EXTINCTION AND PERMANENCE OF TWO-NUTRIENT AND ONE-MICROORGANISM CHEMOSTAT MODEL WITH PULSED INPUT

A chemostat model with periodically pulsed input is considered. By using the Floquet theorem, we find that the microorganism eradication periodic solution (u1 (t),v ∗ 1 (t),0) is globally asymptotically stable if the impulsive period T is more than a critical value. At the same time we can find that the nutrient and microorganism are permanent if the impulsive period T is less than the critical value.


Introduction
A chemostat is a piece of laboratory apparatus frequently used for culturing microorganisms.It can be used for representing all kinds of microorganism systems such as lake, waste-water treatment and reaches for commercial production of the advantage of being easily implementable in a laboratory, and hence the model has been studied by more and more people.In its simplest form, the system approximates conditions for plankton growth in lakes, where the limiting nutrients such as silica and phosphate are supplied from streams draining the watershed.Chemostat with period inputs are studied in [5,6,14], those with periodic washout rate in [3,9] and those with periodic input and washout in [11].We all know that nutrients are input into lakes and lakes are washed out when rain is falling.In fact, raining is not continuous.It occurs seasonally or in regular pulses.Thus, it is natural to describe this case in impulsive differential equations.
Impulsive differential equations are suitable for the mathematical simulation of evolutionary process in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (i.e., jumps) in their values.Recently, equations of this kind are found in almost every domain of applied science.Numerous examples are given in Bainov's and his collaborator's book [1,8].Some impulsive differential equations have been recently introduced in population dynamics in relation to impulsive birth [12,15], impulsive vaccination [4,13], chemotherapeutic treatment of disease [7,10], and population ecology [2].

A two-nutrient and one-microorganism chemostat model
In this paper, we consider the dynamics of two-nutrient and one-microorganism chemostat with pulsed input: where S(t) and R(t) denote the concentration of nutrient, and x(t) denotes the concentration of the microorganism at time t.S 0 and R 0 represent that input concentration of the nutrient.Q (0 < Q < 1) is referred to as the dilution rate.μ 1 and μ 2 denote the predation constants of predator, δ 1 and δ 2 show yield term, and T is the period of pulse.The aim of this work is to study the dynamical behaviors of two nutrients and one microorganism with pulsed input, and investigate how the impulsive perturbation affects the dynamical behaviors of unforced continuous system.
The variables in the above system may be rescaled by measuring u=S/S 0 , v = δ 2 /(δ 1 S 0 )R, w = x/δ 1 S 0 , and t = Qt, then we have the following system: where This paper is arranged as follows.In Section 2, we introduce some useful notations and definitions.In Section 3, by using Floquet theorem for the impulsive equation, smallamplitude perturbation skills and techniques of comparison, we get the local stability and global asymptotic stability of the microorganism eradication periodic solution.In Section 4, we show that the system is permanent if the impulsive period is less than some critical value.In Section 5, we give a brief discussion.

Preliminaries
In this section, we will give some definitions, notations, and some lemmas which will be useful for our main results. Let the map defined by the right-hand side of the first three equations of system (1.2).Let V : X. Song and Z. Zhao 3 Definition 2.1.Let V ∈ V 0 , then for (t,x) ∈ (nT,(n + 1)T)] × R 3 + , the upper-right derivative of V (t,x) with respect to the impulsive differential system (1.2) is defined as , for all t ≥ 0. If all species of the system are permanent, then the system is called permanent.
2) subject to ω(0 + ) ≥ 0 for all t ≥ 0, and further suppose where g : t,y)→(nT + ,y) g(t, y) exists, ψ n : R + → R + is nondecreasing.Let r(t) be the maximal solution of the scalar impulsive differential equation Consider the following system: Lemma 2.5.System (2.4) has a positive periodic solution (u * (t),v * (t)) and for every solution 4 A two-nutrient and one-microorganism chemostat model

Extinction
In this section, we study the stability of the microorganism-free periodic solution as a solution of the full system (1.2).Firstly, we present the Floquet theory for the linear Tperiodic impulsive equation: Then we introduce the following conditions: ) there exists a q ∈ N such that Let Φ(t) be a fundamental matrix of (3.1), then there exists a unique nonsingular matrix M ∈ C n×n such that By equality (3.3) there corresponds to the fundamental matrix Φ(t) the constant matrix M which we call the monodromy matrix of (3.1) (corresponding to the fundamental matrix of Φ(t)).
All monodromy matrices of (3.1) are similar and have the same eigenvalues.The eigenvalues μ 1 ,...,μ n of the monodromy matrices are called the Floquet multipliers of (3.1).

Permanence
First, we show that all solutions of (1.2) are uniformly ultimately bounded.Theorem 4.1.There exists a constant M > 0 such that u(t) ≤ M, v(t) ≤ M, w(t) ≤ M for each positive solution ω(t) = (u(t),v(t),w(t)) of (1.2) with t large enough.
Proof.Define a function V (t,ω(t)) = u + v + w, then V (t,ω(t)) ∈ V 0 and the upper-right derivative of V (t,ω(t)) along a solution of (1.2) is described as X. Song and Z. Zhao 7 we obtain By the definition of V (t,ω(t)), we obtain that each positive solution of (1.2) is uniformly ultimately bounded.
Next we give the conditions of permanence.
From system (1.2), let According to Lemma 2.4, we have u(t) ≥ m 1 and v(t) ≥ m 2 for t large enough.
In the following, we want to find m 3 such that w(t) ≥ m 3 for t large enough.We will do it in the following two steps for convenience.

Discussion
In this paper, we have investigated the model for a chemostat with two nutrients and one microorganism and periodically pulsed substrate.We have proved that microorganism eradication periodic solution (u * (t),v * (t),0) is globally asymptotically stable if T > AQ + Bp, which is showed in Figure 5.1.We can see that the variables u(t), v(t) oscillate in a stable periodical cycle, in contrast with w(t) rapidly decrease to zero.At the same time we also have proved the system (1.2) is permanent if T < AQ + Bp, which is simulated in Figure 5.2.The variables u(t), v(t), w(t) oscillate in a stable periodical cycle, respectively.So we can find that T = AQ + Bp is a threshold.In fact, when the period of pulses is less than the threshold, the nutrients and microorganism coexist.If the period is more than the threshold, the microorganism will become extinct.
If we replace the pulse input in system (1.2) with continuous input, the system (1.2) becomes there also exists a microorganism eradication equilibrium for system (5.1), that is, (Q/T, p/T,0), which is globally asymptotically stable if T > AQ + Bp, see the appendix.The result is the same as our system (1.2), the result is simulated in Figure 5.3.We can obtain impulsive input effect that is the same as the continuous input.