THE PERIODIC SOLUTIONS FOR TIME DEPENDENT AGE-STRUCTURED POPULATION MODELS

2000,20B(2): 155-161

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f£$/fmJJ1$tIl

THE PERIODIC SOLUTIONS FOR TIME DEPENDENT AGE-STRUCTURED POPULATION MODELS 1 Zhou Yicang ( PlJl1t) Ma Zhien ( ~~,~ ) Science college, Xi'an Jiaotong University, Xi'an 710049, China Abstract

In this paper, the existence of periodic solutions for a time dependent age-

structured population model is studied. The averaged net reproductive number is introduced as the main parameter to determine the dynamical behaviors of the model. The existence of a global parameterized branch of periodic solutions of the model is obtained by using the contracting mapping theorem in a periodic and continuous function space. The global stability of the trivial equilibrium is studied and a very practical stability criteria for the model is obtained. The dynamics of the linear time-periodic model is similar to that of the linear model.

Key words Age-structure, population model, periodic solution, net reproductive number 1991 MR Subject Classification

1

92D25

Introduction The age-structured population model has been successfully used in population dynanlics[1,2].

The general age-structured population model is described by the following partial differential equations[3] :

au au

-at + -aa.=

-f.-t(a,t)u(a,t),

u(O, t) = no

f

A :!

Al

u(a,O) = uo(a),

b( a,t)u( a, t)da,

°<

a

< A, t > 0,

(la)

t > 0,

(lb)

o < a < A,

(lc)

where 1L(a, t) is the age-specific density of a single age-structured species at time t, f.-t(a, t) and b(a,t) are the age-specific per capital death rate and birth rate of the species at time t, respectively. The real A is the maximum age for any individual in the population, and [Ai' A 2 ] is the fecundity period of females. The function uo(a) is the initial density distribution. The most studies on age-structured population model (1) assume that the vital rates( the death rate and the birth rate) do not depend explicitly on time t. The amount of literature dealing with model (1) in which the vital rates are explicitly time dependent is comparatively 1 Received

Nov.28,1997; revised Oct.28,1998. Project 19641004 supported by the National Natural Science

Foundation of China

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small, In recent year the fact that the vital rates are rarely constant in time has been widely recognized, and considerable attention has been paid to the time explicitly dependent models[4]. In this paper we study the global bifurcation of periodic solutions of the following agestructured population model:

o< a < A,

= no

u(O, t)

j

A2

,B( a) (1 + Ep(t))u(a, t)da,

> 0,

(2a)

a> 0,

(2b)

t

Al

where no and E are constants. Model (2) is a simplification of the general age-structured. population model (1). In general the age-structured population model (2) does not have positive equilibrium solutions, and we have the more difficult challenge of dealing with the periodic solutions if p(t) is a periodic function. There is a growing body of literature dealing with small amplitude periodic solutions for sufficiently small paranleters[5,6]. In this article, we will use the contract mapping theorem in a periodic function space to prove the existence of large periodic solutions for model (2). And we will discuss the stability of the trivial solution for model (2).

2

Basic Assumptions and Preliminaries It will be assumed throughout that: (HI) p(t) is a continuously differentiable periodic function, p(t + 1) = p(t),

IIp(t)1I =

maxtE[O,l] Ip(t) I ::; 1. (H2) fl(a) is a positive continuous function on [0, A), lin1 a--+Aa

Set 7r(a) = exp( - fo fl(s)ds) for 0 ::; a continuous function on [0, A], and 0

J; p(t)dt = 0,

J; fl(s)ds = +00.

< A, and 7r(A) = O. It is obvious that 7r(a) is a

< 1r( a) ::;

1 for 0

< a < A.

(H3) (3(a) is a nonnegative continuous function, and (3(a)

> 0 for A 1 < a < A 2 , (3(a)

=0

for a ::; A 1 or a ~ A 2 , f:1 (3(a)1r(a)da = 1. In order to apply bifurcation theory to model (2) it is necessary to introduce an important and biologically meaningful parameter: "averaged net reproductive number" no. If an individual 2

is born at time t, then n(t)

= no f:

2

1

(3(a)7r(a)(l

+ Ep(t + a))da

=

is the number of offspring

1

produced by the individual over its lifespan. Hence, no fo n(t)dt is the averaged number of offspring produced by all individuals who are born in the time interval [t, t + 1].

Set u(a, t) = 1r(a) + v(a, t), no obtain the equation for v(a, t):

ov ov ot

+ oa

v(O, t) =

i:

= 1 + .>t(E).

By substituting these expressions into (2) we

= -fl(a)v, 2

,B(a)v(a,t)da

+.A(c)(l

+ €p(t))

+ €p(t)

(1 +

1:

(1 + 2

i:

(3a) 2

,B(a)v(a,t)da)

,B(a)v(a,t)da) .

(3b)

157

Zhou & Ma: PERIODIC SOLUTIONS FOR POPULATION MODELS

No.2

The general solution of (3a) is v(a, t)

= 1r(a)B(t -

a), where B(t) is an arbitrary differ-

entiable function. The solution v(a, t) satisfies the condition (3b) if and only if B(t) is solved from the following equation: '

B(t) =

L:' ~(a)7r(a)B(t

(1 + L:' ~(a)7r(a)B(t

- a)da + Ep(t)

+A(E)(l + Ep(t))

(1 + 1:' ~(a)7r(a)B(t

- a)da)

(4)

- a)da) ·

Let pI be the Banach space consisted of continuously differentiable, periodic function h : R -+ R subject to the usual supremum norm Ilh(t)11 = maxo
B(t) =

l

A

~(a)7r(a)B(t -

a)da + h(t).

(5)

Theorem A If (Hl)-(H3) hold. Then equation (5) has a solution B(t) E pI if and only if h(t) E pJ, in which case (5) has a unique solution B(t) E pJ and the operator L : pJ -+ pJ

defined by B = Lh is linear and bounded. By defining

h(t) = Ep(t)

(1 + l:' ~(a)7r(a)B(t

+A(E)

- a)da)

(1 + Ep(t)) (1 + L:' ~(a)7r(a)B(t - a)da)

and applying Theorem A to (4) we see that if B(t) is a solution of (4) in that

A(£) =

-£

1

~2

where

h(E, B(t)) = Ep(t)

J

A2

Al

1£1

and

(7)

.

IIB(t)ll.

j3(a)1r(a)B(t - a)da + h(£, B(t)).

(1 + l:' ~(a)7r(a)B(t

(8)

- a)da)

c JA j3(a)1r(a) fl p(t)B(t - a)dtda Al A Jo 1 (1 + £p(t)) 1 + e fA}2 j3(a)1r(a) fo p(t)B(t - a)dtda 2

-

1

1 + e fA I j3(a)1r(a) fo p(t)B(t - a)dtda

=

pJ then it is necessary

f: 2 j3(a)1r(a) fo p(t)B(t - a)dtda

Clearly, A is well defined by (7) for small Substitute (7) into (4) we have

B(t)

(6)

(j + 1

A2

Al

j3(a)1r(a)B(t - a)da

Since h(£,B(t)) E pJ for any B(t) E pJ, we can define a mapping T: pJ Lh(£, B(t)). Therefore we have a well-defined mapping T if 1£IIIB(t)11 < 1.

-+

pJ

)

.

(9)

by TB(t) =

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The Global Bifurcation of Periodic Solutions

3

From a periodic solution of the integral equation (4) we can have a periodic solution for age-structured population model (3) and (2). Our first result is on existence of the periodic solutions of the integral equation (4).

Theorem 1

Assume that (HI), (H2) and (H3) hold. Then there exists a positive con-

stant Eo and a function ..\(E) defined on [-Eo, Eo], such that the integral equation (4) has a unique nontrivial periodic solution B(t) == B(E)(t) for ..\ == ..\(E) and

lEI:::; Eo.

Proof The existence and uniqueness of the periodic solution of the integral equation (4) is equivalent to the existence and uniqueness of the fixed point of the mapping T. Therefore, we need only to prove the existence of the unique fixed point of the mapping T in the Banach space

pJ.

pJ

Let Xo be positive constant and p(t) E

pJ

X of

X == {f(t)1 f(t) E

be a given function. Define a bounded subset

pJ, Ilf(t)11 :::;

Xo

< I}.

From the assumptions and Theorem A it follows that for any B(t) E X

IITB(t)ll:::; IILIIIEI

[1 + Xo + 1-71 (1 + xo)(l + Icl)] .

From equation (10) it is easy to see that the mapping T maps X into X if is defined by C1

. {I

= mm

(10)

E Xo

1, 2xo'

Xo

211 L II(1 + xo)(l + 4xo)

}

lEI < E1,

E1

·

P6, lEI:::; E1, from the definition ofT and the linearity of L

For any B(t),D(t) E

where

we have

TB(t) - TD(t)

= «t.

(P(t)

L:2

jJ(a)1l"(a) (B(t - a) - D(t - a» da) 1

+ [1 + e Jo1 p(t) J':,

EJo p(t) J:12 (3(a)1r(a) (D(t - a) - B(t - a)) dadt

2

·L

(

(1

+ Ep(t))

·L ((1+c P(t»

jJ(a)1l"(a)B(t - a)dadt]

[1 + e Jo p(t) J':, jJ(a)1l"(a)D(t -

( +j

a)dadt]

1p(t)J:2(3(a)1r(a)D(t-a)dadt

EJo

+ ---1 A2- - - - - - 1 + Efo p(t) fA f3(a)1r(a)D(t - a)dadt ---"-.I-

1

2

jJ(a)1l"(a)(D(t-a) -B(t-a»da).

From (11) we can see that, for any

co = min {c

1,

(11)

lEI < E1,

IITB(t) - TD(t)11 :::; Therefore, if we take that

2

A2 )) (3(a)1r(a)B(t - a)da Al

1

L:

1

IILIIIEIIIB(t) -

IILII(9~12xo)}'

D(t)II(9

+ 12xo).

then from the above analysis it is easy to see

IITB(t) - TD(t)11 ~ 81IB(t) - D(t)ll,

as

lEI < Eo,

Zhou & Ma: PERIODIC SOLUTIONS FOR POPULATION MODELS

No.2

where ()

159

< 1. By the contracting mapping theorem we know that there is one and only one

fixed point B(t) E X. The proof of Theorem 1 completes. Theorem 1 shows that there exists a unique periodic solution B(£)(t) of the integral equation(4) if ;\ == ;\(c) and e E [-£0, co]. From this periodic solution B(£)(t) of (4) we can get a periodic solution v(c)(a,t) == 1r(a)B(c)(t - a) of (3) and a periodic solution u(c)(a,t) == 1r(a) (1 + B(c)(t - a)) of (2). Since Xo < 1 we see that u(c)(a, t) is a positive periodic solution. From the linearity of the equation (2) it is easy to get the following theorem: Theorem 2 Assume that (HI), (H2) and (H3) hold. Then there exists a function ;\(c) defined on [-co, co], such that the age-structured population model (2) has a family of periodic solutions cu(c)(a,t)(c is an arbitrary positive constant) for A == ;\(c) and 1£1:::; co. Theorem 1 and 2 have established the existence of the periodic solutions for (4) and (2), respectively. The lower-order terms in the e expansion of B(t) and ;\(c) can be obtained by the following procedure: Substitute

into (4), equate coefficients of like powers of e on both sides of Equation (4), and solve the resulting recursive linear nonhomogeneous systems for Bj(t). A natural way to study time-periodic solutions of (4) is to assume that

p(t)

=

+00

L

Pm exp(21rimt), where Pm

m=-oo

=

1 1

p(t) exp( -21rimt)dt.

In this case

Bo(t) == 0,

;\1

== 0

- '" B 1 (t ) -L.-J m~O 1 -

A2 == -

l

A2

JA

A2 1

f3(a)1r(a)

. Al

_ '" B 2 (t ) -L.-J

m;to

where

bm ==

1 1

j3(a)1r(a) exp( -21rima)da

11 p(t)B

1(t

'

- a)dtda,

(12)

0

i; exp(21rimt) ' 1 - JA 1213 (a)1r(a) exp( - 21rima )da A

jA2 f3(a)1r(a)B

1(t - a) exp( -21rimt)dadt. o Al Any desired order terms of the periodic solution B (c)(t) and the bifurcation function A(c) can be computed.

4

p(t)

Pm eX P (21ri.m t )

Stability of the Trivial Equilibrium Solution

The stability of the trivial equilibrium solution for time independent age-structured population model is very simple: the trivial equilibrium solution is globally stable if the net reproductive number is less than 1, it is unstable if the net reproductive number is greater than 1,

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and there exists a family of time independent age distribution if the net reproductive number is equal to 1. The dynamics for time dependent age-structured population model (2) is very similar to that of the linear time independent model. The following theorem gives the precise stability results for model (2).

Theorem 3 Assume that (HI), (H2) and (H3) hold. Let E and A(E) be defined in Theorem 1. Then the trivial equilibrium solution of model (2) is globally stable if 0 < A < A(E), and it is unstable if A > A(E) Proof The general solution of model (2) is of the form u(a, t) = 1r(a)B(t - a). The function B( t) is a continuously differentiable function and satisfies the integral equation

r = (1 + A)(1 + €p(t)) lA!

A2

B(t)

(3(a)7f(a)B~t

- a)da.

(13)

From the expression u(a, t) = 1r(a)B(t-a) we see that it is sufficient to prove the global stability of the trivial solution of the equation (13) if 0 < A < A( E). Since the integral equation (4) has a

=

unique solution B(E)(t) for A A(E) with E E [-EO, Eo], the integral equation (13) has a family of periodic solutions cBe(t) = C(B(E)(t) + 1), where Be(t) = B(E)(t) + 1 is a positive periodic function. If 0 < A < A(E), we define U = A(E) - A > O. Assume that B(t) is a positive solution of the integral equation (13) and bl = maxtE[O,A] B(t). The positivity of Be (t) + 1 implies that there exists a positive constant Cl, such that clBe(t) > bl . When A ::; t ::; A + A l

B(t) = (1 + A) (1 + €p(t))

lA2 {3(a)7f(a)B(t - a)da Al

2

:S (1 + A) (1 + €p(t)) fA {3(a)7f(a)c iB€(t - a)da

JA

1

== Cl (1 + A(E)) (1 + Ep(t))

f

A 2

{3(a)7f(a)B€{t - a)da

Al

2

-Cia (1 + €p(t)) fA (3(a)7f(a)B€(t - a)da

JA

1

:S ciB€(t) - cia (1 + €p(t)) Set Ul == min (1 + Ep(t))

L:2 {3(a)7f(a)B€{t - a)da.

J: f3(a)1r(a)B e(t - a)da > 2 1

ClUUl. By repeating we have B(t)

< cl(Be(t) -

(14)

0, then (14) implies that B(t) < clBe(t)-

UUl) for A ~ t ::; 2A. Since Be(t) is bounded

and UUl > 0 we can find a positive constant 0 < , < 1, such that ,Be (t) ::; UUl for any t E [A,2A]. The definition of, implies that B(t) < cl(1 - ,)Be(t) for A ::; t ::; 2A, that is, B(t) < c2Be(t) with C2 == cl(1 - , ) if A ~ t ~ 2A. By repeating the above procedure, the new boundary function C2Be(t) on t E [A,2A] implies that B(t) < c-B, (t) for 2A ::; t ::; 3A, where C3 < c2(1 - ,) == Cl (1 - 'Y)2. The mathematical induction implies that there exists a positive, monotonically decreasing sequence

{Ck}, where Ck == cl(I-,)k, such that B(t) < ckBe(t)

for (k - I)A ~ t ~ kA.

(15)

Set c" == limk-.oo Ck, then c" == O. The inequality (15) and the definition of Ck imply that B(t) < Ck II Be (t)1I < clllBe (t)ll, and limt-.oo B(t) O. In order to make B(t) less than

=

Zhou & Ma: PERIODIC SOLUTIONS FOR POPULATION MODELS

No.2

161

tt)II , where flo =

q£6

any given positive number q, it is sufficient to take maxtE[O,A] IB(t)1 :::; II B

=

mintE[O,l] Be(t) > O. In fact, InaxtE[O,A] IB(t)1 :::; IIJc6(~)1I TIB:1mr B~(t)Be(t) :::; ~Be(t), if we choose C1 to be ~, then for any t ~ 0, B(t) :::; c11IBe(t)11 :::; q. This proves the stability of the trivial equilibrium solution of (13) and (2). The rest part of the theorem can be proved by the similar procedure. Using the same idea we can prove the following theorem: Theorem 4 Assume that the conditions for Theorem 3 hold. If B(t) 2: <5 1 > 0 for

t E [0, A] and A > A(E), then liInt-+oo B(t) = 00. Theorem 3 and 4 demonstrate that A = A(E) is a threshold of the stability. This quantity plays the same role as that of the net reproductive number for the linear time independent age-structured population model.

5

Concluding Remarks

The theorems in this paper establishes the global existence of periodic solutions for the time dependent age-structured population model. The bifurcation of the periodic solutions is global because that the parameter interval [-co, co] is large. These results are more useful than that for the local bifurcation. The periodic solutions bifurcate from the trivial equilibrium solution u(a, t) == 0 at a critical value of the averaged net reproductive number no. The lower order terms of the approximations of the periodic solutions and the bifurcation function A(E) are calculated by the standard perturbation techniques. For e E [-Eo, Eo], we obtain the stability criteria for the trivial equilibrium solutions. We find the threshold value to distinguish that a positive solution of model (2) tends to zero or infinity as time goes to infinity. The conclusion in this paper is a generalization of the results on time independent age-structured population models and the local bifurcation of the periodic solutions. The idea and method used to determine the stability of the trivial equilibrium solution have very natural biological interpretation for the evolution process of a species. These idea and method can easily be used to study more complicated age-structured population models, References 1 Ma Z E. The mathematical modelling and study of the population ecology. Hefei: Anhui Education Press,

1996 2 Webb G F. Theory of nonlinear age dependent population dynamics. New York: Marcel Dekker Inc, 1985 3 Song J, Yu J Y. Population system control. Berlin: Springer-Verlag, 1987 4 Guo B Z, Yao C Z. New results on the exponential stability of non-stationary population dynamics. Acta Mathematica Scientia, 1996, 16(3):330-337 5 Cushing J M. Periodic Mckendrick equations for age-structured population growth. Comp & Maths with

Appls, 1986,12(4/5):513-526 6 Wu Y Y, Zhao L X. Cycling in the population developing equation. Practice, 1995, 15(4):16-24

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