A study on a Ivlev-type Chemostat model with one competitor producing a toxin

In order to illustrate the phenomena microorganisms produce toxins to compete with its competitors, a Chemostat model of two competitors for single nutrient with Ivlev-type functional response is investigated. The existence of its solutions is analysed, and the local asymptotic stability of the model’s equilibria is obtained. By use of the theory of limit systems and Lyapunov-LaSalle invariant principle, it is shown that its equilibria are globally asymptotically stable under some conditions. Our results demonstrate that the principle of competition exclusion is established, and under different conditions, different microorganism can beat the competition.


Introduction
A Chemostat is a common laboratory apparatus which can be used to culture micro-organisms [1,2] , for example, it can be used to model the simple lake in ecology, and it can also be used to model the bioreactor which is the apparatus of manufacturing products in chemical engineering. The basic exploitative competition Chemostat model consists of equations of two microorganisms and single nutrient. In nature, however, in order to gain survival advantage, micro-organisms can produce inhibitors (toxins) to compete with their rivals [3][4][5][6] . Therefore, we also need an equation to describe the toxins.
It is known that functional response is an important factor in the modelling of Chemostat, and there are many different types of functions which have been used in the literature, for example, Holling-type, Tissiet-type, Ratio-dependent-type, Crowley-Martin-type, Ivlev-type functional response, and so on [1] . And it has been proved that they are important and suitable to describe different phenomena in practice. In this article, the Ivlev-type function will be used to account for the functional response, and it is an efficient tool in the modelling of Chemostat and predator-prey systems. Wang et al. [7] constructed a reaction-diffusion predator-prey model and gave detailed analysis of Hopf bifurcation, where the Ivlev-type functional response is used. Using the Ivlev functional response, Gong et al. [8] studied a stochastic single-species Chemostat system and obtained the threshold conditions of the persistence in mean and the extinction of microorganisms. Considering the disease in prey, Shao et al. [9] proposed a predator-prey model with Ivlev-type functional response and obtained the permanence of the system. Liu et al. [10] studied the dynamics properties of a predator-prey model with Ivlev functional response and time delay, they showed that there is a Hopf bifurcation. Base on the above discussion, the Ivlev functional response theory may work in the Chemostat model to describe the phenomena that one of the competitors may produces toxins to compete with its rivals. To author's best knowledge, considering that one competitor may produces a toxin, the Chemostat model with Ivlev-type functional response has not been found. In the paper, the following system is discussed.    c are all positive constants. With the same discussion as [3], in this paper, we assume 01 k  . Moreover, P e   denotes the effect of toxin on the growth of microorganisms, which is different from that was discussed in [3].
From the biological point of view, the initial conditions of (1) are (2) The remainder of the article is organized as follows. Some preliminary results including existence of the equilibria of (1) and ultimate boundedness of the solutions of (1) are given in section 2. In section 3, it is shown that its equilibria are globally asymptotically stable under some conditions. Section 4 concludes the paper.

Preliminary analysis
In this section, we obtain some basic results about the positively invariant set and the equilibria are presented.
is positively invariant with respect to (1) and attracts any solution of (1), moreover Proof. Using the same method as [6], the result is obtained easily. ■ Theorem 2.2 For the existence of the equilibria, we have (i) There is always a washout equilibrium , there is a boundary equilibrium (iii) When  , there is a boundary equilibrium (iv) When

Global asymptotic stability analysis
In this section, let us consider global asymptotic stability of the equilibria of (1). From Theorem 2.1, it is enough to consider (1) on  . For the asymptotical stability of the equilibria, we have the following theorems.  Proof. By use of Lyapunov-LaSalle invariance principle [6] , it is first shown that which implies that we obtain a Lyapunov function of (1) on  , i.e., V x y . Next, the subset G of  is defined as follows and let M be the largest invariant set of (1) in  . Since , By the invariance of M and (1), it easily has that 1 { } . ME  Therefore, it follows from the well-known Lyapunov-LaSalle invariance principle that 1 E is globally attractive. Now the globally asymptotical stability of 2 E is proved with the theory of limit systems. Since