A class of nonlinear diffusion problems in age-dependent population dynamics

Nonlinear Analysis, Theory, Primed in Great Britain.

Methods & Applications,

Vol. 7, No. 5, pp. 501-529. 1983.

0362%546X/83/OSO501-29 SO3.m/0 @I 1983 Pergamon Press Ltd.

A CLASS OF NONLINEAR DIFFUSION PROBLEMS DEPENDENT POPULATION DYNAMICS* S. BUSENBERG~ Dipartimento

IN AGE-

and M. IANNELLI

di Matematica, Libera Universita di Trento, 38050 Povo (TN) Italy (Received 2 June 1982)

Key words and phrases: Nonlinear diffusion, age structure, population models, integro-differential equations, asymptotic behaviour.

1. INTRODUCTION THIS paper we study the existence, the uniqueness and the asymptotic behaviour of a class of nonlinear diffusion problems. These problems are motivated by models of age-dependent population dynamics which originate in the article by Gurtin & MacCamy [ll]. Only a few special cases of these models have been analyzed to date, with the most recent published work being that of MacCamy [17], Gurtin & MacCamy [12] and Langlais [15, 161. The cases treated in [12, 171 are restricted to special simple forms of the birth and death moduli that occur in these models and to a specific form of the nonlinear diffusion mechanism. On the other hand, Langlais [16] considers a different nonlinear diffusion mechanism and obtains a type of generalized solution for general forms of the birth and death moduli. In this work, we consider a general form of the nonlinear diffusion mechanism and we develop a method for establishing the existence and uniqueness of solutions of the resulting equations. Our method allows us to obtain classical solutions even with general forms of the birth and death moduli. We also describe the asymptotic behaviour of the solutions when the time variable becomes large, and apply our results to some specific examples. The equations that we study are motivated as follows. Consider a population that can disperse in a spatial domain R. For simplicity we take Q = J, where J is an open interval in R, but it will be clear from the context that more general regions Q of R” can be considered. Let ~(a, t, x) denote the number of individuals, per unit length and unit age, that are of age a at time t and are located at the position x E J. The total population, per unit length, at time t and at position x is given by IN

P(x, t) =

I0

ccu(a, t,x) da.

The derivation of the dynamic equations for age-dependent populations has been given in several places (see [9, 11, 131, for example), and we proceed directly to the problem we shall

* This work was partially supported by a Visiting Professorship grant of the C.N.R. and by the C.N.R. Grant No. 80.02333.01. t Permanent address: Department of Mathematics, Harvey Mudd College, Claremont, California 91711, U.S.A. 501

502

S. BUSENBERG~II~ M. IANNELLI

consider:

r

cdl +

( 44

0, x>= uo(a,xl.

u, +

p(a, t,x)u

= b(t,x, p, P&x

+ c(t,x, p, p,, PXJU,

1u(O,t,x) = i m @(a,t, x>u(a,t, x) da, We have adopted a notation where subscripts denote partial differentiation with respect to the subscripted variable, and where the independent variables are not shown if their suppression does not lead to ambiguity. Thus, in this notation, u, = &(a, t, x)/da. The specific hypotheses on p, /3, b and c will be given in the sequel where we will also discuss the type of boundary conditions that will be considered. At this point,we note some of the specific cases that can be included in our formulation of the problem. The spatial diffusion term bu, + cu includes operators of the form

[~~(P)Pxlx,

(1.1)

and in particular, it includes the case k[uP,], , k = constant, which was considered by MacCamy in [17]. The additional restriction ,~(a, t, x) = A = constant and P(a, t, x) = De-@‘, with c~ and /3 constants was imposed in [17]. Other cases included by the diffusion term are k[?

1 x

and

kc1 P,, k = constant. P

(1.2)

These are the age-dependent forms of the migratory diffusion terms introduced in [5]. The present formulation (P) does not include terms of the form [kPu,], which are studied by Langlais [15, 161 and which arise from diffusion due to random motion of individuals instead of resulting from total population pressures. The linear diffusion term uXX, in the context of nonlinear age dependent population models, has been studied by Di Blasio [6,7] and by Webb [22]. A discussion of nonlinear diffusion models for age-structured populations is given by Gurtin & MacCamy [ll]. In this paper we illustrate the applicability of our method of analysis by treating in detail the two diffusion mechanisms in (1.2). In a subsequent paper we shall consider other nonlinear diffusion terms. We are able to treat general forms of the birth and death moduli /3 and ,u, mainly because we can transform the problem (P) into an equivalent problem where the age-dependent dynamics is separated from the nonlinear diffusion mechanism. The splitting of the problem is also the main technical step that allows us to obtain the results on asymptotic behaviour. Once this splitting is accomplished, the major technical difficulties result from the nonlinear diffusion terms. The analysis of each specific type of nonlinearity in these terms would ordinarily require its own special methods. We note that this basic step of splitting the problem can also be carried out for appropriate models with vector valued u. The paper is organized as follows. In the next section we perform the formal decomposition of the problem which decouples the population dynamics from the diffusion portion of the problem. In section 3 we analyse the decoupled population dynamics part of the problem. Here, for the sake of clarity in the exposition, we restrict the treatment to the case where ,n and /3 are independent of t and x. The extension of the results of this section to the case where ,U and p depend on the two variables (a, t) is considered in section 5. In section 4 we treat

503

A class of nonlinear diffusion problems in age-dependent population dynamics the diffusion we give

part of the problem

existence

when

and uniqueness

the diffu_sion

results

for

terms

have

the forms

shown

in (1.2),

and

(P).

In section 5 we consider extension of our results that include different boundary conditions and 0 and p that can depend on the two variables (a, t). In section 6 we establish results on the asymptotic behaviour of the solutions of (P) and proceed to apply them to analyse some specific examples. Finally, the appendix contains the development of the technical tools needed in the analysis of the diffusion mechanism treated in section 4. A brief description of the method that is developed here for treating problem (P), has been given in [4]. 2. FORMAL In this section,

REDUCTION

we develop,some

preliminary

r 2.4,+ ua + pcl=

b(P,

Pa)&

OF THE formal + c(P,

PROBLEM

analysis

of the problem:

P,, Px*)u

~(0, t, x) = 0m/3(a)u(a, t, x) da i <

P(t,x)

=

440,x)

i0 =

u(u,t,x)

3

U(U,t,x)+

cc ~(a, t, x) da

(PI uo(a,x) 0 0

as

a-++m,

t E[O, T],x

EJ

without specific boundary conditions at this stage. We impose the following conditions on /j’ and ,u: p and p are continuous and bounded functions on [0, cc), so there exists y > 0 such that s(u) = y + P(u) - ,~(a) satisfies 0 C 6(u) C 2y. We now perform the following sequence of formal transformations u(u, t,x) =

of the problem. Letting:

u(u, t, x) P(LX) ’

then: vz =

u,/P - uP,/P2; v, = up/P; v, = UJP - uP,/P2

and, from (P): vt + Ua + yu = -v(logP),+

b(P, P,) (vx+

m u,(u, t, x) da = b(P, PJP,

+ c(P, P,, PXX)P - yP +

UPJP)

+ c(P, P,, P,,)v.

Also PI=

hence

J0

I0

a G(u)u(u, t, x) da

(2.1)

504

S. BUSENBERG and M. IANNELLI

Combining

(2.1) and (2.2), we get = G(a)u(a,

t, x) da

1 u

(2.3)

and P,=b(P,P,)P,+c(P,P,,P,)PNext,

let @(t, to, x) denote

the solutions

mG(u)u(a, I0

y-

[

of the following

t, x) da

(2.4)

1 P.

problem:

@z= A(& 4); @(to, to, x) =x

(2.5)

where: A(r, 2) = -b(W,

z), P&, 2))

(2.6)

and put w(a, t,x)

= +,

t, G(4 0, x)).

Then, w,+w,+pw=

y-

O”G(u)w(u, i0

[

w

t, x) da 1

and P,=b(P,P,)P,+c(P,P,,P,,)P

-

[

y-

z 6(u)w(u, I0

t, qb(O, t, x)) da

P. 1

Moreover CD w(a, t,x) da = 1;

cs P(O,x)

= PO(X) =

uo(a,x> da

uo(a,x> PO(X)’

w(u, 0, x) = wo(u, x) = -.

Thus, any solution of our original problem (P), for which our formal transformations can be justified, gives rise to a triplet (w, P, c$) which is a solution of the following set of three problems: wt+w,+/kw=

y[

x G(u)w(u, I0

t, x) da

1 w

@(a)w(a, t, x) da

w(a, 0, x) = wo(u,x>

I

w(u, f, x) 2 0, w(u,t,x)--+O

co

I

w(a, t,x) da = 1

0

as

u++~,W’t[O,T].xEJ

(PII

A class of nonlinear diffusion problems in age-dependent

- yP +P

P,=b(P,P,)P,+c(P,P,,P,)P P(O,x) = Pa(x);

4t(t, to,x) =

1 @(to,

20

P(t,x)

population dynamics

&)w(a, Im 0

4

w-4 t, 4) da

505

(W

-b( P(t, #Cl,totx>>, at, 4(t, to,4)) (P3)

to,

x>

=x.

We note that problem (PI)is completely independent of the unknown P, and the x variable appears in this problem only as a parameter. The solution w of (PI) enters in the definition of ( P2), while the solution P of ( P2) enters in the definition of (P3). So, problem (PI)can be studied separately at first, and this is what we proceed to do in section 3. We also note that, in the “small gradient case” (b = 0), problem ( Ps)is trivial and has the solution @(t, to, x) = x, that is, the variables u and w are identical. In this case, we need to deal with the two problems (PI)and (P2) only, and the situation is somewhat simpler. The advantage of this particular reformulation of the problem (P)is due to the separation of the diffusion problem from the age dependent birth-death dynamics equation. Consequently, the questions of existence and uniqueness of solutions can be settled by analysing the properties of the solutions of the diffusion problem (Pz), without having to simultaneously deal with the age-dependent dynamics. We note that difficulties in the treatment of (PI) may arise from its nonlinear character. It will turn out that what we really need is an existence theorem for sufficiently regular solutions of ( Pz). Here is a sketch of the procedure that we will follow in treating the problem (P).We first show that, given sufficiently regular initial data, problem (PI)has a unique regular solution w. When this solution is substituted in (Pz), we are left with a nonlinear partial differential equation which describes the diffusion process in (P).In the “large gradient” case, part of the nonlinearity in the problem ( Pz)comes from the problem (P3) which essentially defines the characteristics in the space variable x. The existence and uniqueness result is completed by showing that, whenever the problems (P2)( P3) have sufficiently regular solutions, then the function u(a, t, x) = P(4 x)w(a, t, @(O,t, 4) is a strict solution of the original problem (P). 3. THE AGE-DEPENDENT

DYNAMIC

PROBLEM

We consider the problem (PI)which describes the age-dependent defining the following sets which we will use in the sequel: E = [0, T] x J;

F =

dynamics. We start by

[0,+ a) x [0,T]; G = [0,+ m) x.f; H =[O,+ m) x [0,T] xJ

and we denote the respective elements of these sets by: (4x);

(44;

(a,x);

(4&x).

We will use the following function spaces: C(A, B), ‘=‘(A, B), L’([O, + m); B) denoting continuous function from A to B, continuously differentiable functions from A to B, and integrable functions from [0, + ~4) to B, where A can be any one of the sets E, F,

506 [O, +

S.BUSENBERGandM. IANNELLI cc),

[0,T] and B can be any one of the functions

P(J). We define

spaces

L’(0, + cc), C(J),

C’(J),

R as follows: R = {w E C(E;L'(O,+ CO));w,, w,,EC(E;L'(O,+ a));~SO}.

Our goal is to prove THEOREM 3.1. such that:

the following

theorem:

Letwo E C'([O, + m);C(J))flC([O,+ w);C*(j))n L'([O, + =);C2(.f)) be

wo(a, x) 3 0,

CD w,,(u,x)

da = 1,

i0

vx EJw&r,x)

-0

as

a-++~

wo(o,x)= 1mP(~)w~(v) da

(3.2)

JO

woaE L’([O,+ WI; c(:J))

!

(3.1)

and

WOa(O, x>+,@)w0@,x>=

Then

there

exists a unique

solution wE

of problem

(Pi).

R IIC'(F;C(J))n C(F;C*(J))

If, in addition,

for some P E J

woX(a, i) = 0

for all

a E [0, + w)

(3.4)

(a, t) E F.

(3.5)

then w,(u,

t,X) = 0 for all

The conditions (3.2) and (3.3) that we impose on the intial data are simply a requirement that these data agree with the dynamic equations and the renewal law in (PI) at t = 0. These conditions are imposed in order to avoid discontinuities along the characteristic a - t = 0. Moreover, the smoothness restrictions on the variable x are imposed on the initial data in order to obtain a sufficiently regular solution w that allows a construction of a strict solution of the initial problem for u. However, from the proof of the theorem it will be clear that the following corollary also holds. 3.2.Let wo E C’([O, + m); C(j)) fl L'((0, + a);C(J))[respectively, w. E C'([O, + ~0); C(j))flC([O,+ m);C'(J))flL1([O, + m);C'(j))] and suppose (3.1)-(3.2).

COROLLARY Then

problem

(PI)

has a unique

solution

w E c?(F,c(J))n c(E; L1(o,+ x)) [respectively,

w E C’(F;

C(J))

fl C(F;

C’(j))

L’(O, + @+I. Before giving the proof of this theorem, For any w E R, define: D(q,

ai,xlw)

= exp

n C( E;

we perform

L'(0,+ =)) such that w, E C(E;

some preliminary

nz = 6(u)w(u, [J-I ffi 0

S, x) da ds

steps.

I

A class of nonlinear

diffusion

problems

= {(R,

4:

in age-dependent

population

507

dynamics

with: (@2, ai>

The function: notations:

((yz, EI,X)+

E r

0 s~,=scv~GTT)

and

XE.!.

D(Lx~, al, x/w) belongs to C’(T; C’(f)).

Moreover,

using the

the following estimates can be seen to hold:

(3.8)

s,x) - @(.,s,x)ll+ llwx(.,.w)-@x(.,s>x)II>ds

+

IlwxC, s, x)

(3.9)

(3.11)

- @Aa, s, x) 11)ds.

Here the constant & depends on y and T only, while the constants K1 and K2, in addition to their dependence on y and T, are monotone increasing functions of the parameters specifically exhibited in the estimates (3.9), (3.10), (3.11).* If we regard the terms in the square bracket on the right-hand side of the first equation of (PI) as given, this equation can be formally integrated along characteristics (see, for example, [lo] where a similar calculation is done in detail) and the result substituted in the second equation of (PI) to obtain the following equation for B(t, x) = w(0, t, x) B(t,x)

= F(t,x)

+

K(t,s,x)B(t

-s,x)

ds,

(t,x)

* All the constants that will appear in the various estimates used in the sequel addition, will be increasing functions of the parameters specifically exhibited.

EE

will depend

(3.12) on y and T, and in

508

!%BUSENBERG and M.

IANNELLI

where, for a given element w E R, and a given initial datum w. satisfying (3.1)-(3.3), K are defined by:

F and

F(t, x) = eyf D(t, 0, x Iw) lffi P(a)TI(a, a - t)wo(u - t, x) da

(3.13)

K(t, s, x) = P(s)II(s) eYsD(t, t - s, x Iw)

(3.14)

with I’I(a, b) = exp [ - [P(S)

ds] ; II(a) = II(u, 0).

Using the estimates (3.6)-(3.11), and employing standard contraction is possible to prove the following result:

mapping methods,

it

3.3. Let W-E R, then there exists one and only one solution B E C([O, T]; C*(J)) rl C1([O, T]; C(J)) of (3.12) with F and K defined by (3.13) and (3.14). If B(t, xjw) denotes such a solution, where we explicitly exhibit the dependence on W, the following estimates hold: PROPOSITION

OcB(t,xlw)cCo IB,(t,+)~

~G(lllwozIll)

(3.15) (3.16)

(1 + /‘~tw,(~)lidsj 0

(3.17)

jB(t,xlw)-B(t,xt~)I~Co~~~w(..s,x) -W,s,x)IIds I&(t,xlw)

- &(t,xl~)

=GGII~~oxIII, Illw,lll)

,s,x)-W(.,s,x)ll+llw,(.,s,x) x I ot~llw(~ I &xk

x I w> - &x0, ss c4(lllwoxlll~ x

(3.18)

-%(.,s,x)(I)

ds

(3.19)

x 1w> I IIIw*lll,

III~XIII~ IIlwoxxlll~

/Ilw~~llI~

Ill~xxlll>

I o’w(.. 3,x) - *( -9s, x>/I+ ll%( .>$7x>- W*(-23,x>II

+ Ilw&.v)

-w*x(~~,x)lI)ds.

(3.20)

Using again a formal integration along the characteristics a - t = constant of the first equation of (PI), we obtain a relation for w which we can use to define a fixed point problem for w in the following way. Let r: R + R be defined as follows: wo(u - t, x)fl(a, (tw>(a,

t,x> = (

a - t) eW(t,

B(t-u,x(w)II(u)ePD(t,t-u,xjw)

0, x Iw)

ifu > t ifast.

(3.21)

A class of nonlinear diffusion problems in age-dependent

509

population dynamics

It is clear that the definition (3.21) of r is meaningful for w E R. It is also easy to show that rw is in R. In order to see this, consider the following formal derivatives:

[w&a - t,x)D(t, 0, xjw) + w@(u- f, x)D.(t, 0, x jw)]II(u, a -t) if (ZW)&,

eyt, u>t

4 x> =

(3.22) 1 [B,(t - a, xlw)D(t,

t -

a,

x Iw)

+

B

(t

-

a,

x (w)D.(t,

t -a,

x Iw)]

n(a)

if [woxx(a - t, x)DO, 0, x lb4 + 2wo*(u - t, xP&

+ wo(u - t,x)Dxx(t, 0, xlw)]II(u,

a - t) eyr,

0, x

eYa,

ast

lb4

if a > t (3.23)

(~w)xx(Q, 4 x) =

[B,(t - a, x(w)D(t, l - a, x Iw) + 2B,(t - a, x Iw)D&, t -a, x Iw) + B(t - a, xIw)DJt,

1

t - a, x Iw)]II(a) eYa,

if a S t.

By

the properties of w. (in particular see (3.2) and the relations that it implies for woXand woXX)tw, (rw),, (tw), belong to C(H). Moreover, using the above expressions, we can also get the following bounds showing that for each (t, x) E E, tw, (zw), and (rw),, belong to L’([O, + a)):

ll~W(~,~,4ll~~O IlWx(

.

(3.24)

,~,~)Il~~~(~Ilwo~/ll) (1 + j-hC,s~lld~) 0

(3.25)

lilwxlll> (1 + j-ll%( ~,w>~l ds). IIWM *>O)li~ffdb’oxlll,Illwoxxlll> 0

(3.26)

Next, note that for any A > T: omIzw(u, t, x) - zw(u, I, X) 1da 6 [lrw(q I

t, x) - rw(u, t, X) I da + 2 eY' Lrw

o(a, x) da.

Now, owing to the integrability of WO,A can be chosen so as to make the second integral as small as we please, while the first integral can be made small because of the continuity of the integrand on [0, A] X E. Frcm this it follows that tw E C(E; L’(0, m)). From analogous estimates, the same is true for (rw), and (rw),. Finally, WC, > 0 yields tw 2 0 because of (3.21) and (3.15). The following estimate, which can be derived in a straightforward manner, will be used in the proof of theorem 3.1.

Ilrw( . >t, x>

- TN * >t, 4

II+ ll(~W)x( .Y&X)-(t~)*(.,t,X)ll+ll(tW),,(.,t,X) -W)*.(.,t,X)ll

=zff3(lIIwoxlII~ IIIwoxxxlII7IIIwxIII~IIl~*llI~lIIw*IIl~III~x*IIo (3.27)

510

S. BUSENBERG and M. IANNELLI

we are now prepared

to give the proof of the theorem:

Proof of theorem 3.1. The proof will consist of showing that tleaves defined convex is a contraction

subset of C([O, T]; L’((0, + a); on that set. To this end, choose

R > ffd/~bdI) Then

choose

numbers

C*(J))) positive

invariant an appropriately and that there exists IZ such that r” numbers R and A4 such that:

ff~(lIIwoxlII)R andM > R - ff~(tIl~oxIl/)’

S and N such that

lllw~~~l/l, R eMT)

S >&(11I~ili, and

Define

the set RO C R by

R0 = {w E R: l/wx( ., t,x)l/ s R eMr,IIwxx(*, t,x)li SS e”“}. Now, from (3.25) and (3.26), if w E Ro, then

Ilw4x(~, t,x)l/

~~~(Illw~~ll()(l

+ R l’eMsds)

From these inequalities, and the fact that (3.27) it follows that for w, ti in RO

c e”‘fJ1(IIlwd/)

t: R + R, it is seen that

(1 + $)

GR

eMt

r: Ro+ Ro. NOW, from

gw- iGIl*

IIz”w - t”WII* =s where

and

IM* = ($JE [II4 *>t,XIII+ Ilw(->t,XIII+ Ilwx(*>t,x>Ill. Thus, if IZ is sufficiently large, r” is a contraction in the norm induced by /I I(*. Since the set R. is closed in this norm, there exists a unique fixed point w E Ro of the mapping r. Thus we have found a solution w E Ro to the equation wo(a - t, x)JYI(a, a - t) eyrD(t, 0, x Iw),

a >t

B(t - a,xlw)n(a)

a St.

(3.28)

w(a, t,x) = eY”D(t, t - a,xlw),

A class of nonlinear

diffusion

problems

in age-dependent

population

dynamics

511

It immediately follows from this equation that w E C’(F; C(j)) fl C(F; C’(j)). Note that condition (3.3) is needed in checking that w is differentiable in a and t. Moreover, by directly substituting this expression for w in the left hand side of the first four equations in PI, it is easy to see that w satisfies the equations. Conversely from the derivation of the fixed point problem for t, every solution of (Pi) with the above regularity, must satisfy the equations. Finally, we show that the above fixed point satisfies Jo”w(a, t, x) da =l. In fact, letting

Q(t,x) = 6” ~(a, t, x) da, then for all x E .!, condition (3.1) of the theorem implies Q(0, x) = 1, while the first and second equations of (PI) imply that Qt exists, and

Qt= [I- Ql I icG(a)w(a,t, x) 0

da.

Thus, Q(t, x) = 1 for all t 3 0, and our claim is established. From the fact that B satisfies (3.12), it is seen that, if we fix A?E j and choose wg and w E R such that wax and w, vanish at X for all (a, t) E F, then B, must vanish at X. From the definition (3.21) of the operator z, it follows that, if wax and w, vanish at X, then (zw), also vanishes at X. This is so, because of the above noted behaviour of B, because of (3.22), and the fact that D,(t, t - a, XIw) vanishes at f whenever w, vanishes there for (a, t) E F. Hence, the closed subset of R given by {w E R : w, = 0 at 2, for (a, t) E F} is invariant under r. Since the fixed point is unique, the conclusion (3.5) of the theorem follows immediately. 4. SEMI-LINEAR

(Pz)

PROBLEMS

Here we treat a class of nonlinear diffusion problems which lead to a semilinear (Pz) problem. The type of diffusion term that we use has been chosen to model population migration where the pressure to migrate depends on the total populations and its gradient, while the flux in each age class is proportional to its relative size in the total population. This type of diffusion is discussed in [5] for a population model that does not involve agedependence. In the small gradient case, we obtain the following problem: u, + u, + p(a)u =

k; P,

~(0, t, x) = o- /3(a)u(u, t, x) da I

(4.1)

P(t, x) = l= ~(a, t, x) da

and in the “large gradient” case, we obtain the problem ut + ua + p(a)u = k

[u(O,t,x),P(t,x),u(a,O,x)

X

(4.2) givenasin(4.1).

512

S. BUSENBERG and M. IANNELLI

The diffusion ,operator in (4.1) and (4.2) is not defined when P = 0, and we treat the nondegenerate case where P > 0 on .!. We assume Neumann conditions on dJ, since this type of data is compatible with the non-degeneracy of the problem: ~,(a, t,x) We seek a nonnegative

=0

for

strict solution,

U E C(F;

x E dJ

and all

(a, t) E F.

that is, we seek a function

C’(J>) n C’(F; C(J))

n L’((0,

(4.3)

u such that

+ m); C([O, z-1; C’(j)))

(4.4)

and u(a,t,x)~OonH,P(t,x)>OonE,

V((t,x) EEu(a,t,x)-+O

as

a-w.

(4.5)

and such that (4.1) (respectively 4.2)) and (4.3) are satisfied. In order to get such a solution, we need to impose the following compatibility conditions on the initial datum u(]: ug E C([O, + 03); C”(j)) uo(u, X) 2 0, PO(X) =

I0

n Cl([O, + m); C(J)) co uo(u,x)

uo(O,x)

=

da >O,uo(u,x)

i

o= B(a>uo(a,x)

Uoa E L’((O, + w); C(J)) =

I

omB(~)[uo&v)

uoX(u,x) = 0 We have the following

THEOREM

one strict, Proof.

theorem

and +

for

x E 8J

+O

+ a); C’(.!)) as

a + + ~0

da

UO@,,~)

lu(a>uo(a,x>l

for problem

4.1. Let u. satisfy conditions non-negative solution.

l-l L’((0,

(4.7)

(4.8) +

,~O>UO(~,X>

da

and all

(4.6)

(4.9) a E [0, + ~0).

(4.10)

(4.1)

(4.6)-(4.10),

then the problem

(4.1) has one and only

Setting wo(a, x) = uo(a, X)/PO(X)

(4.11)

weseethatwo EC’([O, + w),C(j)) II C([O, + w); C’(j)) fl L’((0. + a); C’(j)) and conditions (3.1)-(3.3) and (3.4) f or x E 8J are satisfied. So by theorem 3.1, the problem (PI) has a unique solution which satisfies the Neumann condition (3.5) on dJ. Let w be the solution of (P,), and let A(t, x) = lffi G(u)w(u, t, x) da

(4.12)

Note that because of condition (3.3), the expression for ~,(a, t, x) given by (3.28) belongs to C(E; L’(0, + w)). Th’1s can be seen using the same type of argument as was already employed in section 3, when showing that rw EC(E, L’(0, + m)). From this, and the fact that w E R,

A class of nonlinear diffusion problems in age-dependent

it follows

that: k(t, x) 5 0, A E Ci([O, T]; C(J)) n C([O, T]; CZ(J)).

Now, the problem

(P2) corresponding

to (4.1) has the following

(4.14)

I P(0, x) = PO(x), P,(t, x) = 0 The problem

(4.14) has a unique

strict solution

P E C1([O, T]; C(j))

(4.13)

form:

P, = kP,, - yP + A(t, x)P

Finally

513

population dynamics

for

x E M.

P (see appendix and

fl C([O, T]; C’(J))

Al for the details),

with

P(t, x) >O.

put : u(a, t, x) = P(t, x)w(a,

t, x).

It is easy to verify that ~(a, t, x) is a strict nonnegative solution of (4.1). We need to show that this solution is unique. Let q(a, t, x) be any nonnegative, strict solution of problem (4.1) md define

Q = lz da, t, x> da.

= - +I

Then, from (4.1) we see that Qt exists, and Qt

kQxx

YQ

O” @Ma, t, x>da

0

Q(0) = PO, QX = 0 By the maximum

principle,

it follows

x E CU.

that

Q(t, x) > 0 and hence,

for

(4.15)

for

(t, x) E E

we can define: ~(a, t, x) = q(a, t,

x)/Q@, x>.

It is easily seen that z satisfies problem (PI), hence, z(a, t, x) = ~(a, t, x). Thus (4.15) becomes identical to (4.14), and hence, P = Q. This implies q = WP = u, and the theorem is proved. The treatment of the problem (4.2) is more complicated because we have to deal with Nevertheless, we have the same result in this case problems (P2) and (P 3) simultaneously. also. Let uosatisfyconditions (4.6)-(4.10). Then the problem (4.2) hasone andonly one strict, nonnegative solution. The proof is set up by a fixed point procedure, for which we need some preliminary analysis of the problems (Pz) and (Ps). Let 7j satisfy

THEOREM~.~.

7 E C’([O, q> 0

on

Tl; Go) fl C([O,Tl; C2(J>) E,

qX(t,x) = 0

for

x E aJ

(4.16) and all

t E [0, T]

S. BUSENBERGand M. IANNELLI

514

and consider

the problem P, = kP,, - yP + VP

(4.17)

1 P(0, x) = PO(x), P,(t, x) = 0

for

x E &I

where PO(X) =po” uo(a, x) d a and, from (4.6), (4.7), (4.10) satisfies: POE C’(J), Pox = 0

on

The problem (4.17) has a unique strict solution explicitly show the dependence on yl. PROPOSITION

4.3. The following

estimates

aJ, P,, > 0

on

which we denote

hold for problem

P(t,x/q) IP(L+)/

J. by P(t, xl 11) in order

to

(4.17).

SC”>0

(4.18)

c Gr(llnil=)

(4.19) (4.20)

(4.21)

Jo

- Q(s,.>lx + Ids, where

.) - %(s, %) ds

and in Ildlm =ct~gElvk~!l~ Ids?% = ~~plv~~~~~l.

indicated their dependence on PO and P ox. In the sequel, dependence of constants on POand Pox.

(4.22)

the constants

GrG4

we will continue

we have not

suppressing

the

Proof. (4.18) and (4.19) follow from standard estimates on the problem (4.17). The estimate (4.21) is a consequence of the fact that A = P(t, XIV) - P(t, xl?) is the strict solution of 1 Ar = kA, (A(0,

- YA + VA + (r - rS)P(4)

x) = 0, A,(t, x) = 0

on

CU.

Finally, (4.20) and (4.22) are a consequence of the fact that Q = P,(q) and A = P,(q) P*(q) are the strong solutions (see the appendix Al) of the following respective problems

Qr = kQxx - YQ + rJ’(r7) + rQ i

Q(o, xl = Pox(x), Q(t, x> = 0

A,=kA,-yh+yA-(r-4)P,(~)+(r,-~dP(r)+Ilx(p(r)-P(rS)) A(0, x) = 0, A(t, x) = 0 1 This completes the proof.

on

dJ.

on

&I

-

A class of nonlinear diffusion problems in age-dependent

population dynamics

515

Next, we consider the problem (J’S): @At,to, x) = A(& Q (t, to, x) /~1

(4.23)

@(lo, ro, x) = x where A(t,

zlq) = - k Px(t,zIv),toE[O,T],xE~

fYt>zl4

and P(t, xlq) denotes the solutions of (4.17) for fixed rl and PO satisfying the appropriate conditions that were discussed above. _ _ The problem (4.23) has a unique solution 4 EC’([O, T] X [0, T] X I; J). This follows from the fact that A(t, zlq) is Lipschitz continuous in z, uniformly in t E [0, T] and A(t, 21~) = 0 for z E U. Denote this solution by $(t, to, xlq) in order to indicate the dependence on r~. Now, note that the following estimates hold:

-

$6, .>lm+ MS, a> - Tsx(s,.)lm>ds

A(t, h(t, x)

Irl) = - $ln(f’(t,h(t,x>14) + ;A’(&

where h E C’(E,J). quence of:

h(t, x) Ir) - ;A@,

(4.25)

Y + v(t, h(t, x)) h(t, x) 1v)h(t,

In fact, (4.24), (4.25) follow from (4.18)-(4.22)

h

P(tN,x)lv)

(4.26)

while (4.26) is a conse-

Pt(t>W>x> 1rl) + P& h(t, x>1r> -$ln(f’(t, h(t, x> Iv>>= P(t,W,x)/v)

x).

(t

x>

’’

and (4.17). Then we have the following result. PROPOSITION 4.4.

The following estimates hold for problem (4.23): l$(l,

toA

=sMl

0 s @X(GtoA?)s ~2(IlrIlcc)

IQwO>4d- ~Wo,4$l+ l~tWo,4r> - CPrWo,xl~~l + I&l(t~to> x Irl) - @kit,to,x I$ I + I&(t,to,x Irl) - M, to,x Ifj) (

where f = max(t, to).

(4.27) (4.28)

516

S. BUSENBERG and M. IANNELLI

Proof. The estimate

(4.26) is trivial

because

+ E j. From

(4.23),

it follows

that:

and (4.30) so that by (4.26): MY to,4

r) = exp[ - r@ -

to> + iO’ 4~~ G(s, to, x Iv>> ds]

P(t >

From (4.31), (4.18) and (4.19) it is seen that (4.28) holds. by noting that A(t) = $(t, to, xln) - $(t, to, xlq) satisfies:

$ A(t)

=A@,

@(vo,xlv)Ir)

-A(&

~~~“;6:“, >7

In order to prove

(4.31)

,+

(4.29),

we start

~(~>~o>~lr5)IrS) +A@> dWohIr5)

+

f >CJ9(cto,xlv) + (1 - +#Wo,+/l4) A(0 J,-Az(

da

so that 1’1’ A, ( r, o@ ( r,to,xln)

A(t) =[‘exp[

+ (1 - o)$(~o>~Ir5)j$

x [A@, @Wo,xIv>h)

Now, the following

Ii

‘&(r,

s

dodr

s 0

0

estimate

follows from (4.24)-(4.26)

N(r,t~,xIv) +(I - 4W7to,x14j11?)dr

-A@,

1

@(vo,~/rl)I$] ds.

and proposition

(4.32)

(4.3):

~~~(ll~ll-~ 1141lm~ llVxllm7 Il4dlm).(4.33)

The estimate (4.29) now follows from (4.25), (4.30)-(4.33) and the estimates of proposition (4.3). This completes the proof of proposition (4.4). A direct consequence of the estimates in these propositions is that the mapping: _ _ rl+ G(& to, 44 : C’(-q -+ cm Tl x LO,Tl x J; J) can be extended

to all the rl in the set K = {n E C([O, T]; Cl@)),

n 2 0, rj*(t, x) = 0

on

U]

so that (4.29), (4.30) and (4.31) still hold. We still use $(t, to, xlr]) to denote this extension. Now we are ready for the proof of theorem 4.2. First, we proceed as in the proof of theorem 4.1, that is, we put wo(a, x) = Uo(K X)/PO(X) and consider

w ER fl C’(F, C(J)) n C(F, C’(j))

th e solution

of problem

(PI) with datum

WO.

A class of nonlinear diffusion problems in age-dependent

517

population dynamics

Next, we put A@, x) = irn G(a)w(a, t, x) da. A is such that (4.13) holds. Define the mapping t: KC putting

C([O, T]; C’(J)) + C([O, T];C’(J))

for all rj E K: (4.34)

(V) (t, x) = %? G(O, t>x IV)). This mapping

is constructed

following

the diagram:

rl+P(ln)+

@(IV)+

A@?@(O>0lr)).

In this way, an eventual fixed point will yield the proper E = A(t, $(O, t, xlQ> E C’([O, T]; Cl(j)) so that P( IE) and $( ]E) will solve problems (Pz), (Pj) (in the special case of problem 4.2). The following estimates hold: 0 S (rn)(t, I(~M~~)I l(Ttx In addition

I

TI>(4X)I

o (Ids,

+ lb7 - m&x>I

.) - r(s, .)I= + Ids,

(r~)~(t, x) = 0 on U because (rn),(t,

x) S No

(4.35)

zs N(ll~ll~)

(4.36)

c N2m?llw ll~ll=~ IIllxll=>IhI-> .) - MS> .)I-)

(4.37)

ds.

of

x) = lrn Ww,(a,

t, @(O, t, x in)) da&(O,

t, x In)

and for x E aJ %(a, t, @(O, t, x Irl)) = wx(a, 4 x) = 0. Thus,

t maps the following

closed convex

set of C([O, T]; C’(f)): c No, 11 llxll ccs NdNo)l

Ko = {V E K, ]I& into itself,

where

and

I(r$ =,,sug,

(iq(t, x)1 + I q,(t, x)1)

is

the

norm

in

C([O, T]; C’(J))

and

K =

TN(No, No, ho), NWo)). F rom this it follows that r has a unique fixed point E E Ko. Note that 5 is also the unique fixed point of r in C([O, T]; C’(j)), in fact, due to (4.35) and (4.36), any fixed point of r belongs to Ko. Moreover, due to the regularity properties of A and +, we also have that E E C’(E). Finally put: u(a, t, x) = P(t, x I EM(a, t, 440, t, x In).

(4.38)

518

S. BUSENBERGand M. IANNELLI

This is the desired solution of problem (4.2), as can be easily checked by direct substitution and noting that @ is two times continuously differentiable with respect to X, owing to (4.31). As far as uniqueness is concerned, we may proceed as in the proof of theorem 4.1. In fact, if q(a, t, x) is a solution of the problem (4.2), then Q(t, x) =Jl$ q(a, t, x) da must satisfy:

Qc = kQxx - rQ + 1 which implies

I0

Q(0) = PO, QX = 0

Oc@Ida, on

U

Q > 0 on .!. Put Z.7 cXa)q(a, t, x) da I c= O

Q(t,x)

then Q(t, x) = l’(t, XI C), and moreover, z(a

satisfies

t, x> da

problem

(Pi).

’

the function

t X) = 4(a, t7 #(G 0, xl 03)

33

Qk @,(t>0, x I C)>

Consequently w(u, t, x) = z(a, t, x).

Thus we have f(r, x) = [ and c coincides

with g, the unique

+)w(a,

fixed point

t, $J(O, f, x!<)> da of the mapping

q(a, t, x) = fyt, x IE)w(a, t, 4m and this completes

t defined

in (4.34).

Thus

t, x IEl> = 4% t, x)

the proof. 5. EXTENSIONS

OF THE

PREVIOUS

RESULTS

We consider some extensions of the results of the previous sections. These extensions can be handled with only small changes in the procedure already used. We will treat three different topics: a generalization of the boundary conditions, the case where J = R is unbounded, and the case where /L?and ,u depend on the two variables (a, t). We start by looking at the problem with more general boundary conditions. In section 4 we treated only the Neumann condition (4.3). Our results also hold for mixed conditions of the form pu(u, t, x) + v(x)u,(u,

t, x) = 0

on

aJ

for all (a, t) E F,

(5.1)

where p 3 0 and V(X) = + 1 at the right extreme of the interval .!, and V(X) = - 1 at the left one. Condition (5.1) is still compatible with the nondegeneracy requirement P(t, x) > 0 on E. We first note that (5.1) induces a Neumann condition on w and the condition pP(t, x) + v(x)P,(t, x) = 0 on &7, t E [0, T]. So, the problem (PI) remains unchanged, and theorems 4.1 and 4.2 can be proved in the following form.

A class of nonlinear THEOREM

5.1. Assume

diffusion

problems

that u. satisfies

in age-dependent

population

conditions(4.6)-(4.9)

puo(a, x) + v(~)u&a,

x) = 0

on

519

dynamics

and that

&/ for all a E [0, 00).

(5.2)

strict solution Then the problem (4.1) (respectively (4.2)) h as one and only one nonnegative, (that is, a solution satisfying (4.4) and (4.5)) such that (5.1) holds. The proof of this result is essentially identical to that of theorems 4.1 and 4.2. One need only note that the condition (5.1) guarantees that problem (4.23) still has a global solution. Only trivial changes occur in the treatment of the problem on the whole line (that is, J = (- co, co)). In this case, we have essentially the same result as before.

THEOREM

5.2. Let J = [w and assume

the following

conditions

on ~0:

L’([O, m); C2(J)),

uo 2 0, uo E

PO(x) = lE ~(a, x) da > 0

wo(u, x) = uo(u, x)/PO(x) satisfies the conditions

of theorem

on

(5.3)

J,

(5.4)

3.1 (excluding

condition

(3.4)). (5.5)

strict solution, Then the problem (4.1) (respectively (4.2)) h as a unique nonnegative a solution r.4E C(F; c;(J)> fl C’(F; COV)) fl L’([O, m); NO, Tl; G(J))) such that (4.5) holds. The functions used in this theorem are defined in the same way as in section 3, with the added fact that the subscript zero denotes functions that infinity with the supremum norm. We now consider the non-autonomous, non-homogeneous case, that is, and ,u depend on t, in addition to their dependence on a. Also, b and c may X. So, we consider the problem I

that is,

the beginning of approach zero at the case where /3 depend on t and

ut + u, + ,~(a, t)u = b(t, x, P, J’x)u, + c(t, x, P, P, P&G ‘/I@. t)u(u, t, x) da,

We make the following

assumptions

about

/3 and p are nonnegative

(5.6)

/3 and ,U functions

on Hsuch

that /3, p (5.7)

8, ,LL~, are continuous

and bounded.

In this new situation, the procedure developed in sections 3 and 4 remains unchanged. Some changes in the hypotheses need to be made. In particular, in theorem 3.1, condition (3.2) and

520

S. BUSENBERG and M. IANNELLI

(3.3) have to be replaced by the following:

(54

wo(O,X) = or &(a, O)wo(a, x) dla, I woaE L’([O, m); P(J)) and ~oa(O,x) + ~(O,O)wo(0, x) = lrn P(a, O)[wo,(a, x) + ~(0, O)wo(a,x)1 da

(5.9) -

I

o1 Pt(a, O)wo(a) da.

The conditions (5.8) and (5.9) induce obvious changes on the conditions (4.8) and (4.9) on uo, and theorems 4.1 and 4.2 still hold. 6. STEADY

STATE

SOLUTIONS

AND

LIMITING

BEHAVIOUR

In this section we study the time independent solutions of problem (P) and the limiting behaviour as t+ CQof any solution of (P) when /3’and ,Bdepend on a only. We also apply our results to some particular examples. If ~(a, t, x) = ~(a, x) is independent of t, then so is the total population P(t, x) = P(x) = J 0” ~(a, x) da. Consequently, w(a, t, @(O, t, x)) = ~(a, x)/P(x), x E 1, is also independent of t. If $(O, t, x) + IT as t + cc, then ~(a, X)/P(X) = w(u, m, X), and we are lead to seek the stationary solutions of the problem (Pr). It will be seen that these special solutions of (PI) play a major role in the asymptotic behaviour of the problem (P). They must satisfy the following problem

~(0, x>= i

I

()*P(a>w(a,x>da,

(6.1)

XI

I

w(u,x) da = 1

0

as

w(u,x)+O

a+

w.

We start by noting that, since /3(u)II(u) > 0 and /3(u)II(a) f 0, the equation I

om/3(u)II(u) e-$‘du = 1,

has a unique real solution s = p*. So, p* is the unique real number satisfying I

omp(u)II(a)

e-pea da = 1.

(6.2)

We can now state the following result. THEOREM 6.1.

Let p* be defined by (6.2) and suppose that Jr e-P”TI(a) da < co. Then

(P)

A class of nonlinear diffusion problems in age-dependent

admits the following nontrivial, time-independent, u(a, x) = P(x)II(a)

population dynamics

521

strict solutions

e-p*’

m

e-P*"II(a)

da,

(6.3)

provided P solves the boundary value problem b(P, P,)P,

+ c(P, P,, PXX)P -p *P =O, pP + VP, =0

on

U.

(6.4)

Moreover, if b = 0, the solutions given by (6.3) are the only nontrivial, time-independent, strict solutions of (P). Before proving this theorem, we give the form of the solutions of (6.1). We have the following lemma. LEMMA 6.2. Let p* be given by (6.2). Then w E C’([O, a); C(j)) fl L’([O, w); C(J)) is a nontrivial solution of (6.1), if and only if, Jr e-P*‘TI(a) da < CQ,and

cDe-P*TI(u)

w(u, x) = w,(u) = II(u) e-p*’ Thus, all time-independent

da.

solutions of (PI) are also independent

(63)

of x.

Proof By direct substitution, it is easily seen that w, is a solution of (6.1). One only needs to note that [e-J’*“ll(u)]a = (-p* - p(u)) e -~*“II(u) is integrable because p is bounded, hence, e-P*“II(u) + 0 as a + ~0even when p* s 0. So, we need to show that every sufficiently regular, nontrivial solution of (6.1) has the form (6.5). Let w be such a solution and define A E C(J) by A(x) = So”[p(u) - P(u)lw(u, x) da. F rom the first equation in (6.1) it follows that w is given by

w(u,x) = C(x) exp

[

-

I0

‘,n(s) Q + A(x)u

From the third condition of (6.1), Jo”II(u) exp[A(x)u] C(x) = l/la

I

= C(x)II(u)

exp[A(x)u].

(6.6)

da < 00, and

II(u) exp[A(x)u]

da.

(6.7)

So, C(x) > 0, and the second condition in (6.1) implies that 1 = om~(u)II(u) I

exp[A(x)u]

da.

Hence, A(x) = -p* and A is independent of x. From (6.6) and (6.7) we now see that w(u, x) = w,(u), and the lemma is established. The proof of theorem 6.1 is now straightforward. The fact that (6.3) with condition (6.4) is a solution of (P) follows by direct substitution in (P). Now, if b = 0, that is, in the small gradient case, @(O,t, x) = x, and hence, ~(a, x)/P(x) = w(u, t, c$(O, t, x)) = w(u, t, x) is independent of t. From lemma 6.2, w = w,, and hence, ~(a, x) is given by (6.3). This completes the proof of the theorem. We now consider the behaviour of u as t- 03. This asymptotic behaviour depends on the

522

S. BUSENBERG and M.IANNELLI

specific form of the particular nonlinear diffusion term. However, a general result can be established for the ratio u/P. We shall use the following condition. m eeP*TI(a) da < ~0whenp* < 0, and when p* < 0 I0 either (i) w~(a, x) has compact support in the variable a, (63) or (ii) there exists a> 0 such that, for a > a, p(a) is monotone nondecreasing. 6.3. Suppose that condition (6.8) holds. Then any solution w of (P,) (see theorem 3.1) satisfies

THEOREM

lim w(a, t, x) = wm(a),

t--tm

where wrn is given by (6.5). If

Proof. Let

~(a,

t,

x)

I0

r emP*TI(a) da = CQ, then

lim w(a, r,x) = 0. rF+m

be a solution of (PI). define A E C’(E) A(r> x) = o/r [~(a) -

I

by

P(a)1~(a, t, x>da,

and note that (A(& x) 1< 2y. Now, let f q(a,t,x)

= ~(a, t,X) exp

[

-

I0

A(s,x)

ds

1 ,

and note that q satisfies the problem 91-t qn + p-19= 0, q(0, t, x) = 02 P(a) q(a, r, x) da, I

(6.9)

1 q(a, 0, x) = wo(a, x). This is the classical linear age-dependent renewal equation for which the following asymptotic behaviour is known [3,13] if

wo(a-t)II(ala-t), q(a, t,x) =

1

tda

[C(X) + 0(x, t - a)] IX(a) exp(p*(t

- a)), t > a,

where u(x, t - a) + 0 when t - a+ a, and p* is given by (6.3). Here, C(x) > 0 may depend on X, but p” is constant. From (6.7) we now have wo(a - t)II(aja

- t) exp

f il

A(s,x)

0

w(a,

t,x)

ds

, t ca 1

(6.10)

= [C(x)

+

0(x,

t -

a)]

IX(a)

exp[p*(t

- a) + b(A(s,x)

ds]. t : a.

A class of nonlinear diffusion problems in age-dependent

population dynamics

523

From the condition 1 = J-0”w(a, t, x) da, we obtain A@, x) ds - p*t

wo(a>rI(a + 1 I Ic = ePP*’

t la)

0

+

I0

‘II(u) e-P*“(C(x) + o(x, t -x))

da.

(6.11)

Now, if p* > 0, then Jr e-P*TI(u) da < 00, while the same condition holds for p* =s0 because of (6.8). As was noted in the proof of lemma 6.2, this implies that the derivative of e-P*’ II(u) is integrable, hence, e-P*“II(u) + 0 as a+ ~4. From Lebesgue’s dominated convergence theorem we then have lim ‘e-p*OII(u)o(x, t-u) II 0 f--+W

da s I

I0

a e-P*“II(u) !i;

1o (x, t - a) 1da = 0.

If p* = 0, the same type of reasoning leads to the conclusion lim mwo(u)II(u + tlu) = 0. f-+WI0 So, if p* 2 0, (6.11) implies that f

A(s,x)ds lim exp f’m [ I0

-p*t]

= C(x) lmII(u)

e-f’*adu.

(6.12)

If p* < 0, we start by noting that e-P**

m wrJ(u)rI(u +

[wo(u)/e-P*TI(u)] e-P*(‘ta)H(u + t) da.

tlu) du =

(6.13)

If w. has compact support, the dominated convergence theorem again implies that this term goes to zero as t+ m. If wo does not have compact support, then by (6.8) ~(a) t p* as a-, M. Again from (6.8), Jr e-P*“IT(u) da < 00, hence, -p* - p* < 0. In turn, this implies that e -P*“II(u) is mon otone decreasing if a is sufficiently large, and hence e-P*@+QII(u + t) < e- P*“II(u) if a is sufficiently large. Again by the dominated convergence theorem and by (6.13) we habe m

e-P*’

we(u) II(u + tlu) da+0

as

t--,

m.

From (6.11) it now follows that (6.12) holds when p* < 0, and hence, for all p*. Now, from (6.12) and (6.10) it immediately follows that w(u, t,x)+ w,(u) as tcompletes the proof of the case where Jr e-P*TI(u) da < m.

(6.14)

m. This

Next suppose that J-0”e-P*TI(u) da = CQ.From (6.11) we see that for any N 2 0, and for t > N, we have

t

A(s,x)

N

ds -p*t

1

2

I

II(u) e-J”“[C(x) + 0(x, t - a)] da

0

N

3

I

o II(u) e_P*‘du[C(x)/2],

S. BUSENBERGand M. IANNELLI

524

provided t is large enough. Consequently, exp[Jf,A(s,x) ds +p*t]+ 0 as t+ 00, and from (6.10) we obtain w(a, t, x) + 0 as t+ m. This completes the proof of the theorem. We note that, since Jt w(a, t, x) da = 1, and p(a) - /3(a ) is b ounded, then a direct consequence of theorem 6.3, when (6.8) holds, is lim A(t, x) = Rex lim 0Z [~(a)

I

t-m

= om[p(a) I

-

/%a>1 w(a>t>x>da

- P(a)] w,(a)

da = -p*.

(6.15)

of theorem When Jo” e -“*“II(a) da = w, A(t, x) -+ 0 as t -+ CQ.Another direct consequence is the following remark. Let (P, @) solve the problem (Pl)-(J’s), and suppose that

6.3

lim p(t, @(t, 0, x)) t--tee exists. Then lim ~(a, t, $(t, 0, x)) = w,(a) t-m provided

condition

lim P(t, $(t, 0, x)), r+m

(6.16)

(6.8) holds when p* s 0. If J$ emP*“II(a) da = 0~) then lim ~(a, t, $(t, 0, x)) = 0. t+=

From this we see that the asymptotic behaviour of u is totally determined by the properties of the solutions of ( Pz)-( P3). The conditions (6.8) (i) or (ii) are reasonable from the view-point of the situation that is being modelled. The requirement they imply is that, either the initial population does not have individuals beyond an arbitrarily large maximum age, or that eventually the mortality rate does not decrease when the age increases. We also note that, when p* > 0, the average number of children expected to be born to an individual is greater than one, while if p* < 0 it is less than one, and this number is exactly one when p* = 0. The condition a requirement that the death rate is Jo” e-P*“lT(a) da < 00, when p* s 0, is essentially sufficiently large to concentrate the population at the younger ages, and allow the limit of U/P to be nonzero. The condition (1.8) is not needed if it is assumed that J$” p(s) ds = 00, for some aM > 0, that is, that there exists a maximum possible age aM beyond which nobody survives. We now consider some specific examples for which we describe the asymptotic behaviour. Example

1. Consider

the small gradient b(P, P&X

Since b = 0, the problem

(Pj)

is trivial,

diffusion

problem

of section

4. Here

+ c(P, PX, P*x)u = ;&. and the problem

(Pz) is

P, = Pxx - A(t, x) P, P(0, x) = PO(x) > 0, P, = 0 on U.

+(t, 0, x) =x, so ~(a, t,x)/P(t, x ) -+ w,(a) if (1.8) holds, while the limit is zero if -p* as da = ~4. Now suppose that (1.8) holds. Then, if p* > 0, A(t,x)-+ ~0 and it follows immediately that P(t, x) + CQas t+ a. This follows by noting that there

Now,

So” e -“*“II(a)

t+

(6.17)

A class of nonlinear diffusion problems in age-dependent

population dynamics

525

exists T such that and

PtaPIX+$P,tsT

for

P(T,x)>O,

x EJ.

Hence, if p* > 0, there is no limiting distribution for u(a, t, x). When p* < 0, the same reasoning leads to the conclusion P(t, x) + 0 as t--, co, and hence, u(a, t,x)+ 0 as t+ CQ.If p* = 0, A(t, x) + 0 as t+ CQ,and u(a, t, x)/P@, x) + II(a)/J-o” II(u) da. However, P(t, x) will depend on the initial distribution u. because of the term A (t, x) in (6.17), hence, the behaviour of u(u, t, x) as t+ 00 will depend on the initial datum ua. Example 2. This is a mathematically simple case but is interesting because it corresponds to the age dependent version of the logistic equation when the crowding effects are assumed to influence the death rate but not the birth law. Here, we take bu, + cu = - kPu, k > 0. Of course, there is no diffusion mechanism present, and the variable x is a parameter that can be suppressed. The ( P3) equation is again trivial, while the (Pz) equation now is P( = -kP*

- A(t)P,

If A(t)-+ 0 as t+ ~0, it is seen directly from (6.18) that P-+ 0 as t+ the following explicit integral P(t) = exp[-

[A(s)

(6.18)

P(0) = PO.

ds] Po[l + kPo[exp(-

CA(s)

~1. Moreover,

dsj du]-‘,

(6.18) has

(6.19)

hence,P(t)~OifA(t)~-p*>OandP(r)~p*/kifA(t)-,-p*
;* “
(6.20)

From (6.20) and corollary 6.4 we obtain lim ~(a, t, x) =

I-+rn

p*wm(u)/k, 0,

if p* ~0,

if p* < 0.

(6.21)

Note that if p E CIO, UM) with J-0””p(u) da = ~4, then H(u) = 0 for a 2 uM and the condition Jg H(u) e-Pen da < ~4 is automatically satisfied. Under this additional condition, this example has also been studied by Marcati [ 181. For our final example we consider a problem defined on the whole line. Example

3. Consider the case bu,+ cu = f P,, + k(1- P)u, k > 0.

This corresponds to a situation where there is population migration according to the small gradient mechanism of section 4, and where there is an Allee type of effect regulating the death mechanism. For this type of regulation mechanism, there is intraspecific mutualism at

526

S. BUSENBERG and M. IANNELLI

low population density, while at high densities the usual logistic type of control comes in. A discussion of the Allee effect can be found in May [19] and Watt [21], while a recent description of experimental results and models for this type of response to crowding is given in Bardi [2]. The existence and uniqueness of strict solutions for this case can be proved using the methods of section 4 and 5. For simplicity, we shall treat the special case where ~(a) = P(a) and $0”n(a) da < ~0, hence, p* = 0.The problem (Pz) then is

P,= P,,+ k(1 - P)P, P(0)= PO.

(6.22)

The asymptotic behaviour of u is H(a) lim ~(a, t,~) = Jo”l-I(a) da ;!rnz ‘(” ‘)’ I+ cc Now, when (6.22) is considered as an initial value problem on the real line, and when 0 =SPOG 1, it has solutions obeying 0 c P c 1. Moreover, (6.22) admits travelling wave solutions, that is, solutions of the form

P(t,x) = q(x- at),

(6.23)

where rj~satisfies 0 S 11,S 1, lim t@(X)= 1, lim t/j(x) = 0. *+-Cc .X-m In fact, it has been shown that there exists a0 > 0, such that for a> CQ, (6.22) has a unique (up to translations) solution of the form (6.23) travelling with speed CY.A discussion of the wave-front solution can be found in [ 1, 81. It follows from the above discussion that (P)admits a travelling wave solution of the form u(u, t, X) = w(u, t, x) q(x - at),

(6.24)

and lim ~(a, t, x) = II(u)/im I--Pm

II(u) da.

(6.25)

In particular, if POis an initial datum leading to the solution P(t, x)= q(x - at), and if we take ua(u,x) = P&)II(u)/_f$ II(u) d a in the problem (P),then the solution u is given by ~(a, t, x) = II(u) q(x - ~$1~ l-I(u) da.

(6.26)

Hence, a travelling front solution of the form (6.26) exists for each (~2 CXJ. 7. APPENDIX Here we recall some abstract results and their application to the problems we have met in the previous sections. Let E be a Banach space (endowed with the norm 1.I), and let the linear operator A : DA C E + E be the infinitesimal generator of a strongly continuous semigroup on E. Consider the following Cauchy problem

$ =Au(t) +L(t)u(t)

+ F(t);

u(O) = ug

(Al)

A class of nonlinear diffusion problems in age-dependent

population dynamics

527

where t E [0, T] and F(r) E E, L(Z) E Z(E) Vt E [0, T]. Concerning the existence of solutions to problem (Al) we have THEOREMA~. Let CQE Da, F(.) E Cl([O, T]; E), L(.) E C’([O, T]; X(E)), thentheproblem (Al) hasone andonly one strict solution, that is, there exists a function u E C’([O, T]; E) fl C([O, T]; DA) satisfying the problem (Al). Weaker conditions on uo,f, L lead to the following result THEOREMA.2. Let u0 E E, f(.) E C([O, T]; E), L(.) E C([O, T]; Y(E)). Then (Al) has one and only one strong solution, that is, a function u E C([O, T]; E), such that, there exists a sequence un E C’([O, T]; E) II C([O, T]; Da) with the properties un* u

in

C([O, T];E)

- Au,(f) - L(t)u,(t) -f

%(r)

in

C([O, T]; E)

(A2)

in E. i u,(O)+ug It is obvious that a strict solution is also a strong one. In both cases we have: PROPOSITION A.3. Let u E C([O, T]; E) be the strong solution of (Al), then the following estimate holds: In(r)1 g CecL’~lu~] + r(l’lf(s)l ds)

(A3)

where c is a constant depending only on A, while

IWI IW)

L = ,$PTl

The estimate (A3) leads to continuous dependence of the solutions on the data uO,F and L, in the following form: COROLLARYA.4. Consider the problem = Au”(t) + L”(t)u”(r) + F”(t);

z(t)

u”(0) = u”a,

Let u” be the strong solution of this problem. Assume that ufi-9 ug in

E

F”+

F

in

C([O, T];E)

1 L” + L

in

C((0, T]; Z(E)).

Then u”+ u

in

C([O, T];E).

The previous results can be applied to the following problem which was considered in sections 4 and 5: P, = kP,, + I@, x)P +f(t, x);

P(0) = Pa

(A4)

for (t, X) E [0, T] X J, where J is an interval, with one of the following boundary conditions and spaces: =P(t,jz)

J=(ir,jz),

P(t,jr)

J= (jl,jz),

pP(t,j~) -Px(t,jt)

J=R,

P(t,x)+O

as

=O, =O,

]x/++m,

E =EI=&(~), pP(t,jz)

(As)

+Px(t,j3

=0

with

E =E2=C(J‘)

~30,

E=Es=Ca(R).

(A6)

(A7)

Here 1 and f are such that the mappings L(.); [0, T] + Z(E), (L(t)u) (x) = I(t, x)u(x), and F( .): [0, T]+ (x) =f(f, x) satisfy the conditions of either theorem Al or theorem A2. In fact, the following operators:

EF(f)

Dal = {u E C’(j),

A _

I

A2

E

u = uxx =0

on

aJ} 648)

AIU = ku,

IDal= {UE C*(j),w(h) Azu = ku,

- u,(jl) =0, pu(j2)

+u,(j2)

=o}

528

S. BUSENBERG and M. IANNELLI

(AlO) are generators

of strongly

continuous

semigroups

Ei = CO(~); We omit the description of the meaning following result is used in our analysis:

in the spaces

Ez = C(j);

PROPOSITION A5. Let PO E Da2 with p = 0, and suppose strict solution in Ez, satisfying A6. If moreover &(t,x) then Q(t, x) = Pz(t, x) is the strong

E3 = C,(R),

of strict or strong

solution

solutions

respectively.

with respect

I, f E C’([O, T] x j).

=fJt,x)

= 0

on

to these three different

Then the problem

cases, The

A4, has a unique

U,

(All)

in El of the problem

Q, = kQxx + lQ + g(t, n);

Q(O) = Pox

6412)

with Q satisfying the boundary conditions (A5). Here g = I,P + fi belongs to C([O, T]; Co(j)). Similar results hold in the case p # 0 and in the case J = Iw. The first four results stated above can be easily found in the literature (see for instance [14, 201). It is only worthwhile to sketch the proof of the final part of proposition A5. Let Pt,f”, I” be sequences such that: POE Da:, PO”+ PO and

P&-+Po,

I” E C([O, T]; D,+),ln-+i f” E C([O, Because

and

Tl; D~;),fn-+f

of the fact that AZ: DA: +Da2

g enerates

in

I:+l,

(A13) C([O, T]; ES

in

and C-fx

a strongly

Ez

(A14)

in C([O,Tl; Ed

continuous

semigroup

(A15)

on Da?, the problem

P: = kP,, + 1”P” + f”; P’(O) = Pi has a unique

strict solution P” E C([O, Tl; Da;) n C’([O, Tl; DAJ,

and, owing to corollary

A4, we have P”+

where

P is the (strict)

solution

P

in

C([O, T];Ez)

of (A4) in E2. Now, put Q” = P,“. The known Qn E C([O, T]; DA,) n C1([O,

regularity

of P” implies

Tl; El)

and Q: = kQ:x + 1”Q” + l:P” + f:; Q’(0) = P$ that is, Q” is the (unique) A4 now implies:

strict solution

of this latter PJ=Q”+Q

in E 1, with the boundary

problem in

C([O,

conditions

(A5).

Corollary

Tl; El),

that is, Q = P,.

REFERENCES 1. ARONSON D. G. & WEINBERGER H. F., Nonlinear diffusion in population genetics, combustion and nerve propagation, in Proceedings of the Tulane Program in Partial Differential Equations and Related Topics, Lecture Notes in Biomathematics 446, Springer, New York (1975). 2. BARDI M., An equation of growth of a single species with realistic dependence on crowding and seasonal factors, preprint. 3. BELLMAN R. & COOKE K. L., Differential-Difference Equations, Academic Press, New York (1963). 4. BUSENBERG S. & IANNELLI M., A method for treating a class of nonlinear diffusion problems, Rc. Acad. naz. Lincei, in press.

A class of nonlinear diffusion problems in age-dependent

population dynamics

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5. BUSENBERGS. & TRAVISC., Epidemic models with spatial spread due to population migration, J. math. Biol., in press. 6. DI BLASIOG., Nonlinear age-dependent population diffusion, J. math. Biol. 8, 265-284 (1979). 7. DI BLASIO G., A problem arising in the mathematical theory of epidemics, in Nonlinear Phenomena in the Mathematical Sciences (Edited by V. Lakshmikantham) Academic Press, New York (1982). 8. FIFE P. C., Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics 28, Springer, New York (1979). 9. GURTINM., A system of equations for age-dependent population diffusion, J. theor. Biol. 40, 382-392 (1973). 10. GURTINM. & MACCAMYR. C., Non-linear age-dependent population dynamics, Archs ration. Mech. Analysis 54, 281-300 (1974). 11. GURTINM. & MACCAMYR. C., On the diffusion of biological populations, Math. 12. GURTINM. & MAC&MY R. C., Diffusion models for age-structured populations, (1981). 13. HOPPENSTEADT F., Mathematical Theory of Population Demographics, Genetics Regional Conference Series in Applied Mathematics 20, Society for Industrial and

14. 15. 16. 17. 18. 19. 20. 21. 22.

Biosciences 33, 35-79 (1977). Muth. Biosciences 54, 49-59 and Epidemics,

CBMS-NSF

Philadelphia (1975). KREINS. G., Linear differential equations in Banach spaces, Am. math. Sot. Transl. 29, Providence, R.I. (1971). LANGLAISM., Equations d’evolutions degenertes, preprint. LANGLAISM., Sur une classe de problbmes aux limites degenerees et quelques applications, Doctorate thesis, UniversitC de Bordeaux (1981). MACCAMYR. C., A population model with nonlinear diffusion, J. diff. Eqns 39, 57-72 (1981). MARCATIP., On the global stability of the logistic age-dependent population growth, J. math. Biol. 15, 215-226 (1982). MAY R. M., Stability and Complexity in Model Ecosystems, 2nd edition, Princeton University Press, Princeton (1974). PAZY A., Semi-groups of linear operators and applications to partial differential equations. Lecture Note No. 10, Department of Mathematics, University of Maryland, College Park (1974). WATT K. E. F.. Ecoloav and Resource Mamwement. McGraw-Hill. New York (1968). WEBB G., An gge-structured epidemic mod; with spatial diffusion, Archs ration. hech. Analysis 75, 91-102 (1980).

Applied Mathematics,