CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 11, Number 1, Spring 2003

ANALYSIS OF A CLASS OF MONOD-LIKE

FUNCTIONS THAT LINK SPECIFIC GROWTH

RATE TO DELAYED GROWTH RESPONSE

SEAN ELLERMEYER

ABSTRACT. We analyze a family of functions that pro-vide a link between specific growth rate (x′ (t) /x (t)) and de-layed growth response (DGR) in a class of delay differentialequation models for microbial growth in batch and continu-

ous culture. The connection between specific growth rate anddelayed growth response, which has not been considered in pre-vious studies of these models, is then employed in studying amodel of continuous culture competition between two speciesof microorganisms that are “Monod–equivalent” in that theyhave the same maximal specific growth rate (µm) and half–saturation constant (Kh). It is shown that the species with thesmaller DGR is a superior competitor if the dilution rate of thechemostat is high but that the species with the larger DGR isa better competitor if the dilution rate is low.

1 Introduction A function of the form

(1) f (s) =as

b + s,

for given a > 0 and b > 0, is called a Monod function in honor ofthe French biologist Jacques Monod (1910–1976) who employed suchfunctions in his pioneering studies on quantitative aspects of microbialgrowth [6]. The Monod model for a microorganism being grown inbatch culture and whose growth rate is limited only by the availabilityof a single essential nutrient (substrate) is

s′ (t) =−Y −1µms (t)

Kh + s (t)x (t) , t ≥ 0(2)

x′ (t) =µms (t)

Kh + s (t)x (t) , t ≥ 0(3)

Copyright c©Applied Mathematics Institute, University of Alberta.

69

70 SEAN ELLERMEYER

where s (t) and x (t) are, respectively, the concentrations of the substrateand the microorganism at time t. The derivation of the Monod modelis based on two simple hypotheses:

(M1) If the substrate concentration can be maintained constant (s (t) =s for all t ≥ 0), then the specific growth rate of the culture, definedas x′ (t) /x (t), should be constant for all t ≥ 0. Furthermore, thespecific growth rate depends on s according to a function of theform (1). Thus there exist a > 0 and b > 0 such that if s ∈ [0,∞)is fixed, then

x′ (t)

x (t)=

as

b + sfor all t ≥ 0.

(M2) The rate of substrate consumption at any time t is proportional tothe rate of biomass formation at time t. Thus, there exists c > 0such that

s′ (t) = −cx′ (t) for all t ≥ 0.

Model (2)–(3) is produced from hypotheses (M1) and (M2) by inter-preting the modelling parameters a, b, and c in terms of the followingobservable parameters:

• µm, called the maximal specific growth rate, and defined to be theconstant specific growth rate that results when the substrate concen-tration is maintained in excess for all time

• Kh, called the half–saturation constant, and defined to be the sub-strate concentration at which the specific growth rate is half–maximal

• Y , called the yield constant, and defined to be the ratio of biomassproduced per unit mass of substrate consumed.

By formally setting s = ∞ (interpreted to mean that substrate isavailable in abundance), we obtain from hypothesis (M1) and the def-inition of µm that a = µm. Likewise, by setting s = Kh, we obtainµmKh/ (b + Kh) = µh ≡ µm/2, and hence b = Kh, from (M1) and thedefinition of Kh. Since hypothesis (M2) implies that

x (t) − x (0)

s (0) − s (t)=

1

cfor all t > 0,

we obtain c = Y −1 from the definition of Y .The functions

p (s) =Y −1µms

Kh + s

A CLASS OF MONOD-LIKE FUNCTIONS 71

and

µ (s) =µms

Kh + s

are called, respectively, the per capita substrate uptake function (or thefunctional response) and the specific growth rate function for the Monodmodel. Both p and µ are Monod functions and they are simply relatedto each other via µ = Y p. We make this observation in order to contrastit with the more complicated relationship that exists between µ and pwhen a delayed growth response is taken into account in the modellingprocess.

A model for single substrate dependent batch culture growth thattakes a delayed growth response into account was formulated and ana-lyzed in [2]. This model, which we will refer to as the DGR model, takesthe form

s′ (t) =−Y −1µm exp (µmτ) s (t)

(2 exp (µhτ) − 1) Kh + s (t)x (t) , t ≥ 0(4)

x′ (t) =µm exp (µmτ) s (t − τ)

(2 exp (µhτ) − 1) Kh + s (t − τ)x (t − τ) , t ≥ τ(5)

where s (t) and x (t) are as defined for the Monod model and τ > 0 isa fixed microbe–substrate specific amount of time that is assumed toelapse between the consumption of substrate and the biomass formationthat results from this consumption. A unique positive solution of system(4)–(5) is determined for all t ≥ τ by specifying an initial microbe con-centration, x (t) = x0 (t) > 0, t ∈ [0, τ ], along with an initial substrateconcentration, s (0) = s0 ≥ 0.

The DGR model is derived from two hypotheses:

(H1) Substrate consumption occurs according to a Monod process (justas in the Monod model). Thus there exist a > 0 and b > 0 suchthat

s′ (t) =−as (t)

b + s (t)x (t) for all t ≥ 0.

(H2) Biomass formation at time t occurs at a rate proportional to therate of substrate consumption at time t − τ . Thus there existsc > 0 such that x′ (t) = −cs′ (t − τ) for all t ≥ τ .

Interpretations of the modelling parameters a, b, and c are obtainedfrom hypotheses (H1) and (H2) and the definitions of µm, Kh, and Y as

72 SEAN ELLERMEYER

follows: First, we note that (H1) and (H2) imply that

x′ (t) =cas (t − τ)

b + s (t − τ)x (t − τ) , t ≥ τ .

Setting s (t) = s (constant) for all t ≥ 0 in the above equation producesthe linear delay differential equation

(6) x′ (t) =cas

b + sx (t − τ) , t ≥ τ .

It was proved in Appendix A of [2] that for any given A ≥ 0, everypositive solution of the equation x′ (t) = Ax (t − τ) satisfies

limt→∞

x′ (t)

x (t)= r

where r is the unique solution of r exp (rτ) = A. Thus every positivesolution of equation (6) satisfies

limt→∞

x′ (t)

x (t)= µ (s)

where µ (s) is the unique solution of

µ (s) exp (µ (s) τ) =cas

b + s.

By formally setting s = ∞, we obtain

µm exp (µmτ) = ca.

Likewise, by setting s = Kh, we obtain

µh exp (µhτ) =caKh

b + Kh

.

Thus a = c−1µm exp (µmτ) and b = (2 exp (µhτ) − 1)Kh. In addition,since hypothesis (H2) implies that

x (t) − x (τ)

s (0) − s (t − τ)= c, t ≥ τ ,

we obtain c = Y .

A CLASS OF MONOD-LIKE FUNCTIONS 73

Based on the fact that all positive solutions of equation (5) with fixeds ∈ [0,∞) satisfy limt→∞ x′ (t) /x (t) = µ (s) where µ (s) is defined by

(7) µ (s) exp (µ (s) τ) =µm exp (µmτ) s

(2 exp (µhτ) − 1) Kh + s,

we find it appropriate to define the specific growth rate function for theDGR model to be the function µ : [0,∞) → [0, µm) defined implicitlyfor each s ∈ [0,∞) by equation (7). If τ = 0, then µ is a Monodfunction (and the DGR model is identical to the Monod model), but µis not a Monod function if τ > 0. The per capita substrate consumptionfunction, p, in the DGR model is related to µ according to µ exp (µτ) =Y p.

The fundamental difference in the properties of solutions of the Monodmodel and solutions of the DGR model can be observed by consideringthe situation of excess substrate (s (0) >> 0). In this situation, theMonod model predicts that the culture will begin growing at its max-imal specific growth rate (µm) immediately after inoculation, but theDGR model predicts that the culture will only achieve specific growthrate µm after some time has passed. The latter scenario is more in accor-dance with observations that are typically made in actual experiments,where an initial “lag phase” is usually observed to precede a phase ofapproximately constant exponential growth [3].

The main objective of our present work is to analyze the dependenceof the specific growth rate function, µ, on the response time, τ , for givenvalues of µm > 0 and Kh > 0. With the Monod model as a basis, µm andKh completely determine the specific growth rate of a batch culture andalso determine which of two or more competing species (with differingµm and Kh) will persist in continuous culture while driving the otherspecies to extinction [4]. We will show in Section 2 that, for fixed µm andKh, increasing τ has a “flattening” effect on µ. Thus, a microorganismwith large τ is one that requires a large substrate concentration in orderto achieve maximal specific growth rate but, on the other hand, is ableto maintain an approximately constant (but less than maximal) growthrate over a wide range of substrate concentrations. A result of this, aswe will show in Section 3, is that if two microbial species with identicalµm and Kh, but with different τ , compete for a shared substrate incontinuous culture, then the species with smaller τ has a competitiveadvantage if the removal rate is high but the species with the larger τhas a competitive advantage if the removal rate is low.

74 SEAN ELLERMEYER

2 Properties of the family of specific growth rate functions

For fixed µm > 0, Kh > 0, and τ ≥ 0, the specific growth rate functionµ = µ (s) associated with the DGR model (4,5) is defined by

(8) µ exp (µτ) =µm exp (µmτ) s

(2 exp (µhτ) − 1) Kh + s

where µh = µm/2. Assuming µm > 0 and Kh > 0 to be fixed, we willconsider the properties of the family of functions {µ (τ, ·)}τ≥0

. It willbe shown that each function µ (τ, ·) has qualitative properties similarto those of a Monod function and that limτ→∞ µ (τ, ·) = µh uniformlyon any compact interval [s1, s2] ⊆ (0,∞). The latter fact implies thatamong all microorganisms with given µm and Kh, those with large τhave specific growth rates that remain close to µh over a wide range ofsubstrate concentrations.

For each fixed τ ≥ 0, it is easily seen from the definition (8) thatµ (τ, 0) = 0, µ (τ, Kh) = µh, and lims→∞ µ (τ, s) = µm. Also, since

∂µ

∂s=

µ

µτ + 1

(

1

s−

1

(2 exp (µh) τ − 1) Kh + s

)

> 0 for s ∈ (0,∞)

and

∂2µ

∂s2< −

2

(2 exp (µhτ) − 1)Kh + s·∂µ

∂s< 0 for s ∈ (0,∞) ,

it can be seen that µ (τ, ·) is monotone increasing and concave down on[0,∞). Hence, the functions µ (τ, ·) have the same qualitative character-istics as a Monod function (1).

Next, we wish to show that if 0 < s < Kh, then µ (·, s) is a monotoneincreasing function of τ , and that if s > Kh, then µ (·, s) is a monotonedecreasing function of τ . We also wish to show that limτ→∞ µ (τ, s) = µh

for each fixed s > 0. In order to prove these facts, we introduce anotherfamily {y (τ, ·)}τ≥0

where y is defined for each τ ≥ 0 and s ≥ 0 by

(9) y = y (τ, s) =µmKh exp (µhτ)

(2 exp (µhτ) − 1) Kh + s.

The key properties of this family are

(10) y =Kh

sµ exp ((µ − µh) τ) , τ ≥ 0, s > 0

A CLASS OF MONOD-LIKE FUNCTIONS 75

and

(11)∂y

∂τ= y (µh − y) .

In addition, it can be seen that

(12)∂µ

∂τ=

µ

µτ + 1(µm − µ − y) .

If 0 < s < Kh, then µ < µh (since µ is a monotone increasing functionof s) and, by definition (9), we have y > µh. Thus y > µ and

Kh

sµ exp ((µ − µh) τ) > µ

by property (10). Therefore

µ > µh −1

τln

(

Kh

s

)

and we conclude that limτ→∞ µ = µh. By similar reasoning, it can beshown that if s > Kh, then

µh < µ < µh +1

τln

(

s

Kh

)

and hence that limτ→∞ µ = µh in this case as well.To prove that µ converges to µh in monotone fashion, we first observe

thatµ (0, s) + y (0, s) = µm for all s ≥ 0.

If 0 < s < Kh, then y > µh and ∂y/∂τ < 0 for all τ ≥ 0 by property(11). We claim that, in addition, it must also be the case that µ (τ, s) +y (τ, s) < µm for all τ > 0. If this were not the case, then there wouldexist some τ > 0 such that µ (τ, s)+y (τ, s) ≥ µm and ∂ (µ + y) /∂τ ≥ 0.However, this would yield the contradiction

0 ≤∂ (µ + y)

∂τ=

µ (µm − µ − y) + (µτ + 1) y (µh − y)

µτ + 1< 0.

Since µ (τ, s) + y (τ, s) < µm for all τ > 0, equation (12) implies that∂µ/∂τ > 0 for all τ > 0 and hence that µ (·, s) is a monotone increasing

76 SEAN ELLERMEYER

FIGURE 1: Graphs of typical µ1 = µ (τ1, ·) and µ2 = µ (τ2, ·) withidentical µm and Kh and τ1 < τ2. These graphs were generated withMaple using µm = 2, Kh = 1, τ1 = 2, and τ2 = 10.

function of τ . By similar reasoning, it can be shown that if s > Kh,then µ (·, s) is a monotone decreasing function of τ .

Since µ is a monotone increasing function of s for each fixed τ and amonotone (increasing if 0 < s < Kh and decreasing if s > Kh) functionof τ for each fixed s, we observe that µ (τ, s1) ≤ µ (T, s) ≤ µ (τ, s2) forall τ ≥ 0, T ≥ τ , and s ∈ [s1, s2] where 0 < s1 < µh < s2. Thisimplies that limτ→∞ µ (τ, ·) = µh uniformly on any compact interval[s1, s2] ⊆ (0,∞).

Two typical specific growth rate functions, µ1 (τ1, ·) and µ2 (τ2, ·) withcommon µm and Kh and τ1 < τ2, are illustrated in Figure 1. This figureillustrates the essential difference that arises in using the DGR modelrather than the Monod model in modelling microbial growth. With theMonod model as a basis, two microbial species with the same µm andKh have the same specific growth rates at any substrate concentration.However, with the DGR model as a basis, two species with identicalµm and Kh, but with different τ , have different specific growth rates atevery substrate concentration other than s = 0, s = Kh, and s = ∞. Atsubstrate concentrations s > Kh, the species with the smaller τ has thelarger specific growth rate. At substrate concentrations 0 < s < Kh, thesituation is reversed. An important additional observation that we wishto make is that it is not necessary to formulate the specific growth ratefunction (8) in terms of the half–saturation constant, Kh. We have doneso here in order to be able to draw comparisons between the DGR model

A CLASS OF MONOD-LIKE FUNCTIONS 77

and the Monod model. To be more general, we could choose arbitraryα ∈ (0, 1), define Kα (which we might call the α–saturation constant)according to µ (Kα) = µα ≡ αµm, and consider the class of specificgrowth rate functions {µ (τ, ·)}τ≥0

defined according to

µ exp (µτ) =µm exp (µmτ) s

(α−1 exp ((1− α) µmτ) − 1)Kα + s.

This more general approach allows us to make comparisons between twospecies with the same maximal specific growth rate and α–saturationconstant but different response times. In this case, an analysis similarto the one we have provided in the case α = 1/2 shows that the specieswith the smaller response time has a greater specific growth rate atsubstrate concentrations s > Kα but has a lesser specific growth rate atsubstrate concentrations 0 < s < Kα.

3 Competition in continuous culture Continuous culture ofmicroorganisms is a process in which a well–stirred culture vessel iscontinuously supplied at a constant rate with fresh growth medium con-taining the growth–limiting substrate at a fixed concentration sf , whilethe contents of the culture vessel are simultaneously allowed to flow outof the vessel at the same rate. If F is the common input and output flowrate (volume/time) and V is the (constant) volume of the culture vessel,then D = F/V is called the specific removal rate of the microorganism.The laboratory apparatus that is used in performing continuous cultureis called a chemostat. In contrast to batch culture, a chemostat allowsfor the maintenance of a constant substrate concentration in the culturevessel. For a thorough exposition of the basic mathematical theory ofthe chemostat, the reader should consult [7].

We now consider the scenario of two Monod–equivalent species ofmicroorganisms competing for the same growth–limiting substrate incontinuous culture. By “Monod–equivalent”, we mean that each specieshas the same maximal specific growth rate, µm, and the same half–saturation constant, Kh. However, we assume that each species has adifferent response time. It will be seen that the DGR model (modified toinclude two species in continuous culture) predicts that the species withthe smaller response time has a competitive advantage if the dilutionrate is high but that the species with the larger response time has acompetitive advantage if the dilution rate is low.

The concentrations of the two species, which we will call species 1 andspecies 2, will be denoted by x1 and x2 and their respective response

78 SEAN ELLERMEYER

times will be denoted by τ1 and τ2, with the assumption that τ1 < τ2.The specific growth rate and per capita substrate consumption functionfor species i are, respectively, µi ≡ µ (τi, ·) as defined by (8) and pi =Y −1

i µi exp (µiτi). The DGR model for competition between the twospecies can be written in terms of the µi as

s′ (t) = D (sf − s (t))−(13)

2∑

i=1

Y −1

i µi (s (t)) exp (µi (s (t)) τi) xi (t)

x′

1(t) = µ1 (s (t − τ1))(14)

· exp ((µ1 (s (t − τ1)) − D) τ1) x1 (t − τ1) − Dx1 (t)

x′2(t) = µ2 (s(t − τ2))(15)

· exp ((µ2 (s (t − τ2)) − D) τ2) x2 (t − τ2) − Dx2 (t) .

or, equivalently, in terms the the pi as

s′ (t) = D (sf − s (t)) −2

∑

i=1

pi (s (t)) xi (t)(16)

x′

1(t) = Y1 exp (−Dτ1) p1 (s (t − τ1)) x1 (t − τ1) − Dx1 (t)(17)

x′

2(t) = Y2 exp (−Dτ2) p2 (s (t − τ2)) x2 (t − τ2) − Dx2 (t) .(18)

where sf and D are, respectively, the concentration of substrate in thefresh medium and the specific removal rate (as defined above). Notethat the xi equations each contain factors of exp (−Dτi). These factorsaccount for the substrate stored inside cells that is lost in the washoutwithout contributing to the formation of new biomass in the culturevessel.

If the removal rate is sufficiently low (specifically, D < µm), then thereexist unique substrate concentrations, λ1 and λ2, such that µ1 (λ1) =µ2 (λ2) = D. Since λi is the substrate concentration at which the specificgrowth rate of species i is equal to its specific removal rate, λi is called thebreak–even substrate concentration for species i. If, in addition, the inputconcentration is sufficiently high (specifically, sf > λ1 and sf > λ2),then system (13)–(15) and the equivalent system (16)–(18) have three

A CLASS OF MONOD-LIKE FUNCTIONS 79

equilibrium points: E0 = (sf , 0, 0), E1 =(

λ1, Y1e−Dτ1 (sf − λ1) , 0

)

, and

E2 =(

λ2, 0, Y2e−Dτ2 (sf − λ2)

)

. The equilibrium point E0 correspondsto both competitors being extinct in the culture vessel; whereas theequilibria Ei, i = 1, 2, correspond to organism i being maintained at apositive steady state concentration in the culture vessel with organismj (j 6= i) extinct.

It was proved in [8] that if µm > D, and λi < λj < sf , then everysolution of system (16)–(18) with positive xi component converges to Ei

as t → ∞. Since we are assuming τ1 < τ2, our analysis of the functionsµi carried out in Section 2 shows that λ1 < λ2 if D > µh and λ2 < λ1

if D < µh. Thus, species 1 persists and species 2 becomes extinct ifD > µh, but the situation is reversed if D < µh.

We conclude with some remarks concerning the two model formula-tions (13)–(15) and (16)–(18). In previous studies of DGR competitionmodels with constant delays [1,5,8], the focus has been on model (16)–(18) with the assumption that p is a Monod function or, more generally,that p is a C1 monotone increasing function that is bounded above andsatisfies p (0) = 0. In terms of p, the break–even substrate concen-tration is determined by the condition p (λ) = DeDτ . However, sincethe previous work [1,5,8] did not consider the specific growth rate func-tion associated with p via the response time τ , it has heretofore beenunclear that the condition p (λ) = DeDτ should in fact define λ as thebreak–even substrate concentration. The equivalent condition µ (λ) = Dclearly provides a much clearer intuitive interpretation of the break–evenconcentration. In addition, the condition λi < λj that determines thewinner of a competition implies the condition µj (λi) < µi (λi) = D,which gives the intuitively reasonable conclusion that, at equilibrium,species i is growing at a rate equal to its removal rate and species jis growing at a rate less than its removal rate. However, when trans-lated in terms of functional responses, the condition λi < λj impliespj (λi) < Y −1

j Yi exp (D (τj − τi)) pi (λi), and this condition is difficultto reconcile with intuition. These considerations indicate that it shouldbe advantageous to take specific growth rate functions into account infuture work involving microbial growth and competition models withdelayed growth response.

80 SEAN ELLERMEYER

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3. M. Gyllenberg, (1993) Does time lag of nutrient utilization justify Monod’smodel of bacterial growth?, Bulletin of Mathematical Biology 55 (1993), 487–489.

4. S. B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single–nutrient competition in continuous cultures of microorganisms, SIAM J. Appl.Math. 32 (1977), 366–383.

5. S. B. Hsu, P. Waltman and S. Ellermeyer, A remark on the global asymptoticstability of a dynamical system modeling two species competition, HiroshimaMath. J. 24 (1994), 435–445.

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7. H. L. Smith and P. Waltman, The theory of the chemostat: Dynamics of mi-crobial competition, Cambridge University Press, Cambridge, 1995.

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Department of Mathematics, Kennesaw State University, 1000 Chastain Road,

Box 1204, Kennesaw, GA 30144-5591

E-mail address: sellerme@kennesaw.edu