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Applied Mathematics Letters
Applied Mathematics Letters 16 (2003) 165-171
www.elsevier.com/locate/aml
S t a b l e P e r i o d i c S o l u t i o n s in a D i s c r e t e P e r i o d i c Logistic E q u a t i o n ZHAN ZHOU Department of Applied Mathematics, Hunan University Changsha, Hunan 410082, P.R. China zhanzhou~mail, hunu. edu. c n
XINGFU Zou* Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, N.F., A1C 5S7, Canada xzou~math, mun. c a
(Received July 2001; accepted March 2002) A b s t r a c t - - I n this paper, we consider a discrete logistic equation
x(n + 1) : x(n)exp Jr(n)(1-
x(n) K(n) ] J '
where {r(n)} and {K(n)} are positive w-periodic sequences. Sufficient conditions are obtained for the existence of a positive and globally asymptotically stable w-periodic solution. Counterexamples are given to illustrate that the conclusions in [1] are incorrect. (~) 2003 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - P e r i o d i c solution, Logistic equation, Stability.
1. I N T R O D U C T I O N One of the the basic differential equation models for population growth of a single species is the logistic equation
dx(t) -r(t)x(t)ll-~] dt
'
t>0,
-
(1.1)
where r(.) and K(.) are positive functions in [0, co), representing the intrinsic growtk rate and the carrying capacity, respectively. W h e n K(.) is constant, the dynamics of (1.1) are completely known: every positive solution converges to the positive equilibrium. In m a n y situations, r(t) and K(t) can be assumed to be n o n c o n s t a n t periodic functions with a common period T to reflect the seasonal fluctuations. In such a periodic case, it has been shown t h a t (1.1) has a positive T-periodic solution 2(t) which attracts every positive solution x(t) of (1.1) as t -~ oo. See, e.g., [2-4]. This work was supported by the NNSF of China and NSERC of Canada. *Author to whom all correspondence should be addressed. 0893-9659/03/$ - see front matter ~) 2003 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(02)00175-1
Typeset by .AA,~S-TEX
166
Z. ZHOU AND X. Z o u
In this paper, we consider a discrete analogue of (1.1),
x(n + 1) = x(n) exp r(n)
1
K(n)
'
n e N,
(1.2)
under the assumptions that x(0) > 0, {r(n)} and { g ( n ) } are strictly positive sequences of real numbers defined for n E N = {0, 1 , 2 , . . . }. In addition, there exist positive constants r., r*, K . , and K* such that
O 0), this implies
z(no) < g(no) < g*. Therefore, by the fact that maxxeR z exp[r(1 --x)] = ( l / r ) e x p ( r -
x(no+l)=x(no)exp[r(no)(1 _
K(no)X(n°))] K(no)
exp
K*exp(r*
0, we have
- 1) = u*.
r*
We claim that z(n)
< u *,
for n > no.
In fact, if there exists an integer m _> no ÷ 2 such that x(m) > u*, and letting m* be the least integer between no and m such that x(m*) = maxno no + 2 and x(m*) > x(m* - 1) which implies x(m*) z(n + 1) for n E N. By (1.2), we see t h a t
1 - z(n___._)_) g(n) >_ g . for n E N. Since {x(n)} is nonincreasing and has a lower bound K . , we know limn--.oo x(n) = ~ > K.. Letting n ~ oo in (1.2), we get = lim K(n) n*.
(2.5)
We consider two cases. CASE (i). There exists a positive integer n0 > n* such that x(~0 + 1) < x(~0). Similar to Case 1 in the proof of(2.2), we see t h a t
x(n) > K. exp r*
1
K.
,
n" n >
(2.6)
CASE (ii). z(n + 1) k x(n) for. n _> n*. According to (2.5), we know limn-,oo x(n) = I. Letting n --- oc in (1.~) leads to l i m ~ o ~ K(n) = I. So, l=
lim x ( n ) = ? l ---4 o o
lim K ( n ) > K . > K . n ---¢ o o
--
exp r*
1
--
g .
Combining Cases (i) and (ii), we see that
liminfx(n)>K, exp(r*(1 n-~oo -
]
u*+e) K. /
Since e is arbitrary, we know (2.4) holds. The proof is completed by combining (2.2) with (2.4). REMARK 2.1. Since u*
Applied Mathematics Letters
Applied Mathematics Letters 16 (2003) 165-171
www.elsevier.com/locate/aml
S t a b l e P e r i o d i c S o l u t i o n s in a D i s c r e t e P e r i o d i c Logistic E q u a t i o n ZHAN ZHOU Department of Applied Mathematics, Hunan University Changsha, Hunan 410082, P.R. China zhanzhou~mail, hunu. edu. c n
XINGFU Zou* Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, N.F., A1C 5S7, Canada xzou~math, mun. c a
(Received July 2001; accepted March 2002) A b s t r a c t - - I n this paper, we consider a discrete logistic equation
x(n + 1) : x(n)exp Jr(n)(1-
x(n) K(n) ] J '
where {r(n)} and {K(n)} are positive w-periodic sequences. Sufficient conditions are obtained for the existence of a positive and globally asymptotically stable w-periodic solution. Counterexamples are given to illustrate that the conclusions in [1] are incorrect. (~) 2003 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - P e r i o d i c solution, Logistic equation, Stability.
1. I N T R O D U C T I O N One of the the basic differential equation models for population growth of a single species is the logistic equation
dx(t) -r(t)x(t)ll-~] dt
'
t>0,
-
(1.1)
where r(.) and K(.) are positive functions in [0, co), representing the intrinsic growtk rate and the carrying capacity, respectively. W h e n K(.) is constant, the dynamics of (1.1) are completely known: every positive solution converges to the positive equilibrium. In m a n y situations, r(t) and K(t) can be assumed to be n o n c o n s t a n t periodic functions with a common period T to reflect the seasonal fluctuations. In such a periodic case, it has been shown t h a t (1.1) has a positive T-periodic solution 2(t) which attracts every positive solution x(t) of (1.1) as t -~ oo. See, e.g., [2-4]. This work was supported by the NNSF of China and NSERC of Canada. *Author to whom all correspondence should be addressed. 0893-9659/03/$ - see front matter ~) 2003 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(02)00175-1
Typeset by .AA,~S-TEX
166
Z. ZHOU AND X. Z o u
In this paper, we consider a discrete analogue of (1.1),
x(n + 1) = x(n) exp r(n)
1
K(n)
'
n e N,
(1.2)
under the assumptions that x(0) > 0, {r(n)} and { g ( n ) } are strictly positive sequences of real numbers defined for n E N = {0, 1 , 2 , . . . }. In addition, there exist positive constants r., r*, K . , and K* such that
O 0), this implies
z(no) < g(no) < g*. Therefore, by the fact that maxxeR z exp[r(1 --x)] = ( l / r ) e x p ( r -
x(no+l)=x(no)exp[r(no)(1 _
K(no)X(n°))] K(no)
exp
K*exp(r*
0, we have
- 1) = u*.
r*
We claim that z(n)
< u *,
for n > no.
In fact, if there exists an integer m _> no ÷ 2 such that x(m) > u*, and letting m* be the least integer between no and m such that x(m*) = maxno no + 2 and x(m*) > x(m* - 1) which implies x(m*) z(n + 1) for n E N. By (1.2), we see t h a t
1 - z(n___._)_) g(n) >_ g . for n E N. Since {x(n)} is nonincreasing and has a lower bound K . , we know limn--.oo x(n) = ~ > K.. Letting n ~ oo in (1.2), we get = lim K(n) n*.
(2.5)
We consider two cases. CASE (i). There exists a positive integer n0 > n* such that x(~0 + 1) < x(~0). Similar to Case 1 in the proof of(2.2), we see t h a t
x(n) > K. exp r*
1
K.
,
n" n >
(2.6)
CASE (ii). z(n + 1) k x(n) for. n _> n*. According to (2.5), we know limn-,oo x(n) = I. Letting n --- oc in (1.~) leads to l i m ~ o ~ K(n) = I. So, l=
lim x ( n ) = ? l ---4 o o
lim K ( n ) > K . > K . n ---¢ o o
--
exp r*
1
--
g .
Combining Cases (i) and (ii), we see that
liminfx(n)>K, exp(r*(1 n-~oo -
]
u*+e) K. /
Since e is arbitrary, we know (2.4) holds. The proof is completed by combining (2.2) with (2.4). REMARK 2.1. Since u*
