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The dynamics of hierarchical age-structured populations

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Executive Summary

This page covers J.M.Cushing's analysis of hierarchical age-structured populations. Specifically, it discusses how Cushing proves P.D.E. population models developed by McKendrick, and elaborated on by Gurtin and MacCamy can be reduced to a O.D.E. population model. Cushing formally proves this reduction is possible by defining the P.D.E. and O.D.E. as two separate problems, and then showing a solution to problem two is a solution to problem one, and vice versa. Also, Cushing shows that any solution of the two problems is unique (i.e. has one and only one corresponding initial condition).

Introduction to Models

McKendrick Equations

The McKendrick equations establish an age based population model using continuous population density, birthrate, and death rate functions. Presented below are his three equations:

{\lim _{{h\to 0}}{\frac  {\rho (t+h,a+h)-\rho (t,a)}{h}}+\delta \rho (t,a)=0,a>0,t>0} (1)

\rho (t,0)=\int _{{0}}^{{+\infty }}\beta (t,a)\,{\mathrm  {d}}a,t>0 (2)

\rho (0,a)=\varphi (a),a\geq 0 (3)

The first equation describes the dynamics of the model within its domain, \textstyle {t,a\in (0,+\infty )}. Using an unbounded integral, equation (2) defines the total number of animals born at time t as the total number of animals born across all ages of the population. The initial condition is defined by a population density function dependent only on the age of the animals. Since this is an age dependent model, density is defined as a function of the current time t and age a of the animals.

These equations are the basis of the P.D.E. population model, which one might have inferred after looking at the left most term in equation one. This term is the formal definition of the complete derivative of the two dimensional density function. If the density function is indeed differentiable, we can write the following:

\lim _{{h\to 0}}{\frac  {\rho (t+h,a+h)-\rho (t,a)}{h}}=\partial _{t}\rho +\partial _{a}\rho

Cushing covers differentiability of the model's functions with a series of assumptions of continuity and boundedness.

Gurtin and MacCamy Assumptions

McKendrick never specified the behavior of the birth or death rates \beta and \delta in his equations. Gurtin and MacCamy elaborate on the McKendrick equations by defining those rates as such:

\beta =\beta (a,P)

\delta =\delta (a,P)

P=P(t)=\int _{0}^{{+\infty }}\rho (t,a)\,{\mathrm  {d}}a

Here, the birth rate \beta and the death rate \delta are defined as functions of age and and P. The third equation defines P to be the total population at a given time t.

Problem One

Define Problem One

Here, J.M.Cushing elaborates on the equations already defined by McKendrick and Gurtin and MacCamy. Specifically, Cushing introduces an age hierarchy by defining the following functions:

Y=Y(t,a)=\int _{0}^{a}\rho (t,s)\,{\mathrm  {d}}s

O=O(t,a)=\int _{a}^{{+\infty }}\rho (t,s)\,{\mathrm  {d}}s

The function Y specifies all animals younger than age a at a given time t, where the function O specifies all animals older than age a at a given time t. Further elaborating on Gurtin and MacCamy's definitions of birth and death rate, Cushing redefines them both:

\beta =\beta (t,Y(t,a),O(t,a))

\delta =\delta (t,Y(t,a),O(t,a))

Cushing uses these definitions to rewrite McKendrick's model equations in the following manner:

\lim _{{h\to 0}}{\frac  {\rho (t+h,a+h)-\rho (t,a)}{h}}=-\delta (t,Y(t,a),O(t,a))\rho (t,a),\;\;a>0,t>0

\rho (t,0)=\int _{0}^{{+\infty }}\beta (t,Y(t,a),O(t,a))\rho (t,a)\,{\mathrm  {d}}a,\;t>0

\rho (0,a)=\varphi (a),\;t\geq a

These equations are referred to as problem one by Cushing. They govern the P.D.E model that Cushing reduces to an O.D.E.

Assumptions about Birth and Death Rates

Cushing wants to provide a definition of a solution for problem one that he can later use in his proof regarding the simplification of the P.D.E. model to the O.D.E. model. First, he starts by making assumptions about the the continuity and domains of the birth and death rates. These are crucial assumptions, since they govern the behavior of the population density function and thus the model. Cushing's assumptions read as follows (note that \textstyle {R_{+}=[0,+\infty )}):

  • \textstyle {A_{1}}: \textstyle {\beta ,\delta \in C^{0}(R_{+}^{3},R_{+})} and \exists positive reals \beta _{0},\delta _{0}>0\beta (t,Y,O)\leq \beta _{0},\delta (t,Y,O)\geq \delta _{0}\forall (t,Y,O)\in R_{+}^{3}

What Cushing is saying here is that the birth rates and death rates are continuous on the positive real line, and that they have finite maximum values (all reasonable biological assumptions).

  • \textstyle {A_{2}}: the partial derivatives \partial _{y}\beta (t,Y,O),\partial _{O}\beta (t,Y,O),\partial _{y}\delta (t,Y,O), and \partial _{O}\delta (t,Y,O) exist and are all continuous on R_{+}^{3}
  • \textstyle {A_{3}}:\varphi \in L^{1}(R_{+},R_{+})

This says that the initial condition is integrable along the positive real line.

Solution to Problem One

Since Cushing intends to prove that a solution to problem one will also be a solution to problem two (the O.D.E. model), Cushing provides a very explicit definition of what constitutes a solution for problem one. Using the above assumptions, a solution is defined as follows according to Cushing:


"We define a solution of Problem 1 on an interval \textstyle {[0,T],0<T<+\infty } to be a function \textstyle {\rho :[0,T]\times R_{+}\to R_{+}} for which \textstyle {\lim _{{h\to 0}}(\rho (t+h,a+h)-\rho (t,a))h^{{-1}}} exists \textstyle {\forall (t,a)\in [0,T]\times R_{+},\rho (t,\cdot )\in L^{1}(R_{+})\forall t\in [0,T],P(t)=\int _{0}^{{+\infty }}\rho (t,a)\,{\mathrm  {d}}a\in C^{1}([0,T],R_{+}),} and (6), (7), and (8) are satisfied for \textstyle {(t,a)\in [0,T]\times R_{+}}. A solution on an interval [0,T),0<T\leq +\infty is defined similarly."


Note that (6), (7), and (8) are Cushing's notation for the problems that comprise problem one. Cushing is being very specific with his definition of the solution to problem one. The statement \textstyle {\rho :[0,T]\times R_{+}\to R_{+}} simply means that the solution is a function that maps every possible interval [0,T],T\in R_{+},T\neq 0 onto R_{+}. Cushing then states that the formal definition of the derivative of \rho exists on the the domain of the function. Then Cushing states the function is integrable with respect to t throughout the function's domain. He also specifies that the total population of the model with respect to time is continuous and first differentiable on the domain [0,T].


Problem Two

Define Problem Two

Problem two is the O.D.E population model that Cushing wants to prove is a legitimate reduction of problem one. Problem two consists of the following equations:

P'=B(t,P)-D(t,P)

Y=Q(P,Y)

The model is governed by the following initial conditions:

P(0)=\int _{0}^{{+\infty }}\varphi (u)\,{\mathrm  {d}}u

Y(0,a)=\int _{0}^{a}\varphi (u)\,{\mathrm  {d}}u

Also there can be no animals younger than zero:

Y(t,0)=0

Here, Cushing is defining B and D to be the total birth and death rates at a given time t with a population size P.

B(t,P)\doteq \int _{0}^{P}\beta (t,u,P-u)\,{\mathrm  {d}}u

D(t,P)\doteq \int _{0}^{P}\delta (t,u,P-u)\,{\mathrm  {d}}u

The role of function Q isn't given much explanation by Cushing at this point in the paper. Its origins become much clearer during the proof of Lemma 1. Q is defined as follows:

If a\geq t>0

Q(P,Y)\doteq \int _{0}^{t}B(u,P(u))\exp {\bigg (}-\int _{0}^{{t-u}}{\tilde  {\delta }}(Y,P)(u+\alpha ,\alpha )\,{\mathrm  {d}}\alpha {\bigg )}\,{\mathrm  {d}}u

+\int _{0}^{{a-t}}\varphi (u)\exp {{\bigg (}-\int _{0}^{t}{\tilde  {\delta }}(Y,P)(\tau ,u+\tau )\,{\mathrm  {d}}\tau {\bigg )}}\,{\mathrm  {d}}u

If 0<a<t:

Q(P,Y)\doteq \int _{{t-a}}^{{t}}B(u,P(u))\exp {\bigg (}-\int _{0}^{{t-u}}{\tilde  {\delta }}(Y,P)(u+\alpha ,\alpha )\,{\mathrm  {d}}\alpha {\bigg )}\,{\mathrm  {d}}u

Where

{\tilde  {\delta }}(Y,P)(t,a)\doteq \delta (t,Y(t,a),P(t)-Y(t,a))