# Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay

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## Abstract

In this paper, dynamic behaviors of a turbidostat model with Tissiet functional response, linear variable yield and time delay are investigated. The existence and boundedness of solutions, the local asymptotic stability of its equilibria and the phenomenon of Hopf bifurcation for this system are considered. Using the Liapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Furthermore, based on some knowledge of limit set, we show the necessary and sufficient conditions of permanent of the turbidostat model. Finally, numerical simulations are offered to support our results.

## Keywords

Turbidostat Time delay Hopf bifurcation Stability permanence## 1 Introduction

The turbidostat is an important laboratory apparatus used to culture the microorganisms continuously. It is of both mathematical and ecological interest since its applicability in microbiology and population biology. Therefore, the study of the turbidostat model has been one of the hottest subjects investigated by many mathematical and theoretical biologists [1, 2, 3, 4, 5, 6, 7].

Dynamical behaviors of ecological systems may be affected by many factors such as time delay, variable yield and functional response. It is well known that the time delay occurs naturally in daily life and makes ecological systems have more complex dynamic behaviors. Therefore, ecological systems with time delay have been investigated in recent years (discrete delays [8, 9, 10, 11, 12, 13, 14, 15], neutral delays [16, 17] and impulsive delay [18]). Taking the time delay as a parameter, the stability of the equilibrium may be changed and periodic solutions may occur as the time delay varies. So it is also necessary to consider the impact of the time delay in turbidostat model.

Chemostat models not only with constant yield but also with variable yield [19, 20, 21, 22, 23, 24] are considered. Since actual experiments show that the constant yield cannot explain the oscillatory phenomenon in the chemostat, and the greater the nutrient concentrate is the lower the consuming rate is. However, mostly for the turbidostat model we assume that the yield term is a constant. Based on the facts, a variable yield should be considered for the turbidostat model and the model with variable yield will have more complicated dynamic behaviors than that with constant yield.

Furthermore, the functional response also has an important impact on the behavior of biological dynamical systems [25, 26, 27, 28, 29]. We note the fact that, in biology, very high nutrient concentration may inhibit the growth of microorganisms actually, and the microorganisms will die eventually as the nutrient concentration increasing unlimitedly. So the Tissiet functional response \(\mu(S)= \frac{\mu_{m}Se^{-\frac{S}{k_{i}}}}{k_{m}+S}\) is introduced, where \(\mu_{m}\), \(k_{m}\) and \(k_{i}\) are positive constants.

*t*, respectively. \(S^{0}>0\) presents the input concentration of the nutrient. \(A+BS(t)\)\((A>0,B>0)\) is the linear variable yield. \(\tau\geq0\) is the delay of digestion, and \(d+kx(t)\)\((d>0, k>0)\) is the dilution rate of the turbidostat.

*k*are dropped, and system (1.1) becomes

The organization of this paper is as follows. In next section, we analyze the existence and boundedness of solutions of (1.2) with the initial condition. In Sect. 3, the existence, local stability of the equilibriums and the existence of the local Hopf bifurcation are considered. Using the Liapunov–LaSalle invariance principle, we in Sect. 4 discuss the global asymptotic stability of the washout equilibrium of (1.2). In Sect. 5, the permanence of (1.2) is discussed by some analytic techniques on limit sets of differential dynamical systems. Finally, some discussions and numerical simulations are given to illustrate the theoretical analysis in Sect. 6.

## 2 Existence and boundedness of solutions

In the section, we investigate the existence and boundedness of solutions of (1.2) with the initial condition. The following theorem is achieved.

## Theorem 2.1

*The solution*\((x(t), y(t))\)

*of system*(1.2)

*with the initial condition exists and is positive on*\([0, +\infty)\).

*Furthermore*,

*where*\(v_{1}= \frac{akdA}{akdA+\mu_{m}^{2}}\).

## Proof

*e*. We first show that \(x(t)>0\) for any \(t\in[0, e)\). From the first equation of (1.2) and \(\varphi_{1}(0)>0\), we have

*M*. Hence, we obtain

*T*, \(t>T\), it is easy to see that from system (1.2)

The proof of Theorem 2.1 is thus completed. □

## 3 Local asymptotic stability of equilibriums and Hopf bifurcations

In this section, we will investigate the existence and local stability of the equilibriums of system (1.2) and Hopf bifurcations are induced by delay.

*G*is a positively invariant set with respect to (1.2).

*φ*, we see that \((x(t), y(t))\) is positive on \([0, +\infty)\) from Theorem 2.1. We show \(y(t)\leq1\) for any \(t\geq0\). In fact, if there is a \(t_{2}>0\) such that \(y(t_{2})> 1\), from the Lagrange mean value theorem, we see that \(\dot{y}(t_{3})>0\) and \(y(t_{3})=1\) for some \(t_{3} \in(0, t_{2})\). From the second equation of (1.2), we see that

Therefore, *G* is a positively invariant set with respect to (1.2). It is enough to consider system (1.2) on *G*.

Next, we will consider the existence of the equilibriums of (1.2).

## Theorem 3.1

- (1)
*If*\((\mathrm{H}_{1})\)*and*\((\mathrm{H}_{2})\)*hold*,*then there is no root for*\(f(y)=0\)*on*\([0, 1]\),*i*.*e*.,*system*(1.2)*only has the washout equilibrium*\(E_{0}=(0, 1)\). - (2)
*If*\((\mathrm{H}_{1})\)*and*\((\mathrm{H}_{3})\)*hold*,*then there is a positive root for*\(f(y)=0\)*on*\([0, 1]\),*denoted by*\(y^{*}\),*i*.*e*.,*system*(1.2)*has a unique positive equilibrium*\(E^{*}=(x^{*}, y^{*})\),*where*\(x^{*}=(1-y^{*})(A+Cy^{*})\).

In the following, we will discuss the locally asymptotical stability of the washout equilibrium \(E_{0}=(0, 1)\) of system (1.2).

## Theorem 3.2

*If*\(\frac{\mu_{m}e^{-b}}{1+a}< d\), *then*\(E_{0}\)*is locally asymptotically stable*; *If*\(\frac{\mu_{m}e^{-b}}{1+a}= d\), *then the trivial solution of the linearized system of* (1.2) *about*\(E_{0}\)*is stable*; *if*\(\frac{\mu_{m}e^{-b}}{1+a}> d\), *then*\(E_{0}\)*is unstable*.

## Proof

If \(\frac{\mu_{m}e^{-b}}{1+a}< d\), then \(\lambda_{2}<0\). Hence, \(E_{0}\) is locally asymptotically stable.

If \(\frac{\mu_{m}e^{-b}}{1+a}= d\), then \(\lambda_{2}=0\). Hence, we see that the trivial solution of the linearized system of (1.2) about \(E_{0}\) is stable.

If \(\frac{\mu_{m}e^{-b}}{1+a}> d\), then \(\lambda_{2}>0\). Hence, \(E_{0}\) is unstable.

The proof of Theorem 3.2 is completed. □

## Theorem 3.3

*If*\((\mathrm{H}_{4})\)*and*\((\mathrm{H}_{5})\)*hold*, *then*\(E^{*}\)*is locally asymptotically stable for*\(\tau<\tau_{0}\); \(E^{*}\)* is unstable for*\(\tau>\tau_{0}\); *Hopf bifurcation occurs when*\(\tau=\tau_{j}\), \(j=0,1,2,\ldots\) , *that is*, *a family of periodic solutions bifurcate from the positive equilibrium*\(E^{*}\)*as**τ**passes through the critical values*\(\tau_{j}\), \(j=0,1,2,\ldots\) .

## Proof

*τ*, then we have

*τ*, we can obtain

Let \(\lambda(\tau)=\alpha(\tau)+i\beta(\tau)\) be the root of (3.4) near \(\tau=\tau_{j}\) satisfying \(\alpha(\tau_{j})=0\) and \(\beta(\tau_{j})=w_{0}\). Next, we will prove the transversality condition of a Hopf bifurcation.

*τ*, we have

The proof of Theorem 3.3 is completed. □

## 4 Global asymptotic stability analysis of \(E_{0}\)

In Sect. 3, we have studied the local stability of \(E_{0}\). In this section, we will analyze the global asymptotic stability of \(E_{0}\) by the Liapunov–LaSalle invariance principle. We obtain the following theorem.

## Theorem 4.1

*For any time delay**τ*, *if*\((\mathrm{H}_{1})\)*holds*, *then the washout equilibrium*\(E_{0}\)*is globally asymptotically stable for*\(\frac{\mu _{m}e^{-b}}{1+a}< d\), *and globally attractive for*\(\frac{\mu _{m}e^{-b}}{1+a}= d\).

## Proof

We have shown that \(G=\{\varphi=(\varphi_{1}, \varphi_{2})\in C | \varphi_{1}\geq0, v_{1}\leq\varphi_{2} \leq1\}\) is a positively invariant set with respect to (1.2).

*V*on

*G*as follows:

*G*. Its derivative along the solution of (1.2) satisfies

*G*.

*E*of

*G*as \(E=\{(x(t), y(t))\in G \mid \dot{V}(x, y)|_{\text{(1.2)}}=0 \}\). From (4.2), we see that

Let *M* be the largest invariant set of (1.2) in *E*. Since \(E_{0}=(0, 1)\in M\), *M* is not empty. We discuss the following two cases, respectively.

*G*, the function \(y(t)\) takes local maximum at

*t*. Hence, it must see that \(\dot{y}(t)=0\). From the second equation of system (1.2), we see that

The proof of Theorem 4.1 is completed. □

## 5 Permanence

In this section, we will use the same method as [30] to prove the permanence of (1.2). We have the following theorem.

## Theorem 5.1

*Under the condition*\((\mathrm{H}_{1})\), *for any time delay**τ*, \((\mathrm{H}_{3})\)*is the necessary and sufficient condition for the permanence of* (1.2).

## Proof

*φ*. The proof is divided into two steps.

*G*, it is enough to consider the solution \((x(t), y(t))\)\((t\geq0)\) with the initial function \(\varphi\in G\). From the above discussion, we see that the omega limit set \(\omega(\varphi)\) of \((x(t), y(t))\)\((t\geq0)\) is nonempty, compact, invariant and \(\omega(\varphi)\subset G\).

If \(\liminf_{t\rightarrow+\infty}x(t)= 0\), we will show that there is a contradiction.

*R*in the wider sense. From the invariance of

*G*and Theorem 4.1, we see that \((\bar{x}(t), \bar{y}(t))\in G\) for any \(t\in R\), and that, for any \(\tau\in R\), the function \((\bar{x}(t+\tau), \bar{y}(t+\tau))\) of

*t*is the solution of (1.2) with the initial function \((\bar{x}_{\tau}, \bar{y}_{\tau})\). Here we note that \(\bar{x}(0)=0\) and \(v_{1}\leq \bar{y}(t)\leq1\) for \(t\in R\).

*τ*, we see that

*G*, we see that \((\bar{x}(t), \bar{y}(t))=(0, 1)\) for any \(t\in R\). Thus, the above claim holds.

*n*, we have

The proof of \(\liminf_{t\rightarrow+\infty}x(t)> 0\) is completed.

If (5.1) does not hold, for some initial function sequence \(\{\varphi_{n}\}=\{(\varphi_{1}^{(n)}, \varphi_{2}^{(n)})\}\subset G\) such that \(\varphi_{1}^{(n)}(0)>0\), we see that there is some \(\bar{\varphi}=(\bar{\varphi}_{1}, \bar{\varphi}_{2})\in\omega^{*}\) such that \(\bar{\varphi}_{1}(\theta_{0})=0\) for some \(\theta_{0}\in[-\tau, 0]\). Now, let \((\bar{x}(t), \bar{y}(t))\) be the solution of (1.2) with the initial function *φ̄*. Then, from the invariance of \(\omega^{*}\), we see that \((\bar{x}_{t}, \bar{y}_{t})\in\omega^{*}\) for any \(t\in R\). From \(\bar{\varphi}_{1}(\theta_{0})=0\) and the positivity of all solutions, we easily see that \(\bar{x}(t)=0\) for all \(t\leq\theta_{0}\). Thus, from (1.2), we have \(\bar{\varphi}_{1}(\theta)=0\)\((-\tau\leq \theta\leq0)\) and \(\bar{x}(t)=0\)\((t\in R)\). This implies that \(\bar{x}(t)=0\), \(\bar{y}(t)=h(t)\) for all \(t\in R\), where \(h(t)=1+(\bar{\varphi}_{2}(0)-1)e^{-dt}\).

If \(\bar{\varphi}_{2}(0)<1\), we see that the negative semi-orbit \((\bar{x}_{t}, \bar{y}_{t})\)\((t\leq0)\) is unbounded. This is a contradiction.

*U*of \(E_{0}\) in

*G*such that \(E_{0}\) is the largest invariant set in

*U*. In fact, we choose

*ϵ*and \(\epsilon< \frac{\mu_{m}e^{-b}-d(a+1)}{\mu_{m}e^{-b}-d}\). We shall show that \(E_{0}\) is the largest invariant set in

*U*for some

*ϵ*.

If not, for any sufficiently small *ϵ*, there exists some invariant set *W*\((W\subset U)\) such that \(W\setminus E_{0}\) is not empty. Let \(\varphi=(\varphi_{1}, \varphi_{2})\in W\setminus E_{0}\) and \((x_{t}, y_{t})\) be the solution of (1.2) with the initial function *φ*. Then, \((x_{t}, y_{t})\in W\) for all \(t\in R\).

If \(\varphi_{1}(0)=0\), by the invariance of *W* and Theorem 2.1, we also have the contradiction that \(\varphi=E_{0}\) or that the negative semi-orbit \((x_{t}, y_{t})\)\(t<0\) of (1.2) through *φ* is unbounded.

*η*and all large \(t\geq t_{5}>0\), we see that \(x(t)\geq\eta>0\). Thus, from (5.4)

It is easy to see that the semigroup defined by the solution of (1.2) satisfies the conditions of Lemma 4.3 in [31] with \(M=E_{0}\). Thus, from the lemma, we see that there is some \(\xi=(\xi_{1}, \xi_{2})\) such that \(\xi\in\omega^{*} \cap(W^{s}(E_{0}) \setminus E_{0})\). Here, \(W^{s}(E_{0})\) is the stable set of \(E_{0}\).

If \(\xi_{1}(0)=0\), by the invariance of *M* and Theorem 2.1, we have the contradiction that \(\xi=E_{0}\) or that the negative semi-orbit \((\tilde{x}_{t}, \tilde{y}_{t})\)\((t<0)\) of (1.2) through *ξ* is unbounded.

If \(\xi_{1}(0)>0\), by Theorem 2.1, we see that \(\tilde{x}(t)>0\), \(\tilde{y}(t)>0\) for any \(t>0\). From \(\xi\in \omega^{*} \cap(W^{s}(E_{0}) \setminus E_{0})\), we have \(\lim_{t\rightarrow+\infty}\tilde{x}(t)=0\), \(\lim_{t\rightarrow+\infty}\tilde{y}(t)=1\), which is a contradiction to (5.2). This shows that (5.1) holds. Hence, (1.2) is permanent.

The proof of Theorem 5.1 is completed. □

## 6 Discussion and numerical simulation

We have studied a turbidostat model with Tissiet functional response, linear variable yield and time delay in this paper. Using comparison principle and some knowledge of functional differential equations, we obtain the global existence and boundedness of solutions of (1.2). Furthermore, based on the Liapunov–LaSalle invariance principle, we also obtain the global attraction and global asymptotic stability of the washout equilibrium of (1.2). The results tell us that the time delay is harmless for the local and global stability of the washout equilibrium of (1.2). However, the stability of the positive equilibrium will be changed and Hopf bifurcations will occur with the time delay varying. Finally, we show that the system is permanent if and only if the positive equilibrium \(E^{*}\) exists. Unfortunately, in this paper, we only consider one of the cases of the existence of the positive equilibriums. The other cases shall be left as future work.

## Notes

### Acknowledgements

We are very grateful to the anonymous referees and the editor for their careful reading of the original manuscript and their kind comments and valuable suggestions, which led to truly significant improvement of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11561022; 11701163) and the China Postdoctoral Science Foundation (Grant No. 2014M562008).

### Authors’ contributions

All authors read and approved the manuscript.

## Competing interests

The authors declare that they have no competing interests.

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