Non-linear Age-dependent Population Dynamics
MORTON E. GURTIN & RICHARD C. MACCAMY
1. Introduction
The simplest model of population dynamics is based on the Malthusian law
i' = 5P (5 = constant), (1.1)
where
P(t)
is the total population at time t and 5 is the growth modulus. This
law is clearly inapplicable to situations in which the population competes for
resources
(e.g.,
space and food), for in these situations 5 should depend on the
size of the population: the larger the population, the slower should be its rate
of growth.
To overcome this deficiency in the Malthusian law, VERHULST [1845, 1847]
assumed that
fi=(6o-co0 P)P (5o, COo=constant). (1.2)
For 5 o and COo positive this differential equation has a stable equilibrium point
P= r 6o/COo, and populations with P(0)<~ grow monotonically to ~ as t--* oo.
The solution of (1.2),
P(t)= , (1.3)
1+ (p~0)-1 )
e -a~
has been applied, with remarkable success, to fit the growth curves of various
types of populations, t
The chief disadvantage of the Malthus and Verhulst models is that they yield
no information whatsoever concerning the age distribution of the population,
and, in fact, are based on the tacit assumption that the birth and death processes
are
age-independent.
A model applicable to
age-dependent
population dynamics
was first proposed by LOTKA and YON FO~RSTER. 2 This model is based on the
assumption that
oo
P(t) = I p(a, t) da,
(1.4)
0
1 See, e.g.,
LOTKA [1925], pp. 66-76.
2 The basic ideas behind this model are contained in the work of LOTKA [1925], where (1.7)
and (1.8) are implicit. The partial differential equation (1.6) is due to YON FOERSTER [1959],
although with d=0 it appears earlier in the work of SCt-mRBAUM & RASCH [1957]. See also
FISHER [1958], LOPEZ [1961], TRUCCO [1965], KEYFITZ [1968], CROW & KIMURA [1970], LANG-
IaA/m [1972].
282 M.E. GURTIN t~ R. C. MACCAMY
where
p(a, t)
is the population at time t in the age-interval
(a, a+da).
Consider
the group of individuals who are of age a at time t. If t is increased by h units,
these individuals age by h units; thus
Op(a,
t)=lim
p(a+h, t+h)-p(a, t)
(1.5)
h-~0 h
is the rate at which the population of this group is changing in time. This rate
plus the number
d(a, t)
of individuals (per unit age and time) of age a who die
at time t must equal zero:
Dp+d=O.
(1.6)
One then assumes that
d (a, t) = 2 (a) p (a, t), (1.7)
and that the birth process is described by the "renewal equation"
o'3
p(0, t)= S
fl(a)p(a, t)da.
(1.8)
0
Here
fl(a),
called the
birth-modulus,
is the average number of offsprings (per unit
time) produced by an individual of age a; 2(a), called the
death-modulus,
is the
death-rate at age a per unit population of age a. 1 The system (1.6)-(1.8), supple-
mented by the initial condition
p (a, 0) = ~0 (a), (1.9)
constitute the Lotka-von Foerster model. As we shall see later, p will generally
not be differentiable everywhere; of course, when it is,
Dp=pa+pr
(1.10)
The same objection can be raised to the Lotka-von Foerster model as was
raised previously to the Malthusian law: the birth and death moduli are independ-
ent of the population P. To rectify this we allow fl and 2 to depend on P. Our
theory is therefore based on the following system of equations:
Dp+2(a,
P)p=0,
oo
P(t)= S
p(a, t) da,
(1.11)
0
oo
p(0, t)=
~ fl(a, P) p(a, t) d a,
0
supplemented, of course, by the initial condition (1.9).
The problem (1.9), (1.11) is studied in detail in Section 2. There we lay down
our basic hypotheses: that 2, fl, and q~ be non-negative and sufficiently smooth
1 See,
e.g.,
ANDREWARTHA (~ BIRCH [1954], where curves of
fl(a)
and
~(a)=exp (- i2(~)d~ )
are given for the vole mouse (microtus agrestis) and for the rice weevil (calandra oryzae).
Non-linear Age-dependent Population Dynamics 283
(in a certain precise sense). We then establish the existence of a unique solution
on a sufficiently small time-interval, and we prove that if (in addition to the
above hypotheses) fl is uniformly bounded, then existence and uniqueness hold
for all time. We show further, by example, that some such additional hypothesis
is necessary for global existence. Our solutions are all in the sense of (1.11)1:
p is not required to possess partial derivatives with respect to a and t, but only
the directional derivative (1.5). We discuss conditions which guarantee that the
solution actually be of class C1; these turn out to be compatibility conditions
on the initial data ~0.
All of our results are based on the reduction of the problem (1.9), (1.11) to a
pair of non-linear functional equations for the total population P(t) and the
birth-rate B(t)=p(O, t). These equations are natural extensions of the linear
Volterra integral equation for B(t) which occurs in the Lotka-von Foerster
theory. ~ We study these functional equations by means of a fixed point argument.
In Section 3 we study equilibrium age distributions; that is, solutions p of
(1.1 l) which are independent of time. We show that in our non-linear theory a
greater variety of such solutions is available than in the linear theory of LOTKA
and VON FOERSTER. We also study the stability of these equilibrium age distribu-
tions. In particular, we give conditions under which such distributions are ex-
ponentially asymtotically stable.
Section 4 contains a discussion of some special cases, corresponding to partic-
ular choices of 2 and fl, in which our model reduces to a pair of nonqinear
ordinary differential equations for P(t) and B(t). In particular, we show that
the models of MALTHUS and VERHULST can be obtained in this manner. We
emphasize, however, that even in these simple cases our study provides new in-
formation: it enables us to determine the evolution in time of an initial age
distribution p (a, 0).
The theory described above can clearly be extended to include several inter-
acting species, and we intend to pursue this problem in the near future. A second
interesting generalization of our theory occurs when we take spatial diffusion
into account. 2 The analogs of our present theory in the diffusion problem present
several very challenging questions, and these too we hope to pursue.
2. Existence and Uniqueness
In this section we shall establish an alternative formulation of the population
problem (1.9), (1.11) in terms of a pair of coupled integral equations for the
birth-rate B(t) and the total population P(t), and we shall use these equations
to prove existence and uniqueness. It is dear that our problem is physically
meaningful only if r 2(a, P), and fl(a, P) are non-negative. Also, since we
want the initial total population to be finite, r should belong to LI(IR+). We
collect these assumptions, together with some others of a more technical nature,
in the following hypotheses: a
1 See, e.g., KEvrrrz [1968], Eq. (5.1.1).
2 Simple linear models for diffusion of a single species have already been proposed by SKELLAM
[1951], K~RNER [1959], and GURTI~ [1973].
3 We use the following notation throughout: ~.=(--o0, oo); IR+= [0, oo); CO:B) is the
set of all continuous functions from ,4 to B; C(.4)---- C(A: ~).
284 M.E. GURTIN • R. C. MACCAMY
(H0 ~0~L1 (1R +) is piecewise continuous;
(H2) 2, fl~C(IR+xlR+); )~v(a,P) and
fle(a,P)
exist for all a>0 and
P>0; 2(., P), 2p(., P), fl(-, P), and fie(-, P), as functions of P, belong to
c(~+: L~o(~.+));
(H3) (p>_-0, 2>__0,/~_->0.
For convenience, we assume, once and for all, that (Hx)-(H3) are satisfied.
By a
solution of the population problem up to time T
>0 we mean a non-negative
function p on IR + x [0, T] with the following properties:
Dp
exists on N + x [0, T];
p(., t)eL 1
(IR +) and
00
P(t)= ~
p(a, t)da
(2.1)
0
is continuous for 0 < t < T;
Dp(a, t)+2(a, P(t))p(a,
t)=0 (a>0, 0<t< T),
oo
p(0, t)=
~B(a,P(t))p(a,t)da
(0<t< T), (2.2)
0
p (a, 0) = ~o (a) (a __> 0).
It is important to note that (2.2)2 is not required to hold at t= 0. Indeed, by
(2.2)3 this relation will be satisfied at t = 0 if and only if q) satisfies the compati-
bility condition
oo oo
qg(0)=
~ fl(a, cb)(p(a)da, cb= ~r
(2.3)
0 0
which is simply the requirement that the initial data be consistent with the birth
process. We do not impose the restriction (2.3), because we envisage situations
in which the initial age distribution (p is completely arbitrary.
Let p be a solution up to time T, let (ao, to)~lR + x [0, T], and let
p(h)=p(ao+h, to+h), 2(h)=2(ao+h,P(to+h)).
(2.4)
Then (1.5) and (2.2)~ imply that
d~ q- 2-(h) fi=0, (2.5)
dh
and this equation has the unique solution
h_
--J A(q) dr/
p( ao + h, to + h )=p( ao, to) e ,
(2.6)
giving the values of p at all points on the characteristic through (ao, to) in terms
of the value of p at (ao, to). In particular, if in (2.6) we take (ao, to)= (a-t, 0)
and h = t, we conclude, with the aid of (2.2)3, that
t
-o 5 2(a-t + ~:, P(r)) dr
p(a, t)=q~(a-t)e
for
a>t.
(2.7)
Non-linear Age-dependent Population Dynamics 285
On the other hand, the substitutions (a0, to)=(0, t-a) and h=a in (2.6) lead to
the conclusion that
a
-J ).(~,
P(t-a+at))
dot
p(a, t)=B(t-a)e for t>a, (2.8)
where
B(t)=p(O, t) (2.9)
is the birth-rate. Finally, if we substitute (2.7) and (2.8) into (2.1) and (2.2)2, we
arrive at the following pair of coupled integral equations for P and B:
t oo
e(t) = ~ K(t- a, t;
P) B(a)
da + I L(a, t; P) ~o(a) da,
0 0
t
B(t)= S fl(t-a, P(t))K(t-a, t; P) B(a)da (2.10)
0
oo
+ I fl(a + t, P(t))L(a, t; P) ~p(a) da,
0
where
-- ~ 2(o~+~-t,P(r
K(g, t; P)=e
(0_<~=< t), (2.11)
t
--J 2(T+~, P(r dr
L(~, t; P) = e
The integral equations (2.10) form the basis for the discussion in this section.
Note that K and L are functionals of P. When 2 and fl are independent of P,
(2.10)2 is the classical linear integral equation of LOTKA 1 for the birth-rate B,
while (2.10)i is simply a formula for P.
From (2.7) it follows that discontinuities in ~p propagate along characteristics.
Further, even when ~p is continuous, (2.7) and (2.8) imply that p will be dis-
continuous across the characteristic t=a unless B(0+)=cp(0). By (2.10)2, B(0 +)
is equal to the right-hand side of (2.3). Thus (when ~p is continuous) a necessary
and sufficient condition that p be continuous across t = a is that (2.3) hold. When
(2.3) is not satisfied, B, defined by (2.9), will exhibit a discontinuity at t=O, and
therefore B defined at t = 0 by (2.9) will not agree with B defined at t = 0 by (2.10)2.
To avoid this (technical) difficulty we define B by (2.9) for t>0 and take B(O)=
B(0+).
Theorem
1. Let p be a solution of the population problem up to time T > O.
Then the total population P and the birth-rate B satisfy the integral equations
(2.10) on [0, T]. Conversely, if P and B are non-negative continuous functions that
satisfy (2.10) on [0, T], and if p is defined on lR+x [0, T] by (2.7), (2.8), then p
is a solution of the population problem up to time T.
Proof. We have already established the first portion of the theorem. To
prove the converse assertion, let P>0 and B>0 be continuous functions on
[0, T] consistent with (2.10), and let p be defined on IR + x [0, T] by (2.7), (2.8).
Then p>0 (since (o and fl are non-negative), (2.3)3 holds, p(0, t)=B(t) for t >0,
and p(., t)eL1 (]R +) (since 2, B, and P are continuous and ~P~L1 0R+)). It there-
1 See, e.g., KEY~aTZ [1968], Eq. (5.1.1).
20 Arch. Rat. Mech. Anal., Vol. 54