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Analysis of the Droop Model with Wall Growth in a Chemostat

In this paper, we construct a simple chemostat-based variable yield model of competition between two bacterial strains, one of which is capable of wall growth [14]. In this model we prove the boundedness of solutions, analyze the local stability of equilibria and establish the global stability for the locally stable extinction equilibrium. Furthermore, when the extinction equilibrium becomes unstable, we prove the existence and uniqueness of the positive equilibrium and the uniform persistence of the system.


Introduction and mathematical models
In the mathematical modeling of predator-prey system, we usually assume that the predator's consumption rate C on prey is directly proportional to the growth rate G of predator. We call the ratio of these two rates, G/C, the yield constant γ. In phytoplankton ecology, it has been known that the yield varies depending on the growth rate. Droop [2] is probably the first one to present a variable yield model, or so called "internal" storage model. He proposed the idea that organism consumes nutrient and converts it into internal storage Q (cell quota). When the internal storage Q is below the minimum cell quota Q min , the organism stops growing. When the cell quota is above the minimum cell quota, its growth rate µ(Q) increases with the cell quota. Furthermore, the nutrient uptake rate f (S, Q) increases with nutrient concentration S and decreases with cell quota Q. In the past fifty years, the variable yield models are well supported by experimental data [7].
In [17], the authors consider simple chemostat equations for two competing microorganisms. The equations take the form where S (0) denotes the input concentration of nutrient and D denotes the dilution rate ( flow rate volume ), S(t) is the concentration of nutrient at time t, x i (t) is the concentration of i-th micro-organism at time t; γ i and f i (S) = m i S a i +S are the yield constant and the growth rate for i-th species respectively.
In [14], the authors studied the wall-effect on micro-organisms in a simple chemostat. Let u be the density of planktonic cells in the channel and w be the density of adherent cells on the wall. The equations for the simple chemostat model with wall effect take the form where α is the rate of adhesion, β is the sloughing rate of adherent bacteria, γ is the yield constant, f u (S) := muS au+S is the specific growth rate of planktonic bacteria, f w (S) := mwS aw+S is the specific growth rate of adherent bacteria, δ = A V , which is area-volume ratio, where A and V are the surface area and volume of the growth chamber, respectively [10]. We note that in [14] by scaling the authors assume δ = 1.
Next we construct the following variable-yield model in a simple chemostat with wall growth where Q is the cell quota and f (S, Q) is the uptake rate for the bacteria both in the channel and on the wall. µ 1 (Q) and µ 2 (Q) are the growth rates of u and w respectively. We assume both of planktonic and adherent bacteria share the same cell quota Q and their growth rates share the same Q min , and  Some examples of f (S, Q) and µ i (Q) are given in [7,13,17]: The importance of wall growth was first cited from the work of microbiologists Rolf Freter and his colleagues [3][4][5][6]. Their mathematical models of the phenomenon of colonization resistance in the mammalian gut show that the bacteria wall attachment could play a crucial role in the observed stability of the natural microflora of the gut to invasion by non-indigenous micro-organisms [3]. Freter's model is a crude model of biofilm. A biofilm is simply a layer of material coating a surface. It is a comfortable refuge. Examples of a biofilm include the scum that grows on a rock in a stream, dental plague on teeth, the surface slime that forms on the inside surface of water pipes and a similar coating of surface of the large intestine of mammals. These bacterial layers can have serious negative consequences in many man-made environments. They contaminate medical devices such as contact lenses, implants, catheters and stints, they can contaminate food and medical production facilities, air-conditioning and water circulation systems. Biofilms are notoriously difficult to eradicate once established. For more biological references of biofilms, we refer to [1]. The organization of this paper is as follows. In Section 2, we prove the boundedness of the solutions for the system (1.1). In Section 3, we find the equilibria of the system (1.1) and do their stability analysis, prove the uniform persistence for the system (1.1). Section 4 is the discussion section.

Boundedness of the solutions of (1.1)
In this section we state and prove the boundedness of the solutions for the variable-yield model of simple chemostat with wall growth (1.1). We note that in (1.1) we assume there is no washout for the adherent cells on the wall. Then it is more technical to prove the boundedness of the solutions of (1.1). Before we do it, we need the following lemmas.
Lemma 2.1 is a lemma about the internal storage. The following Remark 2.2 follows from Lemma 2.1.
Hence the internal storage Q is bounded for all t > 0.
Proof. Since u u+δw ≤ u δw , we can consider the upper bound of u δw : Then we obtain Since α > 0, β > 0, the right-hand side of (2.1) have two roots p 1 and p 2 with different signs. Assume p 1 < 0 < p 2 , then (2.1) can be expressed as where Analysis of the Droop Model with Wall Growth in a Chemostat 797 Therefore, from (2.2), it follows that Similarly, we can get Therefore Because u and δw are nonnegative, and Let η 1 = p 2 + , η 2 = 1 1+q 2 + , then we finish the proof of the lemma.
Next, we introduce three notations to prove the boundedness of u(t) and δw(t) for t ≥ 0. Let U = Qu, W = Qw and g(S, Q) = f (S,Q) Q . Then the system (1.1) is converted into Therefore, by the above differential inequalities, This implies that U + δW ≤ S (0) η 2 + , since U and W are nonnegative, then U (t) and W (t) are bounded for t ≥ 0. Now, we have shown that U (t) and W (t) are bounded for t ≥ 0. By Remark 2.2, 0 < Q min ≤ Q(t) ≤ Q 0 for all t ≥ 0, then we obtain the bounedess of solutions u(t) and w(t) of the system (1.1).
3. Stability analysis of the system (1.1) There are at most two equilibria of the system (1.1), namely, the extinction equilibrium E 0 and possibly the coexistence equilibrium E 1 , where E 0 = (S (0) , Q 0 , 0, 0) which always exists and Q 0 satisfies f ( In our stability analysis, we discuss the following: (1) The stability of E 0 .
(2) The existence and uniqueness of E 1 . (We will give the formula of E 1 in (3.8).) (3) Uniform persistence of the system (1.1).
Since the uniform persistence of the system (1.1) is related to the boundary dynamics of (1.1). Therefore, we first need to study the stability of E 0 .

The local stability analysis of extinction equilibrium
First, we do the local stability analysis for E 0 . Let J(E 0 ) be the Jacobian matrix evaluated at E 0 : Let the eigenvalues of J(E 0 ) be λ 1 , λ 2 , λ 3 , λ 4 , where Our basic hypothesis is (H) E 0 is a hyperbolic equilibrium, i.e., λ i = 0 for all i = 1, 2, 3, 4.

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Claim: E 0 is either a locally asymptotically stable equilibrium or a saddle point.

The global stability analysis of extinction equilibrium
In order to establish the global stability, we introduce a set I = {(S, Q, u, w) ∈ R 4 ; S ∈ [0, S (0) ], Q ∈ [Q min , Q 0 ], u > 0, w > 0}. It is easy to verify that I is a positively invariant set for the system (1.1) by Remark 2.2. We will prove the global stability in I by Kamke theorem [8,15].
We recall that a systemẋ = f (x), f : D ⊆ R n → R n is cooperative [15], where D is an open set in R n , if ∂f i ∂x j ≥ 0 for x ∈ D, i, j = 1, 2, . . . , n and i = j.
Theorem 3.1. Under hypothesis (H), if the extinction equilibrium of the system (1.1) E 0 is locally asymptotically stable, then E 0 is globally asymptotically stable in I.
Proof. First, we consider the subsystem in the invariant set I as Obviously, we obtain the following in I that We note that the system (3.4) is a cooperative system.
Consider the linear and cooperative system The initial value of u and δw of the system (3.5) are same as the system (3.3). The eigenvalues of (3.5) are λ 3 and λ 4 .
This completes the proof of Theorem 3.1.

Uniform persistence of the system (1.1)
In this section, we concern about the uniform persistence of the system (1.1) in the interior and apply the theory of uniform persistence [16,17] by investigating boundary dynamical behavior. We shall obtain the uniform persistence by analyzing the stability property of extinction equilibrium. The following Lemma 3.2 gives a necessary and sufficient condition for the uniform persistence of the system (1.1).
andX be the interior of X.
Hence there is no homoclinic orbit in the system (3.9). Therefore, by Lemma 3.2 and Theorem 2.4, the system (1.1) is ρ-uniformly persistent inX, i.e., there exists α > 0 such that u(t)w(t) ≥ α > 0 for t ≥ 0. From Theorem 2.4, it follows that u(t) ≥ α/w max , w(t) ≥ α/u max for all t ≥ T , where u max and w max are upper bounds of u(t) and w(t).