On the Study of Chemostat Model with Pulsed Input in a Polluted Environment

Chemostat model with pulsed input in a polluted environment is considered. By using the Floquet theorem, we find that the microorganism eradication periodic solution 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.Chemostat with period inputs are studied in [1][2][3], those with periodic washout rate in [4,5] and those with periodic input and washout in [6].However, existing theories on chemostat model largely ignore the effects of environmental pollution.
Environmental pollution by various industries and pesticide used in agriculture is one of the most important of present day social and ecological problems.Organisms are often exposed to a polluted environment and take up toxicant.Uncontrolled contribution of pollutant to the environment has led many species to extinction.In order to use and regulate toxic substance wisely, we must assess the risk of the population exposed to toxicant.Therefore, it is important to study the effects of toxicant on populations and to find a theoretical threshold value, which determines permanence or extinction of a population community.
2 Discrete Dynamics in Nature and Society In this paper, we consider the dynamics of the polluted chemostat with pulsed input: where ΔS = S(nT + ) − S(nT), Δx = x(nT + ) − x(nT), Δc = c(nT + ) − c(nT), n ∈ Z + , Z + = {1, 2,...}.S(t) denotes the concentration of the nutrient, and x(t) denotes the concentration of the microorganism at time t.c(t) is the concentration of the toxicant in the organism at time t.r > 0 is the decreasing rate of the intrinsic growth rate associated with the uptake of the toxicant.f represents the exogenous rate of toxicant input into the organism.S 0 represents the input concentration of the nutrient.Q (0 < Q < 1) is referred to as the dilution rate.μ denote the predation constants of the predator.δ shows yield term.
T is the period of the pulse.The aim of this work is to study the dynamical behaviors of the polluted chemostat 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 S = (Q/μ)x 1 , x = (δQ/μ)x 2 , c = (Q/r)x 3 , 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 x > 0}, and let N be the set of all nonnegative integers.Denote by f = ( f 1 , f 2 , f 3 ) the map defined by the right-hand side of the first three equations of system (1.2).Let V : R Z. Zhao and X. Song 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 The solution of system (1.2) is a piecewise continuous function x : R + → R 3 + , x(t) is continuous on (nT,(n + 1)T], n ∈ Z + , and x(nT + ) = lim t→nT + x(t) exists, the smoothness properties of f guarantees the global existence and uniqueness of the solution (1.2), for details see [7,8].
Definition 2.2.The microorganism x 2 of (1.2) is said to be permanent if there exist constants 0 < m < M and T 0 > 0 such that m < x 2 < M for t > T 0 with initial condition x 2 (0) > 0, that is, the system (1.2) is permanent.
We will use a basic comparison result from [7,Theorem 3.1].For convenience, we state it in our notations.
where g : R + × R + → R is continuous in (nT,(n + 1)T] × R + and for y ∈ R + , n ∈ Z + , lim (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 ) Similar result can be obtained when all conditions of the inequalities in the lemma are reversed.Note that if we have some smoothness conditions of g(t,u) to guarantee the existence and uniqueness of the solutions for (2.3), then r(t) is exactly the unique solution of (2.3).
For convenience, we give the basic properties of the following system: (2.4) Lemma 2.6.System (2.4) has a positive periodic solution x * 1 (t) and for every solution x 1 (t) of (2.4) with initial value ).The proof is complete.

Extinction
In the section, we study the stability of the microorganism eradication periodic solution as a solution of the full system (1.2).Firstly, we present the Floquet theory for the linear T-periodic impulsive equation: Then we introduce the following conditions: where PC(R,C n×n ) is a set of all piecewise continuous matrix functions which is left continuous at t = τ k , and ) 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) and 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).
Z. Zhao and X. Song 5 Theorem 3.2.Let ω(t) = (x 1 (t),x 2 (t),x 3 (t)) be any solution of system (1.2), then (x * 1 (t),0,u) is globally asymptotically stable, provided that T > p/(1 + u).Proof.Firstly, we prove locally asymptotically stable.The local stability of the periodic solution (x * 1 (t),0,u) may be determined by considering the behavior of small-amplitude perturbations of the solution.Define where y 1 , y 2 and y 3 are small perturbations.Equation (1.2) can be expanded in a Taylor series: after neglecting higher-order terms, they may be written as where Φ(t) must satisfy with Φ(0) = I, where I is the identity matrix.Hence, the fundamental solution matrix is there is no need to calculate the exact form ( * ) as it is not required in the analysis that follows.The resetting impulsive conditions of (1.2) from the fourth to the sixth become Thus, the monodromy matrix of (3.5) is Let λ 1 , λ 2 , λ 3 be eigenvalues of M, then (3.10) Hence, according to Lemma 3.1, if absolute values of all eigenvalues of M are less than one, then T-periodic solution locally asymptotically stable.Thus, if and only if T > p/(1 + u), the solution (x * 1 (t),0,u) is locally asymptotically stable.The proof is complete.
Then, we have This completes the proof.
Z. Zhao and X. Song 7

Permanence
First, we show that all solutions of (1.2) are uniformly ultimately bounded.
Proof.Define a function V (t,ω(t)) = x 1 + x 2 + x 3 , 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 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), we can see that Considering the comparison let In the following, we want to find m 3 > 0 such that x 2 (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 polluted chemostat with impulsive input.We have proved that microorganism eradication periodic solution (x * 1 (t),0,u) is globally asymptotically stable if T > p/(1 + u), which is showed in Figure 5.1.We can see that the variables x 1 (t), x 3 (t) oscillate in a stable periodical cycle, in contrast x 2 (t) rapidly decrease to zero.At the same time we also have proved that the system (1.2) is permanent if T < p/(1 + u), which is simulated in Figure 5.2.The variables x 1 (t), x 2 (t), x 3 (t) oscillate in a stable periodical cycle, respectively.So we can find that T = p/(1 + u) is a threshold.In fact, when the period of pulses is less than the threshold, the nutrient 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, (p/T,0,u) which is globally asymptotically stable if T > p/(1 + u) (see the appendix).The result is the same as our system (1.2), the result is simulated in Figure 5.3.We can obtain that impulsive input effect is the same as the continuous input.