Journal of Theoretical Biology 276 (2011) 35–41

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Journal of Theoretical Biology

0022-51

doi:10.1

� Corr

E-m

jan.dolfi

gary.mo1 Pr

Univers

journal homepage: www.elsevier.com/locate/yjtbi

Maintenance affects the stability of a two-tiered microbial ‘food chain’?

Aiping Xu a,�,1, Jan Dolfing b, Thomas P. Curtis b, Gary Montague a, Elaine Martin a

a School of Chemical Engineering and Advanced Materials, Newcastle University, Newcastle upon Tyne NE1 7RU, UKb School of Civil Engineering and Geosciences, Newcastle University, Newcastle upon Tyne NE1 7RU, UK

a r t i c l e i n f o

Article history:

Received 10 May 2010

Received in revised form

15 January 2011

Accepted 18 January 2011Available online 1 February 2011

Keywords:

Anaerobic wastewater treatment

Dimensionless transformation

Stability

Maintenance

Syntrophy

93/$ - see front matter & 2011 Elsevier B.V..

016/j.jtbi.2011.01.026

esponding author. Tel.: +44 24 7688 8599.

ail addresses: xuaiping@yahoo.com (A. Xu),

ng@ncl.ac.uk (J. Dolfing), tom.curtis@ncl.ac.u

ntague@ncl.ac.uk (G. Montague), e.b.martin@

esent address: SIGMA, Faculty of Engineerin

ity, CV1 5FB, UK.

a b s t r a c t

Microbial ‘food chains’ are fundamentally different from canonical food chains in the sense that the

waste products of the organisms on one trophic level are consumed by organisms of the next trophic

level rather than the organisms themselves. In the present paper we introduce a generalised model of a

two-tiered microbial ‘food chain’ with feedback inhibition, after applying an appropriate dimensionless

transformation, and investigate its stability analytically. We then parameterised the model with

consensus values for syntrophic propionate degradation compiled by the IWA Task Group for

Mathematical Modelling of Anaerobic Digestion Processes. Consumption of energy for all processes

other than growth is called maintenance. In the absence of maintenance and decay the microbial ‘food

chain’ is intrinsically stable, but when decay is included in the description this is not necessarily the

case. We point out that this is in analogy to canonical food chains where introduction of maintenance in

the description of a stable (equilibrium or limit cycle) predator–prey system generates chaos.

& 2011 Elsevier B.V.. All rights reserved.

1. Introduction

Anaerobic digestion processes are increasingly applied as awaste treatment technology (Van Lier et al., 2001, 2008; Ramirezet al., 2009). One of its advantages is that the process requires noenergy input for oxygenation but rather produces energy in theform of methane. Methanogenic degradation of organic materialis a sequential process (Dolfing, 1988; Schink, 1997). A series ofmicro-organisms is involved in the various conversion steps ofthese compounds into the ultimate products of methane andcarbon dioxide. Typically, the product of one conversion step isthe substrate for the next organism in the chain; each organismlives off the waste product(s) of its predecessor. This paradigmshows some resemblance to the classical concept of a food chain,the difference being that it is not the organisms themselves butrather their products that serve as a food source for the organismsat the next trophic level (Dolfing and Prins, 1996). Surprisinglylittle is known about the stability of these ‘food chains’ (Shenet al., 2007). We therefore seek to investigate the stability of thesemicrobial ‘food chains’ at a fundamental level and contrast thefindings to what is known about the stability of canonicalfood chains. To this end we constructed a generalised model ofa two-tiered microbial food chain. The model is essentially a

All rights reserved.

k (T.P. Curtis),

ncl.ac.uk (E. Martin).

g and Computing, Coventry

stripped down version of anaerobic digestion model no. 1 (ADM1),the consensus model supported by the IWA Task Group for Mathe-matical Modelling of Anaerobic Digestion Processes (2002), andincludes options for decay, which is the consumption of energy forall processes other than growth, and feedback inhibition, a salientcharacteristic of metabolic interactions in methanogenic microbialcommunities, where hydrogen and acetate both play this role,although with vastly different kinetic characteristics. The model ismade dimensionless in a way that allows us to investigate decayeffects without running into scale imbalance problems, and analysedfor bifurcations. According to the Routh–Hurwitz criteria the non-trivial steady state is possibly unstable when decay is included.However, simulation results with ADM1 consensus values indicatethat the possibly unstable state is always stable as long as it ismeaningful, that is, all its state values are non-negative.

2. The model

Anaerobic digestion has been the subject of considerableresearch effort, not least in the area of modelling (Mosey, 1983;Siegrist et al., 1993). The complexity of the process calls for complexmodels, or so it seems. The current ADM1 consensus model (IWATask Group for Mathematical Modelling of Anaerobic DigestionProcesses, 2002) has a staggering 32 dynamic state variables andsuccessfully captures the necessary dynamics of the process.However, this model is far too complex to permit mathematicallyanalysis of its nonlinear dynamics (Shen et al., 2007). In order tomake such studies possible, we have therefore reduced the model toits very backbone, a two-tiered microbial ‘food chain’ with feedback

Fig. 1. Propionate degradation with hydrogen and acetate inhibitions.

Table 1Parameters for a model of a two-tiered microbial ‘food chain’ with feedback

inhibition, consisting of a propionate degrader and a hydrogenotrophic methanogen.

The parameters are consensus values listed by IWA Task Group for Mathematical

Modelling of Anaerobic Digestion Processes (2002) for high rate mesophilic systems.

Parameters Nominal values Units

km,pro 13 COD/COD/d

Ks,pro 0.3 kg COD/m3

Ypro 0.04 COD/COD

kdec,pro 0.02 1/d

km,H2 35 COD/COD/d

Ks,H2 2.5�10�5 kg COD/m3

YH2 0.06 COD/COD

kdec,H2 0.02 1/d

KI,H2 3.5�10�6 kg COD/m3

A. Xu et al. / Journal of Theoretical Biology 276 (2011) 35–4136

inhibition, which encapsulates the essence of methanogenic degra-dation processes, see Fig. 1. Degradation of propionate is a case inpoint. In methanogenic ecosystems syntrophic propionate degradingbacteria convert propionate into hydrogen and acetate, which arethen converted into methane by two other trophic groups, thehydrogenotrophic and the acetoclastic methanogens (Koch et al.,1983; Schink, 1997). For thermodynamic reasons propionate degra-dation is extremely sensitive to accumulation of hydrogen. Thus inmethanogenic ecosystems propionate degradation is only sustain-able in the presence of hydrogenotrophic organisms (typicallymethanogens) as hydrogen scavengers (Dolfing, 1988). Syntrophicinteractions define the methanogenic degradation of a wide varietyof compounds, including short- and long-chain fatty acids, aminoacids and aromatics (Schink, 1997). Thus our modelling efforts willbe undertaken with propionate as a model substrate. To be able toexecute a thoroughly mathematical analysis we study propionatedegradation with hydrogen production while ignoring acetateproduction. However, the dimensionless transformations introducedlater can be easily extended to the general case. Using the notationalso used in ADM1 the model for a two-tiered ‘food-chain’ withfeedback inhibition can be written as

dSpro

dt¼DðSpro,in�SproÞ�

km,proSpro

Ks,proþSproXproI2

dXpro

dt¼�DXproþYpro

km,proSpro

Ks,proþSproXproI2�kdec,proXpro

dSH2

dt¼�DSH2þ0:43ð1�YproÞ

km,proSpro

Ks,proþSproXproI2�

km,H2SH2

Ks,H2þSH2XH2

dXH2

dt¼�DXH2þYH2

km,H2SH2

Ks,H2þSH2XH2�kdec,H2XH2

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð1Þ

where Spro and Xpro are propionate substrate and biomass concen-trations; SH2 and XH2 are those for hydrogen. 0.43 represents the partwhich goes to hydrogen substrate and the rest (1�0.43 ¼ 0.57)goes to acetate which is left out in our model. Both growth functionstake Monod form and the hydrogen inhibition function takes thenon-competitive one: I2 ¼ 1/(1 + SH2/KI,H2). Other inhibition func-tions can be easily incorporated in this model. Here apart from thetwo environmentally controllable parameters: the inflowing propio-nate concentration Spro,in and the dilution rate D, the others all havebiological meanings, with suggested values for mesophilic high rate(nom 35 1C) taken from IWA Task Group for Mathematical Modellingof Anaerobic Digestion Processes (2002), see Table 1.

We should pay special attention to the significantly differentvalues of the half-saturation rates of hydrogen Ks,H2 and of propio-nate Ks,pro, which decide the scales of their corresponding concen-trations in the chemostat.

Our aim is to study the stability of the model (1) while varyingthe two control parameters: the inflowing propionate concentrationSpro,in and the dilution rate D, or more accurately to obtain thebifurcation diagram of (1). Too many biologically meaningful

parameters hamper in our ability to identify factors that governthe stability of the model. Therefore, our first step is to find asuitable dimensionless transformation.

3. Dimensionless transformations

As pointed out in Gurney and Nisbet (1998), the behaviour of anatural system, where most variables and parameters have units,cannot be affected by the units in which we choose to measure thequantities we use to describe it. By using dimensionless transforma-tion, we can reduce the number of parameters determining thedynamics, saving effort on numerical and analytic investigations.However, there is no unique dimensionless form for the equations.In considering the characteristics of the ADM1 model, which is ‘stiff’due to the scale imbalance of different substrate concentrations (lowhydrogen concentration), we introduce here the following dimen-sionless quantities, adapted from those described in Baltzis andFredrickson (1984) but different from those commonly used for foodchain models (Kot et al., 1992): all concentrations were rescaledsomehow by the inflowing substrate concentration:

t� km,proYprot; a� D

km,proYpro; uf �

Spro,in

Ks,pro

and

s0 �Spro

Ks,pro; x0 �

Xpro

Ks,proYpro

and

s1 �SH2

Ks,H2; x1 �

XH2

Ks,H2YH2

and

f�km,H2YH2

km,proYpro; o� Ks,pro � 0:43ð1�YproÞ

Ks,H2

and

A�kdec,pro

km,proYpro; B�

kdec,H2

km,proYpro

As a result we have the following dimensionless system:

dx0

dt¼�ax0þm0x0�Ax0

dx1

dt¼�ax1þm1x1�Bx1

ds0

dt¼ aðuf�s0Þ�m0x0

ds1

dt ¼�as1þom0x0�m1x1 ð2Þ

A. Xu et al. / Journal of Theoretical Biology 276 (2011) 35–41 37

where

m0 ¼s0

1þs0I2; m1 ¼

fs1

1þs1

I2 ¼1

1þSH2=KI,H2¼

1

1þKs,H2 � s1=KI,H2¼

1

1þs1=KI

with KI � KI,H2=Ks,H2.Utilising this dimensionless transformation we have actually

given the two dimensionless growth functions (propionate andhydrogen equivalents) the same half-saturation rates. Thus wehave addressed the scale imbalance problem and at the same timeintroduced a generalised model, which describes the typical ADM1framework with inhibition feedback and covers a wide range ofmicrobial models.

Regarding to the case study of our two-tiered propionatedegrading ‘food chain’, the parameters of the transformed modelare

f¼ 4:0385, o¼ 4953:6, A¼ B¼ 0:0385, KI ¼ 0:14

From now on we study the stability of the generalised model (2).The steady state equations predict the following possible

steady states (SS) for the system if categorised by the two micro-bial populations:

SS1: x0¼0, x1¼0: trivial solution where both populations arewashed out;

SS2: x0a0, x1¼0: hydrogen population is washed out whilethe host survives;

SS3: x0a0, x1a0: both populations survive.A steady state is called meaningful if and only if all the

concentrations are non-negative and s0ruf . The local stabilityof each steady state will be tested by linearisation around thesteady state values of the variables.

For a system of autonomous ordinary differential ODEs such asEq. (2), the stability of its steady states depends on the signs of thereal parts of the eigenvalues of the corresponding Jacobian matrix(Amundson, 1966). For any n-dimensional ODEs system of theform

dx1

dt¼ f1ðx1,x2, � � � ,xnÞ

dx2

dt¼ f2ðx1,x2, � � � ,xnÞ

^dxn

dt¼ fnðx1,x2, � � � ,xnÞ

8>>>>>>>>><>>>>>>>>>:its Jacobian matrix is the matrix of the partial derivatives of theright hand side with respect to the state variables, that is

J¼@fi

@xj

� �¼

@f1

@x1

@f1

@x2� � �

@f1

@xn

@f2

@x1

@f2

@x2� � �

@f2

@xn

^ ^ & ^@fn

@x1

@fn

@x2� � �

@fn

@xn

0BBBBBBBBBB@

1CCCCCCCCCCA

where all derivatives are evaluated at the steady state. A steadystate is (asymptotically) stable if all eigenvalues have negativereal parts. Otherwise, it is unstable. In the next two sections wewill check the stability of all steady states one by one, with andwithout decay effects, respectively, followed by numerical analy-sis of the ADM1 model as a case study.

4. Stability analysis without decay effects

The equations describing the generalised model, when decayeffects are not taken into account (id est A¼B¼0 in 2), are writtenas the following:

dx0

dt ¼�ax0þm0x0 ð3aÞ

dx1

dt ¼�ax1þm1x1 ð3bÞ

ds0

dt¼ aðuf�s0Þ�m0x0 ð3cÞ

ds1

dt¼�as1þom0x0�m1x1 ð3dÞ

By combining Eqs. (3c) and (3a), we obtain

d

dtðs0ðtÞþx0ðtÞÞ ¼ �aðs0ðtÞþx0ðtÞÞþauf

As a result

s0ðtÞþx0ðtÞ ¼ uf þðs0ð0Þþx0ð0Þ�uf Þe�at

where s0(0) and x0(0) are the initial concentrations of the rescaledpropionate substrate and biomass. It follows that s0ðtÞþx0ðtÞ-uf

as t-1. We may, in other words, study the model’s asymptoticbehaviour along s0+x0¼uf.

With similar reasoning we can obtain the other constraints1�ox0þx1 ¼ 0. Therefore, when asymptotic behaviour is con-cerned we can eliminate two state variables. Our system is thusequivalent to a two-dimensional one:

dx0

dt ¼�ax0þm0x0

dx1

dt ¼�ax1þm1x1

8>><>>:where

m0 ¼s0

1þs0I2; I2 ¼

1

1þs1=KI; m1 ¼

fs1

1þs1

with

s0 ¼ uf�x0; s1 ¼ox0�x1

Its Jacobian matrix is

J¼�aþm0þð�EþoFÞx0 �Fx0

oGx1 �aþm1�Gx1

" #ð4Þ

where

E¼@m0

@s040; F ¼

@m0

@s1o0; G¼

dm1

ds140

By inspecting the conditions under which each steady state ismeaningful and stable, see Appendix A for detailed analysis, onecan conclude that for any pair of values of the operating parameters,a and uf, there is always one, and only one, steady state which ismeaningful and stable.

5. Consideration of decay effects

Consumption of energy for all processes other than growth iscalled maintenance, since the processes involved effectively resultin the maintenance of cell viability. In situations where microbialcells are located in a favourable environment, maintenance canoften be neglected. In other situations, however, a significantportion of the energy-yielding substrate that could be used forgrowth is consumed for maintenance (Fredrickson and Tsuchiya,1977). In the ADM1 model maintenance is taken into account as

A. Xu et al. / Journal of Theoretical Biology 276 (2011) 35–4138

decay. In this section the generalised model (2) is studied, takinginto consideration maintenance effects, which means A40 andB40. This system has the same possible steady states as thoseanalysed in the previous section. Here we focus on the twopositive terms A and B, to evaluate how the stability is affectedby their simultaneous introduction.

The Jacobian matrix for this system is

J¼

a11 0 a13 a14

0 a22 0 a24

a31 0 a33 a34

a41 a42 a43 a44

266664

377775 ð5Þ

where a11 ¼�aþm0�A, a13 ¼ Ex0, a14 ¼ Fx0, a22 ¼�aþm1�B,a24 ¼ Gx1, a31 ¼�m0, a33 ¼�a�Ex0, a34 ¼ �Fx0, a41 ¼om0,a42 ¼�m1, a43 ¼oEx0, a44 ¼�aþoFx0�Gx1, with

E¼@m0

@s040; F ¼

@m0

@s1o0; G¼

dm1

ds1

having the same definitions as before.A detailed study for this case has been performed, see Appendix B.

Analytical conditions for each steady state to be meaningful havebeen derived and the characteristics of the eigenvalues have beenstudied for stability. Analytical expressions of the criteria for localstability have been developed and the Routh–Hurwitz criteria(Amundson, 1966) have been checked wherever necessary. The basicresults of the analysis are: no steady state exhibits multiplicity; thesteady states are mutually exclusive in the sense that for any pair ofvalues for the operating parameters, a and uf, at most one steadystate can be meaningful and stable; Hopf bifurcation can possiblyoccur with the non-trivial steady state.

Remark 1. The steady states depend on the assumption of aspecific functional growth and inhibition. However, our methodfor the stability analysis is still effective for other functional forms aslong as they keep the signs of their derivatives, both for the caseswithout and with decays, since any explicit form is not needed here.

Remark 2. As a special case when only one decay term is takeninto consideration, the system can be reduced into a three-dimensional one. The corresponding Routh–Hurwitz criteria ischecked and it is always satisfied so the non-trivial steady state isalways stable as long as it is meaningful.

6. Some simulation results

In consideration of the authors’ specific interests in anaerobicdigestion in wastewater treatment and the role of ADM1 inits modelling, we apply here its consensus parameters for thetwo-tiered propionate degrading methanogenic ‘food-chain’ to

Feeding concentration of the substrate uf

Dilu

tion

rate

α

0 1 2 3 4 5 6 7 8 9 10

0.10.20.30.40.50.60.70.80.9

1

SS1 stable

SS2 stable

SS3 stable

Fig. 2. Bifurcation diagrams for a two-tiered microbial ‘food chain’ without (a) and with

values listed in Table 1.

numerically check its stability, with maintenance neglected andincluded, respectively. We will present the operating diagrams,which are very generically useful in the case of the chemostat sincewhen one decides to study a particular system in a chemostat all buttwo (the dimensionless dilution rate a and the dimensionlessconcentration of the substrate in the feed uf) of the parameterswhich appear in the generalised model are fixed for they depend onthe organisms and the substrates which have been picked. Theoperating parameters a and uf can vary at our will. The operatingdiagram shows the regions where each steady state is stable.Therefore, this diagram shows how the system behaves when wevary the two control parameters.

6.1. When maintenance effects are neglected

By applying the ADM1 parameters (f¼ 4:0385, o¼ 4954,KI¼0.14) and neglecting the decay terms, which means A¼B¼0,we obtain the bifurcation diagram Fig. 2(a).

The dimensionless holding time is b¼ 1=a. Recall the defini-tion of the border which makes SS2 unstable and at the same timeSS3 meaningful (the blue curve in Fig. 2(a)) F1ðs1Þ ¼ s2

1�ðoðuf þ1ÞþKIðb�1ÞÞs1þoKIðufb�ðuf þ1ÞÞ ¼ 0 with s1 ¼ 1=ðfb�1Þ. Note thatwe can actually solve explicitly uf as a function of b:

uf ¼�1�KIb

2f�oKIf2b2�ofbþo�oKIþKIbþKIfb�KIþ2oKIfb

�ofb�2oKIb2fþoKIbþoKIb

3f2�oKIþo�oKIf

2b2þ2oKIfb

6.2. When maintenance effects are included

With the ADM1 parameters, the two decay rates are equal,that is, A¼B¼0.0385. By applying the ADM1 kinetics (f¼ 4:0385,o¼ 4954, KI¼0.14 and A¼B¼0.0385) we obtain the partition ofthe plane, where different steady states become meaningful,see Fig. 2(b). As discussed before the only possibility of Hopfbifurcation occurs at the steady state SS3 when the two main-tenance effects are included, thus we focus on it to see how toverify its stability, mathematically and numerically and whatneeds to be further addressed, when a specific case is considered.

1.

Dilu

tion

rate

α

000000000

(b)

Steady state SS3: SS3 is the solution of the following algebraicequations:

�ax0þm0x0�Ax0 ¼ 0

�ax1þm1x1�Bx1 ¼ 0

aðuf�s0Þ�m0x0 ¼ 0

�as1þom0x0�m1x1 ¼ 0

8>>>><>>>>:

Feeding concentration of the substrate uf

0 2 4 6 8 10

.1

.2

.3

.4

.5

.6

.7

.8

.91

SS1

SS2

SS3

maintenance effects. The model was parameterised with the ADM1 consensus

A. Xu et al. / Journal of Theoretical Biology 276 (2011) 35–41 39

where

m0 ¼s0

1þs0I2; I2 ¼

1

1þs1=KI; m1 ¼

fs1

1þs1

It is easy to get that s1 ¼ ðaþBÞ=ðf�ðaþBÞÞ. If we denote I2 ¼

1/(1+s1/KI) then s0 ¼ ðaþAÞ=ðI2�ðaþAÞÞ, x0 ¼ aðuf�s0Þ=ðaþAÞ

and x1 ¼ ð�as1þoaðuf�s0ÞÞ=ðaþBÞ. The partition between SS2and SS3 satisfies:

F1ðs1Þ ¼ s21� oðuf þ1ÞþKI

1

aþA�1

� �� �s1

þoKIuf

aþA�ðuf þ1Þ

� �¼ 0

where

s1 ¼1

faþA�1

Moreover, at SS3,

E¼I2

ð1þs0Þ2¼ðI2�ðaþAÞÞ2

I2

F ¼s0

1þs0�

I22

KI

� �¼�ðaþAÞI2

KI

G¼f

ð1þs1Þ2¼ðf�ðaþBÞÞ2

f

2.

Characteristic polynomial at SS3: From (B.2), the characteristicpolynomial at SS3 is

l4þ f1l

3þ f2l

2þ f3lþ f4 ¼ 0

where

f1 ¼ Ex0�oFx0þGx1þ2a

f2 ¼�oAFx0þEx0Gx1þ2Ex0aþa2þAEx0�2aoFx0þ2aGx1þBx1G

f3 ¼ AEx0a�oAFx0a�oa2Fx0þ2Gx1Ex0aþGx1BEx0

þAEx0Gx1þGx1BaþGx1a2þEx0a2

f4 ¼ AGx1BEx0þaGx1BEx0þGx1Ex0a2þAGx1Ex0a

As discussed above, all the polynomial coefficients are onlyfunctions of the two control parameters a and uf.

3.

Stability check: We further numerically check the stability ofSS3. From the Routh–Hurwitz criteria ðf1f2f34 f 2

3 þ f 21 f4Þ we

define a function F ¼ f1f2f3�ðf23 þ f 2

1 f4Þ, which depends only onthe two control parameters a and uf. See Fig. 3 as its surfacedescription. The numeric results show that SS3 is alwaysstable as long as it is meaningful, despite the fact that it canbe theoretically unstable, see Section 5, where the internal

α

u f

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450123456789

10

Domain for meaningful SS3

100

5

10

15

f

Fig. 3. ADM1 stability check at SS3. (a) The part that SS3 is meaningful, duplicated f

constraints of the variables are not taken into account whenconsidering a generalised model.

7. Discussion

The stability of food chains and food webs has been the subject ofnumerous studies and it has been shown that many factors con-tribute in stabilising the system (Gross, 2004; Gross and Feudel, 2006;Gross et al., 2009; May, 1974). Here we point out that maintenanceappears to be a key factor when a two-tiered microbial ’food chain’ isconsidered. Kot et al. (1992) have shown that the populations ofthe protozoan predator and bacterial prey in a chemostat with anconstant inflowing substrate, can coexist on a limit cycle, while Kooiand Boer (2003) have numerically shown that similar structuredmodel can generate chaos. The difference between the models usedby the different groups is that, unlike Kooi and Boer (2003), Kot et al.(1992) do not include maintenance. In spite of the fundamentaldifferences between our ‘food chain’ and the classical predator–preydiscerned in macro ecology we observe the same intrinsic effect ofmaintenance on the stability of the food chain. In the absence ofmaintenance our two-tiered ‘food chain’ too is always stable, butwhen maintenance is included in its description our two-tieredgeneralised ‘food chain’ is not necessarily stable in theory. However,using the consensus parameters of ADM1 we have shown innumerical simulations that our model of the methanogenic two-tiered propionate-hydrogen food chain is always stable. Directapplication of symbolic analysis programs, such as Maple or Math-ematica, turned out fruitless. Stiefs et al. (2008) have proposed a moreeffective way to compute and visualise bifurcation surfaces, whichenhance the qualitative understanding of a system and can help toquickly locate more complex bifurcation situations corresponding tobifurcations. Moreover, numerical analysis using programs like AUTO(Doedel et al., 1997) can hopefully help to deal with the problemproposed here like in Kooi et al. (1998). Both will serve as our tools topinpoint the conditions under which our generalised system is proneto becoming unstable. For the operators of anaerobic wastewatertreatment systems our results are good news and bad news. Goodbecause we have shown that the syntrophic associations betweenpropionate degraders and methanogens are inherently stable underrealistic environmental conditions. The bad news though is that it isnot a priori possible to generalise; there may well be other two-tiered‘food chains’ that become unstable under environmentally realisticconditions, which are the authors’ current research interest.

Hess and Bernard (2008) pointed out that the excessivecomplexity of ADM1 makes any advanced mathematical analysiscritical. The present analysis is based on a two-tiered model but itopens up the questions about the stability of longer microbial‘food chains’ and more intricate ‘food webs’. Computational methodscan be taken into consideration. For example Gross et al. (2005)

0 0.1 0.2 0.30.4 0.5

05

x 1015

αuf

rom Fig. 2(a) and (b) the surface of function F for stability check at SS3.

A. Xu et al. / Journal of Theoretical Biology 276 (2011) 35–4140

have demonstrated that canonical long food chains are in generalchaotic, after applying the steady state under consideration tonormalise the ODE system.

Acknowledgements

The authors acknowledge fundings from EPSRC (Grant no.EP/E057462/1) and from the European Commission, which sup-ported this work through ECOSERV, a Marie Curie Excellence Grant(EXT 023469). The authors thank Dr. Thilo Gross for his constructivecomments and helpful discussion, and the other two anonymousreviewers for useful suggestions on the revision of the manuscript.Aiping Xu also thanks Dr. Jonathan W. Pitchford and other collea-gues for providing an excellent working environment to finallyfinish the paper while visiting YCCSA, University of York, UK.

Appendix A. Stability analysis with maintenance neglected

SS1: x0¼0, x1¼0. As a consequence s0 ¼ uf and s1¼0. Thissteady state is always meaningful. Since m0 ¼ uf =ð1þuf Þ andm1 ¼ 0, the Jacobian matrix (4) becomes

J¼�aþ

uf

1þuf0

0 �a

24

35

Its eigenvalues are l1 ¼�aþuf =ð1þuf Þ and l2 ¼�a. SS1 isa node since all eigenvalues are real; for being stable it mustbe l1o0. Therefore, SS1 is meaningful and stable if and only ifa4uf =ð1þuf Þ.

SS2: x0a0, x1¼0. As a result m0 ¼ s0=ð1þs0Þ=ð1þs1=KIÞ ¼ a ands1 ¼ox0. Note that s0 + x0 ¼uf. Thus s0 satisfies the followingquadratic equation:

F0ðs0Þ ¼ s20þ

KI

o1

a�1

� �þð1�uf Þ

� �s0�

KI

o þuf

� �¼ 0 ðA:1Þ

Due to the fact that �ðKI=oþuf Þo0, there is only one uniquepositive real solution for Eq. (A.1). To guarantee that SS2 is mean-ingful all that we need is s0ouf , that is, F0ðuf Þ ¼ KIðuf� aðuf þ1ÞÞ=a=o40, which is equivalent to aouf =ð1þuf Þ. This implies that assoon as SS1 becomes unstable, SS2 becomes meaningful.

On the other hand, s1 satisfies the following quadratic equation:

F1ðs1Þ ¼ s21� oðuf þ1ÞþKI

1

a�1

� �� �s1þoKI

uf

a�ðuf þ1Þ

� �¼ 0

ðA:2Þ

As previous reasoning aouf =ð1þuf Þ, we have oKIðuf =a�ðuf þ1ÞÞ40 and oðuf þ1ÞþKIð1=a�1Þ40. We further check itsdiscriminant D¼ ½oðuf þ1Þþ KIð1=a�1Þ�2�4oKIðuf =a �ðuf þ1ÞÞ ¼½oðuf þ1Þ�KIð1=a�1Þ�2þ4oKI=a40. Consequently there are tworeal positive roots for Eq. (A.2). Since s1oouf and F1ðouf Þ ¼

�o2uf�oKI o0, only the smaller root is meaningful, which wedenote as s1

n.The Jacobian matrix (4) becomes

J¼ð�EþoFÞx0 �Fx0

0 �aþm1

" #

Its eigenvalues are l1 ¼ ð�EþoFÞx0o0 and l2 ¼�aþm1. It isalso a node and SS2 is stable if and only if a4m1, wherem1 ¼fs1=ð1þs1Þ:

a4m1()s1oa

f�a ¼1

f=a�1

which implies F1ð1=ðf=a�1ÞÞo0.

SS3: x0a0, x1a0. As a result m0 ¼ s0=ð1þs0Þ=ð1þs1=KIÞ ¼ aand m1 ¼fs1=ð1þs1Þ ¼ a, which means s1 ¼ a=ðf�aÞ. If we denoteI2 ¼ 1=ð1þs1=KIÞ then s0 ¼ a=ðI2�aÞ, x0 ¼ uf�s0 and x1 ¼�s1þoðuf�s0Þ. So to guarantee that SS3 is meaningful we must have

s140()aof ðA:3aÞ

s040()ao I2 ðA:3bÞ

x040()s0ouf ðA:3cÞ

x140()s1ooðuf�s0Þ ðA:3dÞ

First we consider (A.3a) and (A.3b):

ao I2 ¼1

1þs1=KI¼

KI

KIþs1()s1o

KIð1�aÞa ¼ KI

1

a�1

� �

Then from (A.3c):

s0 ¼a

I2�a¼

aKI

KIþs1�a

ouf()s1oKIuf

aðuf þ1Þ�1

� �

So s1oKIðuf =a=ðuf þ1Þ�1Þ. Moreover,

F1 KIuf

aðuf þ1Þ�1

� �� �¼

K2I ð�uf það1þuf ÞÞ

a2ð1þuf Þ2

o0

And from (A.3d):

s1ooðuf�s0Þ()s21� oðuf þ1ÞþKI

1

a�1

� �� �s1þoKI

uf

a �ðuf þ1Þ

� �40

Recall the definition of F1 and s1 ¼ 1=ðf=a�1Þ we haveF1ð1=ðf=a�1ÞÞ40.

The Jacobian matrix (4) becomes

J¼ð�EþoFÞx0 �Fx0

oGx1 �Gx1

" #

Its eigenvalues are given as roots of the equation

l2þ½ðE�oFÞx0þGx1�lþEGx0x1 ¼ 0

Since E40, Fo0 and G40, it is obvious that the two eigen-values l1 and l2 satisfy l1þl2o0 and l1l240. Moreover, thediscriminant of the above equation is D¼ ½ðE�oFÞx0þGx1�

2

�4EGx0x1 and it can be either positive or negative. Therefore,the two eigenvalues can either be real and negative or complexconjugate with negative real parts. Hence as long as it exists it isstable. It can never be l1þl2 ¼ 0, hence Hopf bifurcation cannotoccur from SS3.

Appendix B. Stability analysis with maintenance included

SS1: x0¼0, x1¼0. As a result s0 ¼ uf and s1¼0. This steadystate is always meaningful.

The Jacobian matrix (5) becomes

J¼

�aþm0�A 0 0 0

0 �a�B 0 0

�uf

1þuf0 �a 0

ouf

1þuf0 0 �a

266666664

377777775

Its eigenvalues are l1 ¼�aþuf =ð1þuf Þ�A , l2 ¼�a�Bo0 andl3 ¼ l4 ¼�ao0. SS1 is a node since all eigenvalues are real; for itbeing stable we need l1o0. Therefore, SS1 is meaningful andstable if and only if a4uf =ð1þuf Þ�A.

SS2: x0a0, x1 ¼ 0. Consequently, m0 ¼ s0=ð1þs0Þ=ð1þs1=KIÞ

¼ aþA, ðaþAÞx0 ¼ aðuf�s0Þ and as1 ¼oðaþAÞx0. Just as the casewithout maintenance effects we have the constraint s1 ¼o

A. Xu et al. / Journal of Theoretical Biology 276 (2011) 35–41 41

ðuf�s0Þ. Thus s0 satisfies the following quadratic equation:

F0ðs0Þ ¼ s20þ

KI

o1

aþA�1

� �þð1�uf Þ

� �s0�

KI

o þuf

� �¼ 0

As the same reasoning before, to guarantee that SS2 is mean-ingful all we need is aouf =ð1þuf Þ�A. Note that as long as SS1becomes unstable, SS2 becomes meaningful.

And s1 satisfies the following quadratic equation:

F1ðs1Þ ¼ s21� oðuf þ1ÞþKI

1

aþA�1

� �� �s1þoKI

uf

aþA�ðuf þ1Þ

� �¼ 0

ðB:1Þ

As discussed in Appendix A there are two positive roots for thequadratic Eq. (B.1) but only the smaller one is meaningful.

The Jacobian matrix (5) becomes

J¼

0 0 Ex0 Fx0

0 �aþm1�B 0 0

�ðaþAÞ 0 �a�Ex0 �Fx0

oðaþAÞ �m1 oEx0 �aþoFx0

266664

377775

Its characteristic polynomial is

jlI�Jj ¼ ðlþa�m1þBÞðlþaÞ½l2þðaþðE�oFÞx0ÞlþðaþAÞðE�oFÞx0� ¼ 0

Its eigenvalues are l1 ¼�aþm1�B, l2 ¼�ao0, and the othertwo l3 and l4 are given as roots of the following quadraticequation

l2þ½aþðE�oFÞx0�lþðaþAÞðE�oFÞx0 ¼ 0

So for being stable it must be l1o0, which impliesF1ð1=ðf=ðaþBÞ�1ÞÞo0.

SS3: x0a0; x1a0. So s1 ¼ ðaþBÞ=ðf�ðaþBÞÞ. If we denoteI2 ¼ 1=ð1þs1=KIÞ then s0 ¼ ðaþAÞ=ðI2�ðaþAÞÞ, x0 ¼ aðuf�s0Þ=

ðaþAÞ and x1 ¼ ð�as1þoaðuf�s0ÞÞ=ðaþBÞ. As the case withoutmaintenance effects, to guarantee SS3 is meaningful we musthave F1ð1=ðf=ðaþBÞ�1ÞÞ40.

The Jacobian matrix (5) becomes

J¼

0 0 Ex0 Fx0

0 0 0 Gx1

�ðaþAÞ 0 �a�Ex0 �Fx0

oðaþAÞ �ðaþBÞ oEx0 �aþoFx0�Gx1

266664

377775

Its characteristic polynomial is

jlI�Jj ¼ ðlþaÞ2½l2þ½ðE�oFÞx0þGx1�lþEGx0x1�þABEGx0x1

þAx0ðlþaÞ½ðE�oFÞlþEGx1�þBGx1ðlþaÞðlþEx0Þ ¼ 0

It explicitly indicates how the maintenance affects the sys-tem’s long-term behaviour, which is never seen in any literatureup to our knowledge.

It is obvious that the case, when only one decay (A¼0 andBa0, or, Aa0 and B¼0) or no decay (A¼B¼0), can be regardedas a special case of this one.

For the non-trivial case (A,B40), if we expand the aboveequation as

l4þ f1l

3þ f2l

2þ f3lþ f4 ¼ 0 ðB:2Þ

where

f1 ¼ Ex0�oFx0þGx1þ2a

f2 ¼�oAFx0þEx0Gx1þ2Ex0aþa2þAEx0�2aoFx0þ2aGx1þBx1G

f3 ¼ AEx0a�oAFx0a�oa2Fx0þ2Gx1Ex0aþGx1BEx0þAEx0Gx1

þGx1BaþGx1a2þEx0a2

f4 ¼ AGx1BEx0þaGx1BEx0þGx1Ex0a2þAGx1Ex0a

Since E40, Fo0 and G40, all coefficients fi, i¼1,y,4 arepositive. Therefore, (B.2) cannot have any positive roots. Whenthe Routh–Hurwitz criteria ðf1f2f34 f 2

3 þ f 21 f4Þ are checked one can

see that they are not necessary satisfied. Thus the roots of (B.2)can be real and negative, or pairs of complex conjugate numberswith either negative or positive real parts. The possibility ofhaving (at least) one pair of complex eigenvalues with positivereal parts implies that Hopf bifurcation can originate from SS3.

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