Stabilizing a Periodic Solution in the Chemostat: A Case Study in Tracking

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Stabilizing a Periodic Solution in the Chemostat:

A Case Study in Tracking

Fr´ed´eric Mazenc, Patrick De Leenheer, and Michael Malisoff

Abstract— We study the chemostat model for one species competing for one nutrient. For appropriate choices of the nutrient concentration and time-varying dilution rate, we use a Lyapunov-type analysis to prove the stability of the correspond- ing reference signal for the species concentration. We show that the stability is maintained when the model is augmented by additional species that are being driven to extinction.

Index Terms— Chemostat, species concentration, stability analysis, robustness

I. INTRODUCTION

The chemostat model provides the foundation for much of current research in bio-engineering, ecology, and population biology [2], [4], [5], [6], [14]. In the engineering literature, the chemostat is known as the continuously stirred tank reactor. It has been used for modeling the dynamics of interacting organisms in waste water treatment plants, lakes and oceans. In its basic setting, it describes the dynamics of species competing for one or more limiting nutrients. If there aren species with concentrations xifori = 1, . . . , n and just one limiting nutrient with concentrationS and dilution rate D > 0, then the model takes the form

⎧⎪

⎨

⎪⎩

˙S = D(S0− S) −

n i=1

µi(S)xi/γi

˙xi = xi(µi(S) − D), i = 1, . . . , n

(1)

where µi denotes the per capita growth rate of species i. The functions µi are assumed to depend only on the nutrient concentration, and are zero at zero, continuously differentiable and strictly increasing, although non-monotone functions have been the subject of research as well [6].

The conversion of nutrient into new biomass for each species i happens with a certain yield γi ∈ (0, 1) and the natural control variables in this model are the input nutrient concentrationS0and the dilution rateD. The latter variable is deﬁned as the ratio of the volumetric ﬂow rate F (with units of volume over time) and the reactor volume

Corresponding Author: F. Mazenc. This work was done while the ﬁrst two authors visited Louisiana State University (LSU). They thank LSU for the kind hospitality they enjoyed during this period. The ﬁrst author thanks Claude Lobry and Alain Rapaport for illuminating discussions.

Mazenc is with Projet MERE INRIA-INRA, UMR Analyse des Syst`emes et Biom´etrie INRA, 2, pl. Viala, 34060 Montpellier, France (email: Fred- [email protected]).

De Leenheer is with the Department of Mathematics, University of Florida, 411 Little Hall, PO Box 118105, Gainesville, FL 32611–8105 (email: [email protected]ﬂ.edu). Supported by NSF/DMS Grant 0500861.

Malisoff is with the Department of Mathematics, LSU, Baton Rouge, LA 70803-4918 (email: [email protected]). Supported by NSF/DMS Grant 0424011.

V which is kept constant. Therefore it is proportional to the speed of the pump that supplies the reactor with fresh medium containing the nutrient. The equations (1) are then straightforwardly obtained from writing the mass-balance equations for the total amounts of the nutrient and each of the species, assuming the reactor content is well-mixed. The full model (1) is illustrated in Figure 1.

S(t), xi(t) S0

D D

Fig. 1. Chemostat

In the present work, we focus on the case where there is just one species with concentrationx, in which case the equations (1) take the form

˙S = D(S0− S) − µ(S)x/γ

˙x = x(µ(S) − D) (2)

(but see Section V for robustness results for models involving several species). We assume thatS0> 0 is constant and that the per capita growth rateµ is a Monod function (which is also known as a Michaelis-Menten function) taking the form

µ(S) = mS

a + S, (3)

for certain positive constantsm and a that we specify later.

One can check readily that the system (2) leaves the domain of interestX := (0, ∞) × (0, ∞) positively invariant.

Since S0 will remain ﬁxed, it is useful to rescale the variables to reduce the number of parameters. Using the change of variables

S =¯ S

S0, ¯x = x

S0γ, ¯µ( ¯S) = µ(S0S)¯

and dropping bars, we eliminate the parameters S0 and γ and so obtain the new dynamics

˙S = D(1 − S) − µ(S)x

˙x = x(µ(S) − D) (4)

on the state spaceX . Our main result will be for the model (4). We choose and stabilize a periodic reference signal for

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the species concentrationt → xr(t) in (4). We show that this can be achieved for an appropriately chosen time-periodic dilution rateD(t).

The rest of this paper is structured as follows. In the next section, we brieﬂy review the literature focusing on what makes our approach different. In Section III, we obtain the error equations for the reference signal we wish to track, and we precisely state the stability problem we are solving.

We prove our stability result in Section IV. In Section V, we show that the stability is maintained when there are additional species that are being driven to extinction. We validate our results in Section VI using a numerical example.

We conclude in Section VII with suggestions for future research.

II. REVIEW OF THELITERATURE AND

COMPARISON WITHOURRESULTS

The behavior of system(1) is well understood for cases whereS0andD are positive constants, and for cases where n = 2 and either one of the control variables is held ﬁxed while the other is periodically time-varying. See [8], [13]

for periodic variation ofS0 and [1] for periodic variation of D. See also the standard reference [14] on chemostats for a review.

When both S0 andD are constants, the so-called “competitive exclusion principle” holds, stating that at most one species survives. Mathematically this translates into the statement that system (1) has a steady state with at most one nonzero species concentration, which attracts almost all solutions; see [14]. It is this result that has triggered much research to explain the discrepancy between the (theoretical) competitive exclusion principle and the observation that in real ecological systems, many species coexist.

The results on the periodically-varying chemostat men- tioned above should be seen as attempts to explain this paradox. They involve chemostats with n = 2 species, and their purpose is to show that an appropriate periodic forcing for eitherS0(t) or D(t) can make the species coexist, usually in the form of a (positive) attractive periodic solution. Few results on coexistence of n > 2 species are available. An exception is [12], where a periodic functionS0(t) is designed (while D is kept ﬁxed) such that the resulting system has a (positive) periodic solution with an arbitrary number of coexisting, periodically varying species. Unfortunately, the stability properties of this solution are not known.

More recent work has explored the use of state-dependent but time invariant feedback control of the dilution rate D to prove coexistence; see [2], [4] for monotone growth rate functions in the n = 2 species case, and [3] for the n = 3 species case. The paper [6] considers feedback control when the growth rate functions are non-monotone. See also [10]

where coexistence is studied via a bifurcation analysis with the dilution rate as the control.

All the results discussed so far are more general than what we will show for our model(4) in the sense that they are all valid for situations with n > 1 species. This is because the main purpose of these papers is to investigate environmental

conditions under which the competitive exclusion principle fails, and several species can coexist.

Here, we will not be considering any coexistence prob- lems, although we intend to do so in the future. Our main concern is to provide a proof of stability of a periodic solution which is based on a Lyapunov-type analysis. The advantage of having a Lyapunov function is that it can be used to investigate the robustness properties of the periodic solution with respect to perturbations. As an illustration we will show that the stability of the periodic solution is robust with respect to additional species that are being driven to extinction. These features set our work apart from the known results on periodically forced chemostat models since these do not rely on the construction of a Lyapunov function. In fact, it seems that Lyapunov functions have only been used rarely to prove stability in the chemostat. An exception is Theorem4.1 in [14], and more recently the work of [7]. See also [11], [15] where weak Lyapunov functions are used in conjunction with suitable variants of the LaSalle Invariance Principle.

Finally we point out that closely related to our results is [5] where a single-species chemostat with a continuous and bounded (but otherwise arbitrary) function S0(t) and constant dilution rate is investigated; there it is shown that two positive solutions converge to each other. However, the proof is not based on a Lyapunov function.

III. REFERENCETRAJECTORY TO BESTABILIZED

In this section, we choose the dilution rateD = D(t) that will give rise to the reference trajectory we wish to stabilize.

To this end, we use the new variablez = S + x to transform (4) into the system

˙z = D(t)[1 − z]

˙x = [µ(z − x) − D(t)]x (5) evolving on the state space

X:= {(z, x) : x > 0, z − x > 0} .

We next select a periodic reference trajectory(xr(t), zr(t)) and a function D(t) for which (5) is satisﬁed. We choose zr(t) ≡ 1 and so need

˙xr(t) = [µ(1 − xr(t)) − D(t)]xr(t), i.e., sincexr is never zero,

D(t) = −˙xr(t)

xr(t)+ µ(1 − xr(t)). (6) We chooseµ as in (3) and require m > 4a + 1. This gives

D(t) = −˙x_r(t)

xr(t)+ m(1 − xr(t))

a + 1 − xr(t) , (7) assumingxr< 1 everywhere. Choosing the reference trajectory

(xr(t), zr(t)) =

1 2+1

4cos(t), 1

(8)

(3)

yields the choice

D(t) = sin(t)

2 + cos(t)+ m(2 − cos(t))

4a + 2 − cos(t) (9) for the dilution rate. In particular,

D(t) ≥ −1 + m

4a + 1 =: D > 0, (10) by our choice of m. See Figures 2 and 3 for the graphs of xr(t) and D(t) for m = 10 and a = ¹₂.

5 10 15 20 25 30

0.4 0.5 0.6 0.7

Fig. 2. Graph of Reference Trajectory Componentxr(t) From (8) Plotted Against Timet

5 10 15 20 25 30

3.5 4.5 5 5.5 6

Fig. 3. Graph of Dilution RateD(t) for the Chemostat From (9) Plotted Against Timet

We wish to solve this stability problem:

(SP) Given any trajectory(x, z) : [0, ∞) → X for (5) and the dilution rateD(t) from (9), and choosing µ as in (3) with m > 4a + 1, show that the corresponding deviation

(˜x(t), ˜z(t)) := (x(t) − xr(t), z(t) − zr(t)) of(x, z) from our reference trajectory (8) asymptotically approaches(0, 0) as t → +∞.

Remark 1: Our solution of (SP) only uses the condition m > 4a +1 to ensure that D is bounded below by a positive constant. One can easily check that

D(t) ≥ −2

3 + m

4a + 1 (11)

for allt. To see why, ﬁrst note that D(t) is bounded below bymin{Θ(x) : −1 ≤ x ≤ 1} where

Θ(x) := −

√1 − x²

2 + x + m(2 − x) 4a + 2 − x

by taking x = cos(t). For −¹₂ ≤ x ≤ 1, we have Θ(x) ≥

−²₃ +_4a+1^m . On the other hand, Θ(x) ≤ 0 for −1 < x ≤

−¹₂. We conclude that (11) holds everywhere. It follows that our solution of (SP) remains valid under the assumption that m > ²₃(4a + 1) since this in conjunction with (11) gives a uniform positive lower bound onD(t).

We will solve (SP) by performing a Lyapunov-type analysis on the corresponding error equations. Notice for later use that problem (SP) is equivalent to the following one:

(SP) Given any trajectory (S, x) : [0, ∞) → X for (4) and the dilution rate D(t) from (9), and choosing µ as in (3) with m > 4a + 1, show that the corresponding deviation

( ˜S(t), ˜x(t)) := (S(t) − Sr(t), x(t) − xr(t)) of(S, x) from the reference trajectory

(Sr(t), xr(t)) :=

1 2−1

4cos(t),1 2 +1

4cos(t)

asymptotically approaches(0, 0) as t → +∞.

We will use the latter formulation of the problem in our analysis of a multi-species model in Section V.

IV. SOLUTION TOSTABILITYPROBLEM(SP) In what follows, all (in)equalities should be understood to hold wherever they make sense, and we omit the arguments t and so on in our functions whenever this would not lead to any confusion. Recall that for the system (5), we always havex > 0 and z > 0. Introducing the new variable

ξ = ln(x) (12)

transforms the model (5) into

⎧⎨

⎩

˙z = D(t)[1 − z]

˙ξ = µ(z − e^ξ) − D(t) (13) so the reference signal(xr(t), zr(t)) satisﬁes

⎧⎨

⎩

˙zr(t) = D(t)[1 − zr(t)]

˙ξ_r(t) = µ(z_r(t) − e^ξ^r^(t)) − D(t) (14) with ξr(t) = ln(xr(t)). Subtracting these shows that the transformed error signal

(˜z(t), ˜ξ(t)) := (z(t) − z_r(t), ξ(t) − ξ_r(t)) solves ⎧

⎨

⎩

˙˜z = −D(t)˜z

˙˜ξ = µ(z − e^ξ) − µ(z_r(t) − e^ξ^r^(t)) . (15)

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The equations (15) should be regarded merely as expressions for the time derivatives of the deviations ˜ξ(t) = ξ(t) − ξr(t) and ˜z(t) = z(t) − zr(t) computed for X-valued pairs (x(t), z(t)) that solve (5), rather than as a dynamical system.

Therefore, for our purposes, it will not be necessary to precisely specify the state space for (15), which is actually a proper subset ofR². We choose any growth rate of the form

µ(s) = ms

a + s, m > 4a + 1. (16) Then

˙˜ξ = m(z − e^ξ)

a + z − e^ξ −m(zr(t) − e^ξ^r^(t)) a + zr(t) − e^ξ^r^(t)

= m

(z − e^ξ)(a + zr(t) − e^ξ^r^(t)) (a + z − e^ξ)(a + zr(t) − e^ξ^r^(t))

− (z_r(t) − e^ξ^r^(t))(a + z − e^ξ) (a + z − e^ξ)(a + zr(t) − e^ξ^r^(t))

= ma ˜z + (−e^ξ+ e^ξ^r^(t)) (a + z − e^ξ)(a + z_r(t) − e^ξ^r^(t))

= ma ˜z − e^ξ^r^(t)(e^ξ^˜− 1)

(a + z − e^ξ)(a + zr(t) − e^ξ^r^(t)) .

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Hence, we perform our Lyapunov-type analysis on the triangular system

⎧⎨

⎩

˙˜z = −D(t)˜z

˙˜ξ = ma ˜z − e^ξ^r^(t)(e^ξ^˜− 1)

(a + z − e^ξ)(a + zr(t) − e^ξ^r^(t)). (18) To this end, we ﬁrst ﬁnd the time derivative of

L1(˜ξ) = e^ξ^˜− 1 − ˜ξ. (19) along the trajectories of (18) to be

˙L1= ma(e^ξ^˜− 1) ˜z − e^ξ^r^(t)(e^ξ^˜− 1) (a + z − e^ξ)(a + zr(t) − e^ξ^r^(t))

= ma (e^ξ^˜− 1)˜z − e^ξ^r^(t)(e^ξ^˜− 1)² (a + z − e^ξ)(a + z_r(t) − e^ξ^r^(t)) .

(20)

Observe that

ξr(t) = ln(xr(t)) = ln

1 2+1

4cos(t)

. Substituting into (20) gives

˙L1 = ma(e^ξ^˜− 1)˜z −₁

2+¹₄cos(t)

(e^ξ^˜− 1)² (a + z − e^ξ)(a + z_r(t) − e^ξ^r^(t))

≤ ma¹⁶¹(e^ξ^˜− 1)²+ 4˜z²−₁

2+¹₄cos(t)

(e^ξ^˜− 1)² (a + z − e^ξ)(a + zr(t) − e^ξ^r^(t))

≤ ma 4˜z²−₁₆¹(e^ξ^˜− 1)²

(a + z − e^ξ)(a + z_r(t) − e^ξ^r^(t)) . For(x, z) ∈ X, the inequality

(a + z − e^ξ)(a + z_r(t) − e^ξ^r^(t)) ≥ a²

is satisﬁed sinceξ = ln(x). It follows that

˙L₁ ≤ − ma(e^ξ^˜− 1)²

16(a + z − e^ξ)(a + zr(t) − e^ξ^r^(t)) +4m

a ˜z².

We next exploit the stability of the˜z system. The derivative of

L2(˜z) = 4m

aD˜z², (21)

whereD is the lower bound on D(t) in (10), satisﬁes

˙L2= −8m

aDD(t)˜z²≤ −8m

a ˜z². (22) It follows that the time derivative of

L3(˜z, ˜ξ) := L₁(˜ξ) + L₂(˜z) (23) along the trajectories of (18) satisﬁes

˙L3 ≤ − ma(e^ξ^˜− 1)²

16(a + z − e^ξ)(a + zr(t) − e^ξ^r^(t))

−4m a ˜z²

≤ − ma(e^ξ^˜− 1)²

16(a + z − e^ξ)(a + 1) −4m a ˜z²

≤ − ma(e^ξ^˜− 1)²

16(a + 2 + ˜z²)(a + 1)−4m a ˜z²

(24)

where the last inequality follows because z − e^ξ = ˜z + 1 − e^ξ ≤ 2 + ˜z²

(by separately considering the cases where˜z ≥ 1 and ˜z < 1).

The right hand side of (24) is negative definite. Moreover, one can prove that L3 is positive definite and radially un- bounded by observing thatL₁(˜ξ) < 0 for ˜ξ < 0, L₁(˜ξ) > 0 for ˜ξ > 0, and L1(0) = 0. It is tempting to surmise from (24) and the structure ofL3 thatL3 is a Lyapunov function for (18) since then we could use standard Lyapunov function theory to conclude that(˜ξ(t), ˜z(t)) asymptotically converges to zero. However, such an argument would not be technically correct, since the state space of (18) is notR² (because the original system (5) that produced our error equations (18) is only defined onX). Instead, we argue as follows.

Observe that, for anyt ≥ 0, integrating the last inequality of (24) over[0, t] gives

L3(˜z(t), ˜ξ(t)) − L3(˜z(0), ˜ξ(0)) ≤

−

_t

0

ma(e^ξ(l)^˜ − 1)²

16(a + 1)(a + 2 + ˜z²(l))+4m a ˜z²(l)

dl . (25) It follows that, for allt ≥ 0,

L1(˜ξ(t)) ≤ L₃(˜z(0), ˜ξ(0)) . (26) Thereforeξ(t) = ˜ξ(t)+ξr(t) is a bounded function. By (18),

˜z(t) is a bounded function as well. We deduce that ˜ξ and

˜z are uniformly continuous, since their time derivatives (18) are bounded. Reapplying (25) therefore implies

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_+∞

0

ma(e^ξ(l)^˜ − 1)²

16(a + 1)(a + 2 + ˜z²(l)) +4m a ˜z²(l)

dl

is ﬁnite. It follows from Barbalat’s lemma [9, p.323] that

t→+∞lim

ma(e^ξ(t)^˜ − 1)²

16(a + 1)(a + 2 + ˜z²(t))+4m a ˜z²(t)

= 0

and therefore

t→+∞lim ˜ξ(t) = 0 , lim

t→+∞˜z(t) = 0.

This establishes our stability condition in (SP).

Remark 2: Notice that− ˙L3andL3are both bounded below by quadratics of the form˜c|(˜z, ˜ξ)|²along the trajectories of (18), since the trajectories are bounded. From this fact, one can deduce that the trajectories ˜ξ(t) and ˜z(t) converge exponentially to zero.

V. STABILITY IN THE PRESENCE OF SEVERAL SPECIES

As we noted in Section III, the argument from Section IV shows the stability of the reference trajectory

(Sr(t), xr(t)) =

1 2 −1

4cos(t),1 2+1

4cos(t)

(27) for the original single species dynamic (4). We next show that this stability property of (27) is maintained when the model (4) is augmented to include additional species that are being driven to extinction, in the following sense.

Consider the augmented system

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

˙S = D(t)(1 − S) − µ(S)x −ⁿ

i=1

νi(S)yi

˙x = x(µ(S) − D(t))

˙y_i = y_i(ν_i(S) − D(t)), i = 1, . . . , n

(28)

whereµ is as in (16) as before, and for i = 1, 2, . . . , n, νi

is continuous and increasing and satisﬁes νi(0) = 0. The variablesyi represent the levels ofn additional species. We choose D and D as in (9) and (10), and we assume that νi(1) < D for i = 1, 2, . . . , n. We show that the error between any componentwise positive solution of (28) and

(S_r, xr, 0, . . . , 0)

=

1 2−1

4cos(t),1 2 +1

4cos(t), 0, . . . , 0

converges exponentially to the zero vector ast → +∞.

To this end, notice that in the coordinates

˜z = S + x − S_r(t) − x_r(t)

˜ξ= ln(x) − ln(xr(t)),

the system (28) becomes

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

˙˜z = −D(t)˜z(t) −ⁿ

i=1

νi(S)yi

˙˜ξ = ma ˜z − e^ξ^r^(t)(e^ξ^˜− 1) (a + z − e^ξ)(a + zr(t) − e^ξ^r^(t))

˙yi = yi(νi(S) − D(t)), i = 1, . . . , n

, (29)

by (18) where ˜z and ˜ξ can take both positive and negative values. Using the calculations of the previous section, we deduce that the derivative ofL3, deﬁned in (23), along the trajectories of (29) satisﬁes

˙L3 ≤ − ma(e^ξ^˜− 1)²

16(a + 1)(a + 2 + ˜z²)−4m a ˜z²

−8m aD˜zⁿ

i=1

νi(S)yi

= − ma(e^ξ^˜− 1)²

16(a + 1)(a + 2 + ˜z²)−4m a ˜z²

−2

√√m a ˜z

4√

√ m aD

n i=1

νi(S)y_i

≤ − ma(e^ξ^˜− 1)²

16(a + 1)(a + 2 + ˜z²)−3m a ˜z²

+16m aD²

_n

i=1

νi(S)yi

₂

(30)

using the relationJ²+ K²≥ −2JK for real values J and K. On the other hand, since νi(1) < D for each i, the form of the dynamics for S and the nonnegativity of µ and the νi’s along our componentwise positive trajectories imply that there existε > 0 and T ≥ 0 such that

(i) S(t) ≤ 1 + ε for all t ≥ T ; and (ii) νi(1 + ε) < D for all i = 1, 2, . . . , n.

We deduce that, for alli = 1, 2, . . . , n and for all t ≥ T , 1

2 d

dty²_i ≤ (νi(S(t)) − D(t))y²_i

≤ (ν_i(1 + ε) − D)y_i².

(31)

It follows that for allt ≥ T , we have 1

2 d

dty²_i ≤ −δy_i², (32) where

δ := D− max

i=1,...,nνi(1 + ε) > 0.

Therefore, eachyi(t) converges exponentially to zero. Next notice that along each pair(˜ξ(t), ˜z(t)), the function

∆(˜ξ, ˜z) := ma(e^ξ^˜− 1)² 16(a + 1)(a + 2 + ˜z²)

(6)

is positive if and only if ˜ξ = 0. Choose the constant A := 16mn²

aδ .

Using (30) and (32), and omitting the argumentt from S(t) andyi(t) and so on, it follows that the time derivative of

L4(˜z, ˜ξ, y₁, ..., yn) = L₃(˜z, ˜ξ) + Aⁿ

i=1

y²_i (33)

along the trajectories of (29) satisﬁes

˙L₄ ≤ −∆(˜ξ, ˜z) −3m a ˜z² +16m

aD²

_n

i=1

νi(S)y_i

₂

+ Aⁿ

i=1

d dty²_i

≤ −∆(˜ξ, ˜z) −3m a ˜z² +16m

aD²

_n

i=1

νi(S)y_i

₂

− 2Aδⁿ

i=1

y²_i

≤ −∆(˜ξ, ˜z) −3m a ˜z² +16mn²

aD²

n i=1

ν_i²(S)y²_i − 2Aδ

n i=1

y_i²

≤ −∆(˜ξ, ˜z) −3m a ˜z²+

16mn²

a − 2Aδ

_n

i=1

y_i²

= − ma(e^ξ^˜− 1)²

16(a + 1)(a + 2 + ˜z²)−3m a ˜z²

−16mn² a

n i=1

y_i²

providedt > T where T is chosen to satisfy (i)-(ii) above.

(The third inequality follows because for any nonnegative values ak, we get ak ≤ (_n

i=1a²_i)^1/2 which we sum and square to get (_n

i=1ai)² ≤ n²_n

i=1a²_i.) One can easily conclude by invoking arguments similar to those used in the previous section.

VI. SIMULATION

To validate our convergence result, we simulated the dynamics (5) with the initial valuesx(0) = 2 and z(0) = 3, the parameters m = 10 and a = ¹₂, and the reference trajectory xr = ¹₂ +¹₄cos(t). This resulted in the plot of x(t) against time in Figure 4. Our simulation shows that x(t) closely tracks the reference trajectory xr(t) pictured in Figure 2 and so validates our ﬁndings.

VII. CONCLUSIONS

The chemostat is a useful framework for modeling species competing for nutrients. For the special case of one species competing for one nutrient and a suitable time varying dilution rate, we proved stability of an appropriate reference trajectory. Moreover, we found that the stability was maintained even if the model is augmented with other species

5 10 15 20 25 30

0.4 0.6 0.8 1.2 1.4

Fig. 4. Graph of State Trajectory Componentx(t) for the Chemostat Plotted Again Timet

that are being driven to extinction. We conjecture that our Lyapunov approach can be used to establish stability in more general chemostat models in which there are disturbances acting on the species dynamics or where the input nutrient concentration is also time varying.

VIII. ACKNOWLEDGEMENT

The authors thank the anonymous referee for the detailed comments on an earlier version of this paper.

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