds
+ ∫_{T}^{t} ν(t, t − s, P′(t))Λ(t, s, P′)|B(s) − B′(s)|ds
≤ ∫_{T}^{t} ∫_{s}^{t} ν̄μ_{1}(τ, τ − s)|P (τ) − P′(τ)|dτB(s)ds
+ ∫_{T}^{t} ν_{1}(t, t − s)|P (t) − P′(t)|B(s)ds + ∫_{T}^{t} ν̄|B(s) − B′(s)|ds.

We obtain from (3.40) that

|B(t) − B′(t)| ≤ ∫_{T}^{t} ∫_{−∞}^{T} ν̄μ_{1}(τ, τ − s)B(s)ds|P (τ) − P′(τ)|dτ
+ ∫_{T}^{t} ∫_{T}^{τ} ν̄μ_{1}(τ, τ − s)B(s)ds|P (τ) − P′(τ)|dτ
+ ∫_{T}^{t} ν_{1}(t, t − s)B(s)ds|P (t) − P′(t)| + ∫_{T}^{t} ν̄|B(s) − B′(s)|ds
= f(t, T) + ∫_{T}^{t} ν̄|B(s) − B′(s)|ds,  (3.41)

where

f(t, T) = ∫_{T}^{t} [ ∫_{−∞}^{T} ν̄μ_{1}(τ, τ − s)B(s)ds + ∫_{T}^{τ} ν̄μ_{1}(τ, τ − s)B(s)ds]
·|P (τ) − P′(τ)|dτ + ∫_{T}^{t} ν_{1}(t, t − s)B(s)ds|P (t) − P′(t)|,

whenever B|_{(−∞,T]} ∈ L_{1}((−∞, T]). Clearly,

f(t, T) = h(t, T)|P (t) − P′(t)|,  (3.42)

where

h(t, T) = ∫_{T}^{t} [ ∫_{−∞}^{T} ν̄μ_{1}(τ, τ − s)B(s)ds + ∫_{T}^{τ} ν̄μ_{1}(τ, τ − s)B(s)ds]dτ
+ ∫_{T}^{t} ν_{1}(t, t − s)B(s) ds.

Using Lemma 2.1.1 on (3.41), we get

|B(t) − B′(t)| ≤ f(t, T) + ∫_{T}^{t} ν̄f(τ, T)e^{ν̄(t−τ)}dτ.  (3.43)

Substitute |B(t) − B′(t)| on (3.39), we obtain that

|P (t) − P′(t)| ≤ ∫_{T}^{t} [[∫_{−∞}^{T} μ_{1}(τ, τ − s)B(s) ds + ∫_{T}^{τ} μ_{1}(τ, τ − s)B(s) ds]
·|P (τ) − P′(τ)| + f(τ, T) + ∫_{T}^{τ} ν̄f(u, T)e^{ν̄(τ−u)}du]dτ
= ∫_{T}^{t} g(τ, T)|P (τ) − P′(τ)|dτ,  (3.44)

where

g(τ, T) = ∫_{−∞}^{T} μ_{1}(τ, τ − s)B(s)ds + ∫_{T}^{τ} μ_{1}(τ, τ − s)B(s)ds
+h(τ, T) + ∫_{T}^{τ} ν̄h(u, T)e^{ν̄(τ−u)}du.

By using Lemma 2.1.1 on (3.44), we deduce that

|P (t) − P′(t)| = 0, i.e. P (t) = P′(t) on [T, ∞).  (3.45)

From (3.42), (3.43) and (3.45), we conclude that

B(t) = B′(t) on [T, ∞).
■

Remark 3.6.1 Equation (3.42) can easily verify with the hypotheses of μ, ν and B that

||f|| ≤ κ||P − P′|| with κ ∈ (0, 1)

on a closed subinterval of t ≥ T. It then follows from equation (3.44) that

||P − P′|| ≤ κ_{0}||P − P′|| with κ_{0} ∈ (0, 1)

which implies that

(1 − κ_{0})||P − P′|| ≤ 0

on a closed subinterval of t ≥ T. Since κ_{0} < 1, this implies that P (t) = P′(t) on t ≥ T. Hence P is the unique fixed point on t ≥ T.

2. We can write from equation (3.2) that

P (t) = ∫_{−∞}^{T} exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ}B(s)ds
+ ∫_{T}^{t} exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ}B(s)ds
= φ(B,P)_{T}(t) + ∫_{T}^{t} exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ}B(s)ds (3.46)

where

φ(B,P)_{T}(t) = ∫_{−∞}^{T} exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ}B(s)ds.

By hypothesis φ(B,P)_{T} = φ(B′,P′)_{T}, we get from (3.39) that

|P (t) − P′(t)| ≤ ∫_{T}^{t} ∫_{T}^{τ} μ_{1}(τ, τ − s)B(s) ds|P (τ) − P′(τ)|dτ
+ ∫_{T}^{t} |B(s) − B′(s)| ds  (3.47)

and from (3.41), we obtain

|B(t) − B′(t)| ≤ ∫_{T}^{t} ∫_{T}^{τ} ν̄μ_{1}(τ, τ − s)B(s)ds|P (τ) − P′(τ)|dτ
+ ∫_{T}^{t} ν_{1}(t, t − s)B(s)ds|P (t) − P′(t)| + ∫_{T}^{t} ν̄|B(s) − B′(s)|ds
= ψ(t, T)|P (t) − P′(t)| + ∫_{T}^{t} ν̄|B(s) − B′(s)|ds,  (3.48)

where

ψ(t, T) = ∫_{T}^{t} ∫_{T}^{τ} ν̄μ_{1}(τ, τ − s)B(s)dsdτ + ∫_{T}^{t} ν_{1}(t, t − s)B(s)ds.

By using same techniques as 1. (using Lemma 2.1.1 on (3.48) and substitute |B(t) − B′(t)| on (3.47) and again using Lemma 2.1.1 on (3.47)), we can show that

B(t) = B′(t) and P (t) = P′(t) on [T, ∞).
■

3. From (3.13), define

φ_{T} (t) = ∫_{0}^{∞} exp{−∫_{T}^{t} μ(τ, τ + s′ − T, P (τ))dτ}φ_{T} (s′)ds′.

By hypothesis φ_{T} = φ(B,P)_{T}, we obtain from (3.2), (3.13) and (3.46) that

P (t) = ∫_{0}^{∞} exp{− ∫_{T}^{t} μ(τ, τ + s′ − T, P (τ))dτ}φ_{T} (s′) ds′
+ ∫_{T}^{t} exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ} B(s) ds  (3.49)

and similarly from (3.1) and (3.12), we get

B(t) = ∫_{0}^{∞} ν(t, t + s′ − T, P (t)) exp{− ∫_{T}^{t} μ(τ, τ + s′ − T, P (τ))dτ}φ_{T} (s′) ds′
+ ∫_{T}^{t} ν(t, t − s, P (t)) exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ} B(s) ds. (3.50)

Since φ_{T} = φ(B,P)_{T}, the equations (3.1) and (3.2) are equivalent to the equations (3.12) and (3.13). It follows from Theorem 3.5.1, that (B, P) ∈ C(T)^{2} have a solution of equations (3.49) and (3.50) (by using Schauder-Tychonov fixed point theorem). ■

Proof of Theorem 3.2.2.

As we stated in Section 3.4, when φ ∈ L_{1}^{+}(IR_{+}), μ ∈ M and p ∈ C(φ) there exists a unique solution B satisfying

B(t) = φ(−t) exp{∫_{t}^{0} μ(τ, τ − t, p(τ))dτ},  (3.51)

is determined the population at time t ≤ 0 of age 0. The birth modulus is not specified for t ≤ 0. Using (2.10) and (3.6), we obtain

φ(−t) = ∫_{−t}^{∞} ν(t, s + t, p(t))φ(s)
× exp[∫_{t}^{0}{μ(τ, τ + s, p(τ)) − μ(τ, τ − t, p(τ))}dτ]ds. (3.52)

Setting

ν(t, s + t, p(t)) = \frac{−φ′}{φ} exp[∫_{t}^{0}\{μ(τ, τ − t, p(τ)) − μ(τ, τ + s, p(τ))\}dτ] (3.53)

where t ≤ 0 and s + t, p ≥ 0.

Clearly, every ν defined on (3.53) satisfies the equation (3.52), hence also the equation (3.1). It follows that ν ∈ N (φ, μ, p).

It is immediate to see that the initial age distribution φ is absolutely continuous with respect to Lebesgue measure for this ν.

Second part of Theorem 3.2.2. Since

P ∈ C(φ) = {P ∈ C : P (0) = ∫_{0}^{∞} φ(a)da}.

We can write equations (3.1) and (3.2) by

P (t) = P (0) + ∫_{0}^{t} exp{− ∫_{s}^{t} μ(τ, τ − s, P (τ))dτ}B(s)ds  (3.54)

and

B(t) = B(0)+∫_{0}^{t} ν(t, t−s, P (t)) exp{− ∫_{s}^{t} μ(τ, τ −s, P (τ))dτ}B(s)ds (3.55)

where

P (0) = ∫_{0}^{∞} φ(a)da

and

B(0) = ∫_{0}^{∞} ν(0, a, P (0))φ(a)da.

It follows from Theorem 3.5.1, if we consider T = 0, there exists a continuous solution of equations (3.54) and (3.55) on IR_{+}. We can also show that there exists a unique continuous solution of equations (3.54) and (3.55) on IR_{+} by using contraction mapping principle. ■

3.7 Weak GMC Model

In this section, we consider equation (3.1) when it holds almost surely with respect to Lebesgue measure (weak GMC model in Definition 3.1.3). We establish that if ν is uniformly bounded there exists absolutely continuous solution of equation (3.1).

We consider the following problem for fixed P:

Theorem 3.7.1 Let (μ, ν) ∈ M × N. There exists λ > 0 and a solution for

B(t) = ∫_{−∞}^{t} λν(t, t − u, P (t)) exp{− ∫_{u}^{t} μ(τ, τ − u, P (τ))dτ}B(u)du a.s.

(3.56)

Proof. Define

T f(t) = ∫_{t}^{1} ν(−\frac{1}{u}, −\frac{1}{u}+\frac{1}{t}, P (−\frac{1}{u})) exp\{− ∫_{− \frac{1}{t}}^{− \frac{1}{u}} μ(τ, τ+\frac{1}{t}, P (τ))dτ\}\frac{1}{u^{2}} f(u)du

for a continuous function f ∈ C([0, 1]). Since

|T f(t)| ≤ ∫_{t}^{1} \overline{ν} exp\{(\frac{1}{u} − \frac{1}{t})\underline{μ}\}\frac{1}{u^{2}} ∥f∥du
≤ \frac{\overline{ν}}{\underline{μ}}[1 − exp(1 − \frac{1}{t})\underline{μ}]∥f∥.

T maps bounded continuous functions to functions of this type.

Fix 0 < T < 1. Then the restriction of T 1 to [0, T] satisfies

inf_{0≤t≤T} T 1(t) ≥ D

for some D > 0, hence

∫ T 1dρ ≥ D

for every probability measure ρ on [0, T]. The probability measure on [0, T], endowed with a metric for weak convergence is convex and weakly compact (see Walters [15, Theorem 6.5]). The map

L(ρ) = \frac{T^{∗}ρ}{∫ T 1dρ}

is a continuous map on the space of all probability measures on [0, T] into itself, where

∫_{0}^{T} f(t)dT^{∗}ρ(t) = ∫_{0}^{T} T f(t)ρ(dt)

for f ∈ C([0, T]). By Schauder-Tychonov theorem, there exists a fixed point ρ of this map. Let λ^{−1} = ∫ T 1dρ, so T^{∗}ρ = λ^{−1}ρ. Replacing ν by λν, we may assume that T^{∗}ρ = ρ. We consider ρ as a measure on [0, 1].

Let f ∈ C([0, 1]), f(t) = 0 for t ≥ T. Then by definition T f(t) = 0 for t ≥ T and

∫_{0}^{1} f(t)dT^{∗}ρ(t) = ∫_{0}^{T} f(t)dT^{∗}ρ(t) = ∫_{0}^{T} f(t)dρ(t) = ∫_{0}^{1} f(t)dρ(t) = ∫_{0}^{1} T f(t)dρ(t)

= ∫_{0}^{1} ∫_{t}^{1} ν( − \frac{1}{u}, −\frac{1}{u} + \frac{1}{t}, P (−\frac{1}{u})) exp\{− ∫_{− \frac{1}{t}}^{− \frac{1}{u}} μ(τ, τ + \frac{1}{t}, P (τ))dτ\}\frac{1}{u^{2}} f(u)duρ(dt)

= ∫_{0}^{1} ∫_{0}^{u} ν( − \frac{1}{u}, −\frac{1}{u} + \frac{1}{t}, P (−\frac{1}{u})) exp\{− ∫_{− \frac{1}{t}}^{− \frac{1}{u}} μ(τ, τ + \frac{1}{t}, P (τ))dτ\}ρ(dt)\frac{1}{u^{2}} f(u)du

= ∫_{0}^{T} ∫_{0}^{u} ν( − \frac{1}{u}, −\frac{1}{u} + \frac{1}{t}, P (−\frac{1}{u})) exp\{− ∫_{− \frac{1}{t}}^{− \frac{1}{u}} μ(τ, τ + \frac{1}{t}, P (τ))dτ\}ρ(dt)\frac{1}{u^{2}} f(u)du.

Therefore, ρ = T^{∗}ρ is absolutely continuous on [0, T] with density

ψ(u) = \frac{1}{u^{2}} ∫_{0}^{u} ν(− \frac{1}{u}, −\frac{1}{u} + \frac{1}{t}, P (−\frac{1}{u})) exp\{− ∫_{− \frac{1}{t}}^{− \frac{1}{u}} μ(τ, τ + \frac{1}{t}, P (τ))dτ\}ρ(dt).

The function B_{0}(u) = u^{2}ψ(u) satisfies almost everywhere

B_{0}(u) = ∫_{0}^{u} ν( − \frac{1}{u}, −\frac{1}{u} + \frac{1}{t}, P (−\frac{1}{u})) exp\{− ∫_{− \frac{1}{t}}^{− \frac{1}{u}} μ(τ, τ + \frac{1}{t}, P (τ))dτ\}\frac{1}{t^{2}} B_{0}(t)dt
= ∫_{−∞}^{− \frac{1}{u}} ν( − \frac{1}{u}, −\frac{1}{u} − s, P (−\frac{1}{u})) exp\{− ∫_{s}^{− \frac{1}{u}} μ(τ, τ − s, P (τ))dτ\}B_{0}(−\frac{1}{s})ds.

Finally, setting B(x) = B_{0}(−\frac{1}{x}) we obtain for u ∈ (−∞, −T^{−1}) almost surely that

B(u) = ∫_{−∞}^{u} ν(u, u − s, P (u)) exp{− ∫_{s}^{u} μ(τ, τ − s, P (τ))dτ}B(s)ds. (3.57)

Since B(−T^{−1}) ≠ 0, we can apply the method from Theorem 3.5.1 to extend the solution to all of IR. ■

Clearly,

T^{∗}ρ = λ^{−1}ρ ⟺ T f = λ^{−1}f.

Corollary 3.7.1 1. If λ^{−1} > 1 ⇒ T f > f, that is, population grows exponentially.

2. If λ^{−1} < 1 ⇒ T f < f, that is, population vanishes.

3. If λ^{−1} = 1 ⇒ T f = f, that is, population constant.

It follows from the definition of λ^{−1}.

Corollary 3.7.2 Let ρ be a probability measure on [0, 1]. Then ∫ T 1dρ ≥ 0 with equality if and only if ρ denotes the point mass in 1.

Proof. By definition T 1(t) ≥ 0 with equality if and only if t = 1. ■ According to this corollary the map L(ρ) is continuous for every probabil-

ity measure ρ which is not the point mass ϵ_{1} in 1. Let M_{1}([0, 1]) denote the set of probability measures on the unit interval.

Corollary 3.7.3 Let M_{0} ⊂ M_{1}([0, 1]) be a T -invariant subset M_{0} ⊂ M_{1}([0, 1]) such that inf{T^{∗}m([0, 1]) : m ∈ M_{0}} > 0, then the map

L : conv(M_{0}) → conv(M_{0})

is well defined on the weak^{∗}-closed convex hull of M_{0} and has a fixed point ρ on this space.

Proof. Let c be a lower bound for ∫ T 1dm, m ∈ M_{0}. Since the closed convex hull of M_{0} is T -invariant, T is well defined on it and continuous, since ϵ_{1} does not belong to this closed convex hull. This can easily be deduced from the fact that inf ∫ T 1dm ≥ c for m ∈ conv(M_{0}). Therefore the theorem of Schauder-Tychonov applies. ■

Example 3.7.1 If \underline{ν}/\overline{μ}(1 − exp\{−\overline{μ}κ/(κ + t)t\}) ≥ 1 for t ≤ d, then for any measure m, T^{∗}m has at least the same mass on [0, d] as m, provides the population of age ≥ κ has mortality rate 0. In this case the assumptions of the corollary are satisfied.

Proposition 3.7.1 If \underline{ν} ≥ \overline{μ} on (−∞, T], then there exists a measure ρ so that inf_{n≥1} ∫ T^{n}1dρ > 0.

Proof. By definition of T

∫ T^{n}1dρ ≥ \underline{ν}^{n} ∫_{0}^{1} ∫_{t}^{1} ∫_{u_{1}}^{1} ... ∫_{u_{n−1}}^{1}(u_{1}...u_{n})^{−2} exp\{−\overline{μ}(\frac{1}{t} − \frac{1}{u_{n}})\}du_{n}...du_{1}ρ(dt).

For t, let I_{n}(t) denote the integrand of the integral on the right hand side. Then

I_{n}(t) ≥ \overline{μ}^{−1}(I_{n−1}(t) − (\frac{1}{t} − 1)^{n−1}),

hence for t ≥ \frac{3\overline{μ}}{3\overline{μ}+1}

I_{n}(t) ≥ \overline{μ}^{−1}I_{n−1}(t) − \overline{μ}^{−n}3^{−n+1}.

Iterating yields

I_{n}(t) ≥ \overline{μ}^{−n}(1 − ∑_{k=1}^{n−1} 3^{−k}) ≥ \frac{1}{2}\overline{μ}^{−n}.

Thus

∫ T^{n}1dρ ≥ \frac{\underline{ν}^{n}}{2\overline{μ}^{n}} ρ([\frac{3\overline{μ}}{3\overline{μ} + 1}, 1]),

and choosing ρ appropriate, the proposition follows. ■

Random Solutions

We study the effects of random fluctuating environment, it may effect the system (i.e. natality and mortality moduli). Since the environment changes due to many factors, we can approximate by noise (e.g. a Gaussian white noise), although we don’t know the exact behaviour of the noise term (see Arnold [25]). We consider random variations on natality and mortality moduli.

4.1 Perturbations

Let {X_{n}}_{n=1}^{∞} be a sequence of F-measurable random functions on a probability space (Ω, F, IP) with values in C. Let the expection of X_{n} be denoted by

IEX_{n}(t) = μ̄_{0}(t) < ∞, ∀t ∈ IR.

We also define a sequence of random functions Y_{n}(t−s) instead of ν_{n}(t, t− s, P) for all n ≥ 1. Its expection

IEY_{n}(t) = ν̄_{0}(t) < ∞, ∀t ∈ IR.

We can write equations (3.1) and (3.2) in random environment for s ≤ t as follows:

B_{n}(t, ω) = ∫_{−∞}^{t} Y_{n}(t − s, ω) exp{−∫_{s}^{t} X_{n}(τ − s, ω)dτ} B_{n} (s, ω) ds (4.1)

and

P_{n}(t, ω) = ∫_{−∞}^{t} exp{−∫_{s}^{t} X_{n}(τ − s, ω)dτ} B_{n} (s, ω) ds.  (4.2)

For solving the problem, we need certain regularity conditions on random functions X_{n} and Y_{n}.

Definition 4.1.1 The random functions {X_{n}} converge almost surely (a.s.) to μ̄_{0}(t) if IP {t, ω : lim_{n→∞} |X_{n}(t, ω) − μ̄_{0}(t)| = 0} = 1. Let us denote

X_{n} ⟶^{a.s.} IEX_{n} = μ̄_{0}.

Definition 4.1.2 The solution f_{t}(μ̄_{0}, ν̄_{0}) of equations (4.1) and (4.2) corresponding to (μ̄_{0}, ν̄_{0}) is called stable in almost surely in C if

IP {ω : lim_{n→∞} ||f_{t}(X_{n}(ω), Y_{n}(ω)) − f_{t}(μ̄_{0}, ν̄_{0})|| = 0} = 1

holds for all sequences (X_{n}, Y_{n})_{n∈IN} ∈ C for all ω ∈ Ω and X_{n} ⟶^{a.s.} μ̄_{0} and

Y_{n} ⟶^{a.s.} ν̄_{0} for n → ∞.

4.2 Convergence Theorem

In this section, we examine the integral equations (4.1) and (4.2) for existence of solution under the assumptions X_{n} ∈ L_{∞}^{+}(IR) and Y_{n} ∈ L_{∞}^{+}(IR), then the solution of the equations (4.1) and (4.2) are converging almost surely to the solution (B, P) of

B(t) = ∫_{−∞}^{t} ν̄_{0}(t − s) exp{−∫_{s}^{t} μ̄_{0}(τ − s)dτ} B (s) ds  (4.3)

and

P (t) = ∫_{−∞}^{t} exp{−∫_{s}^{t} μ̄_{0}(τ − s)dτ} B (s) ds.  (4.4)

Theorem 4.2.1 Assume that X_{n} ∈ L_{∞}^{+}(IR) and Y_{n} ∈ L_{∞}^{+}(IR) hold. If X_{n} ⟶^{a.s.} μ̄_{0} and Y_{n} ⟶^{a.s.} ν̄_{0} (n = 1, 2, ...), then the pointwise solution (B_{n}, P_{n}) of equations (4.1) and (4.2) under (X_{n}, Y_{n}) are converging almost surly on C to the solution (B, P) of equations (4.3) and (4.4) under (μ̄_{0}, ν̄_{0}).

Proof. For any random functions P_{n} and s ≤ t, we obtain from equation (4.2) that

P_{n}(t, ω) = ∫_{−∞}^{T} exp{− ∫_{s}^{T} X_{n}(τ − s, ω)dτ} exp{− ∫_{T}^{t} X_{n}(τ − s, ω)dτ}B_{n}(s, ω)ds
+ ∫_{T}^{t} exp{− ∫_{s}^{t} X_{n}(τ − s, ω)dτ} B_{n}(s, ω) ds.  (4.5)

Let Λ : IR^{2} × Ω → IR_{+} be denoted for s ≤ t by

Λ(s, t, ω) = exp\{− ∫_{s}^{t} X_{n}(τ − s, ω) dτ\} ≤ exp\{−\underline{X}(t − s)\} ≤ 1,

and for s ≤ T by

Λ(s, T, ω) = exp\{− ∫_{s}^{T} X_{n}(τ − s, ω) dτ\} ≤ exp\{−\underline{X}(T − s)\} ≤ 1,

where

\underline{X} = inf_{u, ω} X_{n}(u, ω) > 0.

Define Λ_{T} : IR^{2} × Ω → IR_{+} for T ≤ t by

Λ_{T} (s, t, ω) = exp{− ∫_{T}^{t} X_{n}(τ − s, ω) dτ}.

For s ≤ t, Λ satisfies the following inequality:

|Λ(s, t, ω) − Λ(s, t, μ̄_{0})|
= | exp{− ∫_{s}^{t} X_{n}(τ − s, ω)dτ} − exp{− ∫_{s}^{t} μ̄_{0}(t) dτ}|
≤ ∫_{s}^{t} |X_{n}(τ − s, ω) − μ̄_{0}(t)|dτ,  (4.6)

where we have used the fact that |e^{x} − e^{y}| ≤ |x − y| for all x, y ≤ 0. We get from (4.4) and (4.5) for P_{n}, P ∈ C

|P_{n}(t, ω) − P (t)| ≤ ∫_{−∞}^{T} Λ(s, T, ω)|Λ_{T} (s, t, ω) − Λ_{T} (s, t, μ̄_{0})|B_{n}(s, ω) ds
+ ∫_{T}^{t} |Λ(s, t, ω)B_{n}(s, ω) − Λ(s, t, μ̄_{0})B(s)| ds
= I + II  (4.7)

where

I ≤ ∫_{−∞}^{T} Λ(s, T, ω) · ∫_{T}^{t} |X_{n}(τ − s, ω) − μ̄_{0}(t)|dτ B_{n}(s, ω) ds
≤ ∫_{−∞}^{T} ∫_{T}^{t} |X_{n}(τ − s, ω) − μ̄_{0}(t)|dτ B_{n}(s, ω) ds

and

II ≤ ∫_{T}^{t} |Λ(s, t, ω) − Λ(s, t, μ̄_{0})| B_{n}(s, ω) ds + ∫_{T}^{t} Λ(s, t, μ̄_{0})|B_{n}(s, ω) − B(s)|ds
≤ ∫_{T}^{t} ∫_{s}^{t} |X_{n}(τ − s, ω) − μ̄_{0}(t)| dτB_{n}(s, ω) ds + ∫_{T}^{t} |B_{n}(s, ω) − B(s)|ds.

We obtain from (4.7) that

|P_{n}(t, ω) − P (t)| ≤ ∫_{T}^{t} ∫_{−∞}^{T} |X_{n}(τ − s, ω) − μ̄_{0}(t)|B_{n}(s, ω) ds dτ

+ ∫_{T}^{t} ∫_{T}^{τ} |X_{n}(τ − s, ω) − μ̄_{0}(t)|B_{n}(s, ω) dsdτ

+ ∫_{T}^{t} |B_{n}(s, ω) − B(s)|ds  (4.8)

From (4.1) and (4.3), we can write,

|B_{n}(t, ω) − B(t)|

≤ ∫_{−∞}^{T} Y_{n}(t − s, ω)Λ(s, T, ω)|Λ_{T} (s, t, ω) − Λ_{T} (s, t, μ̄_{0})|B_{n}(s, ω)ds

+ ∫_{T}^{t} |Y_{n}(t − s, ω)Λ(s, t, ω)B_{n}(s, ω) − ν̄_{0}(t)Λ(s, t, μ̄_{0})B(s)|ds
= III + IV  (4.9)

where

III ≤ ∫_{−∞}^{T} Y_{n}(t − s, ω) ∫_{T}^{t} |X_{n}(τ − s, ω) − μ̄_{0}(t)|dτ B_{n}(s, ω) ds
≤ ∫_{−∞}^{T} ∫_{T}^{t} Ȳ |X_{n}(τ − s, ω) − μ̄_{0}(t)|dτ B_{n}(s, ω)ds

and

Ȳ = sup_{u,ω} Y_{n}(u, ω).

Also,

IV ≤ ∫_{T}^{t} Y_{n}(t − s, ω)|Λ(s, t, ω) − Λ(s, t, μ̄_{0})|B_{n}(s, ω)ds
+ ∫_{T}^{t} |Y_{n}(t − s, ω) − ν̄_{0}(t)|Λ(s, t, μ̄_{0})B_{n}(s, ω)ds
+ ∫_{T}^{t} ν̄_{0}(t)Λ(t, s, μ̄_{0})|B_{n}(s, ω) − B(s)|ds
≤ ∫_{T}^{t} ∫_{s}^{t} Ȳ |X_{n}(τ − s, ω) − μ̄_{0}(t)|dτB_{n}(s, ω)ds
+ ∫_{T}^{t} |Y_{n}(t − s, ω) − ν̄_{0}(t)|B_{n}(s, ω)ds + ∫_{T}^{t} ν̄_{0}(t)|B_{n}(s, ω) − B(s)|ds.

From (4.9), we get

|B_{n}(t, ω) − B(t)| ≤ ∫_{T}^{t} ∫_{−∞}^{T} Ȳ |X_{n}(τ − s, ω) − μ̄_{0}(t)|B_{n}(s, ω)ds dτ

+ ∫_{T}^{t} ∫_{T}^{τ} Ȳ |X_{n}(τ − s, ω) − μ̄_{0}(t)|B_{n}(s, ω)ds dτ

+ ∫_{T}^{t} |Y_{n}(t − s, ω) − ν̄_{0}(t)|B_{n}(s, ω)ds
+ ∫_{T}^{t} ν̄_{0}(t)|B_{n}(s, ω) − B(s)|ds.  (4.10)

By hypotheses X_{n} ⟶^{a.s.} μ̄_{0} and Y_{n} ⟶^{a.s.} ν̄_{0}, equation (4.10) can be written

|B_{n}(t, ω) − B(t)| ≤ ∫_{T}^{t} ν̄_{0}(t)|B_{n}(s, ω) − B(s)|ds.  (4.11)

Using Lemma 2.1.1 on (4.11), we deduce that

|B_{n}(t, ω) − B(t)| = 0, ∀t ≥ T, ∀ω ∈ Ω and n ≥ n_{0}(t).

Hence

B_{n} ⟶^{a.s.} B.

Substitute |B_{n}(t, ω) − B(t)| = 0 and X_{n} ⟶^{a.s.} μ̄_{0} on (4.8), we conclude that

|P_{n}(t, ω) − P (t)| = 0, ∀t ≥ T, ∀ω ∈ Ω and n ≥ n_{0}(t).

Hence

P_{n} ⟶^{a.s.} P.

■

Remark 4.2.1 It follows from Theorem 4.2.1 that if X_{n} ⟶^{a.s.} μ̄_{0} and Y_{n} ⟶^{a.s.}

ν̄_{0} then B_{n} ⟶^{P} B and P_{n} ⟶^{P} P, also B_{n} ⟶^{d} B and P_{n} ⟶^{d} P.

Corollary 4.2.1 Suppose that the assumptions of Theorem 4.2.1 are satisfied. Then the equations (4.1) and 4.2) have a stable solution in almost surely in C.

The proof follows from Theorem 4.2.1.

Bibliography

[1] F. R. Sharpe and A. J. Lotka, A Problem in Age Distribution, Philosophical Magazine, 21 (1911), 435-438.

[2] A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925.

[3] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1958.

[4] D.R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1980.

[5] W.V. Lovitt, Linear Integral Equations, McGraw-Hill Book Company, New York, 1924.

[6] A. M. Bruckner, J. B. Bruckner and B.S. Thomson, Real Analysis, Prentice-Hall, New Jersey, 1997.

[7] J. H. Pollard, Mathematical Models for the Growth of Human Populations, Cambridge University Press, Cambridge, 1973.

[8] N. Keyfitz, Introduction to the Mathematics of Population , Addison-Wesley, 1968.

[9] H. L. Langhaar, General Population Theory in the Age-time Continuum, J. Franklin Inst., 293 (1972), 199-214.

[10] W. Feller, On the Integral Equation of renewal Theory, Ann. Math. Stat., 12 (1941), 243-267.

[11] M. E. Gurtin and R. C. MacCamy, Non-linear Age-dependent Population Dynamics, Arch. Rat. Mech. Anal., 54 (1974), 281-300.

[12] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991.

[13] Md. Shahidul Islam and A.F.M. Khodadad Khan, A Nonlinear Population Model in the Age-time Continuum, Dhaka Univ. J. Sci., 45(1) (1997), 47-56.

[14] Md. Shahidul Islam and A.F.M. Khodadad Khan, On a Pair of Integral Equations Arising in Nonlinear Population Dynamics in the Age-time Continuum-I, Ganit: J. Bang. Math. Soc., 16 (1996), 21-34.

[15] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, Berlin-New York, 1979.

[16] S. C. Chu and F. T. Metcaf, On Grownwall’s Inequality , Proc. Amer. Math. Soc., 18(3) (1967), 439-440.

[17] E. Sinestrari, Non-linear Age-dependent Population Growth, J. Math. Biology, 9 (1980), 331-345.

[18] C.O.A. Sowunmi, Female Dominant Age-dependent Deterministic Population Dynamics, J. Math. Biology, 3 (1976), 9-17.

[19] G.D. Blasio, Nonlinear Age-dependent Population Growth with History Dependent Birth Rate, Mathematical Biosciences, 46 (1979), 279-291.

[20] M. Chipot, On the Equations of Age-dependent Population Dynamics, Arch. Rat. Mech. Anal., 82 (1983), 13-25.

[21] K. Gopalsamy, Age-specific Coexistence in Two-species Competition, Mathematical Biosciences, 61 (1982), 101-122.

[22] A. Haimovici, On the Growth of a Population Dependent on Ages and Involving Resources and Pollution, Mathematical Biosciences, 43 (1979), 213-237.

[23] T. Kostova and F. Milner, Nonlinear Age-dependent Population Dynamics with Constant Size, SIAM J. Math. Anal., 22(1) (1991), 129-137.

[24] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, New York, 1985.

[25] L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998.

Curriculum Vitae

Name Md. Shahidul Islam 11.09.1967 Born in Noakhali, Bangladesh 1982 Secondary School Certificate (SSC)

Joyag High School, Noakhali, Bangladesh

1984 Higher School Certificate (HSC)

Noakhali College, Noakhali, Bangladesh

1989 Bachelor of Science (Mathematics) Department of Mathematics Dhaka University, Bangladesh

1992 Master of Science (Mathematics)

Department of Mathematics Dhaka University, Bangladesh

1992-1994 Lecturer of Mathematics Department

Sylhet University, Bangladesh

1994-1998 Lecturer of Mathematics Department

Dhaka University, Bangladesh

1999 Master of Science (Mathematics)

Institute for Mathematical Stochastics Georg-August-University, Göttingen, Germany

2002 Ph.D. student of Prof. Dr. Manfred Denker Institute for Mathematical Stochastics Georg-August-University, Göttingen, Germany