FUNCTIONAL
DIFFERENTIAL
EQUATIONS
VOLUME 11
2004, NO 1-2
PP. 69 76
EXISTENCE AND UNIQUENESS OF SOLUTIONS OF A
PERTURBATION OF
THE MCKENDRICK-VON FOESTER EQUATION
D. GREENHALGH' AND I. GYORII
Abstract. We consider a perturbation of the classical McKendrick-Von Foester equation originally discussed by Boulanger [1]. As we are dealing with population densities it
is more natural to express the equations as integral equations. We establish existence and
uniqueness of solutions under weaker conditions than previously.
AMS(MOS) subject classification. 35A05, 45G99, 47H10.
Key Words. McKendrick-Von Foester equation; integral equations; existence and
uniqueness.
1. Introduction. Boulanger [1] investigated the following population
dynamical model with age interaction:
(1)
with boundary condition
(2)
p,(t, 0)
f'
b(a)p,(t, a)da
and initial condition
(3)
p,(O, a) =
rf>(a).
' Department of Statistics and Modelling Science, University of Strathclyde, Livingstone
Tower, 26, Richmond Street, Glasgow Gl lXH, U.K.
t Department of Mathematics and Computing, University of Veszprem, P.O. Box 158,
Egyetem str. 10, 8201 Veszprem, Hungary. This research was partially supported by
Hungarian National Foundation for Scientific Research Grant No. T031935.
69
70
D. GREENHALGH AND I. GYORI
The consistency condition
¢(0) =
(4)
f' b(a)¢>(a)da
must also be satisfied. Here Jt,(a) = t-to(a) -
R
E,
where t-to(a) is such that
=fa'"' b(a)e- fo" i'o(/;;)d<da = 1
and t-to(a) 2: E for all a. Boulanger also assumed that j3,(a, s)
E7J(a, s)
where 7J is a function ~+ x ~+. p,(t, a) represents the density with respect
to age of the total number of individuals in the population at time t. This
means that the total number of individuals between ages A 1 and A2 at time
t is
!
A,
A,
p,(t, a)da.
Jt,(a) is the death rate at age a, ¢>(a) is the density of the initial age distribution and b( a) is the birth rate. /3, (a, s) is a term corresponding to interaction
between generations limiting population growth. When E = 0 these equations
reduce to the classical linear McKendrick-Von Foester (McK-VF) equation
for the growth of a population [6, 7]. R is the reproduction number for the
corresponding classical linear McK-VF equation. In his paper Boulanger investigated the approximation of the solutions of the linear McK-VF equation
by the solutions of the full equations with interaction term as E goes to zero.
If the increase in mortality due to crowding is just the total population,
one would obtain the nonlinear age-dependent model treated in Gurtin and
McCamy [3].
Boulanger gives a set of conditions and asserts that these imply that
(1)-(4) have a unique solution for p,(a, t). As previously pointed out [2] we
disagree with this and claim that much stronger conditions are necessary.
However our conditions, based on results in Webb [8], are very strong and
as we are dealing with population densities it is more natural to express
(1)-(3) as integral equations. By doing this we shall establish existence and
uniqueness of solutions under weaker conditions. As p, (t, a) is a population
density it is natural to look for solutions f : [0, oo) -> ~in the Banach space
£l[O,oo) with norm
II f II =
f' I f(a) Ida.
A similar idea to show existence and uniqueness of solutions was taken for
the less general model in Gurtin and McCamy's paper [3].
EXISTENCE AND UNIQUENESS
71
2. Results. We shall suppose that b(a), f3,(a,s), p,,(s) and <P(a) satisfy
the following conditions:
CONDITIONS A
(i) b( a) is measurable and bounded on finite intervals;
(ii) Jt"'' b(a)<f>(a- t)e- J:_,M,(<)df.da is bounded above independently oft by
Mo say;
(iii) f],(a, s) is measurable and bounded absolutely by M 1 say;
(iv) p,,(s) is integrable on [O,a) for all a> 0;
(v) qJ E U[O,oo).
In particular Conditions A(i) and A(ii) are automatically satisfied if b(a)
is absolutely bounded and </> E £ 1 [0, oo) but they are weaker than assuming
that b( a) is absolutely bounded. In fact as b( a) represents a birth rate it
is biologically reasonable to assume that it is bounded, but we shall prove
the existence and uniqueness of solutions to integral equations for p,(t, a)
corresponding to (1)-(3) under the less restrictive Conditions A.
As in [3] we start the proof by making the transformations
p,(t, a)= p,(t, a)efa" ~'·(f.)df., [3,(a, s) = f],(a, s)e- f,' i'•<E)df.,
b,(a) = b(a)e- fa" ~'•(f.)df. and ¢,(a) = <f>(a)efo" J',(<)df.,
equations (1)- (3) reduce to
ap,(t, a)
aa
(5)
+
ap,(t, a)
at
-
-p,
f"-
f],(a,s)p,(t,s)ds,
0
(6)
p,(t,O) -
{" b,(a)p,(t, a)da,
(7) and
p,(O, a) =
¢,(a).
The consistency condition (4) becomes
(8)
¢,(0) = { " b,(a)¢,(a)da.
As p,(t, a) is a density
f" [3,(a, s)p,(t, s)ds f" f],(a, s)p,(t, s)ds
=
exists for each a and t using Condition A( iii).
72
D. GREENHALGH AND I. GYORI
LEMMA
1. Suppose that p,(t, a) satisfies {5)-{7). Define B,(t) = p,(t, 0).
Then
(9)
(10)
p,(t, a)
and B,(t)
-
{ ¢,(a- t)e- f~fo:li:(a-t+a,s)p,(a,s)dsda,
B,(t _ a)e- fo fo f3,(a,s)p,(t-a+a,s)dsda,
-
lot
b,(a)B,(t- a)e- fo'
a 2: t,
t > a,
fo= ji,(a,s)fJ,(t-a+",s)dsda da
+ /,oo b,(a)¢,(a- t)e- J; fo= ji,(a-t+,,s)p,(,,s)dsda da.
Moreover both of the double integrals in {9) and {10} exist.
Proof. Write x = t - a and y = a then
~p,(t, a) =
(11)
-p,(t, a)
fooo {J,(a, s )p,(t, s )ds,
and as previously observed the integral in (11) exists. Therefore
(12)
~ [-log,p.(x + y, y)] = fooo {J,(y, s)p,(x + y, s)ds.
If a 2: t then integrating (12) between -x and y we deduce that
p,(x + y, y)
= p,(O, -x)e- J~" fooo jj,("' ,s)p,(t-a+"' ,s)dsd"'
and the above double integral exists. Equivalently
p,(t, a) =¢,(a_ t)e- J; Jt ji,(a-t+,,s)p,(",s)dsd"
using (7). Similarly if t 2: a then integrating (12) between 0 andy
p,(t, a)
= B,(t- a)e- fo' fo= l!,(,,s)p,(t-a+a,s)dsd"
proving (9). (10) follows immediately, noting that B,(t) = p,(t, 0) and using
(6). The proof is complete. D
Hence if p,(t, a) satisfies (5)-(7) (so in particular is differentiable with
respect to a and t) then it satisfies (9)-(10). On the other hand if both of
the double integrals in (9) exist, p,(t, a) satisfies (9)-(10) and the left-hand
side of (5) exists then p,(t, a) satisfies (5)-(7).
LEMMA 2. Consider the renewal equation
(13)
B,(t)
=
t
b,(a)B,(t- a)da + F(t),
73
EXISTENCE AND UNIQUENESS
where F(t) = Jt"'' b,(a)J;,(a- t)da for 0 ::::; t ::::; T. Fix T with 0 < T < oo.
Suppose that I b,(s) I :<; Kr and I F(t) I :<; Mr on [0, T]. Then (13) has a
unique solution on [0, T].
Proof. This is a straightforward modification of the classical Banach
fixed point argument for the existence and uniqueness of solutions of a linear
Volterra integral equation of the second kind (see Theorem 1.1 on p.87 of [5]
and Remark 2.1.5 of [4]). D.
Consider (9)-(10). Define J;(u) = J;,(u) = ¢(u)efo" {'.,(l;)dt; and B(u) to be
the unique solution of the renewal equation (13). Our first result is to show
that if 0 ::::; T' ::::; T0 and T' is sufficiently small equations (9)-(10) have a
unique solution fortE [0, T0 ] with
and A > 0 and
M =
Bl
[O,T)
pi
E Ll(T, A) for all T';:: T > 0
[O,T]x[O,AJ
E £1 (T) for all T' ;:: T > 0. Consider the metric space
{(p, B) : pi
· [O,T]x[O,A]
is measurable for all T > 0 and A > 0,
o::::; B(t) ::::; B(t),
o : : ; p(t, a) :S {
and
Bl
[0,7']
ii: ~ ~i:
a ;:: t,
t
>a,
is measurable for all T >
o}.
M is a complete metric space with metric
II
(p, B)
II= max[
sup B(t), sup
O:;t'!,T
O<;,t<;,T
roo p(t, a)e- J; 1'->(i;)d(da].
Jo
To show that this metric is well-defined note that if (p, B) E M and A ;:: t
then by splitting the first integral at t E [0, A) and bounding each part
separately
Ap(t,a)e- Jora f'.,(l;)dt;da :S M 2 = loTo B(u)du
.
+ looo ¢(u)du < oo .
.loo
0
0
Hence for 0 :<; t :S T 0 , ./~A p,(t, a)e- .foa p.,(t;)dt;da is positive, monotone increasing in A and bounded above by M2 and II (p, B) II:<; max(M2 , M3 ) forT::::; T0
where M3 = SUPo<t<To B(t).
For t :<; T ~-To consider .fJ .fC: E,(a- t + J, s)p(J, s)dsdJ. This exists for any A and t and by splitting the inner integral at J if J E [0, A]
it can be bounded above by M 4 = M 1 M 2 T0 . Similarly by considering
.foa JC: E,(J,8)p(t- a+ J,s)dsdJ with T0 ;:: T;:: t;:: a and splitting the
74
D. GREENHALGH AND I. GY6Rl
inner integral at t - a+ a if t - a+ a E [0, A] this integral can be bounded
above by M4.
So we can define a map S : M -+ M by
(14)
Sp(t, a)
(15)
and SB(t)
¢,(a- t)e- J; fo= jj,(a--t+a,s)p(a,s)dsda'
{ B(t _ a)e- J; fo= jj,(a,s)p(t-a+a,s)dsda,
-
l
a 2': t,
t >a,
b,(a)B(t- a)e- fo" J;:" jj,(a,s)p(t-a+a,s)dsda da
+ {'" b,(a)J;,(a _ t)e- J; fo= jj,(a-t+a,s)p(a,s)dsda da.
Both the double integrals in (14) exist and are at most M 4 •
0
Clearly
and
0::; SB(t) ::;
l
< Sp(t a) < { ~(a-t),
-
'
-
B(t- a),
a 2': t,
t >a,
b,(a)B(t- a)da + [ ' b,(a)J;,(a- t)da = B(t).
So S : M -+ M is well-defined.
Let d be the distance induced on M by the metric II . II· We shall show
that if T is sufficiently small there is a constant K < 1 such that
For 0 ::; t ::; T, using the triangle inequality,
I SBl(t)- SB2(t) I
(16)
:::::
b,(a) I Bl(t- a)- B2(t- a) I e- fo" fooo jj,(a,s)p,(t-a+a,s)dsda da
(17)
(18)
l
+l
b,(a)B2(t _ a)e- fo" fo= iJ,(a,s)p,(t-a+a,s)dsda
11-
e- fo" fooo jj,(a,s)(p,(t-a+a,s)-p,(t-a+a,s))dsdalda
+ {xo b,(a)J;,(a _ t)e- J; fooo jj,(a-t+a,s)p,(a,s)dsda
11-
e- J; fo= jj,(a-t+a,s)(p,(a,s)-p,(a,s))dsdalda.
(16) can be bounded above by MsTd where M5 = supaE[O,To)
can be bounded above by
M3M5 lo{t
I b(a) I·
·
I eJor" Jor= tJ,(a,s)ip,(t-a+a,s)-p,(t-a+a,s)idsda1 Ida,
(17)
75
EXISTENCE AND UNIQUENESS
I :S
using the inequality 11 -e-x
elxl - l. Now for 0
:S u :S a :S t :S T,
fooo E,(u, s) I pz(t- a+ u, s)- p1 (t- a+ u, s) Ids
= fooo f3(a,s) I P2(t-a+u,s) -p1 (t-a+a,s) I e-f;p,,(f.ldEds,
:S M 1d.
Hence (17) is at most M3 M5 (eM,dT -1)T :S (2M1M_,M5 T)dT ifT is small
enough. A similar argument shows that (18) can be bounded above by
2MoM1dT if Tis small enough. So
I SB1(t)- SB2(t) I <
(M5 + 2M1M3M5T + 2M0 MJ)dT,
< K d, where K < 1 if T is small enough.
fooo I Sp 1(t,a)- Sp2 (t,a) I e- J;p,,(E)dEda
:S lo{' I BI (t- a)- Bz(t- a) I e- J."J.ooo o p,(<r,s)p,(t-a+<r,s)dsdu e- J."
o p,,(E)dEda.
Similarly
rt
+ lo B 2(t _a) e- f."0 J.oo
0 p,(<T,s)p,(t-a+o,s)dsdo
11 _ e- J; fooo ~,(o,s)(p,(t-a+o,s)-p,(t-a+<r,s))dsdol
+
e- j 0" p,,(E)dEda
J,(a- t)e- J.'o J.oo
o p,(a-t+<r,s)p,(<r,s)dsd<r
1
t 00
11 _ e- .r; .{
00
0
~,(a-t+<r,s)(p,(o,s)-p,(<Y,s))dsdol e- j~" p,,(E)dEda,
:S [1 + 2M1 M3T + 2M1 fooo ¢( u )du] dT,
< K d,
where K
if T is sufficiently small,
< 1, if T is small enough.
Hence if T is small enough (T :S T' say where 0 :S T' :S T 0 ) S is a
contraction operator on M with respect to the metric d and hence has a
unique fixed point (p*, B*) which satisfies (9) - (10) on 0 :S t :S T'. Now
define a second metric space
M
1
= M n { (p, B) : (p, B) = (p*, B') on [0, T']},
a complete metric space with metric
II
(p, B)
II= max [
sup
T'S:.t"!;2T'
B(t),
sup
1'''St':5;2T'
roo p(t, a)e- J; jl,(E,)dEda].
Jo
76
D. GREENHALGH AND I. GYORI
This metric is well defined as before and has associated distance d 1 on M 1 .
A similar argument to before shows that Sis a contraction operator on M 1
with respect to d 1 and has a unique fixed point (pj, Bj) on [T', 2T']. From
(9)-(10) (pj,Bi) = (p*,B*) at t = T'. Define (p,B) to be (p*,B*) on [O,T']
and (pj, Bi) on [T', 2T']. (p, B) is the unique fixed point of S defined on
[0, 2T']. Continuing in this way we deduce that S has a unique fixed point
(py0 , Br0 ) defined on [0, To] for any To > 0. A similar argument to above
shows that if T01 ::; To2 then (pr0 , Br02 ) restricted to [0, To 1 ] agrees with
(fYr01 , Br01 ) there. Hence we can define (p, B) on [0, oo) to be this unique
value. (p, B) is the unique fixed point of S on [0, oo).
By making the transformation x = t- a, y = a it is straightforward that
the linear partial derivative in (5) exists, if the partial differential operator
in (1) is understood as
8
[ 8t
8] u(t,a) = dhu(t+h,a+h).
d
+ 8a
(No differentiability in time and space separately can be inferred.) This
completes the proof of the existence and uniqueness of solutions to (1 )-( 4)
under Conditions A.
3. Conclusions. We have looked at existence and uniqueness of solutions of a perturbation of the McKendrick-Von Foester equation [1]. By
transforming the variables and working with the corresponding integral equations (9)-(10) we established existence and uniqueness of solutions under
much weaker conditions.
REFERENCES
[1] E.N. Boulanger, Small perturbations in nonlinear age-structured population equations, Journal of Mathematical Biology, 32 (1994), 521-533.
[2] D. Greenhalgh, I. Gyori, and I. Kovacsviilgyi, On an age-dependent population dynamical model with a small parameter, Mathematical and Computer Modelling,
31 4-5 (2000), 63-72.
[3] M.E. Gurtin, and R.C. McCamy, Population dynamics with age dependence, In: Nonlinear analysis and mechanics: Heriot Watt Symposium III, Pitman, Boston, 1979.
[4] W. Hackbusch, Integral equations: theory and numerical treatment, Birkhauser, Basel,
1995.
[5] R.K. Miller, Nonlinear Volterra integral equations, W.A. Benjamin, Menlo Park, 1971.
[6] A. G. McKendrick, Applications of mathematics to medical problems, Proceedings of
the Edinburgh Mathematical Society, 44 (1926), 98-130.
[7] H. Von Foester, Some remarks on changing populations. The kinetics of cellular proliferations, Grune and Stratton, New York, 1959.
[8] G.F. Webb, Theory of nonlinear age-dependent population dynamics, Dekker, New
York, 1985.