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Existence, uniqueness, and global stability of positive solutions to the food-limited population model with random perturbation
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MATHEMATICAL
AND
aCIENCE ~_~DIRECTe ELSEVIER
COMPUTER MODELLING
Mathematical and Computer Modelling 42 (2005) 651-658 www.elsevier.eom/loeate/mcm
Existence, Uniqueness, and Global Stability of Positive Solutions to the Food-Limited Population Model with Random Perturbation DAQING J I A N G ,
NINGZHONG SHI AND YANAN Department of Mathematics Northeast Normal University Changchun 130024, Jilin, P.R. China
ZHAO
daqingj iang©vip. 163. com
(Received and accepted March 200/,) Abstract--This
paper discusses a randomized food-limited population model/V(t) = (r +a/~(t)).
N(t)[(K- N(t))/(K + CN(t))] with initial value N(0) = No, and No is a random variable satisfying 0 < No < K. The existence, uniqueness, and global stability of positive solutions of the equation are studied. @ 2005 Elsevier Ltd. All rights reserved.
K e y w o r d s - - R a n d o m i z e d food-limited population model, Existence, Uniqueness, Global stability in mean and mean square, Lyapunov functional.
1. I N T R O D U C T I O N A s i m p l e f o o d - l i m i t e d p o p u l a t i o n m o d e l , b a s e d on o r d i n a r y differential e q u a t i o n s , is u s u a l l y denoted by
N(t) = rN(t) W ~-:N(t)] [K + CX(t)J
'
(1.1)
and models the population density N of a single species whose members compete among themselves for a limited amount of food and living space, where r is called the intrinsicrate of growth and K the carrying capacity, and both are assmned to be positive. Some detailed studies about the model m a y be found in [i]. M a n y authors have obtained a lot of interesting results about the stabilityand Hopf bifurcation of positive solutions for the above system (I.i) with its general case, for example, see [2,3]. The work was supported by NNSF
of China.
0895-7177/05/$ - see front matter (D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2004.03.011
Typeset by AAdS-TEX
652
D. JIANG et al.
However it is important, in practice, to consider a randomized model based on (1.1). This paper discusses the existence, uniqueness, and global stability of positive solutions to the randomized food-limited population model with intensity a 2,
[ K-N(t)]
iV(t) = (r +aJR(t)) N ( t ) Li~-~-C--~(t)j ,
r , K , C > O,
(1.2)
where B(t) is the one-dimensional standard Brownian motion with B(0) = 0, t > 0 with N(0) = No, and No is a random variable satisfying 0 < No < K. It is reasonable to assume that No is independent of B(t). By the Ito interpretation, equation (1.2) is equivalent to
dN(t) = N ( t )
[L~¥C-~-~(t)J K-N(t)] (r dt + a d B ( t ) ).
(1.3)
It is known that the simple population growth randomized model is of the form (see [4,5])
N(t) = (~ + ~(t))
x(t),
t > 0,
~ > 0.
(1.4)
Equation (1.4) is also said to be a risky investment model, where N ( t ) is the price at time t. The existence, uniqueness, and stability of the above model may be depended solutions to the following stochastic differential equation: dx(t) = f ( x ( t ) , t) dt + g(x(t), t) dB(t),
t > 0;
(1.5)
see also [4,5]. If both f and g are continuous functions, and they satisfy the local Lipschitz condition and the linear growth condition, then equation (1.5) has a unique continuous solution x(t) on t _> 0 (cf. [4,5]). We cannot directly apply the existence and uniqueness results of [4,5] to equation (1.3), since we search for positive solutions. The main results about equation (1.3) are given as follows. THEOREM 1. There exists a uniqueness uniformly continuous positive solution to equation (1.3) for any initial value N(O) = No with 0 < No < K. The following theorems are the global stability in mean and mean square of positive solutions to equation (1.3) by using Lyapunov functional. THEOREM 2. Let N(t) be a continuous positive solution to equation (1.3) for any initial value N(O) = No with 0 < No < K. ](fr > (1/2)a 2, then E[N(t)] -+ K when t ~ +oo. THEOREM 3. Let N ( t ) be a continuous positive solution to equation (1.3) for any initial value N(O) = No with 0 < No < K. If r > c~2, then
lira E [(~¢(t) - K) 2] = 0.
t---+4-oo
Here, and in the sequel,"E[f]" shall mean the mathematical expectation of f.
Existence, Uniqueness, and Global Stability
653
2. P R O O F OF MAIN RESULTS The following lemma is needed for the proofs of Theorems 2 and 3. LEMMA 2.1. (See [6].) Let f be a nonnegative function defined on [0, oc) such that f is integrable on [0, oc) and is uniformly continuous on [0, co). Then limt~+oo f(t) = O. Let N(t) be a solution of (1.3). It is obvious that N = 0 and N = K are solutions of (1.3). If N(t) ~ 0 and N(t) ~ K, by Ito's formula, dln ~ - N
=
dlntNl-dlnlK- NI
- [~-
2~r21( d N ) 2 ] - [ / N - K - 2 ( N 1 K ) 2 ( d N )
2]
K ( K - 2N)K d 2 N ( K - N) d N - 2 ~ 5 ~ - ~ - ~ ) 2 ( N ) _ K ( r dt + a dB)
K+CN K
K+CN then there exists a random variable N(t)
_
r-
(2.1)
K ( K - 2N)a 2 dt 2(K + CN) 2 a 2 dt+adB+
C 1 = C l(~y)
1+
K+CN
such that
Clefo(K/(K+CN(s)))[(r_(1/Z)a2)ds+adB(s)+(l+C/2)(N(s)a2/(K+CN(s)))ds] '
K - N(t)
i.e.~ N(t) =
K 1 4- (1/C1)e-[ ffo(K/(K+CN(~)))(~-O/2)~2) ds+c~dB(s)+(l+C/2)(N(s)c~2/(K+CN(s)))as]'
Since N(0) = No and 0 < No < K, then C1 =- No/(K - No), so we yield g(t) = K 1 1 4- (K/No - 1)e -[f~(K/(K+CN(s)))(r-(1/2)a2) ds+c~dB(s)+(l+C/2)(N(s)a2/(K+CN(s)))ds]'
×
(2.2)
and 0 < N(t) < K, PROOF OF THEOREM
t > O.
(2.3)
1. We first prove the existence of positive solutions. Making the change
of variable
eZ(t) N(t) = K 1 + e~(t) '
(2.4)
then x(t) = l n
N(t) K - N(t)
By the same arguments for getting (2.1), (1.3) is reformed as 1 + eZ(t)[( dx(t) = l + ( C + l ) e X ( t )
l a 2 + (1 + C ) r- 2
a2
ex(t) ) 1 l+(l+C)ex(,) dt+adB(t)
on t >_ 0 with initial value x(0) = z0 --: lnNo/(K - No). Then x0 is independent of B(t).
t2.5)
D, J[ANG et at.
654
Let
I+ e~(~) ( !~2+ (i+c) ~2 e~(t) f(m)= l+(C+l)eZ(0 r- 2 l+(l+C)eZ(t) l+e x
~(~) = 1 + (c + 1)e~
) '
(2.6)
OL
Then both f and 9 are bounded and continuous functions, and they satisfy the local Lipschitz condition and the linear growth condition. It is known (cf. [41 or I5]) that equation (2.5) has a unique continuous solution x(t) on t > 0 with initial value x(0) = x0. Set N(t) = K(e~(t)/(1 + eX(t))), then 0 < N(t) < K and N(t) is a positive continuous solution of (1,3), since
dN
_,,
'K =
-
( eX \ l + e :~]
dm+2(1-~eX%a
(<2
N ( K - N) [
2N d~ + K -~-,# (&)~]
~;-{
N(K-N) Ks K +K ON [(r - la2 + (1 + C) +
K +CNJ
2K
N(K - N) K(K + CN)
K+ON Ne2 )
dt + ~ dB]
(2.7)
dt
(r dt + c, dB).
In fact, since the solution N(t) of equation (1.3) satisfies 0 < N(t) < K, we can see N(t)(K N ( t ) ) / ( K + CN(t)} is bounded and satisfies the Lipschitz condition, and for every p > 0, E[NP(t)] < oo. By Mao [7, Lemma 2.4], almost every samp]e path of N(t) is uniformly continuous on t > 0. Next we prove the uniqueness of solutions. Let Nl(t) and ~%(t) be solutions with initial value N~(0) : N2(0) = No and 0 < No < K. It follows from (2.3) that 0 < N{(t) < K, i : 1,2. Then
(C + I)K2
] (rdt+adB(t))
d(N2(t)- N1(t)) = -c(N2(t) - N1(t)) 1 - (K + CN-7~('K-~ CN2(t))J and
(C+ 1)K2 ] (~d~+~dP(~)). N~(t)-N~(t):-~ifo~(N~(~)-N~(~))[1- (K+CN-7~7~TCN~(~))J Let Z(t) = N2(t) - Nl(t). By the Ito isometry,
2 E[Z~(t)]-< pvE[f~z(4 2 + -C7 E
[i-
we obtain
(C+I)K2 ]~&]~ (~;+CN-~(~7 CN2(4)J
if[Z(~) [i- (K + CN,(s))(K (C + ~)K~ + CN2(s))] adB(s)]
: U~E [Jjz(4 - (,c+ CNI(~))(K+ cN=(~))]
(2.s)
Existence, Uniqueness, and Global Stability
+ -czE
<
<-
655
Z2(s) 1 - (K + CN-~-(-K-~ CN2(s))j
62
<
2(C + 2) 2 c~
~
IZ(s)lds
(~t + ~ ) ]o ~E
~
(2.8)(eont.)
E [Z2(s)] ds
C2
[z%)] ds.
So the function u(t) = E[Z2(t)];
t >_O,
satisfies
u(t) < A(t)
A(t) . - 2(c + 27 (~2t + ~:).
~ot u(s) ds,
(2.9)
C2
Let w(t) = f to u(s)ds. Then w'(t) < A(t)w(t). Hence,
w(t) < w(o)J~ A(s)~ Since w(0) = 0, then w(t) = 0 and so u(t) = 0 for all t > 0. Hence,
P[INI(t)-N2(t)I=O ,
for all t e q N [ 0 , ~ ) ] = 1,
where Q denote the rational numbers. By continuity of t ~ tNl(t) - N½(t)] it follows that
P[INl(t) - N2(t)l = O,
for all t E [0, c~)] = 1,
and the uniqueness is proved. Thus we complete the proof of Theorem 1. PROOF OF THEOREM 2. Let N(t) be a solution with initial value N(0) = No and 0 < No < K. It follows from (2.3) that 0 < N(t) < K on t >_ 0. We define a Lyapunov function V(t) by
V(t) = In K - In N(t),
t _> 0.
By Ito's formula, we have
dr(t) = - d l n N ( t ) -
1 -dN(t) -t-~(dN(t)) N(t)
K = - ( r d t + a d B ) ( lk f
N(t) + CN(t) + -~ LK + CN(t)]
K - N(t) [ = - I~-C---~(t) rdt + a d B and
v(t) = v(0) -
u
(2.10)
dt
ct2 K - N(t) ] 2 K + C N ( t ) dt
~ot ~KT b ~N(s) ( ~ ) [~ds + ~dB(s)
a 2 K - N(s)
]
(2.11)
D. J1ANGet al.
656
Taking the expectation of both sides of equation (2.11), we have
E[V(t)] : E[V(0)] -
(
/o ELK r. _-_-s_-_ -<-<> + CN(s)
r - o= 2 I{T-C--~s)
If r > a2/2, since 0 <
Z LK + C N ( , )
N(t)
ds
(r-a2/2)K+rCN(s)+(a2/2)N(s)]
F~-N(,) : Ely(o)] -
)]
K + CN(s)
(2.12)
ds.
< K, so we obtain
dE[V(t)]d~- E [ KK+-cN(t)N(t) (r -
< -Z
a2/2)
(K - N(t))(K +
K K+ rCN(CN(t) t)+ + (a2/2) N(t)
CN(t))2J
(2.13)
-< ~(7 Tb-7 ElK- N(t)]. Inequality (2.13) means that we have
E[V(t)] is decreasing.
Integrating both sides of inequa]ity (2.13),
(r - +c)~ ~2/2)/0 s[v(t)]<_E[v(o)] K(f
t ElK - ~(~)] d~
Since
EV(O) =
E[ln K - In No] < c~,
thus t
E[V(t)]
+ ~(<- +,~/2) C) 2
E[K - N(~)] d~ < Ev(o) < ~ ,
which leads to E[K - N(t)] C Ll[0, ~ ) . It follows from Theorem 1, continuous on [0, oc). By Lemma 2.1 one obtains lira
t--~OO
EIK -
E[K -
(2.14)
N(t)] is uniformly
N(t)] = 0.
Thus we complete the proof of Theorem 2. PROOF
OF THEOREM
3. Let
V(t)
= InK
- InN(t),
t ~ 0. Following
(2.10) and Ito's formula,
we have
d [V2(t)]
=
=
2v(t) dr(t) + (dr(t)) 2
K-N(t) -2(lnZ-lnN(t))~-+--4X~(t)
+~2 [ K- N(t) ]2
~ T ~-~-(i(t)j dt
[~ d t + ~ d B - a22K~--C--~(t)K-N(t)dt]
(2.15)
Existence, Uniqueness, and Global Stability
657
T h u s , we have
E [vb)] :
E Ivy(o)]
I4 - N(s) (r -2E [fot(lnK - lnN(s)) R~-C--'~(s)
+a2E[fot ( K--N(t) \K + CN(t)) Since
1/K <_ ( l n K - l n N ) / ( K -
dElve(t)] dt
: -2E
N) < 1/N
2 K+CN(s)] ds
(2.16)
ds] .
w h e n 0 < N < K , so we have
K - N(t) (r- a2/2) K +rCN(t) + (a2/2) N(t)] K + CN(t)
(ink - lnN(t))/~C-~(t)
3
[
2
+a2E [K +CN(t)J [ K-N(t) (r-a2/2) K" [ _K_2_N(t) ]2 <_-2E (lnK-lnN(t))~2~-C-~(t) K+CN(t) +a2E K+CN(t)J <_2E[K-N(t) ]2(r_~___~)+a2E[_K_--N(t) ]2 /(:~ C-~(t)J [K + CN(t)]
(2.17)
V5 : N(t) 12 <--2(~-~2)E LK+cN(t)J 2(~-~ 2)
_< (17~)-~K~E [(K- N(t))~l. Inequality (2.17) means that we have
E[V2(t)] is decreasing.
Integrating both sides of inequality (2.17),
2 (~ - ~2) 2 f0t E [(K - N(s)) 2] (1-~)-ff-KK
E [V2(t)] < E [V2(0)]
ds.
(2.18)
Since E [V2(0)] : E [ ( l n K - lnN0) 2] < 0% then
2 (~ - ~2) [~
E[V2(t)] + (I~)-ff-~K2 ~, E[(K- N(s)) 2] ds <_E[V2(O)]
lira t--*Oo
2] is uniformly
El(K- N(t)) 2] =0.
T h u s , we c o m p l e t e the proof of T h e o r e m 3.
REFERENCES 1. F.E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology 44, 651-663, (1963). 2. K. Gopalsamy, M.R.S. Kulenovic and G. Ladas, Time lags in a "food-limited" population model, AppL Anal. 31, 225-237, (1988). 3. K. Gopalsamy, M.R.S. Kulenovic and G. Ladas, Environmental periodicity and time delays in a "food-limited" population model, J. Math. Anal. Appl. 147, 225-237, (1990).
658
D. JIANG et al.
4. L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, (1972). 5. A. Friedman, Stochastic Differential Equations and Their Applications~ Volume 2, Academic Press~ San Diego, (1976). 6. I. Barbalat, Systems d'equations differentielles d'osci d'oscillations nonlineaires~ Rev. Rournaine Math. Pures Appl. 4, 267-270, (1959). 7. X. Mao, Stochastic versions of the Lassalle theorem, J. Differential Equations 153, 175-195~ (1999).
MATHEMATICAL
AND
aCIENCE ~_~DIRECTe ELSEVIER
COMPUTER MODELLING
Mathematical and Computer Modelling 42 (2005) 651-658 www.elsevier.eom/loeate/mcm
Existence, Uniqueness, and Global Stability of Positive Solutions to the Food-Limited Population Model with Random Perturbation DAQING J I A N G ,
NINGZHONG SHI AND YANAN Department of Mathematics Northeast Normal University Changchun 130024, Jilin, P.R. China
ZHAO
daqingj iang©vip. 163. com
(Received and accepted March 200/,) Abstract--This
paper discusses a randomized food-limited population model/V(t) = (r +a/~(t)).
N(t)[(K- N(t))/(K + CN(t))] with initial value N(0) = No, and No is a random variable satisfying 0 < No < K. The existence, uniqueness, and global stability of positive solutions of the equation are studied. @ 2005 Elsevier Ltd. All rights reserved.
K e y w o r d s - - R a n d o m i z e d food-limited population model, Existence, Uniqueness, Global stability in mean and mean square, Lyapunov functional.
1. I N T R O D U C T I O N A s i m p l e f o o d - l i m i t e d p o p u l a t i o n m o d e l , b a s e d on o r d i n a r y differential e q u a t i o n s , is u s u a l l y denoted by
N(t) = rN(t) W ~-:N(t)] [K + CX(t)J
'
(1.1)
and models the population density N of a single species whose members compete among themselves for a limited amount of food and living space, where r is called the intrinsicrate of growth and K the carrying capacity, and both are assmned to be positive. Some detailed studies about the model m a y be found in [i]. M a n y authors have obtained a lot of interesting results about the stabilityand Hopf bifurcation of positive solutions for the above system (I.i) with its general case, for example, see [2,3]. The work was supported by NNSF
of China.
0895-7177/05/$ - see front matter (D 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2004.03.011
Typeset by AAdS-TEX
652
D. JIANG et al.
However it is important, in practice, to consider a randomized model based on (1.1). This paper discusses the existence, uniqueness, and global stability of positive solutions to the randomized food-limited population model with intensity a 2,
[ K-N(t)]
iV(t) = (r +aJR(t)) N ( t ) Li~-~-C--~(t)j ,
r , K , C > O,
(1.2)
where B(t) is the one-dimensional standard Brownian motion with B(0) = 0, t > 0 with N(0) = No, and No is a random variable satisfying 0 < No < K. It is reasonable to assume that No is independent of B(t). By the Ito interpretation, equation (1.2) is equivalent to
dN(t) = N ( t )
[L~¥C-~-~(t)J K-N(t)] (r dt + a d B ( t ) ).
(1.3)
It is known that the simple population growth randomized model is of the form (see [4,5])
N(t) = (~ + ~(t))
x(t),
t > 0,
~ > 0.
(1.4)
Equation (1.4) is also said to be a risky investment model, where N ( t ) is the price at time t. The existence, uniqueness, and stability of the above model may be depended solutions to the following stochastic differential equation: dx(t) = f ( x ( t ) , t) dt + g(x(t), t) dB(t),
t > 0;
(1.5)
see also [4,5]. If both f and g are continuous functions, and they satisfy the local Lipschitz condition and the linear growth condition, then equation (1.5) has a unique continuous solution x(t) on t _> 0 (cf. [4,5]). We cannot directly apply the existence and uniqueness results of [4,5] to equation (1.3), since we search for positive solutions. The main results about equation (1.3) are given as follows. THEOREM 1. There exists a uniqueness uniformly continuous positive solution to equation (1.3) for any initial value N(O) = No with 0 < No < K. The following theorems are the global stability in mean and mean square of positive solutions to equation (1.3) by using Lyapunov functional. THEOREM 2. Let N(t) be a continuous positive solution to equation (1.3) for any initial value N(O) = No with 0 < No < K. ](fr > (1/2)a 2, then E[N(t)] -+ K when t ~ +oo. THEOREM 3. Let N ( t ) be a continuous positive solution to equation (1.3) for any initial value N(O) = No with 0 < No < K. If r > c~2, then
lira E [(~¢(t) - K) 2] = 0.
t---+4-oo
Here, and in the sequel,"E[f]" shall mean the mathematical expectation of f.
Existence, Uniqueness, and Global Stability
653
2. P R O O F OF MAIN RESULTS The following lemma is needed for the proofs of Theorems 2 and 3. LEMMA 2.1. (See [6].) Let f be a nonnegative function defined on [0, oc) such that f is integrable on [0, oc) and is uniformly continuous on [0, co). Then limt~+oo f(t) = O. Let N(t) be a solution of (1.3). It is obvious that N = 0 and N = K are solutions of (1.3). If N(t) ~ 0 and N(t) ~ K, by Ito's formula, dln ~ - N
=
dlntNl-dlnlK- NI
- [~-
2~r21( d N ) 2 ] - [ / N - K - 2 ( N 1 K ) 2 ( d N )
2]
K ( K - 2N)K d 2 N ( K - N) d N - 2 ~ 5 ~ - ~ - ~ ) 2 ( N ) _ K ( r dt + a dB)
K+CN K
K+CN then there exists a random variable N(t)
_
r-
(2.1)
K ( K - 2N)a 2 dt 2(K + CN) 2 a 2 dt+adB+
C 1 = C l(~y)
1+
K+CN
such that
Clefo(K/(K+CN(s)))[(r_(1/Z)a2)ds+adB(s)+(l+C/2)(N(s)a2/(K+CN(s)))ds] '
K - N(t)
i.e.~ N(t) =
K 1 4- (1/C1)e-[ ffo(K/(K+CN(~)))(~-O/2)~2) ds+c~dB(s)+(l+C/2)(N(s)c~2/(K+CN(s)))as]'
Since N(0) = No and 0 < No < K, then C1 =- No/(K - No), so we yield g(t) = K 1 1 4- (K/No - 1)e -[f~(K/(K+CN(s)))(r-(1/2)a2) ds+c~dB(s)+(l+C/2)(N(s)a2/(K+CN(s)))ds]'
×
(2.2)
and 0 < N(t) < K, PROOF OF THEOREM
t > O.
(2.3)
1. We first prove the existence of positive solutions. Making the change
of variable
eZ(t) N(t) = K 1 + e~(t) '
(2.4)
then x(t) = l n
N(t) K - N(t)
By the same arguments for getting (2.1), (1.3) is reformed as 1 + eZ(t)[( dx(t) = l + ( C + l ) e X ( t )
l a 2 + (1 + C ) r- 2
a2
ex(t) ) 1 l+(l+C)ex(,) dt+adB(t)
on t >_ 0 with initial value x(0) = z0 --: lnNo/(K - No). Then x0 is independent of B(t).
t2.5)
D, J[ANG et at.
654
Let
I+ e~(~) ( !~2+ (i+c) ~2 e~(t) f(m)= l+(C+l)eZ(0 r- 2 l+(l+C)eZ(t) l+e x
~(~) = 1 + (c + 1)e~
) '
(2.6)
OL
Then both f and 9 are bounded and continuous functions, and they satisfy the local Lipschitz condition and the linear growth condition. It is known (cf. [41 or I5]) that equation (2.5) has a unique continuous solution x(t) on t > 0 with initial value x(0) = x0. Set N(t) = K(e~(t)/(1 + eX(t))), then 0 < N(t) < K and N(t) is a positive continuous solution of (1,3), since
dN
_,,
'K =
-
( eX \ l + e :~]
dm+2(1-~eX%a
(<2
N ( K - N) [
2N d~ + K -~-,# (&)~]
~;-{
N(K-N) Ks K +K ON [(r - la2 + (1 + C) +
K +CNJ
2K
N(K - N) K(K + CN)
K+ON Ne2 )
dt + ~ dB]
(2.7)
dt
(r dt + c, dB).
In fact, since the solution N(t) of equation (1.3) satisfies 0 < N(t) < K, we can see N(t)(K N ( t ) ) / ( K + CN(t)} is bounded and satisfies the Lipschitz condition, and for every p > 0, E[NP(t)] < oo. By Mao [7, Lemma 2.4], almost every samp]e path of N(t) is uniformly continuous on t > 0. Next we prove the uniqueness of solutions. Let Nl(t) and ~%(t) be solutions with initial value N~(0) : N2(0) = No and 0 < No < K. It follows from (2.3) that 0 < N{(t) < K, i : 1,2. Then
(C + I)K2
] (rdt+adB(t))
d(N2(t)- N1(t)) = -c(N2(t) - N1(t)) 1 - (K + CN-7~('K-~ CN2(t))J and
(C+ 1)K2 ] (~d~+~dP(~)). N~(t)-N~(t):-~ifo~(N~(~)-N~(~))[1- (K+CN-7~7~TCN~(~))J Let Z(t) = N2(t) - Nl(t). By the Ito isometry,
2 E[Z~(t)]-< pvE[f~z(4 2 + -C7 E
[i-
we obtain
(C+I)K2 ]~&]~ (~;+CN-~(~7 CN2(4)J
if[Z(~) [i- (K + CN,(s))(K (C + ~)K~ + CN2(s))] adB(s)]
: U~E [Jjz(4 - (,c+ CNI(~))(K+ cN=(~))]
(2.s)
Existence, Uniqueness, and Global Stability
+ -czE
<
<-
655
Z2(s) 1 - (K + CN-~-(-K-~ CN2(s))j
62
<
2(C + 2) 2 c~
~
IZ(s)lds
(~t + ~ ) ]o ~E
~
(2.8)(eont.)
E [Z2(s)] ds
C2
[z%)] ds.
So the function u(t) = E[Z2(t)];
t >_O,
satisfies
u(t) < A(t)
A(t) . - 2(c + 27 (~2t + ~:).
~ot u(s) ds,
(2.9)
C2
Let w(t) = f to u(s)ds. Then w'(t) < A(t)w(t). Hence,
w(t) < w(o)J~ A(s)~ Since w(0) = 0, then w(t) = 0 and so u(t) = 0 for all t > 0. Hence,
P[INI(t)-N2(t)I=O ,
for all t e q N [ 0 , ~ ) ] = 1,
where Q denote the rational numbers. By continuity of t ~ tNl(t) - N½(t)] it follows that
P[INl(t) - N2(t)l = O,
for all t E [0, c~)] = 1,
and the uniqueness is proved. Thus we complete the proof of Theorem 1. PROOF OF THEOREM 2. Let N(t) be a solution with initial value N(0) = No and 0 < No < K. It follows from (2.3) that 0 < N(t) < K on t >_ 0. We define a Lyapunov function V(t) by
V(t) = In K - In N(t),
t _> 0.
By Ito's formula, we have
dr(t) = - d l n N ( t ) -
1 -dN(t) -t-~(dN(t)) N(t)
K = - ( r d t + a d B ) ( lk f
N(t) + CN(t) + -~ LK + CN(t)]
K - N(t) [ = - I~-C---~(t) rdt + a d B and
v(t) = v(0) -
u
(2.10)
dt
ct2 K - N(t) ] 2 K + C N ( t ) dt
~ot ~KT b ~N(s) ( ~ ) [~ds + ~dB(s)
a 2 K - N(s)
]
(2.11)
D. J1ANGet al.
656
Taking the expectation of both sides of equation (2.11), we have
E[V(t)] : E[V(0)] -
(
/o ELK r. _-_-s_-_ -<-<> + CN(s)
r - o= 2 I{T-C--~s)
If r > a2/2, since 0 <
Z LK + C N ( , )
N(t)
ds
(r-a2/2)K+rCN(s)+(a2/2)N(s)]
F~-N(,) : Ely(o)] -
)]
K + CN(s)
(2.12)
ds.
< K, so we obtain
dE[V(t)]d~- E [ KK+-cN(t)N(t) (r -
< -Z
a2/2)
(K - N(t))(K +
K K+ rCN(CN(t) t)+ + (a2/2) N(t)
CN(t))2J
(2.13)
-< ~(7 Tb-7 ElK- N(t)]. Inequality (2.13) means that we have
E[V(t)] is decreasing.
Integrating both sides of inequa]ity (2.13),
(r - +c)~ ~2/2)/0 s[v(t)]<_E[v(o)] K(f
t ElK - ~(~)] d~
Since
EV(O) =
E[ln K - In No] < c~,
thus t
E[V(t)]
+ ~(<- +,~/2) C) 2
E[K - N(~)] d~ < Ev(o) < ~ ,
which leads to E[K - N(t)] C Ll[0, ~ ) . It follows from Theorem 1, continuous on [0, oc). By Lemma 2.1 one obtains lira
t--~OO
EIK -
E[K -
(2.14)
N(t)] is uniformly
N(t)] = 0.
Thus we complete the proof of Theorem 2. PROOF
OF THEOREM
3. Let
V(t)
= InK
- InN(t),
t ~ 0. Following
(2.10) and Ito's formula,
we have
d [V2(t)]
=
=
2v(t) dr(t) + (dr(t)) 2
K-N(t) -2(lnZ-lnN(t))~-+--4X~(t)
+~2 [ K- N(t) ]2
~ T ~-~-(i(t)j dt
[~ d t + ~ d B - a22K~--C--~(t)K-N(t)dt]
(2.15)
Existence, Uniqueness, and Global Stability
657
T h u s , we have
E [vb)] :
E Ivy(o)]
I4 - N(s) (r -2E [fot(lnK - lnN(s)) R~-C--'~(s)
+a2E[fot ( K--N(t) \K + CN(t)) Since
1/K <_ ( l n K - l n N ) / ( K -
dElve(t)] dt
: -2E
N) < 1/N
2 K+CN(s)] ds
(2.16)
ds] .
w h e n 0 < N < K , so we have
K - N(t) (r- a2/2) K +rCN(t) + (a2/2) N(t)] K + CN(t)
(ink - lnN(t))/~C-~(t)
3
[
2
+a2E [K +CN(t)J [ K-N(t) (r-a2/2) K" [ _K_2_N(t) ]2 <_-2E (lnK-lnN(t))~2~-C-~(t) K+CN(t) +a2E K+CN(t)J <_2E[K-N(t) ]2(r_~___~)+a2E[_K_--N(t) ]2 /(:~ C-~(t)J [K + CN(t)]
(2.17)
V5 : N(t) 12 <--2(~-~2)E LK+cN(t)J 2(~-~ 2)
_< (17~)-~K~E [(K- N(t))~l. Inequality (2.17) means that we have
E[V2(t)] is decreasing.
Integrating both sides of inequality (2.17),
2 (~ - ~2) 2 f0t E [(K - N(s)) 2] (1-~)-ff-KK
E [V2(t)] < E [V2(0)]
ds.
(2.18)
Since E [V2(0)] : E [ ( l n K - lnN0) 2] < 0% then
2 (~ - ~2) [~
E[V2(t)] + (I~)-ff-~K2 ~, E[(K- N(s)) 2] ds <_E[V2(O)]
lira t--*Oo
2] is uniformly
El(K- N(t)) 2] =0.
T h u s , we c o m p l e t e the proof of T h e o r e m 3.
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658
D. JIANG et al.
4. L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, (1972). 5. A. Friedman, Stochastic Differential Equations and Their Applications~ Volume 2, Academic Press~ San Diego, (1976). 6. I. Barbalat, Systems d'equations differentielles d'osci d'oscillations nonlineaires~ Rev. Rournaine Math. Pures Appl. 4, 267-270, (1959). 7. X. Mao, Stochastic versions of the Lassalle theorem, J. Differential Equations 153, 175-195~ (1999).