Denote
$$\Gamma=\bigl\{ (S,x)\in R^{2}_{+}:S+x=S^{0}\bigr\} . $$
For the convenience of demonstration of the main results in this section, we give the following two remarks.
Remark 1
From (2.1), we know that Γ is a nonnegative invariant set for turbidostat stochastic model (1.1), which is essential characteristic for our theoretical analysis in the following section.
Remark 2
Yuan et al. [30] pointed out that if \(m\leq D\) (the maximum growth rate is less than or equal to washout rate), microorganism in the system must be washed out. Moreover, this conclusion can also be found in [2] (Chapter 1 and Section 4 of Chapter 2) and [7]. If \(m\leq d+kx\), the microorganism must be washed out in model (1.1). Thus we always assume \(m>d+kx\) in this paper, which means \(m>d\).
From a biological point of view and the mechanism of turbidostat, if the washout rate (the output constant of turbidostat) is larger than the maximum growth rate (the yield constant of turbidostat), there is no microorganism in the culture vessel. In other words, if the maximum growth rate is smaller than the washout rate, the population will be extinct. The proof in this section is based on \(m>d+kx\).
On the basis of the positive invariant set \(\Gamma=\{(S,x)\in R^{2}_{+}:S+x=S^{0}\}\), we only need to investigate the following system:
$$ \mathrm{d}x= \biggl[\frac{m(S^{0}x)^{2}}{a+(S^{0}x)^{2}}(d+kx) \biggr]x\,\mathrm {d}t+ \frac{\alpha(S^{0}x)^{2}x}{a+(S^{0}x)^{2}}\,\mathrm{d}B(t), $$
(3.1)
with the initial value \(x(0)=x_{0}\in(0,S^{0})\). We need the following definition and lemma in order to determine the main results.
Definition 3.1
([42])

(I)
The microorganism in system (1.1) is persistent if
$$\lim _{t\rightarrow\infty}\frac{1}{t} \int^{t}_{0}x(s)\,\mathrm{d}s\geq \zeta $$
for some constant \(\zeta>0\);

(II)
Microorganism in system (3.1) is stochastically persistent in the turbidostat if, for any \(\epsilon\in(0,1)\), there are positive constants \(B_{1}=B_{1}(\epsilon)\) and \(B_{2}=B_{2}(\epsilon)\) such that, for any initial value \(x_{0}\in R_{+}\),
$$\lim _{t\rightarrow\infty}\inf P\bigl(x(t)\leq B_{1}\bigr)>1\epsilon \quad\text{and}\quad \lim _{t\rightarrow\infty}\inf P\bigl(x(t)\geq B_{2}\bigr)>1\epsilon. $$
Lemma 3.1
([43])
Let
\(f\in C[[0,\infty)\times\Omega,(0,\infty)]\). If there exist positive constants
\(\lambda_{0}\)
and
λ
such that
$$\log f(t)\geq\lambda t\lambda_{0} \int^{t}_{0}f(s)\,\mathrm{d}s+F(t), \quad\textit{a.s.} $$
for all
\(t\geq0\), where
\(F\in C[[0,\infty)\times\Omega,R]\)
and
\(\lim _{t\rightarrow\infty}\frac{F(t)}{t}=0\), a.s. Then
$$\lim _{t\rightarrow\infty}\inf\frac{1}{t} \int^{t}_{0}f(s)\,\mathrm {d}s\geq\frac{\lambda}{\lambda_{0}},\quad \textit{a.s.} $$
Theorem 3.1
If the breakeven concentration
\(\lambda_{1}< S^{0}\), where
$$\lambda_{1}=\frac{d(a+(S^{0})^{2})}{mS^{0}}+\frac{\alpha^{2}(S^{0})^{3}}{2m(a+(S^{0})^{2})} $$
for any given initial value
\((S_{0},x_{0})\in R^{2}_{+}\), the solution of turbidostat model (1.1) satisfies
$$\lim _{t\rightarrow\infty}\inf\frac{1}{t} \int^{t}_{0}x(s)\,\mathrm {d}s\geq\frac{maS^{0}}{(a+(S^{0})^{2}(2mS^{0}+ka))} \bigl(S^{0}\lambda_{1}\bigr)>0, \quad\textit {almost surely}, $$
which means the microorganism in system (1.1) is persistent.
Proof
Define a function \(V(x(t))=\ln{x(t)}\). Applying Itô’s formula, we have
$$ \mathrm{d}V=\mathcal{L}V\,\mathrm{d}t+\frac{\alpha S^{2}}{a+S^{2}} \,\mathrm{d}B(t), $$
(3.2)
where
$$\begin{aligned} \mathcal{L}V&=\frac{mS^{2}}{a+S^{2}}(d+kx)\frac{1}{2}\frac{\alpha ^{2}S^{4}}{(a+S^{2})^{2}} \\ &\geq\frac{m(S^{0})^{2}}{a+(S^{0})^{2}}\frac{2mS^{0}}{a}xdkx\frac{1}{2} \frac {\alpha^{2}(S^{0})^{4}}{(a+(S^{0})^{2})^{2}} \\ &= \biggl[\frac{m(S^{0})^{2}}{a+(S^{0})^{2}}d\frac{1}{2}\frac{\alpha ^{2}(S^{0})^{4}}{(a+(S^{0})^{2})^{2}} \biggr] \biggl(\frac{2mS^{0}}{a}+k\biggr)x.\end{aligned} $$
Integrating (3.2) from 0 to t, we obtain
$$\begin{aligned} \ln{x(t)}\ln{x(0)}\geq {}&\biggl[\frac{m(S^{0})^{2}}{a+(S^{0})^{2}}d\frac {1}{2} \frac{\alpha^{2}(S^{0})^{4}}{(a+(S^{0})^{2})^{2}} \biggr]t \\ & \biggl(\frac{2mS^{0}}{a}+k\biggr) \int^{t}_{0}x(s)\,\mathrm{d}s+ \int^{t}_{0}\frac{\alpha S^{2}(s)}{a+S^{2}(s)}\,\mathrm{d}B(s),\end{aligned} $$
which means
$$\begin{aligned} \frac{\ln{x(t)}}{t}\geq{}&\frac{mS^{0}}{a+(S^{0})^{2}}\bigl(S^{0} \lambda_{1}\bigr)\\ &\biggl(\frac {2mS^{0}}{a}+k\biggr)\frac{1}{t} \int^{t}_{0}x(s)\,\mathrm{d}s+\frac{1}{t}M(t)+ \frac {1}{t}\ln{x_{0}}, \end{aligned}$$
where \(M(t)=\int^{t}_{0}\frac{\alpha S^{2}(s)}{a+S^{2}(s)}\,\mathrm{d}B(s)\) is a local continuous martingale with \(M(0)=0\). Define
$$Y_{t}=\langle M,M\rangle_{t}= \int^{t}_{0}\frac{\alpha^{2}S^{4}}{(a+S^{2})^{2}}\,\mathrm{d}t $$
is the quadratic variation process and \(Y_{t}\leq (\frac{\alpha (S^{0})^{2}}{a+(S^{0})^{2}} )^{4}t\). Therefore,
$$\lim _{t\rightarrow\infty}\sup\frac{\langle M,M\rangle_{t}}{t}\leq \biggl(\frac{\alpha(S^{0})^{2}}{a+(S^{0})^{2}} \biggr)^{4}< \infty. $$
Using the strong principle of large number, we obtain
$$\lim _{t\rightarrow\infty}\frac{M(t)}{t}=0, \quad\text{almost surely}, $$
and
$$\lim _{t\rightarrow\infty}\frac{\ln{x_{0}}}{t}=0, \quad\text{almost surely}. $$
If \(\lambda_{1}< S^{0}\), we can derive the following result by Lemma 3.1:
$$\lim _{t\rightarrow\infty}\inf\frac{1}{t} \int^{t}_{0}x(s)\,\mathrm {d}s\geq\frac{maS^{0}}{(a+(S^{0})^{2}(2mS^{0}+ka))} \bigl(S^{0}\lambda_{1}\bigr)>0, \quad\text{almost surely}. $$
This completes the proof of Theorem 3.1. □
Consider the following timehomogeneous stochastic equation:
$$\mathrm{d}X(t)=b\bigl(X(t)\bigr)\,\mathrm{d}t+\alpha\bigl(X(t)\bigr)\,\mathrm{d}B(t) \quad\text{with } X(0)\in R_{+}. $$
Lemma 3.2
([44])
Let
\(X(t)\)
be a timehomogeneous solution of the above onedimensional timehomogeneous stochastic equation on
\(E_{1}\) (onedimensional Euclidean space). Assume that:

(I)
In the domain
\(U\subset E_{1}\)
and some neighborhood thereof, the diffusion
\(\alpha(X)\)
is bounded away from zero;

(II)
If, for all
\(X\in E_{1}\setminus U\), the mean time
\(\tau_{X}\)
at which a path emerging from
X
reaches the set
U
is finite, and
\(\sup _{X\in K}E(\tau_{X})<\infty\)
for every compact subset
\(K\subset E_{1}\).
Then the Markov process
\(X(t)\)
has a stationary distribution
\(\pi(x)\).
Theorem 3.2
If the breakeven concentration
\(\lambda_{2}< S^{0}\), where
$$\lambda_{2}=\frac{(a+(S^{0})^{2})d}{mS^{0}}+\frac{\alpha^{2}(S^{0})^{3}}{m(a+(S^{0})^{2})}, $$
for any given initial value
\((S_{0},x_{0})\in R^{2}_{+}\), the microorganism
\(x(t)\)
is stochastically persistent in the turbidostat and system (3.1) has a stationary distribution.
Proof
Define a \(C^{2}\)function \(V:R_{+}\rightarrow R_{+}\) for any \(p\in(0,1)\) as follows:
$$V(x)=\frac{1}{x^{p}(t)}, \quad p\in(0,1). $$
Applying Itô’s formula, one can obtain
$$\begin{aligned} \mathrm{d}V={}&\frac{p}{x^{p+1}} \biggl\{ \biggl[\frac {m(S^{0}x)^{2}}{a+(S^{0}x)^{2}}(d+kx) \biggr]x\,\mathrm{d}t +\frac{\alpha(S^{0}x)^{2}x}{a+(S^{0}x)^{2}}\,\mathrm{d}B(t) \biggr\} \\ & +\frac{p(p+1)\alpha^{2}(S^{0}x)^{4}x^{2}}{2x^{p+2}(a+(S^{0}x)^{2})^{2}}\,\mathrm {d}t \\ ={}&{}\frac{p}{x^{p}} \biggl\{ \frac{m(S^{0}x)^{2}}{a+(S^{0}x)^{2}}dkx\frac {(p+1)\alpha^{2}(S^{0}x)^{4}}{2(a+(S^{0}x)^{2})^{2}} \biggr\} \,\mathrm{d}t \\ & \frac{p\alpha(S^{0}x)^{2}}{x^{p}(a+(S^{0}x)^{2})}\,\mathrm{d}B(t),\end{aligned} $$
which implies that
$$ \begin{aligned}[b] \mathrm{d}V={}&{}\frac{p}{x^{p}} \biggl\{ \frac{m(S^{0})^{2}}{a+(S^{0})^{2}}d\frac {(p+1)\alpha^{2}(S^{0})^{2}}{2(a+(S^{0})^{2})^{2}} \biggr\} \,\mathrm{d}t+F(t)\,\mathrm {d}t \\ & \frac{p\alpha(S^{0}x)^{2}}{x^{p}(a+(S^{0}x)^{2})}\,\mathrm{d}B(t), \end{aligned} $$
(3.3)
where
$$\begin{aligned} F(t)={}&\frac{p}{x^{p}} \biggl\{ \frac{m(S^{0})^{2}}{a+(S^{0})^{2}}\frac {m(S^{0}x)^{2}}{a+(S^{0}x)^{2}} \frac{(p+1)\alpha^{2}(S^{0})^{4}}{2(a+(S^{0})^{2})^{2}} +\frac{(p+1)\alpha^{2}(S^{0}x)^{4}}{2(a+(S^{0}x)^{2})^{2}}+kx \biggr\} \\ ={}&\frac{p(2maS^{0}xmax^{2})}{x^{p}(a+(S^{0})^{2})(a+(S^{0}x)^{2})}\frac {p(p+1)\alpha^{2}}{2x^{p}} \biggl\{ \biggl[\frac{(S^{0})^{2}}{a+(S^{0})^{2}} \biggr]^{2} \biggl[\frac{(S^{0}x)^{2}}{a+(S^{0}x)^{2}} \biggr]^{2} \biggr\} \\ & +\frac{pkx}{x^{p}} \\ \leq{}&\frac{2pmaS^{0}x}{x^{p}(a+(S^{0})^{2})(a+(S^{0}x)^{2})}+\frac{pkx}{x^{p}} \\ \leq{}& \biggl[\frac{2pmaS^{0}}{a(a+(S^{0})^{2})}+kp \biggr]\bigl(S^{0} \bigr)^{1p}.\end{aligned} $$
Let \(\theta=p [\frac{m(S^{0})^{2}}{a+(S^{0})^{2}}d\frac{(p+1)\alpha ^{2}(S^{0})^{4}}{2(a+(S^{0})^{2})^{2}} ]\), then we can choose p small enough such that \(\theta>0\). Multiplying (3.3) by \(e^{\theta t}\) and taking an integration from 0 to t, we obtain
$$ \begin{aligned}[b] \frac{1}{x^{p}(t)}&=e^{\theta t} \frac{1}{x^{p}(0)}+ \int^{t}_{0}F(s)e^{\theta (ts)}\,\mathrm{d}s \int^{t}_{0}\frac{p\alpha (S^{0}x(t))^{2}}{x^{p}(a+(S^{0}x(t))^{2})}\,\mathrm{d}B(s) \\ &\leq\frac{1}{x^{p}(0)}+\frac{1}{\theta} \biggl[\frac {2pmaS^{0}}{a(a+(S^{0})^{2})}+kp \biggr] \bigl(S^{0}\bigr)^{1p}M(t), \end{aligned} $$
(3.4)
where \(M(t)=\int^{t}_{0}\frac{p\alpha(S^{0}x)^{2}}{x^{p}(a+(S^{0}x)^{2})}\,\mathrm {d}B(s)\) is a continuous martingale with \(M(0)=0\). Taking expectation on both sides of (3.4), we conclude that
$$\begin{aligned} E \biggl[\frac{1}{x^{p}(t)} \biggr]&=\frac{1}{x^{p}(0)}+ \int ^{t}_{0}E\bigl(F(s)\bigr)e^{\theta(ts)} \,\mathrm{d}sE\bigl(M(t)\bigr) \\ &\leq\frac{1}{x^{p}(0)}+\frac{1}{\theta} \biggl[\frac {2pmaS^{0}}{a(a+(S^{0})^{2})}+kp \biggr] \bigl(S^{0}\bigr)^{1p}.\end{aligned} $$
Let \(B_{1}=S^{0}\), we have the following equality:
$$P\bigl(x(t)\leq B_{1}\bigr)=P\bigl(x(t)\leq S^{0}\bigr)=1 \geq1\epsilon, $$
on the positive invariant set Γ. Moreover, applying Chebyshev’s inequality [6], we obtain
$$\begin{aligned} P\bigl(B_{2}\leq x(t)\bigr)&=P\biggl(\frac{1}{B_{2}^{p}}\geq \frac{1}{x^{p}(t)}\biggr)=1P\biggl(\frac {1}{B_{2}^{p}}\leq\frac{1}{x^{p}(t)} \biggr) \\ &\geq1B_{2}^{p}E \biggl[\frac{1}{x^{p}(t)} \biggr] \\ &\geq1B_{2}^{p} \biggl\{ \frac{1}{x^{p}(0)}+ \frac{1}{\theta} \biggl[\frac {2pmaS^{0}}{a(a+(S^{0})^{2})}+kp \biggr]\bigl(S^{0} \bigr)^{1p} \biggr\} .\end{aligned} $$
We can choose \(B_{2}\) such that \(B_{2}^{p} \{\frac{1}{x^{p}(0)}+\frac {1}{\theta} [\frac{2pmaS^{0}}{a(a+(S^{0})^{2})}+kp ](S^{0})^{1p} \} <\epsilon\), which implies that
$$P\bigl(B_{2}\leq x(t)\bigr)\geq1\epsilon. $$
Therefore, the microorganism is stochastically persistent in the turbidostat. Next we prove that system (3.1) has a stationary distribution. Let \(\varepsilon>0\) be a small enough number and U be a bounded open subset with a regular boundary such that
$$\bigl\{ x:\varepsilon\leq x\leq S^{0}\varepsilon\bigr\} \subset U \subset\bar {U}\subset\bigl(0,S^{0}\bigr), $$
where Ū represents the closure of U. Define a \(C^{2}\)function \(V:R_{+}\rightarrow R_{+}\) as
$$V\bigl(x(t)\bigr)=\frac{1}{px^{p}(t)}+\frac{1}{S^{0}x(t)} $$
for any \(p\in(0,1)\). Then apply Itô’s formula to get
$$\begin{aligned} \mathrm{d}V={}&{}\frac{1}{x^{p}} \biggl\{ \frac {a(S^{0}x)^{2}}{a+(S^{0}x)^{2}}dkx \frac{(p+1)\alpha^{2}}{2}\frac {(S^{0}x)^{4}}{(a+(S^{0}x)^{2})^{2}} \biggr\} \,\mathrm{d}t \\ & + \biggl\{ \frac{1}{(S^{0}x)^{2}} \biggl[\frac {m(S^{0}x)^{2}x}{a+(S^{0}x)^{2}}(d+kx)x \biggr]+ \frac{1}{(S^{0}x)^{3}}\frac{\alpha ^{2}(S^{0}x)^{4}x^{2}}{(a+(S^{0}x)^{2})^{2}} \biggr\} \,\mathrm{d}t \\ & + \biggl\{ \frac{1}{x^{p+1}}+\frac{1}{(S^{0}x)^{2}} \biggr\} \frac{\alpha (S^{0}x)^{2}x}{a+(S^{0}x)^{2}}\,\mathrm{d}B(t) \\ :={}&\mathcal{L}V\,\mathrm{d}t+ \biggl\{ \frac{1}{x^{p+1}}+\frac {1}{(S^{0}x)^{2}} \biggr\} \frac{\alpha(S^{0}x)^{2}x}{a+(S^{0}x)^{2}}\,\mathrm{d}B(t),\end{aligned} $$
where
$$\begin{aligned} \mathcal{L}V\leq{}&{}\frac{1}{x^{p}} \biggl\{ \frac {m(S^{0})^{2}}{a+(S^{0})^{2}}d \frac{(p+1)\alpha^{2}}{2}\frac {(S^{0})^{4}}{(a+(S^{0})^{2})^{2}} \biggr\} + \biggl[\frac{2pmaS^{0}}{a(a+(S^{0})^{2})}+kp \biggr]\bigl(S^{0}\bigr)^{1p} \\ & +\frac{mS^{0}}{a}+\frac{\alpha^{2}(S^{0})^{3}}{a^{2}}\frac{dx}{(S^{0}x)^{2}}.\end{aligned} $$
Use the inequality \(\lambda_{2}< S^{0}\) and \(p\in(0,1)\) to check that, for sufficiently small \(\varepsilon>0\),
$$\mathcal{L}V(x)\leq1 \quad \text{for all } x\in\bigl(0,S^{0}\bigr) \setminus U, $$
which yields that (II) in Lemma 3.2 holds. It is easy to check that the diffusion \(\sigma(x)=\frac{\alpha(S^{0}x)^{2}x}{a+(S^{0}x)^{2}}\) in system (3.1) is bounded away from zero for \(x\in(0,S^{0})\). Therefore, system (3.1) has a stationary distribution. This completes the proof of Theorem 3.2. □
Theorem 3.3
If
\(\frac{\alpha^{2}}{4}+kS^{0}< d\frac{m(S^{0})^{2}}{a+(S^{0})^{2}}\), then for any initial condition
\((S_{0},x_{0})\in R^{2}_{+}\), the microorganism
\(x(t)\)
will be extinct with probability one in the turbidostat.
Proof
Defining a \(C^{2}\)function \(V(x(t))=\ln{x(t)}\), we obtain the following equality by Itô’s formula:
$$ \mathrm{d}V= \biggl[\frac{mS^{2}}{a+S^{2}}(d+kx)\frac{\alpha ^{2}S^{4}}{2(a+S^{2})^{2}} \biggr]\,\mathrm{d}t+\frac{\alpha S^{2}}{a+S^{2}}\,\mathrm{d}B(t). $$
(3.5)
By equation (3.5), we define
$$h(S)=\frac{mS^{2}}{a+S^{2}}dkx\frac{\alpha^{2}S^{4}}{2(a+S^{2})^{2}}. $$
For \(h(S)\), we can obtain
$$ \begin{aligned}[b] h(S)&\leq\frac{\alpha^{2}aS^{2}}{(2\sqrt{a}S)^{2}}+ \frac {m(S^{0})^{2}}{a+(S^{0})^{2}}d+kS^{0} \\ &=\frac{\alpha^{2}}{4}+\frac{m(S^{0})^{2}}{a+(S^{0})^{2}}d+kS^{0}. \end{aligned} $$
(3.6)
By equations (3.5) and (3.6), we see that
$$ \begin{aligned}[b] \ln{x(t)}\ln{x_{0}}&= \int^{t}_{0}h(S)\,\mathrm{d}t+ \int^{t}_{0}\frac{\alpha S^{2}}{a+S^{2}}\,\mathrm{d}B(t) \\ &\leq \biggl[\frac{\alpha^{2}}{4}+\frac{m(S^{0})^{2}}{a+(S^{0})^{2}}d+kS^{0} \biggr]t+ \int^{t}_{0}\frac{\alpha S^{2}}{a+S^{2}}\,\mathrm{d}B(t), \end{aligned} $$
(3.7)
which yields the inequality
$$ \frac{\ln{x(t)}}{t}\leq\frac{\alpha^{2}}{4}+\frac {m(S^{0})^{2}}{a+(S^{0})^{2}}d+kS^{0}+ \frac{\ln{x_{0}}}{t}+\frac{1}{t}M(t), $$
(3.8)
where \(M(t)=\int^{t}_{0}\frac{\alpha S^{2}}{a+S^{2}}\,\mathrm{d}B(t)\) is a local continuous martingale with \(M(0)=0\). If \(\frac{\alpha^{2}}{4}+\frac{m(S^{0})^{2}}{a+(S^{0})^{2}}d+kS^{0}<0\), then taking the supremum and limit for (3.8), we get
$$ \lim _{t\rightarrow\infty} \text{sup}\frac{\ln{x(t)}}{t}\leq\frac{\alpha^{2}}{4}+ \frac {m(S^{0})^{2}}{a+(S^{0})^{2}}d+kS^{0}< 0,\quad \text{almost surely}. $$
(3.9)
That is, the microorganism \(x(t)\) in the vessel will exponentially tend to zero. The proof of Theorem 3.3 is completed. □