Periodic solution for the stochastic chemostat with general response function

Physica A 486 (2017) 378–385

Contents lists available at ScienceDirect

Physica A journal homepage: www.elsevier.com/locate/physa

Periodic solution for the stochastic chemostat with general response function Liang Wang a,b , Daqing Jiang a,c,d, * a

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, PR China Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada c College of Science, China University of Petroleum (East China), Qingdao 266580, PR China d Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia b

highlights • A stochastic chemostat model with periodic dilution rate and general response function is firstly proposed. • Two kinds of sufficient conditions are given for the existence of the nontrivial positive periodic solution. • The conditions for the existence of the periodic solution are more general than that in pre-existing papers.

article

a b s t r a c t

info

Article history: Received 19 January 2017 Received in revised form 1 May 2017 Available online 5 June 2017

This paper addresses a stochastic chemostat model with periodic dilution rate and general class of response functions. The general functional response is assumed to satisfy two classifications of conditions, and these assumptions on the functional response are relative weak that are valid for many forms of growth response. For the chemostat with periodic dilution rate, we derive the sufficient criteria for the existence of the stochastic nontrivial positive periodic solution, by utilizing Khasminskii’s theory on periodic Markov process. © 2017 Elsevier B.V. All rights reserved.

Keywords: Stochastic chemostat General response function Periodic Markov process Nontrivial periodic solution

1. Introduction The chemostat is a basic piece of laboratory apparatus which consists of a series of vessels: the feed bottle, the culture vessel and the collecting vessel which are connoted by pumps [1]. Chemostat is utilized for the continuous culture of microorganisms, and occupies a central place both in mathematical and theoretical ecology. In recent decades, modeling and researching of the chemostat has been an increasingly active field, and attracted great attention from mathematics and ecology [2–6]. One classic chemostat model with single species and single substrate that described by a system of ordinary differential equations can be written as the following form

⎧ ⎨

S ′ (t) = D(S 0 − S(t)) −

⎩

1

δ

p(S(t))x(t),

(1.1)

x (t) = −Dx(t) + p(S(t))x(t), ′

where S(t), x(t) are the concentrations of nutrient and microbe at time t, respectively; S 0 is the original input concentration of nutrient and D represents the volumetric flow rate of the mixture of nutrient and microorganism, i.e. the common dilution

*

Corresponding author at: School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, PR China. E-mail addresses: [email protected] (L. Wang), [email protected] (D. Jiang).

http://dx.doi.org/10.1016/j.physa.2017.05.097 0378-4371/© 2017 Elsevier B.V. All rights reserved.

L. Wang, D. Jiang / Physica A 486 (2017) 378–385

379

rate. The term 1δ p(S) denotes the uptake rate of substrate of the microbe population. We assume that p(S) represents the per-capita growth rate of the species and δ is a growth yield constant. The growth response function p : R+ → R+ is generally assumed to satisfy p is continuously differentiable,

(1.2)

p(0) = 0,

(1.3)

p(S) > 0

for S > 0.

Butler et al. [7] investigated the global dynamics of a multiple competing species chemostat model with a general class of functions describing nutrient uptake. In the single-species case, we can derive from the study in [7] that there exists a uniquely defined positive real number 0 < λ ≤ ∞ such that p(S) < D for 0 < S < λ; p(S) > D for S > λ. Here λ represents the break-even concentration of the substrate for the species x(t). If λ < S 0 , the solution of system (1.1) satisfies lim S(t) = λ,

t →∞

lim x(t) := x∗ = δ (S 0 − λ),

t →∞

while when p(S 0 ) ≤ D, there exists a boundary equilibrium E0 = (S 0 , 0) which is asymptotically stable (E0 is also called the washout equilibrium). In other words, in the case of any monotone uptake functions (such as the Monod functional response), when p(S 0 ) > D, then the critical point Eλ = (λ, x∗ ) is globally asymptotically stable. Results in the case of multiple competing microbial populations are obtained in [8–10] which prove that the competitive exclusion principle holds in competition chemostat models. In the traditional chemostat equations, two parameters are under the control of the experimenter, the concentration of the input nutrient S 0 and the dilution rate D (the pump rate). The chemostat can be applied in the waste water treatment process and industrial engineers, thus it is sensible to vary the dilution rate D with time. Moreover, biological populations are always subject to fluctuations that occur in a periodic phenomenon. Butler et al. [2] investigate this modification in a m S0 competitive chemostat with periodic dilution rate D(t) and Monod growth response. They find that if a +i S 0 > ⟨D⟩θ (D(t)

∫θ

i

is a continuous θ -periodic function and ⟨D⟩θ = θ1 0 D(s)ds is the mean value), then there are positive θ -periodic solutions (S(t), xi (t)) which are exponentially asymptotically stable for the system with the absence of one competitor. In the aforementioned studies, the chemostat models are described by the system of ordinary differential equations. This is valid only at the macroscopic scale, i.e. the stochastic effects can be neglected or averaged out, on the basis of the law of large numbers. However, the real environment is full of stochasticity, biological models are inevitably affected by environmental noises, which is an important component. Environmental noises will disturb the steady-state of the deterministic system either by directly acting on the population densities or affecting the parameter values. Recently, many stochastic biological models have been developed [11–16]. On the other hand, in many instances, environmental stochasticity also has a critical influence on the nature growth of the biotic populations. Especially for the chemostat, according to [17], every component is inevitably affected by white noise at the microscopic scale. Taking the white noise into account, microorganism systems described by the stochastic differential equations have recently been studied by many researchers [17–22]. In [23], Wang et al. construct a stochastic chemostat with periodic dilution rate and consider the Monod response function. They find the existence conditions for the stochastic nontrivial positive periodic solution and the globally attractive boundary periodic solution, which corresponding to the conclusions in [2]. However, to the best of our knowledge, there is little theoretical results about the nontrivial periodic solution for the stochastic chemostat with periodic dilution rate and general growth responses. So in this paper, we further establish the following stochastic periodic chemostat with general response function:

] [ ⎧ ⎨dS(t) = D(t)(S 0 − S(t)) − 1 p(S(t))x(t) dt + σ (t)S(t)dB (t), 1 1 δ ⎩ dx(t) = [−D(t) + p(S(t))] x(t)dt + σ2 (t)x(t)dB2 (t),

(1.4)

where B1 (t), B2 (t) are independent standard one-dimensional Brownian motions defined on a complete probability space (Ω , F , {Ft }t ≥0 , P) with a filtration {Ft }t ≥0 satisfying the usual conditions (i.e. it is right continuous and F0 contains all Pnull sets), and σi (σi2 > 0, i = 1, 2) are their intensities, respectively. The parameter functions D(t), σi (t) are continuous θ -periodic functions and D(t) > 0. Our aim is to obtain the sufficient criteria for the existence of the stochastic nontrivial positive periodic solution for system (1.4). We establish the stochastic chemostat model (1.4) on the basis of the approach used in [18] to include stochastic effects (readers can also refer to [24, Appendix A] to see the construction of this kind of stochastic model). This paper deals with the stochastic chemostat (1.4) with general class of growth functions p(S). It is for the sake of biological reality that we impose the general assumptions for p(S) above. Again, for the sake of clarity, we make two further classifications of assumptions for the generic nature A1 : p(S) (i) p ∈ C 2 ([0, +∞), [0, +∞)) and S ≤ c, for any S ∈ (0, +∞), where c is a positive constant. ′′ 3 (ii) p (S)S ≥ m0 , for any S ∈ (0, +∞). m0 is a constant which requires no assumption on the sign.

380

L. Wang, D. Jiang / Physica A 486 (2017) 378–385

A2 :

(i) p ∈ C 2 ([0, +∞), [0, +∞)) and S ≤ c, p(S) ≤ c˜ for any S ∈ (0, +∞), where c, c˜ are positive constants. ˜ 0 , for any S ∈ (0, +∞). m ˜ 0 is a constant which requires no assumption on the sign. (ii) p′′ (S) ≥ m For chemostat system, we always consider the following prototypes of response functions that often found in literature. Three prototypes of monotone response functions are (i) Lotka–Volterra: p(S) = mS; IvIev: p(S) = 1 − e−µS . m, µ > 0 are positive constants. , here m is the maximal growth rate of the microbial species and a is the (ii) Michaelis–Menten (Monod): p(S) = amS +S half-saturation (or Michaelis–Menten) constant. p(S) = arctan S is also a growth function that has the same property as Monod. 2 , here m > 0, a > 0 and b is a constant. (iii) Sigmoidal: p(S) = (a+mS S)(b+S) A prototype for a nonmonotone response function is (iv) Monod–Haldane: p(S) = a+SmS , here the term bS 2 is an inhibition. +bS 2 It is easy to verify that these response functions mentioned above satisfy assumptions (A1 ) or (A2 ). Under the above assumptions of p(S) and using the same method as in [25], we can derive that the stochastic system (1.4) has a unique global positive solution. Since the conclusion is easily to obtain and the proof is standard, we present the following theorem concerns the existence and uniqueness of positive solutions for stochastic system (1.4) without proof. p(S)

Theorem 1.1. For any system parameters and any given initial value (S(0), x(0)) ∈ R2+ , there is a unique positive solution (S(t), x(t)) for system (1.4) on t ≥ 0, and the solution will remain in R2+ with probability one, i.e. (S(t), x(t)) ∈ R2+ for t ≥ 0 almost surely. The paper is organized as follows. In Section 2, we present some notations and auxiliary results which are necessary for later discussions. In Section 3, we investigate the sufficient conditions for the existence of the unique θ -periodic solution. Concluding discussion is provided in Section 4 to end this paper. 2. Preliminary In the following we introduce some preliminaries about the Khasminskii’s theory for periodic Markov process (see more detail in [26, Chapter 3]). Definition 2.1. A stochastic process X (t , ω) (here ω is a sample point in space Ω ) with values in Rl , defined for t ≥ 0 on a probability space (Ω , F , P), is called a Markov process if for all A ∈ B (B is the Borel σ -algebra), 0 ≤ s < t, P{X (t , ω) ∈ A|Ns } = P{X (t , ω) ∈ A|X (s, ω)},

a.s.,

where Ns is the σ -algebra of events generated by all events of the form

{X (u, ω) ∈ A} (u ≤ s, A ∈ B). Remark 2.1. It can be proved that there exists a function P(s, x, t , A) defined for 0 ≤ s ≤ t, x ∈ Rl , A ∈ B, which is B-measurable in x for every fixed s, t, A, and which constitutes a measure as a function of the set A, satisfying the condition P{X (t , ω) ∈ A|X (s, ω)} = P{s, X (t , ω), t , A} a.s. One can also prove that for all x, except those possibly from a set N such that P{x(s, ω) ∈ N } = 0, the Chapman–Kolmogorov equation holds: P{s, x, t , A} =

∫ Rl

P(s, x, u, dy)P(u, y, t , A).

(2.1)

The function P{s, x, t , A} is called the transition probability function of the Markov process. Definition 2.2. A stochastic process X (t) (−∞ < t < +∞) is said to be periodic with period θ if for every finite sequence of numbers t1 , t2 , . . . , tn , the joint distribution of random variables X (t1 + h), . . . , X (t2 + h) is independent of h, where h = kθ (k = ±1, ±2, . . .). Remark 2.2. Khasminskii shows in [26] that a Markov process X (t) is θ -periodic if and only if its transition probability function is θ -periodic and the function P0 (t , A) = P{X (t) ∈ A} satisfies the equation P0 (s, A) = for every A ∈ B.

∫ Rl

P0 (s, dx)P(s, x, s + θ, A) ≡ P0 (s + θ, A),

L. Wang, D. Jiang / Physica A 486 (2017) 378–385

381

Consider the following equation

∫

k ∫ ∑

t

b(s, X (s))ds +

X (t) = X (t0 ) + t0

t

σr (s, X (s))dBr (s),

X ∈ Rl ,

(2.2)

t0

r =1

where the vectors b(s, X ), σ1 (s, X ), . . . , σk (s, X ) (X ∈ Rl ) are continuous functions of (s, X ) and satisfy the conditions:

⎧ k ∑ ⎪ ⎪ ⎪ ⎪ | b(s , x) − b(s , y) | + |σr (s, x) − σr (s, y)| ≤ B|x − y|, ⎪ ⎨ r =1

(2.3)

k ⎪ ∑ ⎪ ⎪ ⎪ | b(s , x) | + |σr (s, x)| ≤ B(1 + |x|), ⎪ ⎩ r =1

where B is a constant. Let U be a given open set in Rl and E = I × Rl . Let C 2 denote the class of functions on E which are twice continuously differentiable with respect to x1 , . . . , xk and continuously differentiable with respect to t. Lemma 2.1. Suppose that the coefficient of (2.2) is θ -periodic in t and satisfy condition (2.3) in every cylinder I × U. Suppose further that there exists a function V (t , x) ∈ C 2 in E which is θ -periodic in t, and satisfies the following conditions: inf V (t , x) → ∞ as R → ∞,

(2.4)

|x|>R

LV (t , x) ≤ −1

outside some compact set ,

(2.5)

where the operator L is given by L=

l l ∑ ∂ 1∑ ∂2 ∂ + bi (t , x) + aij (t , x) , ∂t ∂ xi 2 ∂ xi ∂ xj i,j=1

i=1

aij =

k ∑

σri (t , x)σrj (t , x).

r =1

Then there exists a solution of (2.2) which is a θ -periodic Markov process. The Lemma 2.1 has been proved by Khasminskii in his monograph [26, Chapter 3, Page 80]. Readers can refer to this book for details, so we omit the proof here. 3. Statement of main results In this section, we formulate the main results about the existence of the nontrivial positive periodic solution for system (1.4). For convenience, we first introduce some notations which will be∫ needed later. Supposing f (t) is an integrable function t on [0, ∞), denote ⟨f ⟩t the mean value of function f (t), i.e. ⟨f ⟩t = 1t 0 f (s)ds. In addition, if f (t) is a bounded function on u l [0, ∞), let f = supt ∈[0,∞) f (t), f = inft ∈[0,∞) f (t). Theorem 3.1. Supposing Assumption A1 (i) and (ii) hold true. Let λ1 (t) := p(S 0 ) − D(t) − 12 σ22 (t) − constant c1 satisfying

{

c1 > max 0, −

}

m0 2Dl S 0

2

,

c1 0 2 2 S 1 (t). 2

σ

If there exists a

(3.1)

˜ ⟩θ + such that ⟨λ1 ⟩θ > 0, i.e. p(S 0 ) > ⟨D θ -periodic solution.

c1 0 2 S 2

˜ = D(t) + 1 σ 2 (t). Then system (1.4) admits a nontrivial positive ⟨σ12 ⟩θ , here D(t) 2 2

Proof. Since for any initial value (S(0), x(0)) ∈ R2+ there exists a unique global positive solution for system (1.4), we take R2+ as the whole space. It is easy to verify that the coefficients of system (1.4) satisfy conditions (2.3). According to Lemma 2.1, in order to prove Theorem 3.1 it suffices to find a C 2 -function V (t , x) and a closed set U ∈ R2+ such that conditions (2.4) and (2.5) hold. Firstly, we take a constant α ∈ (0, 1) small enough such that (A) Dl − α2 [(σ1u )2 ∨ (σ2u )2 ] > 0. Then choose a positive constant M large enough such that (B) Φ u − M ⟨λ1 ⟩θ ≤ −2. Here function Φ (S , x) will be given later in (C). Now let h(t) be the unique θ -periodic solution of equation h′ (t) = h(t)D(t) − p′ (S 0 ). In fact, we have

∫ t +θ h(t) =

t

exp

{∫

t s

}

D(τ )dτ p′ (S 0 )ds

{ ∫ θ

1 − exp −

0

D(τ )dτ

}

,

t ≥ 0,

382

L. Wang, D. Jiang / Physica A 486 (2017) 378–385 S0

∫θ

and ⟨h′ (t)S 0 ⟩θ = θ 0 h′ (t)dt = S 0 (h(θ ) − h(0)) = 0. Denote V1 = c1 (S − S 0 − S 0 log SS0 ) − h(t)(S + 1δ x) − log x + w1 (t), 1 (S + 1δ x)α+1 , V3 = − log S. Here w1 (t) is a function defined on [0, ∞) satisfying V2 = α+ 1

w ˙ 1 (t) = R1 (t) − ⟨R1 ⟩θ ,

w1 (0) = 0,

(3.2)

where R1 (t) = λ1 (t) + h (t)S . Obviously w1 (t) is a θ -periodic function on [0, +∞). Therefore we get a C -function: 0

′

2

V = MV1 + V2 + V3 ,

(3.3)

and V (t , S , x) is θ -periodic in t which satisfies condition (2.4). Applying the Itô’s formula to calculate LV1 separately in three parts, one can get that

( L

S − S 0 − S 0 log

)

S S0

=

S − S0 S

= −

[

D(t)(S 0 − S) −

D(t)(S − S 0 )2

−

S D(t)(S − S 0 )2

p(S)

δ

1

δ

] p(S)x +

x+

S 0 p(S)

δ

S

S0

1

S0

2

2 x+

σ12 (t) S0

2

2

σ12 (t)

2

+ cS 0 x + σ 2 (t). S δ 2 1 )) ( ) ( ) ( ( 1 1 1 = h(t)D(t) S 0 − S − x + h′ (t) S + x L h(t) S + x δ δ δ ≤ −

1

= (h′ (t) − h(t)D(t))(S − S 0 ) + (h′ (t) − h(t)D(t)) x + h′ (t)S 0 δ 1

= −p′ (S 0 )(S − S 0 ) − p′ (S 0 )x + h′ (t)S 0 , δ L(− log x) = −p(S) + D(t) +

1 2

σ22 (t).

Then LV1 ≤ −p(S) + D(t) +

1

1

σ 2 (t) + p′ (S 0 )(S − S 0 ) + p′ (S 0 )x − h′ (t)S 0 2 2 δ

c1 D(t)(S − S 0 )2

1

c1

2

+ cc1 S 0 x + S 0 σ12 (t) S δ 2 [ ] 1 2 c1 0 2 2 0 = − p(S ) − D(t) − σ2 (t) − S σ1 (t) − h′ (t)S 0 −

2

2

1

+ P1 (S , t) + (p′ (S 0 ) + cc1 S 0 )x, δ where P1 (S , t) = p(S 0 ) − p(S) + p′ (S 0 )(S − S 0 ) −

c1 D(t)(S − S 0 )2

.

S It then follows from the general assumptions (1.2), (1.3) and condition A1 that,

( ) 2 ∂ P1 (S , t) S0 ′ ′ 0 = −p (S) + p (S ) − c1 D(t) 1 − 2 , ∂S S 02 ∂ 2 P1 (S , t) S = −p′′ (S) − 2c1 D(t) 3 . ∂ S2 S Obviously

⏐ ∂ P1 (S , t) ⏐⏐ = 0, ∂ S ⏐S =S 0

∂ 2 P1 (S , t) 1 2 < 3 (m0 − 2c1 Dl S 0 ) < 0. ∂ S2 S

Therefore P1 (S , t) ≤ P1 (S 0 , t) = 0. Hence

[

LV1 ≤ − p(S 0 ) − D(t) −

1

σ22 (t) −

2 1

c1 2

2

]

S 0 σ12 (t) − h′ (t)S 0 +

:= − λ1 (t) − h′ (t)S 0 + (p′ (S 0 ) + cc1 S 0 )x. δ

1

δ

(p′ (S 0 ) + cc1 S 0 )x

L. Wang, D. Jiang / Physica A 486 (2017) 378–385

383

Combining this inequality with (3.2), we obtain that L(V1 + w1 (t)) ≤ −⟨λ1 ⟩θ +

1

δ

(p′ (S 0 ) + cc1 S 0 )x.

(3.4)

In addition,

) )α−1 ( 1 σ12 (t)S 2 + 2 σ22 (t)x2 δ δ 2 δ δ ( ( )α ( )α+1 ( )α−1 )2 1 1 1 α 1 ≤ Du S 0 S + x − Dl S + x + S+ x ((σ1u )2 ∨ (σ2u )2 ) S + x δ δ 2 δ δ )α ( )α+1 ( )( 1 α u2 1 u 2 l u 0 S+ x , = D S S + x − D − (σ1 ) ∨ (σ2 ) δ 2 δ (

L V2 =

S+

L V3 = −

= −

1

D(t)

)α (

x

S Dl S 0 S

1 p(S)

(S 0 − S) +

S D(t)S 0

≤ −

D(t)S 0 − D(t)S −

D(t)

1 2

δ S

+ D(t) + σ12 (t) + 1

+

α

(

S+

1

x

(3.5)

1

σ 2 (t) 2 1 1 p(S)

δ S

x+

)

x

(3.6)

x

c

+ Du + (σ1u )2 + x. 2 δ

Substituting (3.4)–(3.6) into (3.3), thus LV ≤ Φ (S , x) + Ψ (x),

where

(C)

)θ ( )θ +1 ( ⎧ )( 1 α u2 Dl S 0 1 1 ⎪ l u 2 u 0 ⎪ S + x − D − ( σ ) ∨ ( σ ) x − + Du + (σ1u )2 , Φ (S , x) = D S S + ⎨ 1 2 δ 2 δ S 2 [ ] ⎪ 1 c ⎪ ⎩Ψ (x) = M −⟨λ1 ⟩θ + (p′ (S 0 ) + cc1 S 0 )x + x. δ δ

(3.7)

In view of (A), we observe that

Φ (+∞, x) + Ψ u → −∞,

as S → +∞,

Φ (S , +∞) + Ψ (+∞) → −∞, Φ (0+ , x) + Ψ u → −∞,

as x → +∞,

as S → 0+ .

The above cases lead to LV < −1, respectively. According to condition (B)

Φ (S , 0+ ) + Ψ (0+ ) → Φ u − M ⟨λ1 ⟩θ ≤ −2,

as x → 0+ .

Take ε small enough, and let U = [ε, 1ε ] × [ε, 1ε ]. It follows that LV < −1,

(S , x) ∈ R2+ \ U .

This completes the proof. 2

Theorem 3.2. Suppose Assumption A2 (i) and (ii) hold. Let λ2 (t) := p(S 0 ) − D(t) − 12 σ22 (t) − c2 S 0 σ12 (t). If (σ1u )2 < Dl and there exists a constant c2 satisfying

{

c2 > max 0, −

}

˜0 m 2(Dl − (σ1u )2 )

,

(3.8)

˜ ⟩θ + c2 S 0 2 ⟨σ 2 ⟩θ . Then system (1.4) admits a unique nontrivial positive θ -periodic solution. such that ⟨λ2 ⟩θ > 0, i.e. p(S 0 ) > ⟨D 1 Proof. The proof is quite similar to that of Theorem 3.1. Define a function w2 (t), t ∈ [0, ∞) which satisfies

w ˙ 2 (t) = R2 (t) − ⟨R2 ⟩θ ,

w2 (0) = 0,

(3.9)

384

L. Wang, D. Jiang / Physica A 486 (2017) 378–385 (S −S 0 )2

where R2 (t) = λ2 (t) + h′ (t)S 0 . Obviously w2 (t) is a θ -periodic function on [0, +∞). Taking c2 2 instead of c1 (S − S 0 − S 0 2 2 2 S log S 0 ) in V1 . Using the fact that (a + b) ≤ a + b for any a, b ∈ R, by direct calculation we obtain

( L

(S − S 0 )2

)

2

1

= −D(t)(S − S 0 )2 − p(S)(S − S 0 )x + δ

σ12 (t) 2

S0

1

= −D(t)(S − S 0 )2 − p(S)Sx + p(S)x + δ δ ≤ − Dl (S − S 0 )2 + = −(D − (σ l

u 2 1 ) )(S

S0

δ

S2

σ12 (t) 2

(S − S 0 + S 0 )2 2

p(S)x + (σ1u )2 (S − S 0 )2 + S 0 σ12 (t)

− S 0 )2 +

S0

δ

2

p(S)x + S 0 σ12 (t).

Thus LV1 becomes LV1 ≤ −p(S) + D(t) +

1 2

1

σ22 (t) + p′ (S 0 )(S − S 0 ) + p′ (S 0 )x − h′ (t)S 0 δ c2 S 0

2

− c2 (Dl − (σ1u )2 )(S − S 0 )2 + p(S)x + c2 S 0 σ12 (t) δ [ ] 1 2 0 02 2 = − p(S ) − D(t) − σ2 (t) − c2 S σ1 (t) − h′ (t)S 0 2

1

+ P2 (S) + (p (S ) + c˜ c2 S 0 )x, δ ′

0

here P2 (S) = p(S 0 ) − p(S) + p′ (S 0 )(S − S 0 ) − c2 (Dl − (σ1u )2 )(S − S 0 )2 . It follows from the general assumptions (1.2), (1.3) and condition A2 , P2′ (S) = −p′ (S) + p′ (S 0 ) − 2c2 (Dl − (σ1u )2 )(S − S 0 ), P2′′ (S) = −p′′ (S) − 2c2 (Dl − (σ1u )2 ). Obviously P2′ (S)|S =S 0 = 0,

˜ 0 − 2c2 (Dl − (σ1u )2 ) < 0. P2′′ (S) < m

Therefore P2 (S) ≤ P2 (S 0 ) = 0. Hence

[

LV1 ≤ − p(S 0 ) − D(t) −

] 1 2 σ22 (t) − c2 S 0 σ12 (t) − h′ (t)S 0 + (p′ (S 0 ) + c˜ c2 S 0 )x 2 δ

1

1

:= −λ2 (t) − h′ (t)S 0 + (p′ (S 0 ) + c˜ c2 S 0 )x. δ Combining this inequality with (3.9), we obtain that L(V1 + w2 (t)) ≤ −⟨λ2 ⟩θ +

1

δ

(p′ (S 0 ) + c˜ c2 S 0 )x.

By using the same argument of the proof in Theorem 3.1, one can conclude that there is a unique nontrivial positive periodic solution for stochastic system (1.4). 4. Discussion A stochastic chemostat population system with general growth response is formulated in this paper. The stochastic chemostat has periodic dilution rate D(t). On the basis of Khasminskii’s theory for periodic Markov process, it is proved that the stochastic system has a nontrivial periodic solution, under two main classifications of hypothesis of p(S) and some constraint conditions for environmental white noise. The methods and results in this paper may enrich the research of asymptotic behavior in chemostat and help us better understanding the dynamics in stochastic sense. When p(S) is a monotonically increasing function, converging the white noise σ1 (t) to zero, we can define an analogue of break-even concentration ⟨λ⟩θ which is expressed in terms of the system parameters and the intensity of the white noise, i.e. ⟨λ⟩θ satisfies p(⟨λ⟩θ ) = ⟨D + 12 σ22 ⟩θ . Therefore if ⟨λ⟩θ < S 0 , then the stochastic system admits a positive periodic solution, which

L. Wang, D. Jiang / Physica A 486 (2017) 378–385

385

means the microorganism persists in the chemostat. In this sense, our results essentially improve the corresponding result in [2,7,23]. In addition, the assumptions for p(S) are relative weak and valid for many forms of response functions. While the criteria for the existence of the nontrivial positive periodic solution are sufficient but not necessary, we cannot make the assertion that ⟨λ⟩θ is the break-even concentration precisely. It is an open problem that in the case of any monotone uptake functions, whether the existence condition for the boundary periodic solution of the stochastic chemostat and its global attractivity is ⟨λ⟩θ > S 0 or not. For further investigation, we will concentrate on the principle of competitive-exclusion and food chain for multi-group stochastic chemostat system with periodic parameters and general response functions. The research is now underway. Acknowledgment The work was supported by Program for NSFC of China (No: 11371085), and the Fundamental Research Funds for the Central Universities (No: 15CX08011A), and the Scholarship Foundation of China Scholarship Council (No: 201606620033). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

H.L. Smith, P.E. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995. G.J. Butler, S.B. Hsu, P.E. Waltman, A mathematical model of the chemostat with perodic washout rate, SIAM J. Appl. Math. 45 (1985) 435–449. G. Stephanopoulis, G. Lapidus, Chemostat dynamics of plasmid-bearing plasmid-free mixed recombinant cultures, Chem. Eng. Sci. 43 (1988) 49–57. M. Weedermann, Gunog Seo, G.S.K. Wolkowicz, Mathematical model of anaerobic digestion in a chemostat: effects of syntrophy and inhibition, J. Biol. Dyn. 7 (2013) 59–85. M. Weedermann, G.S.K. Wolkowicz, J. Sasara, Optimal biogas production in a model for anaerobic digestion, Nonlinear Dynam. 81 (2015) 1097–1112. T. Zhang, T. Zhang, X. Meng, Stability analysis of a chemostat model with maintenance energy, Appl. Math. Lett. 68 (2017) 1–7. 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. G.S.K. Wolkowicz, Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM J. Appl. Math. 52 (1992) 222–233. B. Li, Global asymptotic behavior of the chemostat: general response functions and different removal rates, SIAM J. Appl. Math. 59 (1998) 411–422. S. Liu, X. Wang, L. Wang, H. Song, Competitive exclusion in delayed chemostat models with differential removal rates, SIAM J. Appl. Math. 74 (2014) 634–648. C. Ji, D. Jiang, N. Shi, Analysis of a predator–prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl. 359 (2009) 482–498. X. Meng, Z. Li, X. Wang, Dynamics of a novel nonlinear SIR model with double epidemic hypothesis and impulsive effects, Nonlinear Dynam. 59 (2010) 503–513. Y. Zhao, D. Jiang, D. O’Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Physica A 392 (2013) 4916–4927. X. Meng, S. Zhao, T. Feng, T. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl. 433 (2016) 227–242. X. Zhang, D. Jiang, T. Hayat, B. Ahmad, Dynamics of a stochastic SIS model with double epidemic diseases driven by Lévy jumps, Physica A 471 (2017) 767–777. T. Zhang, Z. Chen, M. Han, Dynamical analysis of a stochastic model for cascaded continuous flow bioreactors, J. Math. Chem. 52 (2014) 1441–1459. F. Campillo, M. Joannides, I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecol. Model. 222 (2011) 2676–2689. L. Imhof, S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations 217 (2005) 26–53. J. Grasman, M. Gee, Breakdown of a chemostat exposed to stochastic noise volume, J. Eng. Math. 53 (2005) 291–300. C. Xu, S. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Appl. Math. Lett. 48 (2015) 62–68. D. Zhao, S. Yuan, Critical result on the break-even concentration in a single-species stochastic chemostat model, J. Math. Anal. Appl. 434 (2016) 1336– 1345. Q. Zhang, D. Jiang, Competitive exclusion in a stochastic chemostat model with Holling type II functional response, J. Math. Chem. 54 (2016) 777–791. L. Wang, D. Jiang, D. O’Regan, The periodic solutions of a stochastic chemostat model with periodic washout rate, Commun. Nonlinear Sci. Numer. Simul. 37 (2016) 1–13. C. Xu, S. Yuan, Competition in the chemostat: A stochastic multi-species model and its asymptotic behavior, Math. Biosci. 280 (2016) 1–9. A. Gray, D. Greenhalgh, L. Hu, X. Mao, J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (2011) 876–902. R.Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, The Netherlands Rockville, Maryland, 1980.