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Sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat
Applied Mathematics and Computation 339 (2018) 199–205
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat Dianli Zhao∗, Sanling Yuan College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
a r t i c l e
i n f o
a b s t r a c t
Keywords: Stochastic chemostat model Feller property Stationary distribution Extinction Break-even concentration
This paper studies the asymptotic behaviors of one classical chemostat model in a stochastic environment. Based on the Feller property, sharp conditions are derived for the existence of a stationary distribution by using the mutually exclusive possibilities known in [11, 12] (See Lemma 2.4 for details), which closes the gap left by the Lyapunov function. Further, we obtain a sufficient condition for the extinction of the organism based on two noise-induced parameters: an analogue of the feed concentration S∗ and the break-even concentration λ. Results indicate that both noises have negative effects on persistence of the microorganism. © 2018 Elsevier Inc. All rights reserved.
1. Introduction Let S(t) and x(t) be concentrations of the nutrient and the microorganisms at time of the nutrient, D be the washout rates for S, D1 be the removal rate combining the the death rate of the microorganism x, and kμ+SS be the growth rate function. Then, the tory apparatus for the continuous culture of microorganisms) is classically represented equations taking the form (See [1,2])
dS (t ) = D S0 − S (t ) −
t respectively, S0 be the input rate dilution rate of the chemostat and chemostat (a basic piece of laboraas a system of ordinary differential
μS(t )x(t ) μS(t ) dt, dx(t ) = − D1 x(t )dt. k + S (t ) k + S (t )
(1)
Break-even concentration λd = μ−D1 is an important parameter for analyzing (1). In detail, if λd ≥ S0 , the only washout 1 kD
equilibrium E 0 = (S0 , 0 ) is globally asymptotically stable; if λd < S0 , the positive equilibrium E ∗ = (λd ,
D k + λd
S0 −λd
μλd
) exists
and is also globally asymptotically stable, see [3,4] for details. Due to the existence of random effects almost everywhere in the reality, it is natural to address what happens when the stochastic perturbation is taken into account. For instance, Campillo et al. [5] established a set of stochastic chemostat models that are valid at different scales and expound the mechanism to switch from one model to another. Imhof and Walcher [6] introduced a rigorous method to get the stochastic chemostat model by defining a discrete time Markov chain and proving its convergence. The above method has also been used in [7,8] to establish the studied stochastic models. By using the method in [6], a stochastic chemostat model with the classical Monod growth rate function can be written as
∗
Corresponding author. E-mail address: [email protected] (D. Zhao).
https://doi.org/10.1016/j.amc.2018.07.020 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
200
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
follows
0
D S − S (t ) − μkS+(tS)(xt()t ) dt + σ0 S (t )dB0 (t ), μS(t ) − D1 x(t )dt + σ1 x(t )dB1 (t ). k+S (t )
dS (t ) = dx(t ) =
(2)
B0 (t) and B1 (t) are two independent Brownian motions. The parameters in model (2) are all positive. Then for any initial value (S (0 ), x(0 )) ∈ R2+ , model (2) has a uniquely global solution (S (t ), x(t )) ∈ R2+ a.s. for all t ≥ 0 (See [6, Proposition 6]). Our goal is to establish an almost perfect condition for existence of the stationary distribution for (2). Further, we derive the sufficient condition for extinction of the microorganism. Remark 1. Wang and Jiang [8] have studied the stationary distribution for the stochastic chemostat model with general response functions. Except for the linear growth rate, the actual growth rate function will be less than its value at S0 in time average subject to the effects of noises. However the averaged growth rate function can not be applied in the Lyapunov method [8, Lemma 2.1], and the sufficient condition has to be derived under extra conditions, and is not the optimal condition. Different from the above, in this paper we employe the Feller property and mutually exclusive possibilities to derive the condition for existence of the stationary distribution, which can close the gap left by using the Lyapunov method. The method of this paper can be extended to study the stochastic chemostat models with general response functions. 2. Conditions for the stationary distribution To begin with, let’s prepare some basic results. Lemma 2.1 [9, Lemma 4.9]. For any initial value ϕ (0 ) ∈ R+ , consider the equation
dϕ (t ) = D S0 − ϕ (t ) dt + σ0 ϕ (t )dB0 (t ), t > 0. Then (3) has the stationary distribution
ba −(a+1) − b 2 p( x ) = x e x f or x ∈ R+ with a = 2 (a ) σ0 Lemma 2.2. Let ϕ (t) be solution of (3), then lim
t→∞
(3)
ln(k+ϕ (t ) ) t
D+
σ02
, b=
2
2 DS 0
σ02
and (a ) =
∞ 0
t a−1 e−t dt .
= 0 a.s.
Proof. Let ξ (t) be solution of dξ (t ) = −Dξ (t )dt + σ0 ξ (t )dB0 (t ) with ξ (0 ) = 1. By using Itoˆ formula to ln ξ (t) and σ02
then taking integrations, ξ (t ) = e−(D+ 2 )t +σ0 B0 (t ) . Then, by applying the variation-of-constants formula, it yields σ0 sup 2|B0 (s )| σ02 σ02 0 ϕ (t ) = ϕ (0 )e−(D+ 2 )t +σ0 B0 (t ) + DS0 0t e−(D+ 2 )(t−s)+σ0 (B0 (t )−B0 (s)) ds. Then ϕ (t ) ≤ ϕ (0 )eσ0 B0 (t ) + DSσ 2 e 0≤s≤t . Noting sup |B0 (s )|
lim
t→∞
0≤s≤t
t
D+ 20
= 0, we get the desired result.
1 (ln(k t→∞ t
Remark 2. Similar to proof of Lemma 2.2, lim
+ S(t ) + x(t )) ) = 0 a.s.
Remark 3. Note the positivity of (S(t), x(t)) and ϕ (t), then S(t) ≤ ϕ (t) a.s. holds due to the stochastic comparison theorem. Lemma 2.3. Let (S(t), x(t)) be solution of (2) with initial value (S(0 ), x(0 )) ∈ R2+ , then for any p ∈ [1, 1 + that
E [S (t ) + x(t )] p ≤ where K ( p) =
pγ ( p) 2K ( p ) 1 + [S (0 ) + x(0 )] p e− 2 t and lim sup γ ( p) t t→∞
sup{DS0 x p−1 x>0
γ ( p) − 2 x p } and
γ ( p) = [min{D, D1 } −
( p−1 ) 2
t 0
E [S (s ) + x(s )] p ds ≤
it holds
2K ( p ) , γ ( p)
max{σ02 , σ12 }].
Proof. Applying Itoˆ formula to model (2) yields
2 min{D,D1 } ], max{σ02 ,σ12 }
d[S (t ) + x(t )] p = p [S (t ) + x(t )] p−1 DS0 − DS (t ) − D1 x(t ) +
( p − 1) 2
[S (t ) + x(t )]
p−2
σ02 S2 (t ) + σ12 x2 (t ) dt
+ p[S (t ) + x(t )] p−1 (σ0 S (t )dB0 (t ) + σ1 x(t )dB1 (t ))
( p − 1) ≤ p DS0 [S (t ) + x(t )] p−1 − min {D, D1 } − max σ02 , σ12 [S (t ) + x(t )] p dt 2 + p[S (t ) + x(t )] p−1 (σ0 S (t )dB0 (t ) + σ1 x(t )dB1 (t ))
γ ( p) p p−1 ≤ p K ( p) − [S (t ) + x(t )] dt + p[S (t ) + x(t )] (σ0 S (t )dB0 (t ) + σ1 x(t )dB1 (t )) 2
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
201
and
d[S (t ) + x(t )] p e
pγ ( p) 2 t
=
pγ ( p) pγ ( p) p pγ ( p) p [S (t ) + x(t )] e 2 t dt + e 2 t d[S (t ) + x(t )] 2
≤ pK ( p)e
pγ ( p) 2 t
dt + e
pγ ( p) 2 t
p[S (t ) + x(t )] p−1 (σ0 S (t )dB0 (t ) + σ1 x(t )dB1 (t )).
Then by taking integrations and taking the expectations, we get
E [S (t ) + x(t )] p ≤ [S (0 ) + x(0 )] p e−
pγ ( p) 2 t
+ pK ( p)
t
e−
pγ ( p) 2
(t−s ) ds
0
≤ [S (0 ) + x(0 )] p e−
pγ ( p) 2 t
+
2K ( p ) . γ ( p)
Further, we can get
lim sup t→∞
1 t
t 0
E [S (s ) + x(s )] p ds ≤ [S (0 ) + x(0 )] p lim sup t→∞
1 t
t
e−
pγ ( p) 2 s
0
ds +
2K ( p ) 2K ( p ) = . γ ( p) γ ( p)
The following result, known as mutually exclusive possibilities, is vital to proving the existence of the stationary distribution. It was proved by Stettner (1986 [11]), and stated and used in the proof of Theorem 4.5 of Meyn and Tweedie (1993 [12]). Lemma 2.4 (Mutually exclusive possibilities, [11,12]). Let X(t) ∈ Rd be a Feller process, then either an invariant probability measure π ( · ) exists, or
1 ν t
t
P (s, x, C )ν (dx )ds = 0,
lim sup
t→∞
0
(4)
for any compact set C ∈ Rd , where the supremum is taken over all initial distributions ν on Rd and P (t, x, c ) is the probability for X (t ) ∈ C with X (0 ) = x. (μ
t
S (s )
Remark 4. From (2), x(t ) = x(0 )e 0 k+S(s) t 0 lim 1t 0 k+ϕϕ(s()s ) ds ≤ k+S S0 < 1, it’s clear that
σ2 ds−(D1 + 21 )t )+σ1 B1 (t )
. Since
lim 1 t→∞ t
t 0
ϕ (s )ds = S0 and lim sup 1t t→∞
t
S (s ) 0 k+S(s ) ds
≤
t→∞
x(t ) → 0 a.s. as t → ∞, i f
μ ≤ D1 +
σ12 2
.
So, in the following, we always assume that μ > D1 +
σ12 2
.
∞ For convenience, we denote P (S ) = kμ+SS and F (a, b, k ) = 0 P (S∗ ) = μF (a, b, k ) and P (λ ) = D1 +
σ12 2
b x ba x−(a+1) e− x dx. k+x (a )
Let S∗ and λ be constants such that
.
λ < S∗ ,
Theorem 2.5. If then for any initial value (S (0 ), x(0 )) ∈ R2+ , model (2) admits the unique stationary distribution, denoted by π ( · , · ), and it has ergodic property. Proof. From (2) and (3), by Itoˆ formula,
D S0 − S (t )
d ln (k + S (t )) =
k + S (t )
D k + S0
=
k +
and
S (t ) S0 − k + S (t ) k + S0
S (t ) k + S (t )
2 dt +
1 μS(t )x(t ) σ02 − − k + S (t ) k + S (t ) 2
σ0 S(t ) dB0 (t ) k + S (t )
S (t ) k + S (t )
2
dt
σ0 S(t ) dB0 (t ), k + S (t )
d ln (k + ϕ (t )) =
1 μS(t )x(t ) σ02 − − k + S (t ) k + S (t ) 2
D k + S0 k
S0 ϕ (t ) − k + ϕ (t ) k + S0
Then, integrating the above two equations shows
(5)
−
σ02 2
ϕ (t ) k + ϕ (t )
2 dt +
σ0 ϕ (t ) dB0 (t ). k + ϕ (t )
(6)
D k + S0 t σ 2k S (s ) k + ϕ (t ) S (s ) ϕ (s ) ϕ (s ) 0 ln =− − 1+ + ds k + S (t ) k k + ϕ (s ) k + S (s ) k + ϕ (s ) k + S (s ) 2D k + S 0 0
202
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
+ Similarly,
ln
t 0
μS(s )x(s ) ds + σ0 ( k + S ( s ) )2
t
0
S (s ) k + ϕ (0 ) ϕ (s ) − dB0 (s ) + ln . k + ϕ (s ) k + S (s ) k + S (0 )
(7)
σ2 μS ( s ) − D1 + 1 ds + σ1 B1 (t ) k + S (s ) 2 0 t t σ2 μS ( s ) μϕ (s ) μϕ (s ) = − D1 + 1 ds − − ds + σ1 B1 (t ). k + ϕ (s ) 2 k + ϕ (s ) k + S (s ) 0 0
x(t ) = x (0 )
t
From (6)−(7) and Remark 3,
ln
x(t ) μk k + ϕ (t ) ln − ≥ x (0 ) k + S (t ) D k + S0
(8)
t σ2 μk μS(s )x(s ) μϕ (s ) − D1 + 1 ds + σ1 B1 (t ) − ds k + ϕ (s ) 2 D k + S0 0 (k + S (s ))2 0 t S (s ) μkσ0 μk k + ϕ (0 ) ϕ (s ) ln − − d B0 ( s ) − . 0 0 k + ϕ s k + S s k + S (0 ) ( ) ( ) D k+S D k+S 0
t
Then, based on Remark 2−3 and Lemma 2.2, we obtain
lim sup t→∞
1 x(t ) 1 k + S (t ) + x(t ) ln ≤ lim sup ln ≤ 0, t x (0 ) t x (0 ) t→∞
and
0 ≤ lim
t→∞
1 k + ϕ (t ) 1 ln ≤ lim [ln (k + ϕ (t )) − ln k] = 0 a.s. t→∞ t t k + S (t )
Applications of the strong law of large number show that
lim
σ1 B1 (t ) t
t→∞
= 0 and lim
t→∞
μk 1 0 t→∞ t D (k+S )
Moreover, lim
1 lim inf t→∞ t
t
0
ln
k+ϕ ( 0 ) k+S ( 0 )
1 t
t 0
S (s ) ϕ (s ) − dB0 (s ) = 0 a.s. k + ϕ (s ) k + S (s )
= 0. Thus we have that
D k + S0 μS(s )x(s ) 1 t μS ( s ) μϕ (s ) ds ≥ lim inf − ds t→∞ t 0 kμ k + ϕ (s ) k + S (s ) (k + S(s ))2 D k + S0 σ2 1 t μϕ (s ) ≥ lim − D1 + 1 t→∞ t 0 kμ k + ϕ (s ) 2 0 D k+S = [P (S∗ ) − P (λ )] > 0 a.s. kμ
(9)
Denote 1 = { (s, x )|s ≥ h, and, x ≥ h}, 2 = { (s, x )|x ≤ h} and 3 = { (s, x )|s ≤ h}. By choosing h > 0 be small such that μ k
2 μD S 0 )h k min{D,D1 }
(1 +
lim inf t→∞
1 t
0 ≤ D(2kk+μS ) [P (S∗ ) − P (λ )], we apply (9) and Lemma 2.3 to show that
t
E 0
t 1 t 1 μS(s )x(s ) μS(s )x(s ) μS(s )x(s ) I ds ≥ lim inf E ds − lim sup E I ds
1 2 2 t→∞ t 0 t t→∞ 0 (k + S (s ) ) ( k + S ( s ) )2 ( k + S ( s ) )2 1 t μS(s )x(s ) − lim sup E I
3 ds t 0 t→∞ (k + S(s ))2 D k + S0 μ μ 1 t ≥ E (x(s ) )ds [P (S∗ ) − P (λ )] − h − 2 h lim sup kμ k t 0 k t→∞ D k + S0 μ 2 μD S 0 ≥ 1+ h [P (S∗ ) − P (λ )] − kμ k k min {D, D1 } D k + S0 ≥ [P (S∗ ) − P (λ )]. 2kμ
Let q = n0 > 2 be positive integer such that p = we get
lim inf t→∞
1 t
t
E 0
n0 n0 −1
∈ ( 1, 1 +
2 min{D,D1 } ), max{σ02 ,σ12 }
then by the inequality ab ≤
p 1 t 1 μS (s )x(s ) 1 μS(s )x(s ) I ds ≤ lim inf E + I ds
1 t→∞ t 0 p (k + S (s ))2 n0 n0 1 ( k + S ( s ) )2
ap p
+
bq q
f or a, b > 0,
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
≤ lim inf t→∞
1 t
t
E
1 n0
0
I n 0 1
ds +
1 p
μ p k
203
lim sup t→∞
1 t
t 0
E (x p (s ) )ds,
(10)
p D ( k+S 0 ) 2K p where ϖ > 0 is a constant satisfying pγ (( p)) kμ ≤ 4kμ [P (S∗ ) − P (λ )]. By (10) and Lemma 2.3, 1 lim inf t→∞ t
t
E I 1 ds ≥n0
0
D k + S0
n0
≥
2kμ
0
D k + S n0 n0 4kμ
2K ( p ) [P (S ) − P (λ )] − pγ ( p) ∗
μ p
k
[P (S∗ ) − P (λ )].
(11)
Further, we denote 4 = {(s, x )|s ≥ H, or, x ≥ H } and D¯ = { (s, x )|h ≤ s ≤ H, and, h ≤ x ≤ H}, where H > h is a positive constant such that
1 2DS 0 H ( min{D,D1 }
D ( k+S 0 )n0 n0 8kμ
+ S(0 ) + x(0 )) ≤
E I 4 ≤ P (S (t ) ≥ H ) + P (x(t ) ≥ H ) ≤ and then it follows that
1 lim sup t t→∞
t
t→∞
1 t
t
8kμ
E ID¯ ds ≥ lim inf t→∞
0
≥
2 DS 0 + S (0 ) + x (0 ) min {D, D1 }
This, together with (11), shows us that
lim inf
E (S (t ) + x(t )) 1 ≤ H H
D k + S 0 n0 n0
E I 4 ds ≤
0
[P (S∗ ) − P (λ )]. By the moment inequality,
1 t
t
0
[P (S∗ ) − P (λ )].
E I 1 ds − lim sup t→∞
D k + S 0 n0 n0 8kμ
1 t
(12)
t
E I 4 ds
0
[P (S∗ ) − P (λ )].
D ( k + S 0 )n 0 n 0 Hence, we find a compact set D¯ ⊂ R2+ and a positive constant ε [P (S∗ ) − P (λ )] such that 8kμ
lim inf t→∞
1 t
t 0
P s, (S (0 ), x(0 )), D¯ ds ≥ ε .
(13)
Next, we show the Feller property of model (2). Assume that (S(t), x(t)) and S˜(t ), x˜(t ) be solution of (2) with initial value
2 (S(0 ), x(0 )) ∈ R2+ and S˜(0 ), x˜(0 ) ∈ R2+ respectively, denote Q (t ) = (S(t ) + x(t )) − S˜(t ) + x˜(t ) and Q¯ (t ) = S(t ) − S˜(t ) + (x(t ) − x˜(t ))2 , then
2 2 2 2μkx(t ) S(t ) − S˜(t ) 2μS˜(t ) + d S (t ) − S˜(t ) = − 2D − σ02 S (t ) − S˜(t ) − (x(t ) − x˜(t )) S(t ) − S˜(t ) dt k + S˜(t ) (k + S(t )) k + S˜(t ) 2 +2σ0 S (t ) − S˜(t ) dB0 (t ),
2
dQ 2 (t ) = −2Q (t ) D S (t ) − S˜(t ) + D1 (x(t ) − x˜(t ) ) + σ02 S (t ) − S˜(t )
+ σ12 (x(t ) − x˜(t ) )2 dt
+2σ0 Q (t ) S (t ) − S˜(t ) dB0 (t ) + 2σ1 Q (t )(x(t ) − x˜(t ) )dB1 (t ). In view of the Ho¨lder inequality, we take expectations to show
2
E S (t ) − S˜(t )
2
2 0
t
2
E S (s ) − S˜(s ) ds +
t
0
and
E Q 2 (t ) = Q 2 (0 ) − 2(D + D1 ) +
2μS˜(s ) − 2D − σ E (x(t ) − x˜(s )) S(s ) − S˜(s ) ds ˜ k + S (s ) 0 0 t t 2 2 ≤ S (0 ) − S˜(0 ) + 4μ − 2D + σ02 E S (s ) − S˜(s ) ds + 4μ E (x(s ) − x˜(s ) )2 ds, ≤ S (0 ) − S˜(0 )
σ − 2D 2 0
≤ Q 2 (0 ) +
t 0
t 0
0
S (s ) − S˜(s ) (x(s ) − x˜(s ) )ds
2
E S (s ) − S˜(s ) ds +
σ02 + 2D + 4D1
t 0
σ12 − 2D1 2
E S (s ) − S˜(s ) ds
0
t
E (x(s ) − x˜(s ) )2 ds
204
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
+
σ12 + 4D + 2D1
t 0
E (x(s ) − x˜(s ) )2 ds.
Denote ρ = max 3 4μ − 2D + σ02 + 2 σ02 + 2D + 4D1 , 12μ + 2 σ12 + 4D + 2D1
. Noting that
2 Q¯ (t ) = S (t ) − S˜(t ) + (x(t ) − x˜(t ) )2 2 2 = S (t ) − S˜(t ) + Q (t ) − S (t ) − S˜(t ) 2 2 ≤ S (t ) − S˜(t ) + 2 Q 2 (t ) + S (t ) − S˜(t ) 2 ≤ 3 S (t ) − S˜(t ) + 2Q 2 (t ),
we have
2
E Q¯ (t ) ≤ 3 S (0 ) − S˜(0 )
+ 2Q 2 ( 0 ) + ρ
t
E Q¯ (s )ds.
0
By the Gronwall inequality, there is a deterministic and positive function ρ (t ) = eρ t such that
2
E Q¯ (t ) ≤ 3 S (0 ) − S˜(0 )
ρ (t ).
+ 2Q 2 ( 0 )
For every bounded Lipschitzian function f on R2+ with f (S, x ) − f S˜, x˜ ≤ L
E f (S(t ), x(t )) − E f S˜(t ), x˜(t ) ≤ E f (S(t ), x(t )) − f S˜(t ), x˜(t ) ≤ L E Q¯ (t ) ≤ L 5Q¯ (0 )ρ (t ) → 0
2
S − S˜
+ (x − x˜)2 ,
as (S (0 ), x(0 )) → S˜(0 ), x˜(0 ) . Then it follows from [10, Theorem 5.1], that the Feller property holds. By (13), we know that (4) is impossible. Combining this with the Feller property of (S(t), x(t)), Lemma 2.4 (See also [11] or Page 530–531 [12]) implies that model (2) has a stationary distribution, denoted by π ( · , · ). Uniqueness of the stationary distribution can be proved by following Step II [13, Page 659] completely. The proof is completed. Remark 5. Based on Theorem 2.5, by essentially applying the same technical as the proof of Theorem 5.1 of Hasminskii (1980) [14], for any π −integrable function f : R2+ → R
P
lim
T →∞
1 T
T 0
f (S (t ), x(t ))dt =
∞
0
∞ 0
f (s, x )π (ds, dx )
= 1.
3. Extinction Theorem 3.1. If λ > S∗ , then the microorganism in the chemostat will go extinct, i.e., lim x(t ) = 0 a.s. t→∞
Proof. In view of (8) and Remark 3, we have that
lim sup t→∞
1 x(t ) ln = t x (0 ) =
μ lim sup t→∞
1 t
t 0
S (s ) ds − k + S (s )
μF (a, b, k) − D1 +
Then λ > S∗ yields the desired result.
2
σ1 2
D1 +
σ12 2
≤ μ lim
t→∞
1 t
0
t
σ2 ϕ (s ) ds − D1 + 1 k + ϕ (s ) 2
= P (S∗ ) − P (λ ).
4. Concluding remarks Before summarizing the main results, we firstly show a comparison result: S∗ < S0 . Remark 6. Based on (6), we get
D k + S0 k + ϕ (t ) ln = k + ϕ (0 ) k
S0 t− k + S0
Note the fact that
lim
t→∞
1 k + ϕ (t ) 1 ln = 0 and lim σ0 t→∞ t t k + ϕ (0 )
t
0
0
t
2 t σ2 t ϕ (s ) ϕ (s ) ϕ (s ) ds − 0 ds + σ0 dB0 (s ) a.s. k + ϕ (s ) 2 0 k + ϕ (s ) 0 k + ϕ (s ) ϕ (s ) dB0 (s ) = 0 a.s., k + ϕ (s )
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
then
S0 1 − lim k + S0 t→∞ t
Applying the inequality
lim
t→∞
1 t
t 0
t 0
1 t
kσ 2 ϕ (s ) 1 0 lim ds = 0 t→∞ k + ϕ (s ) t 2D k + S t ϕ (s ) 0 k+ϕ (s ) ds ≥
t 0
t 0
t 0
2
( k+ϕϕ(s()s) ) ds ≥ ( 1t
ϕ (s ) k + ϕ (s )
2 ds.
(14)
t ϕ (s ) 2 0 k+ϕ (s ) ds ) , we show that
2D k + S 0 ϕ (s ) S0 2 DS 0 ds ≥ = 0 k + ϕ (s ) 2D k + S0 + kσ02 k + S 2D k + S0 + kσ02
and
1 lim t→∞ t
1 t
205
ϕ (s ) k + ϕ (s )
2 ds ≥
2
2 DS 0
2D k + S0 + kσ02
In view of the definition of S∗ , by (14)
kσ 2 S∗ S0 0 − ≥ ∗ 0 k+S k+S 2D k + S 0
2 DS 0
2D k + S0 + kσ02
.
2 .
It follows that S∗ < S0 . This paper mainly derives sharp conditions for existence of the stationary distribution by using the property of Feller process, which close the gap left by the Lyapunov function. Further results on extinction of the microorganism indicate that the derived sufficient conditions for existence of the stationary distribution is almost necessary. Subject to the effects of the noises, S∗ is less than the counterpart parameter Sd (Sd = S0 ) of the deterministic model, and λ is larger than the counterpart λd for the deterministic model, which leads to the conclusion that both noises have negative effects on persistence of the microorganism in the chemostat. Acknowledgements The authors would like to thank the anonymous referees for very helpful comments and suggestions. This work was supported by NSFC(No.11671260) and Shanghai Leading Academic Discipline Project (No. XTKX2012). References [1] J. Monod, La technique de culture continue, theorie et applications, Ann. Inst. Pasteur 79 (1950) 390–410. [2] A. Novick, L. Szilard, Description of the chemostat, Science 112 (1950) 715–716. [3] G.J. Butler, G.S.K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math. 45 (1985) 138–151. [4] S.B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math. 34 (1978) 760–763. [5] F. Campillo, M. Joannides, I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecol. Model. 222 (2011) 2676–2689. [6] L. Imhof, S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equ. 217 (2005) 26–53. [7] C. Xu, S. Yuan, Competition in the chemostat: a stochastic multi-species model and its asymptotic behavior, Math. Biosci. 280 (2016) 1–9. [8] L. Wang, D. Jiang, A note on the stationary distribution of the stochastic chemostat model with general response functions, Appl. Math. Lett. 73 (2017) 22–28. [9] Y. Cai, Y. Kang, W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput. 305 (2017) 221–240. [10] R. Bhattacharya, E. Waymire, Stochastic processes with applications, in: Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, JohnWiley & Sons, New York, NY, USA, 1990. [11] L. Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes, Technical Report, LCDS 86–18, Brown University, Providence, RI, 1986. April. [12] S.P. Meyn, R.L. Tweedie, Stability of Markovian processes III: Foster−Lyapunov criteria for continuous-time processes, Adv. Appl. Probab. 25 (1993) 518–548. [13] J. Tong, Z. Zhang, J. Bao, The stationary distribution of the facultative population model with a degenerate noise, Stat. Probab. Lett. 83 (2013) 655–664. [14] R.Z. Has’minskii, Stochastic stability of differential equations, in: Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat Dianli Zhao∗, Sanling Yuan College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
a r t i c l e
i n f o
a b s t r a c t
Keywords: Stochastic chemostat model Feller property Stationary distribution Extinction Break-even concentration
This paper studies the asymptotic behaviors of one classical chemostat model in a stochastic environment. Based on the Feller property, sharp conditions are derived for the existence of a stationary distribution by using the mutually exclusive possibilities known in [11, 12] (See Lemma 2.4 for details), which closes the gap left by the Lyapunov function. Further, we obtain a sufficient condition for the extinction of the organism based on two noise-induced parameters: an analogue of the feed concentration S∗ and the break-even concentration λ. Results indicate that both noises have negative effects on persistence of the microorganism. © 2018 Elsevier Inc. All rights reserved.
1. Introduction Let S(t) and x(t) be concentrations of the nutrient and the microorganisms at time of the nutrient, D be the washout rates for S, D1 be the removal rate combining the the death rate of the microorganism x, and kμ+SS be the growth rate function. Then, the tory apparatus for the continuous culture of microorganisms) is classically represented equations taking the form (See [1,2])
dS (t ) = D S0 − S (t ) −
t respectively, S0 be the input rate dilution rate of the chemostat and chemostat (a basic piece of laboraas a system of ordinary differential
μS(t )x(t ) μS(t ) dt, dx(t ) = − D1 x(t )dt. k + S (t ) k + S (t )
(1)
Break-even concentration λd = μ−D1 is an important parameter for analyzing (1). In detail, if λd ≥ S0 , the only washout 1 kD
equilibrium E 0 = (S0 , 0 ) is globally asymptotically stable; if λd < S0 , the positive equilibrium E ∗ = (λd ,
D k + λd
S0 −λd
μλd
) exists
and is also globally asymptotically stable, see [3,4] for details. Due to the existence of random effects almost everywhere in the reality, it is natural to address what happens when the stochastic perturbation is taken into account. For instance, Campillo et al. [5] established a set of stochastic chemostat models that are valid at different scales and expound the mechanism to switch from one model to another. Imhof and Walcher [6] introduced a rigorous method to get the stochastic chemostat model by defining a discrete time Markov chain and proving its convergence. The above method has also been used in [7,8] to establish the studied stochastic models. By using the method in [6], a stochastic chemostat model with the classical Monod growth rate function can be written as
∗
Corresponding author. E-mail address: [email protected] (D. Zhao).
https://doi.org/10.1016/j.amc.2018.07.020 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
200
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
follows
0
D S − S (t ) − μkS+(tS)(xt()t ) dt + σ0 S (t )dB0 (t ), μS(t ) − D1 x(t )dt + σ1 x(t )dB1 (t ). k+S (t )
dS (t ) = dx(t ) =
(2)
B0 (t) and B1 (t) are two independent Brownian motions. The parameters in model (2) are all positive. Then for any initial value (S (0 ), x(0 )) ∈ R2+ , model (2) has a uniquely global solution (S (t ), x(t )) ∈ R2+ a.s. for all t ≥ 0 (See [6, Proposition 6]). Our goal is to establish an almost perfect condition for existence of the stationary distribution for (2). Further, we derive the sufficient condition for extinction of the microorganism. Remark 1. Wang and Jiang [8] have studied the stationary distribution for the stochastic chemostat model with general response functions. Except for the linear growth rate, the actual growth rate function will be less than its value at S0 in time average subject to the effects of noises. However the averaged growth rate function can not be applied in the Lyapunov method [8, Lemma 2.1], and the sufficient condition has to be derived under extra conditions, and is not the optimal condition. Different from the above, in this paper we employe the Feller property and mutually exclusive possibilities to derive the condition for existence of the stationary distribution, which can close the gap left by using the Lyapunov method. The method of this paper can be extended to study the stochastic chemostat models with general response functions. 2. Conditions for the stationary distribution To begin with, let’s prepare some basic results. Lemma 2.1 [9, Lemma 4.9]. For any initial value ϕ (0 ) ∈ R+ , consider the equation
dϕ (t ) = D S0 − ϕ (t ) dt + σ0 ϕ (t )dB0 (t ), t > 0. Then (3) has the stationary distribution
ba −(a+1) − b 2 p( x ) = x e x f or x ∈ R+ with a = 2 (a ) σ0 Lemma 2.2. Let ϕ (t) be solution of (3), then lim
t→∞
(3)
ln(k+ϕ (t ) ) t
D+
σ02
, b=
2
2 DS 0
σ02
and (a ) =
∞ 0
t a−1 e−t dt .
= 0 a.s.
Proof. Let ξ (t) be solution of dξ (t ) = −Dξ (t )dt + σ0 ξ (t )dB0 (t ) with ξ (0 ) = 1. By using Itoˆ formula to ln ξ (t) and σ02
then taking integrations, ξ (t ) = e−(D+ 2 )t +σ0 B0 (t ) . Then, by applying the variation-of-constants formula, it yields σ0 sup 2|B0 (s )| σ02 σ02 0 ϕ (t ) = ϕ (0 )e−(D+ 2 )t +σ0 B0 (t ) + DS0 0t e−(D+ 2 )(t−s)+σ0 (B0 (t )−B0 (s)) ds. Then ϕ (t ) ≤ ϕ (0 )eσ0 B0 (t ) + DSσ 2 e 0≤s≤t . Noting sup |B0 (s )|
lim
t→∞
0≤s≤t
t
D+ 20
= 0, we get the desired result.
1 (ln(k t→∞ t
Remark 2. Similar to proof of Lemma 2.2, lim
+ S(t ) + x(t )) ) = 0 a.s.
Remark 3. Note the positivity of (S(t), x(t)) and ϕ (t), then S(t) ≤ ϕ (t) a.s. holds due to the stochastic comparison theorem. Lemma 2.3. Let (S(t), x(t)) be solution of (2) with initial value (S(0 ), x(0 )) ∈ R2+ , then for any p ∈ [1, 1 + that
E [S (t ) + x(t )] p ≤ where K ( p) =
pγ ( p) 2K ( p ) 1 + [S (0 ) + x(0 )] p e− 2 t and lim sup γ ( p) t t→∞
sup{DS0 x p−1 x>0
γ ( p) − 2 x p } and
γ ( p) = [min{D, D1 } −
( p−1 ) 2
t 0
E [S (s ) + x(s )] p ds ≤
it holds
2K ( p ) , γ ( p)
max{σ02 , σ12 }].
Proof. Applying Itoˆ formula to model (2) yields
2 min{D,D1 } ], max{σ02 ,σ12 }
d[S (t ) + x(t )] p = p [S (t ) + x(t )] p−1 DS0 − DS (t ) − D1 x(t ) +
( p − 1) 2
[S (t ) + x(t )]
p−2
σ02 S2 (t ) + σ12 x2 (t ) dt
+ p[S (t ) + x(t )] p−1 (σ0 S (t )dB0 (t ) + σ1 x(t )dB1 (t ))
( p − 1) ≤ p DS0 [S (t ) + x(t )] p−1 − min {D, D1 } − max σ02 , σ12 [S (t ) + x(t )] p dt 2 + p[S (t ) + x(t )] p−1 (σ0 S (t )dB0 (t ) + σ1 x(t )dB1 (t ))
γ ( p) p p−1 ≤ p K ( p) − [S (t ) + x(t )] dt + p[S (t ) + x(t )] (σ0 S (t )dB0 (t ) + σ1 x(t )dB1 (t )) 2
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
201
and
d[S (t ) + x(t )] p e
pγ ( p) 2 t
=
pγ ( p) pγ ( p) p pγ ( p) p [S (t ) + x(t )] e 2 t dt + e 2 t d[S (t ) + x(t )] 2
≤ pK ( p)e
pγ ( p) 2 t
dt + e
pγ ( p) 2 t
p[S (t ) + x(t )] p−1 (σ0 S (t )dB0 (t ) + σ1 x(t )dB1 (t )).
Then by taking integrations and taking the expectations, we get
E [S (t ) + x(t )] p ≤ [S (0 ) + x(0 )] p e−
pγ ( p) 2 t
+ pK ( p)
t
e−
pγ ( p) 2
(t−s ) ds
0
≤ [S (0 ) + x(0 )] p e−
pγ ( p) 2 t
+
2K ( p ) . γ ( p)
Further, we can get
lim sup t→∞
1 t
t 0
E [S (s ) + x(s )] p ds ≤ [S (0 ) + x(0 )] p lim sup t→∞
1 t
t
e−
pγ ( p) 2 s
0
ds +
2K ( p ) 2K ( p ) = . γ ( p) γ ( p)
The following result, known as mutually exclusive possibilities, is vital to proving the existence of the stationary distribution. It was proved by Stettner (1986 [11]), and stated and used in the proof of Theorem 4.5 of Meyn and Tweedie (1993 [12]). Lemma 2.4 (Mutually exclusive possibilities, [11,12]). Let X(t) ∈ Rd be a Feller process, then either an invariant probability measure π ( · ) exists, or
1 ν t
t
P (s, x, C )ν (dx )ds = 0,
lim sup
t→∞
0
(4)
for any compact set C ∈ Rd , where the supremum is taken over all initial distributions ν on Rd and P (t, x, c ) is the probability for X (t ) ∈ C with X (0 ) = x. (μ
t
S (s )
Remark 4. From (2), x(t ) = x(0 )e 0 k+S(s) t 0 lim 1t 0 k+ϕϕ(s()s ) ds ≤ k+S S0 < 1, it’s clear that
σ2 ds−(D1 + 21 )t )+σ1 B1 (t )
. Since
lim 1 t→∞ t
t 0
ϕ (s )ds = S0 and lim sup 1t t→∞
t
S (s ) 0 k+S(s ) ds
≤
t→∞
x(t ) → 0 a.s. as t → ∞, i f
μ ≤ D1 +
σ12 2
.
So, in the following, we always assume that μ > D1 +
σ12 2
.
∞ For convenience, we denote P (S ) = kμ+SS and F (a, b, k ) = 0 P (S∗ ) = μF (a, b, k ) and P (λ ) = D1 +
σ12 2
b x ba x−(a+1) e− x dx. k+x (a )
Let S∗ and λ be constants such that
.
λ < S∗ ,
Theorem 2.5. If then for any initial value (S (0 ), x(0 )) ∈ R2+ , model (2) admits the unique stationary distribution, denoted by π ( · , · ), and it has ergodic property. Proof. From (2) and (3), by Itoˆ formula,
D S0 − S (t )
d ln (k + S (t )) =
k + S (t )
D k + S0
=
k +
and
S (t ) S0 − k + S (t ) k + S0
S (t ) k + S (t )
2 dt +
1 μS(t )x(t ) σ02 − − k + S (t ) k + S (t ) 2
σ0 S(t ) dB0 (t ) k + S (t )
S (t ) k + S (t )
2
dt
σ0 S(t ) dB0 (t ), k + S (t )
d ln (k + ϕ (t )) =
1 μS(t )x(t ) σ02 − − k + S (t ) k + S (t ) 2
D k + S0 k
S0 ϕ (t ) − k + ϕ (t ) k + S0
Then, integrating the above two equations shows
(5)
−
σ02 2
ϕ (t ) k + ϕ (t )
2 dt +
σ0 ϕ (t ) dB0 (t ). k + ϕ (t )
(6)
D k + S0 t σ 2k S (s ) k + ϕ (t ) S (s ) ϕ (s ) ϕ (s ) 0 ln =− − 1+ + ds k + S (t ) k k + ϕ (s ) k + S (s ) k + ϕ (s ) k + S (s ) 2D k + S 0 0
202
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
+ Similarly,
ln
t 0
μS(s )x(s ) ds + σ0 ( k + S ( s ) )2
t
0
S (s ) k + ϕ (0 ) ϕ (s ) − dB0 (s ) + ln . k + ϕ (s ) k + S (s ) k + S (0 )
(7)
σ2 μS ( s ) − D1 + 1 ds + σ1 B1 (t ) k + S (s ) 2 0 t t σ2 μS ( s ) μϕ (s ) μϕ (s ) = − D1 + 1 ds − − ds + σ1 B1 (t ). k + ϕ (s ) 2 k + ϕ (s ) k + S (s ) 0 0
x(t ) = x (0 )
t
From (6)−(7) and Remark 3,
ln
x(t ) μk k + ϕ (t ) ln − ≥ x (0 ) k + S (t ) D k + S0
(8)
t σ2 μk μS(s )x(s ) μϕ (s ) − D1 + 1 ds + σ1 B1 (t ) − ds k + ϕ (s ) 2 D k + S0 0 (k + S (s ))2 0 t S (s ) μkσ0 μk k + ϕ (0 ) ϕ (s ) ln − − d B0 ( s ) − . 0 0 k + ϕ s k + S s k + S (0 ) ( ) ( ) D k+S D k+S 0
t
Then, based on Remark 2−3 and Lemma 2.2, we obtain
lim sup t→∞
1 x(t ) 1 k + S (t ) + x(t ) ln ≤ lim sup ln ≤ 0, t x (0 ) t x (0 ) t→∞
and
0 ≤ lim
t→∞
1 k + ϕ (t ) 1 ln ≤ lim [ln (k + ϕ (t )) − ln k] = 0 a.s. t→∞ t t k + S (t )
Applications of the strong law of large number show that
lim
σ1 B1 (t ) t
t→∞
= 0 and lim
t→∞
μk 1 0 t→∞ t D (k+S )
Moreover, lim
1 lim inf t→∞ t
t
0
ln
k+ϕ ( 0 ) k+S ( 0 )
1 t
t 0
S (s ) ϕ (s ) − dB0 (s ) = 0 a.s. k + ϕ (s ) k + S (s )
= 0. Thus we have that
D k + S0 μS(s )x(s ) 1 t μS ( s ) μϕ (s ) ds ≥ lim inf − ds t→∞ t 0 kμ k + ϕ (s ) k + S (s ) (k + S(s ))2 D k + S0 σ2 1 t μϕ (s ) ≥ lim − D1 + 1 t→∞ t 0 kμ k + ϕ (s ) 2 0 D k+S = [P (S∗ ) − P (λ )] > 0 a.s. kμ
(9)
Denote 1 = { (s, x )|s ≥ h, and, x ≥ h}, 2 = { (s, x )|x ≤ h} and 3 = { (s, x )|s ≤ h}. By choosing h > 0 be small such that μ k
2 μD S 0 )h k min{D,D1 }
(1 +
lim inf t→∞
1 t
0 ≤ D(2kk+μS ) [P (S∗ ) − P (λ )], we apply (9) and Lemma 2.3 to show that
t
E 0
t 1 t 1 μS(s )x(s ) μS(s )x(s ) μS(s )x(s ) I ds ≥ lim inf E ds − lim sup E I ds
1 2 2 t→∞ t 0 t t→∞ 0 (k + S (s ) ) ( k + S ( s ) )2 ( k + S ( s ) )2 1 t μS(s )x(s ) − lim sup E I
3 ds t 0 t→∞ (k + S(s ))2 D k + S0 μ μ 1 t ≥ E (x(s ) )ds [P (S∗ ) − P (λ )] − h − 2 h lim sup kμ k t 0 k t→∞ D k + S0 μ 2 μD S 0 ≥ 1+ h [P (S∗ ) − P (λ )] − kμ k k min {D, D1 } D k + S0 ≥ [P (S∗ ) − P (λ )]. 2kμ
Let q = n0 > 2 be positive integer such that p = we get
lim inf t→∞
1 t
t
E 0
n0 n0 −1
∈ ( 1, 1 +
2 min{D,D1 } ), max{σ02 ,σ12 }
then by the inequality ab ≤
p 1 t 1 μS (s )x(s ) 1 μS(s )x(s ) I ds ≤ lim inf E + I ds
1 t→∞ t 0 p (k + S (s ))2 n0 n0 1 ( k + S ( s ) )2
ap p
+
bq q
f or a, b > 0,
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
≤ lim inf t→∞
1 t
t
E
1 n0
0
I n 0 1
ds +
1 p
μ p k
203
lim sup t→∞
1 t
t 0
E (x p (s ) )ds,
(10)
p D ( k+S 0 ) 2K p where ϖ > 0 is a constant satisfying pγ (( p)) kμ ≤ 4kμ [P (S∗ ) − P (λ )]. By (10) and Lemma 2.3, 1 lim inf t→∞ t
t
E I 1 ds ≥n0
0
D k + S0
n0
≥
2kμ
0
D k + S n0 n0 4kμ
2K ( p ) [P (S ) − P (λ )] − pγ ( p) ∗
μ p
k
[P (S∗ ) − P (λ )].
(11)
Further, we denote 4 = {(s, x )|s ≥ H, or, x ≥ H } and D¯ = { (s, x )|h ≤ s ≤ H, and, h ≤ x ≤ H}, where H > h is a positive constant such that
1 2DS 0 H ( min{D,D1 }
D ( k+S 0 )n0 n0 8kμ
+ S(0 ) + x(0 )) ≤
E I 4 ≤ P (S (t ) ≥ H ) + P (x(t ) ≥ H ) ≤ and then it follows that
1 lim sup t t→∞
t
t→∞
1 t
t
8kμ
E ID¯ ds ≥ lim inf t→∞
0
≥
2 DS 0 + S (0 ) + x (0 ) min {D, D1 }
This, together with (11), shows us that
lim inf
E (S (t ) + x(t )) 1 ≤ H H
D k + S 0 n0 n0
E I 4 ds ≤
0
[P (S∗ ) − P (λ )]. By the moment inequality,
1 t
t
0
[P (S∗ ) − P (λ )].
E I 1 ds − lim sup t→∞
D k + S 0 n0 n0 8kμ
1 t
(12)
t
E I 4 ds
0
[P (S∗ ) − P (λ )].
D ( k + S 0 )n 0 n 0 Hence, we find a compact set D¯ ⊂ R2+ and a positive constant ε [P (S∗ ) − P (λ )] such that 8kμ
lim inf t→∞
1 t
t 0
P s, (S (0 ), x(0 )), D¯ ds ≥ ε .
(13)
Next, we show the Feller property of model (2). Assume that (S(t), x(t)) and S˜(t ), x˜(t ) be solution of (2) with initial value
2 (S(0 ), x(0 )) ∈ R2+ and S˜(0 ), x˜(0 ) ∈ R2+ respectively, denote Q (t ) = (S(t ) + x(t )) − S˜(t ) + x˜(t ) and Q¯ (t ) = S(t ) − S˜(t ) + (x(t ) − x˜(t ))2 , then
2 2 2 2μkx(t ) S(t ) − S˜(t ) 2μS˜(t ) + d S (t ) − S˜(t ) = − 2D − σ02 S (t ) − S˜(t ) − (x(t ) − x˜(t )) S(t ) − S˜(t ) dt k + S˜(t ) (k + S(t )) k + S˜(t ) 2 +2σ0 S (t ) − S˜(t ) dB0 (t ),
2
dQ 2 (t ) = −2Q (t ) D S (t ) − S˜(t ) + D1 (x(t ) − x˜(t ) ) + σ02 S (t ) − S˜(t )
+ σ12 (x(t ) − x˜(t ) )2 dt
+2σ0 Q (t ) S (t ) − S˜(t ) dB0 (t ) + 2σ1 Q (t )(x(t ) − x˜(t ) )dB1 (t ). In view of the Ho¨lder inequality, we take expectations to show
2
E S (t ) − S˜(t )
2
2 0
t
2
E S (s ) − S˜(s ) ds +
t
0
and
E Q 2 (t ) = Q 2 (0 ) − 2(D + D1 ) +
2μS˜(s ) − 2D − σ E (x(t ) − x˜(s )) S(s ) − S˜(s ) ds ˜ k + S (s ) 0 0 t t 2 2 ≤ S (0 ) − S˜(0 ) + 4μ − 2D + σ02 E S (s ) − S˜(s ) ds + 4μ E (x(s ) − x˜(s ) )2 ds, ≤ S (0 ) − S˜(0 )
σ − 2D 2 0
≤ Q 2 (0 ) +
t 0
t 0
0
S (s ) − S˜(s ) (x(s ) − x˜(s ) )ds
2
E S (s ) − S˜(s ) ds +
σ02 + 2D + 4D1
t 0
σ12 − 2D1 2
E S (s ) − S˜(s ) ds
0
t
E (x(s ) − x˜(s ) )2 ds
204
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
+
σ12 + 4D + 2D1
t 0
E (x(s ) − x˜(s ) )2 ds.
Denote ρ = max 3 4μ − 2D + σ02 + 2 σ02 + 2D + 4D1 , 12μ + 2 σ12 + 4D + 2D1
. Noting that
2 Q¯ (t ) = S (t ) − S˜(t ) + (x(t ) − x˜(t ) )2 2 2 = S (t ) − S˜(t ) + Q (t ) − S (t ) − S˜(t ) 2 2 ≤ S (t ) − S˜(t ) + 2 Q 2 (t ) + S (t ) − S˜(t ) 2 ≤ 3 S (t ) − S˜(t ) + 2Q 2 (t ),
we have
2
E Q¯ (t ) ≤ 3 S (0 ) − S˜(0 )
+ 2Q 2 ( 0 ) + ρ
t
E Q¯ (s )ds.
0
By the Gronwall inequality, there is a deterministic and positive function ρ (t ) = eρ t such that
2
E Q¯ (t ) ≤ 3 S (0 ) − S˜(0 )
ρ (t ).
+ 2Q 2 ( 0 )
For every bounded Lipschitzian function f on R2+ with f (S, x ) − f S˜, x˜ ≤ L
E f (S(t ), x(t )) − E f S˜(t ), x˜(t ) ≤ E f (S(t ), x(t )) − f S˜(t ), x˜(t ) ≤ L E Q¯ (t ) ≤ L 5Q¯ (0 )ρ (t ) → 0
2
S − S˜
+ (x − x˜)2 ,
as (S (0 ), x(0 )) → S˜(0 ), x˜(0 ) . Then it follows from [10, Theorem 5.1], that the Feller property holds. By (13), we know that (4) is impossible. Combining this with the Feller property of (S(t), x(t)), Lemma 2.4 (See also [11] or Page 530–531 [12]) implies that model (2) has a stationary distribution, denoted by π ( · , · ). Uniqueness of the stationary distribution can be proved by following Step II [13, Page 659] completely. The proof is completed. Remark 5. Based on Theorem 2.5, by essentially applying the same technical as the proof of Theorem 5.1 of Hasminskii (1980) [14], for any π −integrable function f : R2+ → R
P
lim
T →∞
1 T
T 0
f (S (t ), x(t ))dt =
∞
0
∞ 0
f (s, x )π (ds, dx )
= 1.
3. Extinction Theorem 3.1. If λ > S∗ , then the microorganism in the chemostat will go extinct, i.e., lim x(t ) = 0 a.s. t→∞
Proof. In view of (8) and Remark 3, we have that
lim sup t→∞
1 x(t ) ln = t x (0 ) =
μ lim sup t→∞
1 t
t 0
S (s ) ds − k + S (s )
μF (a, b, k) − D1 +
Then λ > S∗ yields the desired result.
2
σ1 2
D1 +
σ12 2
≤ μ lim
t→∞
1 t
0
t
σ2 ϕ (s ) ds − D1 + 1 k + ϕ (s ) 2
= P (S∗ ) − P (λ ).
4. Concluding remarks Before summarizing the main results, we firstly show a comparison result: S∗ < S0 . Remark 6. Based on (6), we get
D k + S0 k + ϕ (t ) ln = k + ϕ (0 ) k
S0 t− k + S0
Note the fact that
lim
t→∞
1 k + ϕ (t ) 1 ln = 0 and lim σ0 t→∞ t t k + ϕ (0 )
t
0
0
t
2 t σ2 t ϕ (s ) ϕ (s ) ϕ (s ) ds − 0 ds + σ0 dB0 (s ) a.s. k + ϕ (s ) 2 0 k + ϕ (s ) 0 k + ϕ (s ) ϕ (s ) dB0 (s ) = 0 a.s., k + ϕ (s )
D. Zhao, S. Yuan / Applied Mathematics and Computation 339 (2018) 199–205
then
S0 1 − lim k + S0 t→∞ t
Applying the inequality
lim
t→∞
1 t
t 0
t 0
1 t
kσ 2 ϕ (s ) 1 0 lim ds = 0 t→∞ k + ϕ (s ) t 2D k + S t ϕ (s ) 0 k+ϕ (s ) ds ≥
t 0
t 0
t 0
2
( k+ϕϕ(s()s) ) ds ≥ ( 1t
ϕ (s ) k + ϕ (s )
2 ds.
(14)
t ϕ (s ) 2 0 k+ϕ (s ) ds ) , we show that
2D k + S 0 ϕ (s ) S0 2 DS 0 ds ≥ = 0 k + ϕ (s ) 2D k + S0 + kσ02 k + S 2D k + S0 + kσ02
and
1 lim t→∞ t
1 t
205
ϕ (s ) k + ϕ (s )
2 ds ≥
2
2 DS 0
2D k + S0 + kσ02
In view of the definition of S∗ , by (14)
kσ 2 S∗ S0 0 − ≥ ∗ 0 k+S k+S 2D k + S 0
2 DS 0
2D k + S0 + kσ02
.
2 .
It follows that S∗ < S0 . This paper mainly derives sharp conditions for existence of the stationary distribution by using the property of Feller process, which close the gap left by the Lyapunov function. Further results on extinction of the microorganism indicate that the derived sufficient conditions for existence of the stationary distribution is almost necessary. Subject to the effects of the noises, S∗ is less than the counterpart parameter Sd (Sd = S0 ) of the deterministic model, and λ is larger than the counterpart λd for the deterministic model, which leads to the conclusion that both noises have negative effects on persistence of the microorganism in the chemostat. Acknowledgements The authors would like to thank the anonymous referees for very helpful comments and suggestions. This work was supported by NSFC(No.11671260) and Shanghai Leading Academic Discipline Project (No. XTKX2012). References [1] J. Monod, La technique de culture continue, theorie et applications, Ann. Inst. Pasteur 79 (1950) 390–410. [2] A. Novick, L. Szilard, Description of the chemostat, Science 112 (1950) 715–716. [3] G.J. Butler, G.S.K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math. 45 (1985) 138–151. [4] S.B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math. 34 (1978) 760–763. [5] F. Campillo, M. Joannides, I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecol. Model. 222 (2011) 2676–2689. [6] L. Imhof, S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equ. 217 (2005) 26–53. [7] C. Xu, S. Yuan, Competition in the chemostat: a stochastic multi-species model and its asymptotic behavior, Math. Biosci. 280 (2016) 1–9. [8] L. Wang, D. Jiang, A note on the stationary distribution of the stochastic chemostat model with general response functions, Appl. Math. Lett. 73 (2017) 22–28. [9] Y. Cai, Y. Kang, W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput. 305 (2017) 221–240. [10] R. Bhattacharya, E. Waymire, Stochastic processes with applications, in: Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, JohnWiley & Sons, New York, NY, USA, 1990. [11] L. Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes, Technical Report, LCDS 86–18, Brown University, Providence, RI, 1986. April. [12] S.P. Meyn, R.L. Tweedie, Stability of Markovian processes III: Foster−Lyapunov criteria for continuous-time processes, Adv. Appl. Probab. 25 (1993) 518–548. [13] J. Tong, Z. Zhang, J. Bao, The stationary distribution of the facultative population model with a degenerate noise, Stat. Probab. Lett. 83 (2013) 655–664. [14] R.Z. Has’minskii, Stochastic stability of differential equations, in: Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.