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Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay
- Yong Yao^{1, 2},
- Zuxiong Li^{1}Email author,
- Huili Xiang^{1} and
- Hailing Wang^{1}
https://doi.org/10.1186/s13662-018-1566-2
© The Author(s) 2018
Received: 9 January 2018
Accepted: 16 March 2018
Published: 27 March 2018
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 [1–7].
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 [8–15], 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 [19–24] 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 [25–29]. 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.
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
Proof
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.
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
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.
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).
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.
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
If \(\liminf_{t\rightarrow+\infty}x(t)= 0\), we will show that there is a contradiction.
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.
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.
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.
Declarations
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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