Average break-even concentration in a simple chemostat model with telegraph noise

Nonlinear Analysis: Hybrid Systems 29 (2018) 373–382

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Average break-even concentration in a simple chemostat model with telegraph noise✩ Chaoqun Xu a , Sanling Yuan a, *, Tonghua Zhang b a b

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC. 3122, Australia

article

info

Article history: Received 17 January 2017 Accepted 23 March 2018

Keywords: Chemostat model Telegraph noise Markov chain Average break-even concentration

a b s t r a c t In this paper, we aim to investigate the effect of telegraph noise on the continuous culture of microorganism in a chemostat. With the help of a finite-state Markov chain, we first construct a regime-switching chemostat model and then establish conditions for extinction and persistence of the microorganism. It is shown that the particular outcome of the chemostat is completely determined by a defined average break-even concentration λA : The microorganism becomes extinct in the chemostat when λA > S 0 , the input substrate concentration; it will persist when λA < S 0 . In the case of persistence, we also investigate some further dynamic behaviors of the solution including the recurrent level and eventually existent domain. The theoretical results are illustrated by simulations at the end of the paper. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Let S(t) be the concentration of substrate and x(t) the concentration of microorganism in a chemostat at time t. Then the system of ordinary differential equations describing the continuous culture of microorganism in the chemostat takes the following form [1]:

⎧ mS(t)x(t) dS(t) 0 ⎪ ⎪ ⎨ dt = (S − S(t))D − a + S(t) , ( ) dx(t) mS(t) ⎪ ⎪ ⎩ = − D x(t), dt a + S(t)

(1.1)

where S 0 is the input substrate concentration, D is the washout rate. Monod functional response function mS /(a + S) represents the substrate uptake rate of the microorganism, m is the maximal growth rate of microorganism, and a is called half-saturation constant, and furthermore all the parameters in (1.1) are positive. The global dynamics of the above system has been studied in [1,2]. In the case that m ≤ D, there only exists a washout equilibrium E 0 = (S 0 , 0) and it is globally asymptotically stable. In the case that m > D, the unique equilibrium E 0 is globally asymptotically stable when the breakeven concentration λ ≡ aD/(m − D) ≥ S 0 ; a positive equilibrium E ∗ = (λ, S 0 − λ) appears and it inherits the global stability from E 0 when λ < S 0 . Biologically speaking, if the maximal growth rate is not greater than the washout rate, the ✩ Research is supported by the National Natural Science Foundation of China (11671260, 11271260), Shanghai Leading Academic Discipline Project (XTKX2012), and Hujiang Foundation of China (B14005). Corresponding author. E-mail address: [email protected] (S. Yuan).

*

https://doi.org/10.1016/j.nahs.2018.03.007 1751-570X/© 2018 Elsevier Ltd. All rights reserved.

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C. Xu et al. / Nonlinear Analysis: Hybrid Systems 29 (2018) 373–382

microorganism invariably becomes extinct in the chemostat. If the maximal growth rate is greater than the washout rate, the destiny of the microorganism is completely determined by its break-even concentration. For more studies on chemostat models, one can refer to Refs. [3–7] and the references cited therein. Due to the continuous fluctuation in environment, the continuous culture of microorganism in the chemostat is neither autonomous nor deterministic. Hence, Imhof and Walcher [8] pointed that the consideration of stochastic effects is helpful in a proper understanding of a biological system, and a thorough study of stochastic chemostat models is justified. Recently, different versions of stochastic chemostat models have been proposed [9–14]. For example, in view of the fact that the maximal growth rate of microorganism is influenced by white noise, we replaced deterministic parameter m in model (1.1) ˙ [9], resulting model by a random variable m + α B(t)

] [ ⎧ α S(t)x(t) mS(t)x(t) 0 ⎪ ⎪ dt − dB, dS(t) = (S − S(t))D − ⎨ a + S(t) a + S(t) ( ) ⎪ mS(t) α S(t)x(t) ⎪ ⎩dx(t) = − D x(t)dt + dB, a + S(t) a + S(t)

(1.2)

where B(t) is a standard Brownian motion, nonnegative constant α represents the intensity of the white noise. We found that if m ≤ D, the microorganism becomes extinct in chemostat; if m > D, for model (1.2) with small noise, there exists a stochastic break-even concentration

λ˜ = λ +

(S 0 )2 α 2 2(a + S 0 )(m − D)

which can be seen as a critical value determining the persistence or extinction of the microorganism in the chemostat. This result was then improved by Zhao and Yuan [10]. Except for the white noise, telegraph noise is another type of environmental noise [15,16]. It consists of sudden instantaneous switching between two or more sets of parameter values in the underlying system corresponding to two or more different environments or regimes. This switching usually cannot be described by the traditional deterministic or Itô type stochastic models. But, some researchers pointed out that the sudden instantaneous switching existed in biological system could be formulated by a continuous time finite-state Markov chain [16–20]. For example, Takeuchi et al. [16] studied a Lotka–Volterra model with telegraph noise, in which the switching is governed by a two-state Markov chain. They showed the significant effect of telegraph noise on the system dynamics: Both subsystems develop periodically, but the switching system becomes neither permanent nor dissipative. In Ref. [17], Gray et al. considered the effect of telegraph noise on the prevalence of disease and proposed a regime-switching SIS epidemic model. They proved that the disease would go extinct almost surely if T0S < 1, while remain persist almost surely if T0S > 1. Assume that the continuous culture of microorganism in the chemostat may switch between two different regimes (mark as 1 and 2) due to the variability of the environment, where regimes 1 and 2 represent the ‘‘good’’ and ‘‘bed’’ environments, respectively. There is a rather possible situation that the microorganism will persist in the chemostat with regime 1 (i.e., λ1 < S 0 ) and extinct in the chemostat with regime 2 (i.e., λ2 > S 0 ). Then two natural questions arise:

• Will the microorganism in the regime-switching chemostat persist or extinct? • What does the destiny of microorganism in the regime-switching chemostat quantificationally depend on the probability distribution of the switching process? Motivated by the above equations and published works, we will consider a regime-switching chemostat model corresponding deterministic model (1.1) to investigate the effect of the telegraph noise on dynamics of the chemostat. More precisely, in Section 2, we propose the regime-switching chemostat model and prove the global existence of the positive solution of the model. An average break-even concentration and the threshold dynamics are shown in Section 3. In Section 4, we show the asymptotic properties of the solution when the microorganism persists in the chemostat. Finally, numerical simulations and discussions are presented in the last section to close the paper. 2. Regime-switching chemostat model Throughout this paper, let (Ω , F , {Ft }t ≥0 , P) be a complete probability space with a filtration {Ft }t ≥0 , which is increasing and right continuous and Ft contains all P-null sets. In the real-world environment, some environmental factors (such as temperature and PH value) may undergo an abrupt shift between different regimes due to the telegraph noise. It is worth noting that the maximal growth rate of microorganism is usually changed by the environmental factors. For instance, the maximal growth rate of microorganism at the high temperature will be much different from that at the low temperature. In this paper, we consider a case in which the maximal growth rate m may experience sudden instantaneous switching due to the telegraph noise. We then model the switching by a right-continuous Markov chain r(t) taking values in finite-state space S = {1, 2, . . . , n}, resulting the following regimeswitching chemostat model:

⎧ mr(t) S(t)x(t) dS(t) 0 ⎪ ⎪ ⎨ dt = (S − S(t))D − a + S(t) , ( ) dx(t) mr(t) S(t) ⎪ ⎪ ⎩ = − D x(t). dt a + S(t)

(2.1)

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Denoting the generator matrix of Markov chain r(t) by Q = (qij )n×n [21,22], one can obtain

P{r(t + ∆t) = j|r(t) = i} =

{

qij ∆t + o(∆t), if i ̸ = j, 1 + qii ∆t + o(∆t), if i = j,

for sufficiently small ∆t > 0, where qij is the transition rate from state i to state j, i, j ∈ S, and qij satisfies qii = − j̸=i qij . We further assume qij > 0 for i ̸ = j. It follows from Definition 5.5 in [22] that every state can be reached from every other state, i.e., the Markov chain r(t) is irreducible. Besides, we know from [22] that almost every sample path of r(·) is a right continuous step function with a finite number of sample jumps in any finite subinterval of [0, ∞). More precisely, there is a sequence {τk }k≥0 of finite-valued Ft -stopping times such that 0 = τ0 < τ1 < · · · < τk → ∞ almost surely and

∑

r(t) =

∞ ∑

r(τk )I[τk ,τk+1 ) (t),

k=0

where IC is the indicator function of set C . Since finite-state Markov chain r(t) is irreducible, we know that r(t) is ergodic and there exists a unique stationary distribution Π = (π1 , π2 , . . . , πn ) for it, where Π satisfies

Π Q = 0,

n ∑

πi = 1 and πi > 0, i = 1, 2, . . . , n.

i=1

To investigate the dynamical behavior of regime-switching model (2.1), we first show the global existence of the positive solution. Theorem 2.1. For any given initial value (S(0), x(0)) ∈ R2+ , there is a unique solution (S(t), x(t)) to regime-switching model (2.1) ¯ + such that (S(t), x(t)) ∈ R2+ for all t ∈ R¯ + almost surely. on R Proof. For any sample path of the Markov chain, without loss of generality, we assume that it has initial value r(τ0 ) = 1. Then for t ∈ [τ0 , τ1 ), we have

⎧ dS(t) m1 S(t)x(t) ⎪ = (S 0 − S(t))D − , ⎨ dt ( ) a + S(t) dx(t) m1 S(t) ⎪ ⎩ = − D x(t). dt a + S(t) Obviously, this model has a unique solution on [τ0 , τ1 ), and the solution is positive. By continuity, this solution is uniquely determined on t = τ1 as well, and (S(τ1 ), x(τ1 )) ∈ R2+ . We further consider model (2.1) for t ∈ [τ1 , τ2 ),

⎧ dS(t) m2 S(t)x(t) ⎪ = (S 0 − S(t))D − , ⎨ dt ( ) a + S(t) dx(t) m2 S(t) ⎪ ⎩ = − D x(t). dt a + S(t) Similar to the above case, the solution of model (2.1) is uniquely determined on [τ1 , τ2 ], and it is positive. Repeating this ¯ + and the solution remains within R2+ procedure, we can conclude that model (2.1) has a unique solution (S(t), x(t)) on R almost surely. Remark 2.1. Simple calculation shows that regime-switching model (2.1) has a positive invariant set

{ } Γ = (S(t), x(t)) ∈ R2+ : S(t) + x(t) = S 0 . Then it is sufficient to study the equation for x(t) dx(t) dt

( =

mr(t) (S 0 − x(t)) a + S 0 − x(t)

) − D x(t)

(2.2)

with initial value x(0) ∈ (0, S 0 ). In the following sections, we will concentrate on discussing the dynamical behavior of regime-switching model (2.2). 3. Threshold dynamics Different from previous studies, where it has been shown that the dynamical behavior of the deterministic simple chemostat model is completely determined by the break-even concentration λ [1,2] and the destiny of microorganism in the chemostat influenced by white noise is completely determined by the stochastic break-even concentration λ˜ [9,10], for

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C. Xu et al. / Nonlinear Analysis: Hybrid Systems 29 (2018) 373–382

regime-switching model (2.2), we will use another type of threshold called average break-even concentration, to determine whether the microorganism will become extinct or persist in the chemostat. Rewrite (2.2) as d ln x(t)

mr(t) (S 0 − x(t))

=

dt

− D.

a + S 0 − x(t)

Integrate it from 0 to t and then divide t on both sides of the resulting equation. One obtains ln x(t) t

ln x(0)

−

t

=

t

∫

1 t

mr(s) (S 0 − x(s)) a+

0

S0

− x(s)

ds − D ≤

S0 a+

1

S0

t

t

∫

mr(s) ds − D.

(3.1)

0

By the ergodic property of the Markov chain, we know that 1

lim

t →∞

t

∫

t

mr(s) ds = 0

n ∑

πi mi ≜ ⟨mr ⟩, a.s.

(3.2)

i=1

Combining Eqs. (3.1) and (3.2), we have lim sup

ln x(t) t

t →∞

≤

S 0 ⟨mr ⟩ a+

S0

−D=

S 0 (⟨mr ⟩ − D) a+

S0

−

aD a + S0

, a.s.

If ⟨mr ⟩ − D ≤ 0, then lim sup

ln x(t) t

t →∞

≤−

aD a + S0

< 0, a.s.,

which implies the microorganism becomes extinct in the chemostat almost surely. Thus we always assume ⟨mr ⟩ − D > 0 in the rest of this paper. Define an average break-even concentration

λA =

aD

(3.3)

⟨m r ⟩ − D

for the microorganism. We next show that the destiny of microorganism in the chemostat is completely determined by λA , namely, the following conclusion: Theorem 3.1. For the regime-switching model (2.2), we have that (i) if λA > S 0 , then the solution of model satisfies lim sup

ln x(t) t

t →∞

≤

⟨m r ⟩ − D 0 (S − λA ) < 0, a.s., a + S0

namely, the microorganism becomes extinct exponentially in the chemostat almost surely; (ii) if λA < S 0 , then the solution has the property that lim inf x(t) > 0, a.s., t →∞

that is, the microorganism will persist in the chemostat almost surely. Proof. It follows from (3.1) and (3.2) that lim sup t →∞

ln x(t) t

≤

S 0 (⟨mr ⟩ − D) a + S0

−

aD a + S0

=

⟨mr ⟩ − D 0 (S − λA ), a.s., a + S0

which implies Conclusion (i). In order to prove Theorem 3.1(ii), we first denote

{ } Ω1 = ω ∈ Ω : lim inf x(t) = 0, lim sup x(t) = 0 t →∞

t →∞

and

{ } Ω2 = ω ∈ Ω : lim inf x(t) = 0, lim sup x(t) > 0 . t →∞

t →∞

Next, we show that P(Ωi ) = 0, i = 1, 2. If this is not true, then P(Ω1 ) > 0 or P(Ω2 ) > 0. Suppose first P(Ω1 ) > 0, then for any ω ∈ Ω1 , we have limt →∞ x(t) = 0. Then for any ε1 ∈ (0, S 0 − λA ), there is a positive number T1 = T1 (ω) such that

C. Xu et al. / Nonlinear Analysis: Hybrid Systems 29 (2018) 373–382

377

x(t) ≤ ε1 for all t ≥ T1 . For any t > T1 , we know that ln x(t) t

ln x(0)

−

t

= =

t

∫

1 t

mr(s) (S 0 − x(s)) a + S 0 − x(s)

0 T1

∫

1 t

mr(s) (S 0 − x(s)) a+

0

ds − D

S0

− x(s)

ds +

t

∫

1 t

mr(s) (S 0 − x(s)) a + S 0 − x(s)

T1

ds − D,

resulting ln x(t) t

≥ =

T1

∫

1 t

mr(s) (S 0 − x(s)) a+

0

S 0 − ε1 a+

S0

S0 1

− x(s) ∫ t

− ε1 t

ds +

S 0 − ε1 a+

S0

1

∫

t

mr(s) ds +

− ε1 t

T1

ln x(0) t

−D

mr(s) ds − D + β1 (t),

0

where

β1 (t) =

T1

∫

1 t

mr(s) (S 0 − x(s))

ds −

a + S 0 − x(s)

0

S 0 − ε1

1

a + S 0 − ε1 t

T1

∫

mr(s) ds + 0

ln x(0) t

.

Let t → ∞, we have limt →∞ β1 (t) = 0 and the following conclusion holds for ω ∈ Ω1 for which Eq. (3.2) is true: lim inf

ln x(t)

t →∞

t

≥

(S 0 − ε1 )⟨mr ⟩ a+

S0

− ε1

−D=

⟨m r ⟩ − D 0 (S − λA − ε1 ) > 0. a + S 0 − ε1

Thus, x(t) → ∞ as t → ∞ which clearly contradicts with the fact ω ∈ Ω1 . Therefore, we have P(Ω1 ) = 0. Similarly, one can prove P(Ω2 ) = 0. Otherwise, for any ω ∈ Ω2 , we have lim inft →∞ x(t) = 0 and lim supt →∞ x(t) > 0. Choose ε2 small enough such that

{

}

0 < ε2 < min S 0 − λA , lim sup x(t) , t →∞

we know that lim inf x(t) < ε2 < lim sup x(t). t →∞

t →∞

There are two positive numbers t0 and T2 = T2 (ω) with t0 > T2 such that if x(t0 ) ≥ ε2 , then there exists t1 > t0 such that x(t1 ) < ε2 ; if x(t0 ) < ε2 , then there exists t2 > t0 such that x(t2 ) ≥ ε2 . Let t ∗ > T2 be the first time that x(t) drops beneath ε2 . From the above analysis, we have that x(t) is bounded below by an increasing unbounded function when x(t) < ε2 , and thus we can conclude that there exists a finite time interval ∆t such that x(t) must rise up to the level ε2 at most t ∗ + ∆t. It follows from dx(t) dt

( =

mr(t) (S 0 − x(t)) a + S 0 − x(t)

) − D x(t) ≥ −Dx(t)

that x(t) ≥ x(t ∗ ) exp(−D∆t) for all t > t ∗ . Then lim inf x(t) ≥ x(t ∗ ) exp(−D∆t) > 0, t →∞

contradicting ω ∈ Ω2 . To sum up,

{

}

P ω ∈ Ω : lim inf x(t) = 0 = P(Ω1 ) + P(Ω2 ) = 0. t →∞

The second conclusion of Theorem 3.1 is thus proved. 4. Further asymptotic properties In this section, we will focus on analyzing some further asymptotic properties for regime-switching model (2.2) in the case when λA < S 0 (i.e., the microorganism persists in the chemostat). Theorem 4.1. If λA < S 0 , then the solution of regime-switching model (2.2) satisfies lim inf x(t) ≤ S 0 − λA ≤ lim sup x(t), a.s. t →∞

t →∞

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C. Xu et al. / Nonlinear Analysis: Hybrid Systems 29 (2018) 373–382

Proof. If the assertion, lim inft →∞ x(t) ≤ S 0 −λA almost surely, is not true, we can find a sufficiently small number ε3 ∈ (0, λA ) such that P(Ω3 ) > 0, where

{ } Ω3 = ω ∈ Ω : lim inf x(t) > S 0 − λA + 2ε3 . t →∞

Then, for any ω ∈ Ω3 , there is a sufficiently large number T3 = T3 (ω) such that for all t > T3 x(t) ≥ S 0 − λA + ε3 and for any t > T3 one has ln x(t) t

−

ln x(0) t

=

1

T3

∫

t

mr(s) (S 0 − x(s)) a+

0

S0

− x(s)

ds +

1

t

∫

t

mr(s) (S 0 − x(s)) a + S 0 − x(s)

T3

ds − D,

and then ln x(t) t

≤ =

1

T3

∫

t

mr(s) (S 0 − x(s)) a+

0

S0

λ A − ε3 1 a + λA − ε3 t

− x(s) ∫ t

ds +

λA − ε3 1 a + λ A − ε3 t

t

∫

mr(s) ds +

ln x(0) t

T3

−D

mr(s) ds − D + β2 (t),

0

where

β2 (t) =

1

T3

∫

t

mr(s) (S 0 − x(s)) a+

0

S0

− x(s)

ds −

λ A − ε3 1 a + λ A − ε3 t

T3

∫

mr(s) ds +

ln x(0) t

0

.

Let t → ∞, we have limt →∞ β2 (t) = 0 and lim sup

ln x(t)

≤

t

t →∞

λ A − ε3 (⟨mr ⟩ − D) ε3 ⟨mr ⟩ − D = − < 0, A a + λ − ε3 a + λA − ε3

which implies that limt →∞ x(t) = 0, contradicting with the fact ω ∈ Ω3 . Thus, lim inft →∞ x(t) ≤ S 0 − λA almost surely. Similarly, one can obtain that lim supt →∞ x(t) ≥ S 0 − λA almost surely. If not, we can find a sufficiently small number ε4 ∈ (0, (S 0 − λA )/2) such that P(Ω4 ) > 0, where

} { Ω4 = ω ∈ Ω : lim sup x(t) < S 0 − λA − 2ε4 . t →∞

Then for any ω ∈ Ω4 , there exists a sufficiently large number T4 = T4 (ω) such that x(t) ≤ S 0 − λA − ε4 for all t > T4 and for any t > T4 , ln x(t) t

−

ln x(0) t

=

1

T4

∫

t

0

mr(s) (S 0 − x(s)) a + S 0 − x(s)

ds +

1 t

t

∫

mr(s) (S 0 − x(s)) a + S 0 − x(s)

T4

ds − D.

One then can have ln x(t) t

λ A + ε4 1 ≥ ds + t 0 a + S 0 − x(s) a + λ A + ε4 t ∫ t A λ + ε4 1 = mr(s) ds − D + β3 (t), a + λ A + ε4 t 0 1

∫

T4

mr(s) (S 0 − x(s))

∫

t

mr(s) ds + T4

ln x(0) t

−D

where

β3 (t) =

1

T4

∫

t

mr(s) (S 0 − x(s)) a+

0

S0

− x(s)

ds −

λ A + ε4 1 a + λ A + ε4 t

T4

∫

mr(s) ds + 0

ln x(0) t

.

Let t → ∞, we have limt →∞ β3 (t) = 0 and lim inf t →∞

ln x(t) t

≥

λ A + ε4 (⟨mr ⟩ − D) ε4 ⟨mr ⟩ − D = > 0, a + λ A + ε4 a + λA + ε4

which implies that x(t) → ∞ as t → ∞, contradicting ω ∈ Ω4 . Thus, lim supt →∞ x(t) ≥ S 0 − λA almost surely. Remark 4.1. Theorem 4.2 indicates that the concentration of microorganism will reach the neighborhood of the level S 0 −λA infinitely many times in the chemostat almost surely.

C. Xu et al. / Nonlinear Analysis: Hybrid Systems 29 (2018) 373–382

379

Theorem 4.2. Assume λA < S 0 and define λˆ = min{λ1 , λ2 , . . ., λn }, λˇ = max{λ1 , λ2 , . . ., λn }, where λi = aD/(mi − D), i = 1, 2, . . . , n. The following statements hold almost surely: (i) If λˆ ≤ λˇ < S 0 , then

ˆ S 0 − λˇ ≤ lim inf x(t) ≤ lim sup x(t) ≤ S 0 − λ. t →∞

t →∞

(ii) If λˆ < S 0 ≤ λˇ , then

ˆ 0 < lim inf x(t) ≤ lim sup x(t) ≤ S 0 − λ. t →∞

t →∞

Proof. We first prove conclusion (i) by contradiction. To this end, suppose that lim supt →∞ x(t) > S 0 − λˆ . Then there exist two positive numbers t3 and t4 with t3 < t4 such that S 0 − λˆ < x(t3 ) < x(t4 ), and x(t) is strictly monotonic increasing ⏐in the open interval (t3 , t4 ). On one hand, we can choose t5 ∈ (t3 , t4 ), not a jump dx(t) point of the Markov chain, such that dt ⏐t > 0; On the other hand, notice 5

⏐ ⏐ mr(t5 ) (S 0 − x(t5 )) 1 dx(t) ⏐ − D, ⏐ = ⏐ = t5 dt x(t) dt t5 a + S 0 − x(t5 ) and without loss of generality, assume that r(t5 ) = j. Then d ln x(t) ⏐

mr(t5 ) (S 0 − x(t5 )) a+

S0

− x(t5 )

−D<

mj λˆ a + λˆ

Since x(t5 ) > 0, we can conclude that

−D≤

mj λj a + λj

− D = 0.

⏐ ⏐ < 0 which is a contradiction. Thus, we have lim supt →∞ x(t) ≤ S 0 − λˆ .

dx(t) dt t5

Similar arguments yield lim inft →∞ x(t) ≥ S 0 − λˇ and conclusion (ii), which are omitted for simplification of discussion. Remark 4.2. Theorem 4.2 suggests that the concentration of microorganism will enter the region (0 ∨ (S 0 − λˇ ), S 0 − λˆ ) in finite time, and will always stay in this region almost surely once entered. 5. Numerical simulations and discussions There are many environment fluctuations that could affect the continuous culture of microorganism in the chemostat, such as temperature and PH value. To describe the phenomenon that the maximal growth rate of microorganism exhibits random fluctuation to a great extent, in Ref. [9], we have constructed a stochastic simple chemostat model of Itô type. For this model, we obtained the threshold dynamics by defining a stochastic break-even concentration λ˜ , and found that the white noise plays a negative role on the persistence of microorganism In this paper, based on the fact that the maximal growth rate of microorganism may experience abrupt change in different regimes due to the telegraph noise, we proposed a regime-switching simple chemostat model with help of a finite-state Markov chain. Our main result (i.e., Theorem 3.1) shows that whether the microorganism will become extinct or persist in the chemostat is completely determined by a critical value λA called average break-even concentration. Specifically, the microorganism becomes extinct in the chemostat when the average break-even concentration is more than the input substrate concentration, and it will persist when the average break-even concentration is less than the input substrate concentration. Besides, from the expression of λA , we can find that the destiny of microorganism strictly depends on not only the parameters in different regimes, but also the probability distribution of the switching process. In the case that the microorganism is persistent, we analyzed some further asymptotic properties for regime-switching model (2.2) to discover the growth behavior of microorganisms in the chemostat. Theorem 4.1 indicates that the concentration of microorganism in the chemostat will oscillate around the value S 0 − λA , and will reach the neighborhood of the level S 0 − λA infinitely many times almost surely. Theorem 4.2 reflects that the concentration of microorganism will enter the region (0 ∨ (S 0 − λˇ ), S 0 − λˆ ) in finite time, and will always stay in this region almost surely once entered. Obviously, this interval is determined by the smallest and largest break-even concentrations in different environment regimes. To verify the theoretical results obtained in this paper, we next give some numerical simulations for model (2.2) with two-state Markov chain. Let us first define the parameters as S 0 = 1, D = 1.2, a = 0.6, m1 = 1.6, m2 = 3. Simple calculations show that

λ1 =

aD m1 − D

= 1.8 > S 0 , λ2 =

aD m2 − D

= 0.4 < S 0 .

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C. Xu et al. / Nonlinear Analysis: Hybrid Systems 29 (2018) 373–382

Fig. 1. Solutions of the two subsystems of model (2.2) with initial value x(0) = 0.5, parameters S 0 = 1, D = 1.2, a = 0.6, m = 1.6 (left) and m = 3 (right).

In that case, the one subsystem with state 1 predicts the microorganism will become extinct in the chemostat, while the other one subsystem with state 2 predicts the microorganism will persist in the chemostat. Results of this simulation are presented in Fig. 1. By using

λA =

aD

π1 m1 + π2 m2 − D

= S 0 , π1 + π2 = 1,

we can obtain a critical value of the probability of the Markov chain taking value 1,

π1∗ =

m2 − D m2 − m1

−

aD S 0 (m2

− m1 )

= 0.7714,

and find that

π1 > π1∗ H⇒ λA > S 0 , π1 < π1∗ H⇒ λA < S 0 . Choose the probability distribution of the Markov chain as (π1 , π2 ) = (0.9, 0.1) which satisfies π1 > π1∗ . Using the definition of λA in (3.3), we deduce that

λA =

aD

π1 m1 + π2 m2 − D

= 1.3333 > S 0 .

By Theorem 3.1(i), we can conclude that the microorganism will become extinct in the chemostat. The simulation in Fig. 2 supports this result clearly. Further choose (π1 , π2 ) = (0.1, 0.9) which satisfies π1 < π1∗ , we calculate λA = 0.4 < S 0 . By Theorem 3.1(ii), we expect that the microorganism will persist in the chemostat, which is illustrated by Fig. 3. In order to illustrate Theorems 4.1 and 4.2, we take the parameters to be S 0 = 1, D = 0.8, a = 0.6, m1 = 2, m2 = 3, π1 = 0.7, π2 = 0.3. Simple calculations show that

λ1 = 0.4, λ2 = 0.2182 and λA = 0.32. By using Theorem 4.1, we expect that the solution of regime-switching model (2.2) satisfies lim inf x(t) ≤ S 0 − λA = 0.68 ≤ lim sup x(t), a.s. t →∞

t →∞

Namely the concentration of microorganism in the chemostat will oscillate around the value 0.68, and will reach the neighborhood of the level 0.68 infinitely many times. This result is illustrated by Fig. 4. Besides, by using Theorem 4.2(i), we know that for any initial value x(0) ∈ (0, 1), the solution of regime-switching model (2.2) satisfies 0.6 = S 0 − λ1 ≤ lim inf x(t) ≤ lim sup x(t) ≤ S 0 − λ2 = 0.7818, a.s. t →∞

t →∞

This result is supported by Fig. 5 clearly. The results obtained in the present paper can enrich the research of asymptotic behavior in chemostat model, and provide reliable mathematical basis for the biological researchers in the areas such as microbial fermentation culture and

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381

Fig. 2. Left: Solution of model (2.2) with initial value x(0) = 0.5, parameters S 0 = 1, D = 1.2, a = 0.6, m1 = 1.6 and m2 = 3; Right: The corresponding Markov chain with initial value r(0) = 1, probability distribution (π1 , π2 ) = (0.9, 0.1).

Fig. 3. Left: Solution of model (2.2) with initial value x(0) = 0.5, parameters S 0 = 1, D = 1.2, a = 0.6, m1 = 1.6 and m2 = 3; Right: The corresponding Markov chain with initial value r(0) = 1, probability distribution (π1 , π2 ) = (0.1, 0.9).

Fig. 4. Left: Solution of model (2.2) with initial value x(0) = 0.55, parameters S 0 = 1, D = 0.8, a = 0.6, m1 = 2 and m2 = 3; Right: The corresponding Markov chain with initial value r(0) = 1, probability distribution (π1 , π2 ) = (0.7, 0.3).

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Fig. 5. Left: Solution of model (2.2) with initial value x(0) = 0.55, parameters S 0 = 1, D = 0.8, a = 0.6, m1 = 2 and m2 = 3; Right: The corresponding Markov chain with initial value r(0) = 1, probability distribution (π1 , π2 ) = (0.7, 0.3).

development, and microbial treatment of waste water. Notice that regime-switching ecological models with white noises have been analyzed by some authors [23,24]. Then there is an interesting problem that whether there also exists a break-even concentration for the simple chemostat model with telegraph noise and white noise simultaneously? Besides, in Ref. [25], we have proven the competitive exclusion principle (CEP) in a competition chemostat model with white noise. Does the CEP still holds or the coexistence can be achieved in the competition chemostat model with telegraph noise is other problem needing further research. References [1] H.L. Smith, P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995. [2] T.C. Gard, A new Liapunov function for the simple chemostat, Nonlinear Anal.-Real 3 (2) (2002) 211–226. [3] S.R. Hansen, S.P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science 207 (1980) 1491–1493. [4] S.B. Hsu, T.K. Luo, P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor, J. Math. Biol. 34 (2) (1995) 225–238. [5] G.S.K. Wolkowicz, H. Xia, S. Ruan, Competition in the chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math. 57 (5) (1997) 1281–1310. [6] S. Liu, X. Wang, L. Wang, H. Song, Competitive exclusion in delayed chemostat models with differential removal rates, SIAM J. Appl. Math. 74 (3) (2014) 634–648. [7] S. Yuan, T. Zhang, Dynamics of a plasmid chemostat model with periodic nutrient input and delayed nutrient recycling, Nonlinear Anal.-Real 13 (5) (2012) 2104–2119. [8] L. Imhof, S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations 217 (1) (2005) 26–53. [9] C. Xu, S. Yuan, An analogue of break-even concentration in an simple stochastic chemostat model, Appl. Math. Lett. 48 (2015) 62–68. [10] D. Zhao, S. Yuan, Critical result on the break-even concentration in a single-species stochastic chemostat model, J. Math. Anal. Appl. 434 (2) (2016) 1336–1345. [11] 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. [12] T. Zhang, Z. Chen, M. Han, Dynamical analysis of a stochastic model for cascaded continuous flow bioreactors, J. Math. Chem. 52 (5) (2014) 1441–1459. [13] C. Xu, S. Yuan, T. Zhang, Stochastic sensitivity analysis for a competitive turbidostat model with inhibitory nutrients, Int. J. Bifurcation Chaos 26 (10) (2016) 1650173. [14] X. Ji, S. Yuan, H. Zhu, Analysis of a stochastic model for algal bloom with nutrient recycling, Int. J. Biomath. 9 (6) (2016) 1650083. [15] M. Slatkin, The dynamics of a population in a markovian environment, Ecology 59 (2) (1978) 249–256. [16] Y. Takeuchi, N.H. Du, N.T. Hieu, K. Sato, Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment, J. Math. Anal. Appl. 323 (2) (2006) 938–957. [17] A. Gray, D. Greenhalgh, X. Mao, J. Pan, The SIS epidemic model with Markovian switching, J. Math. Anal. Appl. 394 (2) (2012) 496–516. [18] N.H. Du, R. Kon, K. Sato, Y. Takeuchi, Dynamical behavior of Lotka–Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math. 170 (2) (2004) 399–422. [19] D. Greenhalgh, Y. Liang, X. Mao, Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching, Physica A 462 (2016) 684–704. [20] W. Zhao, J. Li, T. Zhang, X. Meng, T. Zhang, Persistence and ergodicity of plant disease model with markov conversion and impulsive toxicant input, Commun. Nonlinear Sci. Numer. Simul. 48 (2017) 70–84. [21] W.J. Anderson, Continuous-Time Markov Chains, Springer-Verlag, Berlin-Heidelberg, 1991. [22] L.J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology, second ed., CRC Press, New York, 2010. [23] Y. Zhao, S. Yuan, T. Zhang, The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching, Commun. Nonlinear Sci. Numer. Simul. 37 (2016) 131–142. [24] J. Bao, J. Shao, Permanence and extinction of regime-switching predator–prey models, SIAM J. Math. Appl. 48 (1) (2016) 725–739. [25] C. Xu, S. Yuan, Competition in the chemostat: A stochastic multi-species model and its asymptotic behavior, Math. Biosci. 280 (2016) 1–9.