- Email: [email protected]
Positive periodic solutions of a delayed model in population
Applied Mathematics
Letters PERGAMON
Applied
Mathematics
Letters
16 (2003)
561-565 www.elsevier.nl/locate/aml
Positive Periodic Solutions of a Delayed Model in Population CHANG- JIAN ZHAO Department of Mathematics,Shanghai University Shanghai 200436, China and Binzhou Teachers College Shandong, 256604, P.R. China
L. DEBNATH Department of Mathematics, University of Texas-Pan American Fklinburg, TX 785392999, U.S.A. KE WANG Department of Mathematics, Northeast Normal University Chang Chun 130024, P.R. China (Received November Abstract-using a delayed @ 2003
model Elsevier
2001; accepted March 2002)
the theory of coincidence degree, the existence of positive in population is proved. A new result is obtained. Some related Science Ltd. All rights reaervd
Kevords+oincidence
degree,
Positive
periodic
solution,
Fredholm
periodic results
solutions for are improved.
operator.
1. INTRODUCTION We consider the following logistic equation: A+) = N(t)
u(t) - 2 (
bi(t)N (t - q(t)) - J’
where a E C(W,W); bi E C(W, (0,oo)); ri E C(W, IV) c E C(W- , W+), and with an initial condition N(s) = 4(s) 2 0,
c(s - t)N(s) ds
,
(1)
-00
i=l
are T-periodic continuous functions and
4 Eq-aw+)
4(O) > 0,
’
(2)
The periodic solutions of some special cases of system (1) have been studied extensively. example, the qualitative behaviour of a logistic equation with several delays IQ’(t) = Iv(t) (u - &v(t
-T;))
For
(3)
i=l
This work is supported University of Texas-Pan
by National American.
0893-9659/03/S - see front PII: SO89S9659(03)00037-5
matter
Natural
Sciences
@ 2003 Elsevier
Foundation
Science
Ltd.
of China
All rights
and
reserved.
the
Faculty
5pe=t
Research
by 4&W
Council,
562
C.-J. ZHAO
et al.
and some of its generalizations have been studied, where a, bi, Ti are nonnegative constants. For a recent study of equation (3), we refer to [l-6] and references cited therein. Recently, Zhang [7] h as considered the following periodic delay logistic equation in the form: A(t) = r(t)N(t)
(
1 - N@&;(t))
>
,
where r, k E C(lR, (0, oo)), 7 E C(R, [0, oo)), and T, k, r are T-periodic functions. In particular, he proved that there exist T-periodic solutions of equation (4) if T(t) E mT, m E Z+. The logistic delay equation (1) is a natural generalization of equations (3) and (4). Hence, from mathematical and physical points of view, the study of equation (1) is of specific interest. So the main purpose of this article is to study the existence of positive periodic solutions of equation (1)) by using the theory of coincidence degree.
2. EXISTENCE
OF POSITIVE
PERIODIC
SOLUTIONS
In order to study the existence of a positive periodic solution, we first use the following ideas and results (see [S]). Let X, 2 be normed vector spaces, L : Dom L c X - 2 be a linear operator, and N : X -+ 2 be a continuous operator. The operator L will be called a Fkedholm operator of index zero if dim Ker L = codimIm L < +oo and Im L is closed in 2. If L is a Fredholm operator of index zero, there exist continuous projectors P : X + X and Q : 2 + 2 such that Im P = Ker L, Ker Q = Im L = Im(l-Q). It follows that L 1 Dom LnKer P : (I- P)X + Im L is invertible. We denote the inverse of that operator by K. If n is an open bounded subset of X, the operator N will be called Gcompa& on fi , if QN(n) is bounded and K(I - Q)N : fi + X is compact. Since Im Q is isomorphic to Ker L, there exist isomorphisms J : Im Q + Ker L. LEMMA 1. Let L be a Redholm operator of index zero and let N be L-compact on a. Suppose (1) for each X E [0, 11, every solution x of Lx = XNx is such that x # bR; (2) QNx # 0 for each x E 80 n Iker L; (3) Deg{JQN, R n Ker L, 0) # 0. Then the equation Lx = Nx has at least one solution lying in Dom L n 6. Our main result is given in the following theorem. THEOREM 2. The initial value problem (1),(2) h as at least one positive T-periodic solution, if the following conditions are s&i&d: (1) there exists a constant A4 > 0 such that Jo’ a(t) dt = M, (2) CE”=, b(t) < +m, (3) J_“, c(u) du < +oo. .PRooF. Since NW = N(O) . exp ( jOt ( a(s) - 2
b&)N(s
- TV)
- s’
c(r - s)N(T)dT
-m
i=l
then N(t) > 0. Making the change of variable N(t) = exp{x(t)}, equation (1) is refqrmulated as k(t) = a(t) - 2 hi(t) exp{x(t - q(t))} i=l
- /I,
c(s - t) exp{x(s)} ds.
,
Positive
Periodic
563
Solutions
Let X = 2 = {x E C(W, W) : x(t + T) = x(t)} and jlxll = s~p~elc,~l Ix(t)l. both Banach spaces when they are endowed with the norm 11. II. Let
Then X and 2 are
Lx = k, Nx = a(t) - Fbi(t)
exp(x(t - ri(t))}
- J’
c(s - t) exp{x(s)} ds,
-CO
i=l T
Px = f
I0
Qz = ; lT
x(t) dt,
2 E x,
z(t) dt,
z E ‘7.
Hence, Ker L = {x E X : x E R}, ImL={.ztZ:~Tz(t)dt=O}. Suppose xn E X and n is a natural number. Let Px, = 0, iim,,, Since lim Px, = P lim xn = PZ n--r00 n-wxl
xn = 3.
and Px, = 0, therefore, P?=O, i.e., Im L is closed in 2. On the other hand, dimKer L = codimIm L = 1. Hence, L is a F’redholm operator of index zero. Since for all i(t) E Im L, we have
Q@(t)) =; I T
i(t) dt = $(x(T)
- x(0)) = 0
0
and for all x(t) E X, we have Q(I-Q)x=Q(x-Qx)=Qx-Q2x=0. Hence, ImP = KerL,
ImL=KerQ=Im(l-Q).
Obviously, L 1Dom L f~ Ker P : (I - P)X -+ Im L is one-to-one, consequently, it is invertible, we denote the inverse of that operator by K. Since, z E Im L, there exists k(t) E Dom L such that k(t)
Integrating
both sides of (6) over s from 0 to
= t,
2.
(6)
we obtain t
x(t)
= x(0) +
I0
Z(S) ds
(7)
because 1 TO
T I
x(t) dt = 0.
(8)
564
C.-J.
et al.
ZHAO
It follows from (7) and (8) that T
t
JS 0
0
t
K(z) = J 0
z(s) ds dt.
Moreover,
K(I - Q)Nz =
U(S)
- 2
hi(s) exp{x(s -
- J’
TV)}
- 5
C(T
- s) exp{z(T)} d7
ds
--oo
i=l
hi(s) exp(x(s - Ti(s))} - J’
C(T
- s) exp{z(T)} d7
--M
i=l
a(s) - 2
bi(s)exp(z(s
- TV)}
- lSW C(T - s) exp{z(r)}
dr
i=l
If R is an open bounded subset of X, clearly, QN(fi) is a bounded set and K(I - Q)N : fl+ is compact (see [8]). Hence, N is L-compact on a. On the other hand, we consider the operator equation Lx = XNx, X E (0, l] and find that
i(t) = x a(t) - 2 hi(t)exp{x(t- Ti(t))} - St ~(3- t>~P{4s)l ds , -00
i=l
/+E (O,ll. (9)
Suppose that x(t) E X is a solution of equation (9) for a certain X E (0, 11. Integrating of (9) over t from 0 to T, yields 2 i=l
hi(t) q(t))) t) exp(x(t -
+ It
From (9) and (lo), we have T
T I*(t)1
J0
dt
I J 0
la(t)
I dt +
both sides
) J 2b(t) exp{x(t c(s -
exp{x(s)} ds
dt =
T a(t) dt =: M > 0. (10)
0
-m
G(t))}
i=l
t + J
-ccl
c(s -
t)
exp{x(s)} ds dt = M + M*, ‘1
where M* = /,’ la(t)1 dt. Since x(t) is a continuous function defined on X, then there exist E, 11E [0, T] such that
40 = t~~~TlxI(t)~ 477)= t$o=Tlz(t). Hence,
where
Consequently, there exist to E [O,T] such that Ix(t0)l
I In
X
Positive
Periodic
565
Solutions
Furthermore,
T I
I@o)l
+
/
[k(t)1 dt 5 K + M + M* =: K*. 0
Clearly, ~~x~~ 5 K*, where K, K* are independent of A. Let H = K* + H’, where H’ > 0 is taken so large that K < H’. Let fl = {z < H}. Therefore, Lx # XNx for any . E X : ~~x~~ X E (O,l],z E DomLnaR. In view of the fact that
QNx=$l’(a(t) - 2
hi(t) exp{x(t - pi)}
clearly, we have QNx # Moreover, from ImQ Lemma 1, equation (5) that (l),(2) has at least
- /’
dt,
c(s - t) exp{x(s)} ds -cc2
i=l
1
0, for z E 80 n Ker L and llzll = H. = Ker L and let J = I, then Deg{QN, R n Ker L, 0) # 0. By using has at least a T-periodic solution in S?. Consequently, we easily see a positive T-periodic solution. The proof is complete.
REFERENCES 1. B.S. Chen and Y.Q. Liu, On the stable periodic solutions of single species molds with hereditary effects, Mathematics Applicata 12, 42-46, (1999). 2. G. Seifert, On a delay-differential equation for single species population variations, Nonlinear Ana$sis TA4A 9, 1051-1059, (1987). 3. S.M. Lenhart and C.C. Travis, Global stability of a biological model with time delay, Pruc. Amer. Math. Sot. 96, 75-78, (1986). 4. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, (1993). 5. I. Gyori and G. Ladss, Oscillation Theory of Delay Diflerential Equations, Oxford Science, Oxford, (1991). 6. J.S. Yu, Global attractivity of zero solution for a class of functions and its applications, Science in China (Series A) 26, 23-33, (1996). 7. G.B. Zhang and K. Gopalsamy, Global attractivity and oscillations in a periodic delay-logistic equation, J. Math. Anal. Appl. 150, 274-283, (1990). 8. R.E. Gaines and J.L. Mawhin, Coincidence Degree, and Non-Linear Differential Equations, New York, Springer-Verlag, (1977).
Letters PERGAMON
Applied
Mathematics
Letters
16 (2003)
561-565 www.elsevier.nl/locate/aml
Positive Periodic Solutions of a Delayed Model in Population CHANG- JIAN ZHAO Department of Mathematics,Shanghai University Shanghai 200436, China and Binzhou Teachers College Shandong, 256604, P.R. China
L. DEBNATH Department of Mathematics, University of Texas-Pan American Fklinburg, TX 785392999, U.S.A. KE WANG Department of Mathematics, Northeast Normal University Chang Chun 130024, P.R. China (Received November Abstract-using a delayed @ 2003
model Elsevier
2001; accepted March 2002)
the theory of coincidence degree, the existence of positive in population is proved. A new result is obtained. Some related Science Ltd. All rights reaervd
Kevords+oincidence
degree,
Positive
periodic
solution,
Fredholm
periodic results
solutions for are improved.
operator.
1. INTRODUCTION We consider the following logistic equation: A+) = N(t)
u(t) - 2 (
bi(t)N (t - q(t)) - J’
where a E C(W,W); bi E C(W, (0,oo)); ri E C(W, IV) c E C(W- , W+), and with an initial condition N(s) = 4(s) 2 0,
c(s - t)N(s) ds
,
(1)
-00
i=l
are T-periodic continuous functions and
4 Eq-aw+)
4(O) > 0,
’
(2)
The periodic solutions of some special cases of system (1) have been studied extensively. example, the qualitative behaviour of a logistic equation with several delays IQ’(t) = Iv(t) (u - &v(t
-T;))
For
(3)
i=l
This work is supported University of Texas-Pan
by National American.
0893-9659/03/S - see front PII: SO89S9659(03)00037-5
matter
Natural
Sciences
@ 2003 Elsevier
Foundation
Science
Ltd.
of China
All rights
and
reserved.
the
Faculty
5pe=t
Research
by 4&W
Council,
562
C.-J. ZHAO
et al.
and some of its generalizations have been studied, where a, bi, Ti are nonnegative constants. For a recent study of equation (3), we refer to [l-6] and references cited therein. Recently, Zhang [7] h as considered the following periodic delay logistic equation in the form: A(t) = r(t)N(t)
(
1 - N@&;(t))
>
,
where r, k E C(lR, (0, oo)), 7 E C(R, [0, oo)), and T, k, r are T-periodic functions. In particular, he proved that there exist T-periodic solutions of equation (4) if T(t) E mT, m E Z+. The logistic delay equation (1) is a natural generalization of equations (3) and (4). Hence, from mathematical and physical points of view, the study of equation (1) is of specific interest. So the main purpose of this article is to study the existence of positive periodic solutions of equation (1)) by using the theory of coincidence degree.
2. EXISTENCE
OF POSITIVE
PERIODIC
SOLUTIONS
In order to study the existence of a positive periodic solution, we first use the following ideas and results (see [S]). Let X, 2 be normed vector spaces, L : Dom L c X - 2 be a linear operator, and N : X -+ 2 be a continuous operator. The operator L will be called a Fkedholm operator of index zero if dim Ker L = codimIm L < +oo and Im L is closed in 2. If L is a Fredholm operator of index zero, there exist continuous projectors P : X + X and Q : 2 + 2 such that Im P = Ker L, Ker Q = Im L = Im(l-Q). It follows that L 1 Dom LnKer P : (I- P)X + Im L is invertible. We denote the inverse of that operator by K. If n is an open bounded subset of X, the operator N will be called Gcompa& on fi , if QN(n) is bounded and K(I - Q)N : fi + X is compact. Since Im Q is isomorphic to Ker L, there exist isomorphisms J : Im Q + Ker L. LEMMA 1. Let L be a Redholm operator of index zero and let N be L-compact on a. Suppose (1) for each X E [0, 11, every solution x of Lx = XNx is such that x # bR; (2) QNx # 0 for each x E 80 n Iker L; (3) Deg{JQN, R n Ker L, 0) # 0. Then the equation Lx = Nx has at least one solution lying in Dom L n 6. Our main result is given in the following theorem. THEOREM 2. The initial value problem (1),(2) h as at least one positive T-periodic solution, if the following conditions are s&i&d: (1) there exists a constant A4 > 0 such that Jo’ a(t) dt = M, (2) CE”=, b(t) < +m, (3) J_“, c(u) du < +oo. .PRooF. Since NW = N(O) . exp ( jOt ( a(s) - 2
b&)N(s
- TV)
- s’
c(r - s)N(T)dT
-m
i=l
then N(t) > 0. Making the change of variable N(t) = exp{x(t)}, equation (1) is refqrmulated as k(t) = a(t) - 2 hi(t) exp{x(t - q(t))} i=l
- /I,
c(s - t) exp{x(s)} ds.
,
Positive
Periodic
563
Solutions
Let X = 2 = {x E C(W, W) : x(t + T) = x(t)} and jlxll = s~p~elc,~l Ix(t)l. both Banach spaces when they are endowed with the norm 11. II. Let
Then X and 2 are
Lx = k, Nx = a(t) - Fbi(t)
exp(x(t - ri(t))}
- J’
c(s - t) exp{x(s)} ds,
-CO
i=l T
Px = f
I0
Qz = ; lT
x(t) dt,
2 E x,
z(t) dt,
z E ‘7.
Hence, Ker L = {x E X : x E R}, ImL={.ztZ:~Tz(t)dt=O}. Suppose xn E X and n is a natural number. Let Px, = 0, iim,,, Since lim Px, = P lim xn = PZ n--r00 n-wxl
xn = 3.
and Px, = 0, therefore, P?=O, i.e., Im L is closed in 2. On the other hand, dimKer L = codimIm L = 1. Hence, L is a F’redholm operator of index zero. Since for all i(t) E Im L, we have
Q@(t)) =; I T
i(t) dt = $(x(T)
- x(0)) = 0
0
and for all x(t) E X, we have Q(I-Q)x=Q(x-Qx)=Qx-Q2x=0. Hence, ImP = KerL,
ImL=KerQ=Im(l-Q).
Obviously, L 1Dom L f~ Ker P : (I - P)X -+ Im L is one-to-one, consequently, it is invertible, we denote the inverse of that operator by K. Since, z E Im L, there exists k(t) E Dom L such that k(t)
Integrating
both sides of (6) over s from 0 to
= t,
2.
(6)
we obtain t
x(t)
= x(0) +
I0
Z(S) ds
(7)
because 1 TO
T I
x(t) dt = 0.
(8)
564
C.-J.
et al.
ZHAO
It follows from (7) and (8) that T
t
JS 0
0
t
K(z) = J 0
z(s) ds dt.
Moreover,
K(I - Q)Nz =
U(S)
- 2
hi(s) exp{x(s -
- J’
TV)}
- 5
C(T
- s) exp{z(T)} d7
ds
--oo
i=l
hi(s) exp(x(s - Ti(s))} - J’
C(T
- s) exp{z(T)} d7
--M
i=l
a(s) - 2
bi(s)exp(z(s
- TV)}
- lSW C(T - s) exp{z(r)}
dr
i=l
If R is an open bounded subset of X, clearly, QN(fi) is a bounded set and K(I - Q)N : fl+ is compact (see [8]). Hence, N is L-compact on a. On the other hand, we consider the operator equation Lx = XNx, X E (0, l] and find that
i(t) = x a(t) - 2 hi(t)exp{x(t- Ti(t))} - St ~(3- t>~P{4s)l ds , -00
i=l
/+E (O,ll. (9)
Suppose that x(t) E X is a solution of equation (9) for a certain X E (0, 11. Integrating of (9) over t from 0 to T, yields 2 i=l
hi(t) q(t))) t) exp(x(t -
+ It
From (9) and (lo), we have T
T I*(t)1
J0
dt
I J 0
la(t)
I dt +
both sides
) J 2b(t) exp{x(t c(s -
exp{x(s)} ds
dt =
T a(t) dt =: M > 0. (10)
0
-m
G(t))}
i=l
t + J
-ccl
c(s -
t)
exp{x(s)} ds dt = M + M*, ‘1
where M* = /,’ la(t)1 dt. Since x(t) is a continuous function defined on X, then there exist E, 11E [0, T] such that
40 = t~~~TlxI(t)~ 477)= t$o=Tlz(t). Hence,
where
Consequently, there exist to E [O,T] such that Ix(t0)l
I In
X
Positive
Periodic
565
Solutions
Furthermore,
T I
I@o)l
+
/
[k(t)1 dt 5 K + M + M* =: K*. 0
Clearly, ~~x~~ 5 K*, where K, K* are independent of A. Let H = K* + H’, where H’ > 0 is taken so large that K < H’. Let fl = {z < H}. Therefore, Lx # XNx for any . E X : ~~x~~ X E (O,l],z E DomLnaR. In view of the fact that
QNx=$l’(a(t) - 2
hi(t) exp{x(t - pi)}
clearly, we have QNx # Moreover, from ImQ Lemma 1, equation (5) that (l),(2) has at least
- /’
dt,
c(s - t) exp{x(s)} ds -cc2
i=l
1
0, for z E 80 n Ker L and llzll = H. = Ker L and let J = I, then Deg{QN, R n Ker L, 0) # 0. By using has at least a T-periodic solution in S?. Consequently, we easily see a positive T-periodic solution. The proof is complete.
REFERENCES 1. B.S. Chen and Y.Q. Liu, On the stable periodic solutions of single species molds with hereditary effects, Mathematics Applicata 12, 42-46, (1999). 2. G. Seifert, On a delay-differential equation for single species population variations, Nonlinear Ana$sis TA4A 9, 1051-1059, (1987). 3. S.M. Lenhart and C.C. Travis, Global stability of a biological model with time delay, Pruc. Amer. Math. Sot. 96, 75-78, (1986). 4. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, (1993). 5. I. Gyori and G. Ladss, Oscillation Theory of Delay Diflerential Equations, Oxford Science, Oxford, (1991). 6. J.S. Yu, Global attractivity of zero solution for a class of functions and its applications, Science in China (Series A) 26, 23-33, (1996). 7. G.B. Zhang and K. Gopalsamy, Global attractivity and oscillations in a periodic delay-logistic equation, J. Math. Anal. Appl. 150, 274-283, (1990). 8. R.E. Gaines and J.L. Mawhin, Coincidence Degree, and Non-Linear Differential Equations, New York, Springer-Verlag, (1977).