Global Analysis of Beddington-DeAngelis Type Chemostat Model with Nutrient Recycling and Impulsive Input

In this paper, a Beddington-DeAngelis type chemostat model with nutrient recycling and impulsive input is considered. Except using Floquet theorem, introducing a new method combining with comparison theorem of impulse differential equation and by using the Liapunov function method, the sufficient and necessary conditions on the permanence and extinction of the microorganism are obtained. Two examples are given in the last section to verify our mathematical results. The numerical analysis shows that if only the system is permanent, then it also is globally attractive.


Introduction
The chemostat is an important and basic laboratory apparatus for culturing microorganisms.It can be used to investigate microbial growth and has the advantage that parameters are easily measurable.The chemostat plays an important role in bioprocessing, hence the model has been studied by more and more people.Chemostats with periodic inputs were studied [1,2], those with periodic washout rate [3,4], and those with periodic input and washout [5].In recent years, those with nutrient recycling [6][7][8][9][10] have been investigated and some investing results were obtained.Now many scholars pointed out that it was necessary to consider models with periodic perturbations, since those phenomena might be exposed in many real words.However, there are some other perturbations such as floods, fires and drainaye of sewage which are not suitable to be considered continually.Those perturbations bring sudden changes to the system.Systems with sudden changes are involving in impulsive differential equations which have been studied intensively and systematically [11][12][13].Impulsive differential equations are found in almost every domain of applied sciences.
Recently, many papers studied chemostat model with impulsive effect the Lotka-Volterra type or Monod type functional response.But there are few papers which study a chemostat model with Beddington-DeAngelis functional response, especially a Beddinton-DeAngelis type chemostat with nutrient recycling.The Beddington-DeAngelis functional response is introduced by Beddington and DeAngelis [14,15].It is similar to the wellknown Holling II functional response but has an extra term   B t in the denominator that models mutual interference in species.The model, we consider in this paper, takes the form:

S t x t a S t DS t brx t t nT n Z k A S t Bx t aS t x t x t Dx t rx t t nT n Z A S t Bx t S t S t p t nT n Z x t x t t nT n Z
where S(t),   1 x t represent the concentration of limiting substrate and the microorganism respectively, D is the dilution rate, a is the uptake constant of the microorganism, k is the yield of the microorganism   The organization of this paper is as the following.In Section 2, we introduce some useful notations and lem-mas.In Section 3, we will state and prove the main results on the global asymptotic stability and permanence.In Section 4, we give a brief discussion and the numerical analysis.

Preliminaries
In this section, we will give some notations and lemmas which will be used for our main results.Firstly, for convenience, we set     is left continuous at t = nT and x(t) is continuous at Lemma 1. Suppose is any solution of system (2) with initial solution The proof of Lemma 1 is simple, we omit it here.
In what follows, we give some basic properties about the following system. Clearly, is a positive periodic solution of system (3).Any solution of system (3) is , Hence, we have the following result.Lemma 2. System (3) has a positive periodic solution  , as for any solution u(t) of system (3).Moreover, The proof of Lemma 2 can be found in [16].Lemma 3.There exists a constant M > 0 such that S(t) < M, x(t) < M for each solution of (S(t); x(t)) system (2), for t large enough.
Proof Let (S(t); x(t)) be any solution of system (2) with initial value  for all t¸ 0, where u(t) is the solution of system (3).From Lemma 2, we have Thus, V(t) is ultimately bounded.From the definition of V(t), there exists a constant   such that S(t) < M, x(t) < M for any solution (S(t), x(t)) of system (2), for t large enough.This completes the proof.The solution of system (2) corresponding to x(t) = 0 is called microorganism-free periodic solution.For system (2), if we choose   0 x t  , then system (2) becomes to the following system , ,

S t DS t t nT n Z S t S t p t nT n Z
System (4) has a unique global uniformly attractive positive solution Hence, system (2) has a positive periodic solution V t S t x t   next section, we will study the global asymptotical stability of the microorganism-free periodic solution as a solution of system (2).
   ,0 u t  Then similar to the proof of Lemma 3, we obtain for all where u(t) is the solution of system (3) and Hence, there exists a function By the definition of   V t , we have Then periodic solution of system ( 2) is globally attractive.
  x t ) be any positive solution of system (2).Define a function as follows It follows from the second equation of system (2) that Hence, there exist constants 0   and , such that   x t 0   for all , then from (6) we have For any , we choose an integer such that , then integrating (8) from to t, from (7) we have 0 where In fact, if there exists a such that x t , then there exists a integrating the above inequality from t 2 to t 1 , from (7) we obtain (10).
Obviously, let , then from ( 10) we obtain a contradiction.Hence, This completes the proof.
Then system (2) is permanent.Proof Let (S(t); x(t)) be any solution of system (2) with initial value Lemma 3,   the first equation of system (2) becomes is the solution of the following impulsive system Therefore, we finally obtain This shows that S(t) in system (2) is permanent.
In the following, we want to find a constant , such that Consider the following auxiliary impulsive system from Lemma 2, system (12) has a globally uniformly attractive positive periodic solution Further, for above 2 0   a 0 y M   and M > 0, where M is given in Lemma 3, there is such that for any and  for all t t  , then our go  ssing on the case of , then above t t   , we also have we inequality (16).Particularly, obtain

S t D t n Z A S t S t p t nT n Z
Hence, from the comparison theorem of impulsive differential equations, we have for all , whe the s re y(t) is olution of system (12) with

Further
, we also from (13) have Thus, from system (2), we have From the above discussion, we have   2 lim t x t m   , S(t); x(t)) of and is independent of any solution ( syste (2).This completes the proof.
As a consequence of Theorem 1 and Theorem 2, we have the following corollary.Corollary 1 For system (2), the following conclusions hold.
a) The microorganism-extinction solution

4.
paper, we investigate Beddington-DeAngelis type chemostat with nutrient recycling and impulsive input.We prove that the microorganism-free periodic solution of the system (2) is globally attractive.The necessary and sufficient condition for permanence of system (2) are obtained in this paper.
According to Theorem 1, the microorganism-free periodic solution

Discussion and Numerical Analysis
Then Theorem 1-2 can be state as: If and exp exp that conditions for the system coexist or non-coexist are due to the influences of the impulsive perturbations.
In order to illustrate our mathematical results and investigate the effect of impulsive input nutrient we present the following results of a numerical simulation.

Figure 2 .
Figure 2. (a) Time-series of the nutrient S for permanence and periodic os global attractivity; (b) Time-series of the microorganism population x for global ttractivity.