Soil carbon and nitrogen mineralization: Theory and models - CGD

25 Pages·2009·1.17 MB·English

SoilBiology&Biochemistry41(2009)1355–1379
ContentslistsavailableatScienceDirect
Soil Biology & Biochemistry
journal homepage: www.elsevier.com/locate/soilbio
Soil carbon and nitrogen mineralization: Theory and models across scales
Stefano Manzoni, Amilcare Porporato*
DepartmentofCivilandEnvironmentalEngineering,DukeUniversity,127HudsonHall,Durham,NC27708-0287,USA
a r t i c l e i n f o a b s t r a c t
Articlehistory: In the last 80 years, a number of mathematical models of different level of complexity have been
Received3July2008 developedtodescribebiogeochemicalprocessesinsoils,spanningspatialscalesfromfewmmtothou-
Receivedinrevisedform sandsofkmandtemporalscalesfromhourstocenturies.Mostofthesemodelsarebasedonkineticand
26February2009
stoichiometric laws that constrain elemental cycling within the soil and the nutrient and carbon
Accepted28February2009
exchange with vegetation and the atmosphere. While biogeochemical model performance has been
Availableonline20March2009
previouslyassessedinotherreviews,lessattentionhasbeendevotedtothemathematicalfeaturesofthe
models, and how these are related to spatial and temporal scales. In this review, we consider w250
Keywords:
biogeochemical models, highlighting similarities in their theoretical frameworks and illustrating how
Decompositionmodels
Stoichiometry theirmathematicalstructureandformulationarerelatedtothespatialandtemporalscalesofthemodel
Modelclassification applications.Ouranalysisshowsthatsimilarkineticandstoichiometriclaws,formulatedtomechanis-
Mineralization–immobilizationturnover tically represent the complex underlying biochemical constraints, are common to most models,
Microbialbiomass providingabasisfortheirclassification.Moreover,ahistoricanalysisrevealsthatthecomplexityand
Nutrientlimitation degreeandnumberofnonlinearitiesgenerallyincreasedwithdate,whiletheydecreasedwithincreasing
Heterotrophicrespiration spatial and temporal scale of interest. We also found that mathematical formulations specifically
Decomposition
developed for certain scales (e.g., firstorderdecay rates assumed inyearly time scale decomposition
Spatialandtemporalscales
models)oftentendtobeusedalsoatotherspatialandtemporalscalesdifferentfromtheoriginalones,
possiblyresultingininconsistenciesbetweentheoreticalformulationsandmodelapplication.Itisthus
criticalthatfuturemodelingeffortscarefullyaccount for the scale-dependence of their mathematical
formulations,especiallywhenappliedtoawiderangeofscales.
(cid:2)2009ElsevierLtd.Allrightsreserved.
1. Introduction in natural ecosystems (Waksman et al., 1928; Brady and Weil,
2002).
Aboutthree-fourthoftheorganiccarboncontainedinterrestrial Sincethe1930s,severalmathematicalmodelsatdifferentlevels
ecosystemsandthemajorityoforganicnitrogenarefoundinplant of detail have been developed to quantitatively describe these
residuesandsoilorganicmatter(Schlesinger,1997;Lal,2008).Both processes(e.g.,Tanji,1982;Dewilligen,1991;McGill,1996;Molina
organic carbon and macro-nutrients are mineralized to simple andSmith,1998;BenbiandRichter,2002;Shibuetal.,2006).The
inorganicformsbyahighlydynamiccommunityofmicrobialand number and variety of these models mirror a relentless effort to
faunal decomposers (Bradyand Weil, 2002; Berg and McClaugh- describeandquantifythecomplexnatureofsoilsandtheelemental
erty, 2003; Paul, 2007). Although this process of mineralization cyclingwithinthem.Soilsarespatiallyheterogeneousatmolecular
occurs at the decomposer cell scale and is affected by the soil tocontinentalscales(EttemaandWardle,2002;YoungandCraw-
physicalandbiologicalinteractions(e.g.,climateandvegetation),it ford, 2004), and their temporal dynamics span a wide range of
involves globally a gross release of carbon dioxide to the atmo- scalesgoingfromthehourlyresponsestoenvironmentalfluctua-
sphere in amounts one order of magnitude larger than the tions(Austinetal.,2004;SchwinningandSala,2004)andchanges
anthropogenic emissions (Schlesinger, 1997; Lal, 2008) and inresourcesupply(Zelenevetal.,2000),tothedecadaltimescales
providesmostoftheinorganicnutrientsnecessaryforplantgrowth ofecosystemandclimaticchangesandtheevenlongertimescales
characteristicofsoildevelopment(RichterandMarkewitz,2001).
The extreme variety of biogeochemical processes is further
complicatedbyclimaticandanthropogenicexternalforcingfactors.
The development of a mathematical model generally follows
* Correspondingauthor.Tel.:þ19196605511.
E-mailaddress:[email protected](A.Porporato). threesubsequentsteps(Ulanowicz,1979):i)definitionofthestate
0038-0717/$–seefrontmatter(cid:2)2009ElsevierLtd.Allrightsreserved.
doi:10.1016/j.soilbio.2009.02.031
1356 S.Manzoni,A.Porporato/SoilBiology&Biochemistry41(2009)1355–1379
variables for the scale of interest; ii) identification of inputs, comparemathematicalmodelscomingfromdifferentfields,
outputs,andpossibleinteractions;iii)modelverification.Mostof rangingfrommicrobiologytoecosystemecology.
the previous reviews on biogeochemical modeling primarily C Section4analyzestherelationshipsbetweenmodelformu-
addresstheverificationstep(e.g.,DewilligenandNeeteson,1985; lation and scale, discussing how the different formulations
Andre´nandPaustian,1987;Melilloetal.,1995;Powlsonetal.,1996; areusedacrossoratindividualscales.
Jans-Hammermeister and McGill,1997; Smith et al.,1997; Moor- C Section 5 puts the previous analyses into perspective with
headetal.,1999;Zhangetal.,2008).Fewerworksfocusonthefirst respect to the dominant trends in soil biogeochemical
twocriticalsteps:describingtheeffectsofdifferentchoicesofthe modeling and discusses the main limitations of the current
state variables (e.g., Bolkeret al.,1998; Bruun et al., 2004; Pansu approaches. Based on these conclusions, we provide some
et al., 2004; Fontaine and Barot, 2005) and comparing different guidelinesforfutureresearch.
formulations of the interactions among them (Molina and Smith, C Finally, Appendix A reports the list of the reviewed models
1998;MaandShaffer,2001;McGechanandWu,2001;Planteand andtheirmostimportantmathematicalfeatureswithrespect
Parton, 2005; Manzoni and Porporato, 2007; Wutzler and Reich- tothissynthesis.
stein,2008).
The goal of this review is not to test modeling hypotheses or
assess model performances, but to provide an extensive compar- 2. Historicappraisalofmathematicalstructureand
isonofmathematicalapproachestosoilcarbon(C)andnitrogen(N) complexityinsoilbiogeochemicalmodels
cyclinganddiscussamodelclassificationbasedonkineticlawsand
stoichiometry. The difficulty of classifying soil biogeochemical Soil biogeochemical models describe a system including SOM
modelsaccordingtotheirmathematicalformulationarisesbecause constituents (both passive substrates and active biological
of the large number of possible combinations of different model decomposers), interacting with inorganic compounds, environ-
structures (e.g., number of variables and mass flow architecture) mentalvariables,andsubjecttoexternalinputsandoutputs(Fig.1).
andreactionterms(e.g.,linearvs.nonlinearkineticlaws).However, Here we review how the mathematical description of such
allthesemodelsdealwithprocessescontrolledbysimilarbiogeo- acomplexsystemisframed.Wediscussthegeneralmodelstruc-
chemicalconstraints,andbasicallydescribethetransferofmatter tureandspatialresolution(Sections2.1–2.3),thenumberofvari-
fromorganictoinorganiccompounds.Thismineralizationprocess ables used at different scales (Sections 2.4 and 2.5), and the
can be regarded as a complex, biologically mediated series of presenceofstochasticelements(Section2.6).
reactions, where organic substrates are converted into living
biomass and mineral residues (Swift et al., 1979). It is typically 2.1. Fromcompartmenttocontinuum-qualitymodels
modeled by kinetic laws describing organic matter degradation,
where microbial stoichiometric relationships are employed to Although SOM is an extremely heterogeneous mixture of
regulate the associated C–N balances. In what follows we will compounds (Swift et al., 1979), early mathematical models
comparethemathematicalformulationsusedtodescribethesetwo describedtheprocessesofdecompositionandmineralizationusing
fundamentalaspectsofCandNcycling,withthehopetohighlight simplechemicallyandspatiallylumpedmodels(TableA2;Fig.2a).
possible limitations of current approaches and help future cross- Nikiforoff(1936)wasprobablythefirsttosuggestthattheforma-
disciplinarytheoreticaldevelopments. tion of humus may be described by multiple coupled equations
A simplified mathematical representation of the substrate– each characterizing a pool with different turnover times. Later,
microbial biomass interactions will be presented and guide our Minderman (1968) developed similar ideas to describe the
model classification, where indices of model structure and macroscopicpatternsoforganicmatterdegradationresultingfrom
complexity are included, along with a characterization of key thecompoundeffectsofdifferentsubstrates.Thisidearesultedin
biogeochemical processes (Table A2). About 250 mathematical a number of compartmental models. As noted by Halfon,
models developed during nearly eight decades are reviewed, ‘‘Compartmentalanalysisisaphenomenologicalandmacroscopic
considering both highlycited sources and less known theoretical approachformodelingaphysicochemicalprocess.Acompartment
analysesdescribingthevariousaspectsofsoildynamics,aswellas (orstatevariable[.])isabasicunitoffunctionalinterest’’(Halfon,
soil organic matter submodels embedded in hydrologic or 1979, p. 2). Recent models employ a number of such chemically
ecosystem models. Since model structure and formulations are homogeneous compartments, which interact among them and
expectedtochangewiththescaleandlevelofresolutionneededfor possibly with the microbial biomass (Fig.1). Most soil food web
theparticularapplications(ManzoniandPorporato,2007;Manzoni modelsarealsocompartmentmodels,whereparticularattentionis
et al., 2008b), we also analyze the relationships between model given to the trophic interactions among microbial and faunal
formulationsandthetemporalandspatialscaleofapplication. groups(Huntetal.,1987;Deruiteretal.,1993;Zhengetal.,1999;
Thereviewisorganizedasfollows: Zelenevetal.,2006).
Incompartmentmodels,alltheorganicmattermoleculeswith
C Section 2 describes the general structure of soil biogeo- similarchemicalcharacteristicsordegradabilityareincludedinthe
chemical models, with an emphasis on stochastic compo- samepoolandtheinformationregardingtheageorresidencetime
nents,mathematicalformalisms,andlevelofcomplexity(e.g., of biogeochemical compounds within each compartment (or in
thedimensionofthephase-space),andrelatesthesefeatures general since the introduction of organic matter into the soil
tothescalesofinterest. system)isnotexplicitlytracked.Nevertheless, thedistribution of
C Section3presentsanoverviewofthemathematicalformu- the ages of SOM compounds can be reconstructed knowing the
lations used to characterize two key processes in C and N modelstructureandhowthefluxesamongthepoolsaredefined
cycling in soils, i.e., decomposition of organic matter and (Bruunetal.,2004;Manzonietal.,inpreparation).Incontrastto
nitrogen mineralization and immobilization. Bothprocesses typical compartment models, some biogeochemical models
are analyzed under the common framework of substrate– explicitlytracktheevolutionofanyorganicmatter‘‘cohorts’’(i.e.,
decomposer stoichiometry, thus stressing the role of the ‘‘sets of items of the same age’’, Gignoux et al., 2001) from their
microbial biomass as both an SOM degrading agent and as incorporation into the litter or humus and until their complete
acontrollingfactorofNcycling.Thisapproach allowsusto degradation(Furnissetal.,1982;PastorandPost,1986;Ågrenand
S.Manzoni,A.Porporato/SoilBiology&Biochemistry41(2009)1355–1379 1357
Fig.1. Conceptualschemeofthemaininteractionsbetweensoilsubstratesandorganisms,asrepresentedbyspatiallylumpedandspatiallycontinuousrepresentations.Grayand
blank boxes respectively represent carbon and nitrogen compartments and fluxes. The dashed lines in the SOM boxes (left) qualitatively illustrate howthe substrate and
decomposervariablesaretreatedasacontinuumintheQ-model(ÅgrenandBosatta,1996).
Bosatta,1996; Gignoux et al., 2001). This approach provides one Nsubstratesalongaqualityaxis(Fig.1,dashedlines)analytically
way to interpret tracer studies and other paleoecological dating describe both single and multiple cohort dynamics as well as
techniques(Bruunetal.,2004;Bruunetal.,2005). different substrate–microbe networks through a specific function
Incontrasttothediscreterepresentationofsoilorganicmatter, controllingthequalityevolutionduringmicrobialassimilationand
continuum-quality models (Carpenter, 1981; Bosatta and Ågren, turnover(ÅgrenandBosatta,1996).
1985;ForneyandRothman,2007)basedonadistributionofCand
2.2. Modelingspatialvariability
a
Soilsarehighlyspatiallyheterogeneous,withvariabilityofchemical
and physical properties along the vertical profile (Brady and Weil,
2002), around small-scale biologically active patches and individual
plants,andatthelandscapescale(EttemaandWardle,2002;Youngand
Crawford,2004).Compartmentmodelshavebeeninpartextendedto
account for spatial variability (Fig. 1). While the biogeochemical
processes along vertical soil profiles are addressed by a number of
models,onlyfewattemptanexplicitdescriptionofspatialheteroge-
neitiesfromthesoilaggregatetothelandscapescales.
Some early biogeochemical models already employed either
adiscreterepresentationofsoillayerswithdifferentchemicaland
physicalfeatures(BeekandFrissel,1973;vanVeenandPaul,1981)
b
ora continuousdescriptionof SOM andnutrient dynamics along
the soil profile (O’Brien and Stout, 1978). Today, most soil and
ecosystem models consider different soil layers, often connected
throughthewaterflow,whichdrivestheadvectionoforganicand
inorganic dissolved compounds along the profile (Hansen et al.,
1991;Lietal.,1992;Partonetal.,1993;Frolkingetal.,2001;Garnier
et al., 2001; Grant, 2001; Kirschbaum and Paul, 2002; Liu et al.,
2005;MaggiandPorporato,2007;Wuetal.,2007;Jenkinsonand
Coleman,2008).Somemodelsprovideacontinuousdescriptionof
thesoilprofile(BosattaandÅgren,1996),possiblyalsoconsidering
diffusivetransportduetoslowmixingprocesses(O’BrienandStout,
1978;ElzeinandBalesdent,1995;Bruunetal.,2007).
There are fewer models explicitly describing the spatial
Fig.2. Historictrendsofmodelcomplexityandvariablefunctionaltype.(a)Boxplots dynamicsofwater,organicmatter,ornutrientsatthebacterialcell-
ofthenumberofvariablesinthelitterandonesoillayer(phase-spacedimension, orcolony-scale(Allison,2005;Ginovartetal., 2005;Masse etal.,
PSD);(b)percentageoftotalPSDineachfunctionalclassofvariables:physicalenvi- 2007),intheporespace(LongandOr,2005),withinsoilaggregates
ronment,inorganic,microbial,andorganicsubstratepools.Continuousqualitymodels,
(Leffelaar,1993;ArahandSmith,1989),oraroundaroot(Darrah,
with infinite-dimensional phase-space, are not considered; all models published
1991; Toal et al., 2000; Kravchenko et al., 2004; Raynaud et al.,
before1970aregroupedtogether(mostofthemhaveonevariableonly,sothatthe
meanandthequartilesin(a)coincide). 2006).Horizontalspatialvariabilityattheindividualplantscaleis
1358 S.Manzoni,A.Porporato/SoilBiology&Biochemistry41(2009)1355–1379
generally neglected in soil biogeochemical models, while some comparison we only consider one soil layer in vertically discrete
describe landscape level heterogeneity in SOM and litter inputs models to allow the comparison between models with different
(Walteretal.,2003)orsoilhydraulicproperties(Acutisetal.,2000), vertical resolution, and group the relevant variables according to
the effects of erosion and SOM redistribution (Rosenbloom et al., theirfunctionalrole.Wedistinguishfourgroupsofvariablesbased
2001), or horizontal transport of nutrients (Tonitto and Powell, on their specific function, variables used to describe i) microbial
2006).Overall,thereisalackofspatiallyexplicitmodelsproperly biomass, ii) soil organic matter substrates, iii) mineralization
describing soil carbon and nutrient dynamics at different spatial products(e.g.,ammoniumandnitrate),andiv)thephysicalenvi-
scales. ronment(e.g.,soilmoistureandtemperature).
The summaryof the analysis of these characteristicsfromthe
2.3. Mathematicalformalismandmodelsolution w250modelsclassifiedinTableA2isreportedinFig.2,wherethe
ten-year block averages of the number of variables per soil layer
Some earlier biogeochemical models employed difference (Fig. 2a) and the relative importance of each functional type of
equationsorgeometricseriestocomputeorganicmatterandlitter variablesareshown(Fig.2b).Clearly,theincreaseincomputational
accumulation at the yearly time scale (Salter and Green, 1933; powerinthe1970swasparalleledbyanincreaseinmodeldetail
Nikiforoff, 1936; Jenny et al., 1949; Minderman,1968; Jenkinson (Shafferetal.,2001).Sincethen,themediannumberofvariables
and Rayner, 1977). However, the ordinary differential equation stabilizedaroundfive,whilethevariancedecreased.Thisindicates
(ODE)formalism,introducedinsoilbiogeochemistrybyHeninand that in the 1970s few models had a very large phase-space
Dupuis(1945),allowsthedescriptionofSOMdynamicsincontin- dimension(e.g.,Patten,1972;Hunt,1977;Smith,1979),whilemany
uoustimeandhasbeenpredominantlyusedsincethen.Almostall models were still extremely simple; since the 1980s fewer mini-
compartmentmodelscanberecastintosystemsoffirstorderODEs, malist models have been proposed, leading to relatively smaller
each describing the mass balance of a compartment (possibly variance in PSD (Fig. 2a). In the last decade, however, some
withinacohort)orasoillayer.Thenumberofequationsinthese extremelydetailedmodelsthatincludemultipleelementdynamics
systemsofODEsvariesfrommodeltomodel(Section2.4)asdoes and complex biogeochemical interactions have been presented
theirdegreeofnonlinearity(Section3.2).LinearsystemsofODEs (e.g., Grant, 2001). Today, about 70% of the models have 2–10
withconstantcoefficientscanbesolvedanalytically,regardlessof variables,andmorethan90%havelessthan30variables,including
theircompartmentalorganization(e.g.,Olson,1963;Bolkeretal., all functional types (Fig. 3a). This suggests that in most cases
1998;Ka¨ttererandAndre´n,2001;Nicolardotetal.,2001;Baisden arelativelysmallnumberofvariablesmaybesufficienttodescribe
andAmundson,2003;Saffih-HdadiandMary,2008;Manzonietal., soilCandNdynamics(inagreementwithanalysesbyBolkeretal.,
in preparation), while in general nonlinear models need to be 1998),whileahighnumberofvariablesmay beneededtodetail
solved numerically. Despite the presence of nonlinearities, specific processes (e.g., methanogenesis and denitrification in
a number of models can at least be solved analytically at steady additiontoCandNmineralization).
state (by setting the time derivatives of the ODEs to zero), thus
providing useful information on the asymptotic behavior of the
system (e.g., Loreau,1998; Daufresne and Loreau, 2001; Manzoni
et al., 2004; Fontaine and Barot, 2005; Manzoni and Porporato, a
2007;WutzlerandReichstein,2008).
In contrast to compartment models, those based on the
continuumqualityconcept,involvingbothtimeandorganicmatter
quality as independent variables, are in the form of systems of
partial (and sometimes integro-) differential equations (PDEs)
(Carpenter, 1981; Ågren and Bosatta, 1996). Such equations are
amenable to analytical solution only for suitable choices of the
decomposergrowthanddecayfunctions(ÅgrenandBosatta,1996;
Bosatta and Ågren, 2003), in which case leading to relatively
compact representations of the complex soil dynamics. To our
knowledge, only one model uses delayed differential equations
(DDEs)anddescribesnitrogenmineralizationasapistonflow,i.e.,
mineralizationattimetisassumedequaltotheNfluxenteringthe
soil at the time t(cid:2)Dt (Thornleyet al.,1995). DDEs can be inter- b
preted as a special form of PDEs, and like them can be seen as
equivalenttoinfinite-dimensionalsystemsofODEs.
2.4. Historicevolutionofmodelcomplexity
Afirstmeasureofmodelcomplexity(nottobeconfusedwith
themeasuresofdynamiccomplexity)isanalyzedinthissectionby
considering the number of first order differential equations
necessarytorepresentthedynamicsoflitterandanindividualsoil
layer.Section3willanalyzemodelcomplexityintermsofinternal
nonlinearitiesandfeedbacks.
Thesystemsofequationsdescribingsoilbiogeochemicalmodels
can be interpreted as dynamical systems (Argyris et al., 1994;
Strogatz,2000;Manzonietal.,2004)andthenumberoffirstorder
Fig.3. Percentagesofmodelswithgivennumberofvariables(phase-spacedimension,
ODEsdefinestheso-calledphase-spacedimensionofthedynam-
PSD)(a),andwithgivennumberofdecomposervariables(b).Modelsbasedonpartial
ical system (PSD). When counting the number of ODEs in this anddelaydifferentialequationsareindicatedbyN.
S.Manzoni,A.Porporato/SoilBiology&Biochemistry41(2009)1355–1379 1359
Regardingspecificallythenumberofstatevariablesdevotedto should exist, on the ground that ‘‘short time intervals are most
microbialbiomass,Fig.2bshowsthatithasremainedproportional appropriateforfinelevelsofspatialresolution,andthatlongertime
totheotherfunctionaltypes(e.g.,substratepools),suggestingthat intervalsmaybevalidovercoarserspatialscales’’(McGill,1996,p.
modelerstendtofavorabalancebetweenthelevelofdetailinthe 119). Despite the large scatter, our results confirm this view
passiveandthebiologicallyactivecompartments.Nevertheless,the (Fig.4b),showingahighlysignificantpositivecorrelationbetween
histogramofbiomassvariableshaschangedthroughtime(Fig.3b). spatialandtemporalscales.
Inthe1970s,onlyabout40%ofthemodelsexplicitlyaccountedfor As it will be shown in the next sections, a number of mathe-
the decomposers, but 15% had very detailed microbial models maticalfeaturestendtobelinkedmoretothefieldofapplication
(above 10 variables). In contrast, morerecently, fewer models do than to the spatial and temporal scales at which the models are
notincludeanydynamiccompartmentforthemicrobialbiomass: applied. We denoted these application fields as ‘‘classes’’, and
about70%haveatleastone,themajorityofwhichhaveatmostten, distinguished among models describing small-scale processes
with very few models employing a large number of variables (microbiology,rhizosphere,andaggregatemodels,indicatedbyM),
(Fig. 3b). Also, earlier models did not include variables for the plantresiduedecomposition(L),SOMdynamics(S),coupledsoil–
physical environment, while today many models consider soil plant dynamics(E), and models developed for global-scale appli-
moistureandtemperatureasdynamiccomponents(Fig.2b),often cations (G). As shown in Fig. 5a, most models are in class S, and
bycouplingsoilwaterandheat balance equationstothebiogeo- fewerareinclassesR,L,andG.Theseapplicationfieldsoftencover
chemicalmodel. awiderangeofspatialortemporalscales(Fig.5b).Rhizosphereand
aggregate models are applied at fine spatial and temporal scales,
while litter decomposition and ecosystem models are generally
2.5. Modelcomplexityacrossspatialandtemporalscales
applied at the field scale and at monthly-to-yearly time scales.
Modelsusedforglobalsimulationsandclimatechangeprojections
We analyzed the spatial and temporal scales at which each
are frequently designed for daily or monthly time scales, and
modelisinterpretedintheirtypicalapplications(seeAppendix1
mainly applied on regional domains. This indicates that the
andTableA1fordetails),andcomparedthemtothemodelphase-
temporal resolution is perceived to be more important than the
spacedimension(PSD).Fig.4ashowsaclearinverserelationship
levelofspatialdetailinsuchmodels.Finally,SOMmodels(S)tend
between the average PSD and the temporal scale of the model,
tobeusedatrelativelysmallspatialscales(uptothefieldscale),
which mirrors the necessity to describe in detail (i.e., with large
andatdaily-to-yearlytimescales.
enoughnumberofvariables)highlydynamicsmall-scaleprocesses.
Similarly, Fig. 4c seems to indicate a trend towards lower model
complexity (in terms of PSD) for medium-to-large spatial scales, 2.6. Deterministicvs.stochasticmodels
possibly reflecting the use of few highly aggregated variables at
those scales. In this regard, McGill (1996) hypothesized that Due to the existence of a large number of internal processes,
a correlation between spatial and temporal scales of each model highly intermittent external forcing (e.g., hydroclimatic
% of models
a b
d) 1000 d) 1000 15
e ( e (
cal 100 cal 100
S S 10
al al
or 10 or 10
p p
m m 5
Te 1 Te 1
0
100 101 102 0.01 1 100 10000
PSD Spatial Scale (m)
c
102
D
S
P
101
100
0.01 1 100 10000
Spatial Scale (m)
Fig.4. Scalesandmodelcomplexity.(a)Boxplotsofthephase-spacedimension(PSD)asafunctionoftemporalscaleindays;(b)percentageofmodelswithgivenspatialand
temporalscales,alongwiththetypeIIregressionlinebetweenthem(dot-dashedline;R¼0.45,P<0.0001);(c)boxplotsofPSDasafunctionofspatialscale.Detaileddataforeach
modelarereportedinTableA2.
1360 S.Manzoni,A.Porporato/SoilBiology&Biochemistry41(2009)1355–1379
a manner, it becomes possible to characterize the model results in
100 term of means, distributions around them, and frequency of
extremesinastatisticallymeaningfulway.Hencesuchmodelsmay
help assess the role of climatic variability and quantify uncer-
80 taintiesinmodelprojections(D’Odoricoetal.,2004).Inthissection
wediscusstheuseofstochastictermstoaccountforinternal(i.e.,
structural), spatial, and temporal uncertainties, while we do not
s
el 60 considersensitivityanalysesbasedonMonteCarlosimulations.
d
o Despite the clear need of a stochastic approach, only few
m
of biogeochemicalmodelsemploystochasticcomponents.Amongthe
% 40 few available examples of internal randomness, the Q-model
employs a statistical description of the substrate quality to over-
comelimitationsintheuncertaintyinmodelstructure(Carpenter,
20 1981; Bosatta and Ågren, 1985). The quality of the degraded
substrateisconsideredasaninternalrandomvariabledescribedby
aprobabilitydensityfunction(PDF)evolvingintimeaccordingto
0 dynamicinteractionsofthesubstratewithbioticandabioticfactors
M L S E G
Model class (Ågren and Bosatta, 1996), and in space along the soil profile
according to convective and dispersive mechanisms (Bosatta and
b
Ågren, 1996). As noted before, the use of a PDF to describe the
qualityevolutionintimesubstituteswithaflexibleframeworkthe
commoncompartmentaldeterministicstructure(Fig.1).Moreover,
1000 the continuum decomposability concept has recently been given
a theoretical interpretation based on the physical limitations to
d) substrate diffusion in a random porous medium (Forney and
ale ( 100 Rothman,2007;RothmanandForney,2007).Regardingtheuseof
c internalstochasticcomponentstocontrolsoilfluxes,theonlyother
s
al examplethatwefoundistheindividual-basedmodelbyGinovart
r
po 10 etal.(2005),wherethemaximumnumberofparticlesthatcanbe
m assimilated per unit time and unit cell area is a randomvariable
e
T extracted from a normal distribution. All the other models we
1 reviewed compute the fluxes according to deterministic kinetic
equations.
There are numerous examples in the literature of randomly
distributed parameters to account for spatial variability. In the
0.01 1 100 10000 aggregate-scale model of denitrification developed by Arah and
Spatial scale (m)
Smith (1989), both aggregate radius and oxygen demand are
assumed to be random variables lognormally distributed. Other
M: Aggregate and rhizosphere models modelsbasedondistributedparameters(HuweandTotsche,1995;
Whitehead et al., 1998; Acutis et al., 2000; Walter et al., 2003;
L: Litter models
Botteretal.,2006)accountforsoilheterogeneityatthefieldtothe
S: Soil models
landscapescale.Wenotethatalsothespatiallyexplicitmodelsthat
E: Ecosystem models employ advection diffusion equations can be interpreted as
macroscopic representations of biased randomwalk processes at
G: Global models
theparticlescaleandaccordingtothisinterpretationwouldbelong
to the category of stochastic models (O’Brien and Stout, 1978;
Fig.5. (a)Percentageofbiogeochemicalmodelsindifferentmodelclasses(seealso
Darrah,1991;ElzeinandBalesdent,1995;BosattaandÅgren,1996;
TableA1forcodeexplanation);(b)typicalspatialandtemporalscalesofthemodel
classes.Eachlinemarkstherangesoftemporalandspatialscaleswithinwhich50%of Bruun et al., 2007; Jenkinson and Coleman, 2008). The model by
themodelsineachclasslie(assumingthatthejointpdfofspatialandtemporalscales Walteretal.(2003)accountsfortherandomspatialvariabilityof
islognormal). SOMcontentandlanduse,whichinturndrivesthelitterinputinto
thesoil.Also,thetemporalchangefromonelandusetoanotheris
fluctuations,disturbanceevents,etc.),uncertaintyinstructureand controlledbyaMarkovchaininthismodel.
coupling of state variables, and spatial heterogeneity, a detailed A few models address the stochastic nature of the climatic
descriptionofthesoilsystemwouldinprinciplerequiretheuseof forcing (van Veen et al., 1984; Pastor and Post, 1986; Porporato
an extremely high number of variables. However, given the etal.,2003;Botteretal.,2008;Dalyetal.,2008;Wangetal.,2009),
impossibility of achieving such a detailed modeling, it is highly whichaffectsthroughsoilmoistureandtemperaturetherates of
desirabletoshiftfromhigh-dimensionalmodels(i.e.,largePSDand soil biogeochemical fluxes. In the last four papers, for example,
highly fluctuating variables in space and time) to a suitable rainfallwastreatedasamarkedPoissonprocessintime.Thisallows
combination of deterministic variables and random functions or onetoassesshowtheSOMandmineralcompartmentsrespondto
stochasticprocesses(Katuletal.,2007),therebyprovidingamore intermittentwettingeventsatdifferenttimescalesandtoquantify
parsimonious and balanced representation of the soil system. At thepropagationofhydrodynamicfluctuationsthroughthesystem
thesametimeastochasticrepresentationmakesitpossibletogo components(D’Odorico et al., 2003). The same stochastic rainfall
beyondaspecificsimulationandobtainresultsthatarerepresen- model has been also recently used to drive a simple basin-scale
tative ofawholerangeofenvironmentalconditionsorstructural nitrate transport model (Botter et al., 2006). The statistics of
features,eachleadingtoadifferenttrajectoryforthesystem.Inthis denitrification resulting from stochastic rainfall patterns were
S.Manzoni,A.Porporato/SoilBiology&Biochemistry41(2009)1355–1379 1361
computedbyRidolfietal.(2003)startingfromempiricalrelation-
shipsbetweendenitrificationfluxesandsoilmoisture,butwithno
dynamiccomponent.
3. ModelingdecompositionandNmineralization
Mostmodelsofsoilsubstrate–decomposerinteractionscanbe
castinacommonformintermsofbalanceequations(Section3.1),
Fig. 6. Carbon and nitrogenpools and fluxes in the simplified model of microbial
allowing us to track the historic evolution of the mathematical biomass–substrate interactions (Eqs. (1) and (2)). Continuous and dashed lines
structureofbiogeochemicalmodelsfromthe1930stothepresent represent C and N fluxes; gray and blank compartments are the C and N pools,
day.Tothis purposeweintroduce asimplified theoretical frame- respectively.Themicrobialbiomasscompartmentandrelatedfluxesaredetailedin
Fig.9.Theschemeisusedtodefinethespecificprocessesdiscussedinthetext,andis
work for the substrate–decomposer dynamics to explain the
notmeanttobethemostgeneralmodelstructureforsoilCandNmodels(forexample,
differences and similarities in the formulations of decomposition
itdoesnotconsidermultiplesubstratesanddecomposerpopulations).
(Section 3.2) and nutrient mineralization (Sections 3.3, 3.4;
symbolsareexplained inTable 1, othercodes inTable A1). Using
thisgeneralframeworkwereviewthevariousmodelsintermsof dC ðtÞ
how they logically and historically collocate themselves with dBt ¼ DEC(cid:2)RG(cid:2)RM(cid:2)RO(cid:2)BD; (1)
respecttoit.Weemphasizethelawsregulatingmicrobialstoichi-
ometry, which deal with the combination of the structural and whereDECisthedecompositionflux(discussedinSection3.2),BD
nutritionalelementsallowingthegrowthofdecomposercommu- isthemicrobialdecayterm;RGandRMarerespectivelythegrowth
nities.Wefurtherfocusontheinterplaybetweencarbon,whichis andmaintenancerespirationfluxes,whileROrepresentsCoverflow
the typical energy source for heterotrophs, and nitrogen, often associated with N-limitation. Growth respiration is generally
considered to be the most limiting nutrient for soil and litter assumed to be proportional to DEC (i.e., RG ¼ rDEC) and repre-
decomposers.A similarmathematical frameworkmaybe usedto sentstheonlyrespirationfluxconsideredinmostmodels(denoted
describetheroleofotherlimiting nutrients,suchasphosphorus, byGRWinTableA2).Maintenancerespiration(MNT)dependson
potassium,andsulfur(Smith,1979;Huntetal.,1983;Partonetal., metabolicactivitiesnotrelatedtogrowthandisgenerallymodeled
1988; Bosatta and Ågren,1991; Sinsabaugh and Moorhead,1994; asalinearfunctionofmicrobialbiomass(e.g.,DaufresneandLor-
CherifandLoreau,2009;Manzonietal.,underreview). eau, 2001; Pansu et al., 2004; Moorhead and Sinsabaugh, 2006).
Somemodels,indicatedbyG&MinTableA2,considerbothgrowth
and maintenance respiration in their microbial biomass balance
3.1. Generalmicrobialbiomassbalanceequations equations. Incontrast tomaintenancerespiration,the Coverflow
flux (CO) is defined to accommodate decomposer stoichiometric
Becausemicrobialbiomassisthedriverofmostsoilcarbonand requirements in conditions of nutrient limitation (sensu Schimel
nitrogen cycling, we begin by writing a general form of the and Weintraub, 2003), as discussed in Section 3.4.1. Other meta-
microbialCandNbalanceequations.Weconsiderasimplifiedsoil bolicprocessesleadingtoCoverflow(e.g.,RussellandCook,1995)
systemwhere a single compartment of substrate, including both areseldomaccountedforinsoilmodelsandwillnotbedealtwith
carbon (CS) and nitrogen (NS), is decomposed by the microbial here.ItisimportanttostressthatEq.(1)illustratesinasimplified
biomass(CBandNBaremicrobialcarbonandnitrogenconcentra- way the pathways of C exchange with the decomposer biomass,
tions).WithreferencetoFig.6,andfocusingonlyontheequations where each respiration term plays a distinct functional role. This
relevantforthemicrobialdynamics,wecanwriteforthemicrobial idealizationof themetabolicprocessesoccurringatthecell scale
carbonbalance allows us to classify the different model formulations that have
beenproposed.
ThefluxDECnotonlycontrolsC,butalsodeterminestheorganic
Table1 N decomposition. In fact, N is generallyassumed to follow the C
Explanationofsymbols(seealsoFigs.6and9).
fluxes,asbothelementsarebondintotheorganiccompounds.As
Symbol Description aresult,theamountsofCandNmadeavailablethroughdecom-
BD Microbialbiomassdecay position are proportional to the substrate. While C can only be
CBðNBÞ Microbialcarbon(nitrogen)concentration assimilated and recycled through the microbial biomass, or
CSðNSÞ Substratecarbon(nitrogen)concentration respired(Eq.(1);Figs.1and6),Ndynamicsaremorecomplex.N
ððCC==NNÞÞBCR MCriictircoabliaCl=NC=rNatriaot,idoefinedasðC=NÞB=ð1(cid:2)rÞ can be directly assimilated in organic forms, mineralized to
ðC=NÞIMM Netimmobilizationthreshold,definedashðC=NÞB=ð1(cid:2)rÞ ammoniumandeventuallynitrate,orimmobilizedbythemicrobes
ðC=NÞLIM MineralN-limitationthreshold from these mineral pools. Accounting for all these pathways, the
DEC Carbondecompositionflux(Eqs.(3)–(5)) microbialNbalancecanbewrittenas
IMMgross Grossimmobilization(Eq.(8))
IMMmax Maximumimmobilization dN ðtÞ DEC BD
kS;LIN,kS;MULT, Decompositionratesforlinear,multiplicative,andMichaelis– B ¼ h (cid:2)F(cid:2) ; (2)
kS;MM Mentenformulations(Eqs.(4),(5),and(3),respectively) dt ðC=NÞS ðC=NÞB
KMM Michaelis–Mentenconstant(Eq.(3))
N MineralNconcentration wherethefirsttermistheorganicNassimilation,whichdepends
r FractionofDECthatisrespired on the organic N assimilation efficiency h (i.e., the fraction of
RG Growthrespiration organicNdirectlyassimilatedbythemicrobes,seeSection3.3).The
RM Maintenancerespiration secondterm,F,isthenetNfluxexchangedbetweenthemicrobial
RO Coverflowlosses(Eq.(10))
h OrganicNassimilationefficiency biomassandthemineralNcompartment(Section3.4),whilethe
4MNð4ONÞ Mineral(organic)Ninhibitionfactor(Eqs.(9)and(11), lasttermrepresentstheNlossesduetomicrobialdecay.
respectively) Theoverflowrespirationandmineralizationfluxesaregenerally
F Microbialstoichiometricimbalance(Eq.(7))
defined to ensure a given composition of the microbial biomass,
1362 S.Manzoni,A.Porporato/SoilBiology&Biochemistry41(2009)1355–1379
which in soils tends to vary within a relatively narrow range of microbialbiomassisthusregardedas‘‘asubstrateofdecomposi-
valuesbothacrosssubstratequalities(ClevelandandLiptzin,2007) tion,ratherthanasadecomposer’’(Fangetal.,2005).Accordingto
and in time (Garcia and Rice,1994; Jensen et al.,1997). The next other interpretations, Eq. (4) implicitly assumes that microbial
sections describe the different formulations used to model each activity changes so quickly that microbial biomass itself is never
terminEqs.(1)and(2),andhowthedifferentmodelsbalancetheC alimitingfactor(Paustian,1994;Smithetal.,1998),orthatphysical
andNfluxesaccordingtomicrobialstoichiometry. processes independent of microbial biomass and activity limit
decomposition(RothmanandForney,2007;Kemmittetal.,2008).
3.2. Thedecompositionfunction Whenmicrobialresponsestoenvironmentalstressesorpriming
effectsareinvolved,Eq.(4)istoosimple,andthebioticcomponent
Thedefinitionofthedecompositionflux,DEC,isoneofthemost of decomposition cannot be neglected (Schimel, 2001; Neill and
criticalissuesinsoilCandNmodels,foritembedsmostbioticand Gignoux, 2006). In these cases, Eq. (3) can be simplified by only
abioticcontrolsontheorganicCandNflowtowardsmorestable assuming CS(cid:3)KMM, while still considering the variability of the
formsormineralpools(ManzoniandPorporato,2007;Wutzlerand microbial biomass. We thus obtain the multiplicative model
Reichstein, 2008). Here we follow the more common practice of (MULT),
basingthestoichiometricequationsonorganiccarbon,C ,although
S
DEC ¼ k C ðtÞC ðtÞ; (5)
insomemodelstheyaredefinedasafunctionofN (Daufresneand S;MULT S B
S
Loreau,2001;Raynaudetal.,2006),orbothelements(Smith,1979;
whichincludesthebasiccouplingofbothreactionparticipants(i.e.,
NeillandGignoux,2006).Herewefocusonthedegreeofnonlin-
decomposers and substrate), while being simpler than Eq. (3)
earityinthedecompositionequation,andrestrictourattentionto
(HarteandKinzig,1993;Whitmore,1996b;Porporatoetal.,2003;
decomposition under optimal nutrient conditions, while correc-
Schimel and Weintraub, 2003; Moore et al., 2005). In Eq. (5) the
tionstoDECduetoN-limitationareconsideredinSection3.4.1.
substrate–decomposer coupling is analogous to a Lotka–Volterra
Decomposition can be regarded as an enzyme catalyzed reac-
predator–preyinteraction,anditmaysimilarlygiverisetodamped
tion, where the C flux from the organic substrate degradation
oscillatory dynamics, as sometimes observed in soil variables
dependsonboththesubstrateandtheavailableenzymeconcen-
(ManzoniandPorporato,2007).
tration (Schimel and Weintraub, 2003; Fang et al., 2005; Sinsa-
Finally,anddifferentlyfromtheprevioussimplifications,some
baugh et al., 2008). These enzymes are produced by the modelsassumethatC [K ,thusleadingtoamodellinearwith
S MM
decomposer community, and hence may be assumed to be
respect to the variable C (LINB). This assumption can be made
B
proportionaltotheamountofmicrobialbiomass.Accordingly,the
whentheactuallimitingfactorsfordecompositionarethemicro-
Michaelis–Mentenequation(MM),oftenusedtodescribeenzyme
bialenzymes,andnotthesubstrates,asincaseofrecalcitrantSOM
catalyzedreaction,isarealisticbutparsimoniouschoice(Panikov
(FontaineandBarot,2005).Thisdecompositionformulationisalso
andSizova,1996;BlagodatskyandRichter,1998),
implicit in steady state models where linear microbial or faunal
deathratesareused,andimpliesatop-downcontrolinthedetritus
C ðtÞ
DEC ¼ k C ðtÞ S ; (3) foodchain(Huntetal.,1987;Deruiteretal.,1993).
S;MM B K þC ðtÞ
MM S Given the variety of model types employed in the past, it is
where K is the Michaelis constant and k accounts for the important to emphasize that the choice of the decomposition
MM S;MM
chemical composition of the substrate and the effects of climatic model affects the model ability to describe complex dynamics.
conditionsduringdecomposition. Linear models, although employing multiple pools, behave like
On the one hand, more complex nonlinear models (NL) have pure decay functions and cannot produce fluctuating dynamics
alsobeenadopted,includingnegativefeedbacksduetomicrobial under constant environmental conditions (Bolker et al., 1998;
density (Garnier et al., 2001), or both microbial density and co- Baisden and Amundson, 2003). Manzoni and Porporato (2007)
metabolismeffects(McGilletal.,1981;Huntetal.,1991).Inthese showed that nonlinear models can describe complex dynamics
cases,however,moreparametersareneededandmodelcalibration
becomesmoredifficult.Ontheotherhand,simplifiedversionsof
Eq.(3) havebeen oftenusedaswell. Initsmoredrastic simplifi-
cationitisassumedthateitherthemicrobialpoolisconstantorthe
substrate concentration is much larger or, in contrast, negligible
withrespecttoK .ThusifC isassumedtochangeslowlywith
MM B
respect to the substrate, and CS(cid:3)KMM, Eq. (3) can be approxi-
matedbysimplefirstorderratekinetics(LIN),
DEC ¼ k C ðtÞ; (4)
S;LIN S
wherek isthefirstorderdecayrate.Eq.(4)hasbeenusedsince
S;LIN
the earliest models describing organic C and N decrease in agri-
cultural systems (Jenny, 1941), or litter and humus accretion in
grasslandsandforestedecosystems(Nikiforoff,1936;Olson,1963).
Thesepioneeringworksestablishedtheparadigmoflinear,donor
controlled decomposition rate (Table A2), which today still char-
acterizes most biogeochemical models, as shown in Fig. 7 (Dew-
illigen, 1991; McGill, 1996; Molina and Smith, 1998). The linear
model accounts for the substrate chemistry and ‘‘provides an
excellent first approximation, especially during early stages of Fig.7. Historictrendsof the percentages of models using differentdecomposition
functions(DEC).CONS,constantdecayrate;LINandLINB,firstorderratewithrespect
decay’’(BergandMcClaugherty,2003,p.2);however,itcompletely
tothesubstrateorthemicrobialbiomass,respectively;MULTandMM,multiplicative
neglectstheroleofmicrobialbiomassanditsenzymaticproducts andMichaelis–Mentenformulations,respectively;NL,othernonlineardecomposition
inthedecompositionprocess.Inthesedonor-controlledmodelsthe equations(seealsoTableA1;detailsontheindividualmodelsarereportedinTableA2).
S.Manzoni,A.Porporato/SoilBiology&Biochemistry41(2009)1355–1379 1363
(e.g.,fluctuations)occurringatshorttimescaleswhenthecoupling
between substrates and decomposers is particularly strong, as in
therhizosphere(Zelenevetal.,2000).
Historically,linearmodelshavebeenpredominant,althoughthe
more recent biogeochemical models often adopt nonlinear or
mixed decomposition formulations (Fig. 7). While this tendency
mirrors the criticism against the linear decomposition paradigm
(SchimelandWeintraub,2003;Fangetal.,2005;NeillandGignoux,
2006; Kuzyakov et al., 2009), some recent theoretical consider-
ations and experimental evidence also support the concept of
linear, substrate-controlled decomposition (Forney and Rothman,
2007; Rothman and Forney, 2007; Kemmitt et al., 2008). In any
case,notwithstandingthetrendtowardsmorenonlinearformula-
tions, morethan50% of models developed in the last decade are
basedonfirstorderdecompositionkinetics.Notably,thereisalso
anincreaseintheuseoflinearkineticswithrespecttothemicro-
bialbiomass(LINB),whichallowanalyticaltractability(contraryto
the nonlinear formulations) while accounting for the role of Fig. 8. Historictrends of the percentages of models using different mineralization
microbialactivity.Howdecompositionshouldbemodeledremains pathways(MIN).DIR,MITandPAR:direct,mineralization–immobilizationturnover,
andparallelschemes,respectively;MIX,otherschemeswithsimultaneousminerali-
currently an unresolved issue, and it is likely that the answer is
zation and immobilization; SIMP, simplified mineralization formulations (see also
scale-dependent, as discussed in Section 4, and related to the
TableA1;detailsontheindividualmodelsarereportedinTableA2).
specificpurposeofthemodel.
3.3. Mineralization–immobilizationpathways
tends to behave according to the DIR hypothesis in relatively
homogeneoussoils,whilethemacroscopic-scalebehaviorofmore
Inordertomodelthemineralizationandimmobilizationfluxes
(i.e.,thetermshDECðC=NÞ(cid:2)1andFinEq.(2);seealsoFig.6),two heterogeneoussoilsisbetterrepresentedbythePARscheme.The
S value of h can thus be interpreted as an effect of soil chemical
alternative schemes have been originally proposed. The first
heterogeneity, and the estimated values span the whole range
scheme is based on the Mineralization–Immobilization Turnover
(MIT) hypothesis, which was developed after early 15N tracer between zero and one (Manzoni and Porporato, 2007; Manzoni
etal.,2008b).Asitwillbecomeclearinthenextsections,thisvalue,
studiesdemonstratedtheexistenceofastrongNcyclingbetween
and thus the structure of N cycling, plays a major role in the
organic and the mineral fractions (Kirkham and Bartholomew,
microbial C and N balances when inorganic N availability is
1954,1955;Jansson,1958).ThisschemeassumesthattheorganicN
assumedlimited.AsshowninFig.8,thePARschemeisbecoming
istransferredtothemineralpoolbeforeassimilationbymicrobes.
increasinglymoreused.
Fromamodelingpointofviewthisimpliesthath ¼ 0andthere-
fore that the only N assimilation pathway available is from the
mineral pool through F. Although at the micro-scale the MIT
3.4. Modelingmicrobialstoichiometry
pathway is not physiologically realistic, because the deamination
reactionsresultinginammoniumformationareendogenous(Swift
In this section we first describe the basic equations that are
et al.,1979), such a recycling may be reasonable at the soil-core
commonlyemployedinbiogeochemicalmodelstodefinetheCand
scaleðz10(cid:2)1mÞ,wheretheheterogeneousdistributionofN-rich
NdemandofthemicrobialbiomasswhenðC=NÞ isassumedtobe
and N-poor substrates (the former mineralizing N, the latter B
constant and how they are interpreted to account for a possible
immobilizing it) may drive strong N recycling (Schimel and Ben-
imbalancebetweenorganicCandNavailabilities,withthepossible
nett,2004).
limitingeffectofinorganicN(Sections3.4.1and3.4.2).Finallywe
The second alternative scheme more realistically assumes the
discusstheeffectsofflexiblemicrobialelementalcomposition,i.e.,
existence of some direct assimilation of organic N in low weight
ðC=NÞ sconstantandreviewthemodelsthatadoptthisgeneral-
compounds, namely amino-acids, the N surplus of which is B
ization(Section3.4.3).
released in mineral form. Accordingly, in Eq. (2), h ¼ 1 and all
available organic N is directly assimilated prior to mineralization
3.4.1. Stoichiometryofhomeostaticdecomposers
(Directhypothesis,DIR;Molinaetal.,1983).Foritssimplicityand
Traditionally,inbiogeochemicalmodelsthemicrobialbiomass
realism, the DIR scheme became prevalent in biogeochemical
has been considered tobe strictly homeostatic (i.e., with compo-
modelsduringthe1970s(Fig.8;TableA2),graduallyreplacingthe
sitionindependentofthesubstratecharacteristics;seeSternerand
MITscheme.
Elser (2002), Cleveland and Liptzin (2007)). In our theoretical
In real cases, however, a combination of the DIR and MIT
framework(Section3.1)thistranslatesintoaconstantðC=NÞ ,and
pathwaysistypicallyobservedatthemacroscopiclevel,justifying B
hencedðC=NÞ =dt ¼ 0.UsingEqs.(1)and(2),thisimpliesthatthe
the adoption of models that employ either one or the other B
totalCinputinthemicrobialpoolmustbeequaltothetotalNinput
pathwaydependingonthe decomposergroup(denoted byMIX),
multipliedbyðC=NÞ (seealsoFig.6),thatis
and of the more flexible Parallel scheme (PAR, e.g., Barraclough, B
1997; Garnier et al., 2001). Inparticular, the PAR scheme bridges (cid:2) DEC (cid:3)
DIRandMITbyusinganorganicnitrogenassimilationefficiencyh ð1(cid:2)rÞDEC(cid:2)RO(cid:2)RM ¼ ðC=NÞB hðC=NÞ (cid:2)F : (6)
thatmaytakeanyvaluebetweenzero(MIT)andone(DIR).Thus, S
while a fraction h is directly assimilated, the fraction ð1(cid:2)hÞ is Eq.(6)isextremelygeneraland,asitwillbeshowninthefollowing,
mineralized without assimilation (Eq. (2); Fig. 6). A derivation of can be used to derive specific equations for microbial C and N
the link between h and soil chemical heterogeneity has been balances when either organic C or organic N controls microbial
proposed by Manzoni et al. (2008b), who showed that N cycling growth. Assuming that the decomposition model (DEC) and the
1364 S.Manzoni,A.Porporato/SoilBiology&Biochemistry41(2009)1355–1379
mineralizationscheme(h)havebeenchosen,andthatthemicro- FbecomesnegativeandimmobilizationofmineralNisneededto
bialparametersðC=NÞ andrareknown,therearedifferentpossible compensatefortherelativescarcityoforganicNfromthesubstrate
B
ways (and thus modeling schemes) to ensure constant microbial (hatchedareasinFig.10).LowerorganicNassimilationefficiencies
biomasscompositionwhilethesubstrateNconcentrationchanges. hfavorimmobilization,asmoremineralNisneeded.Whenh ¼ 0
For the sake of simplicity, we will now neglect the maintenance (MIT) the N demand becomes independent of the substrate,
respirationtermR ,inagreementwithsomeobservations(McGill becauseallNneededforgrowthisimmobilizedfromthemineral
M
etal.,1981)andastypicallydoneinbiogeochemicalmodels(Table pool (dotted lines in Fig. 10a). The combinations of organic N
A2), and consider only the C overflow fluxes involved in main- assimilation efficiency h and substrateC=N leading to net miner-
taining the decomposer stoichiometric balance. These simplifica- alizationorimmobilizationarealsoillustratedinFig.10b.Higherh
tions allow deriving from Eq. (6) a number of expressions often allows net mineralization at high ðC=NÞ , while mineral N avail-
S
used in soil models to describe C and N fluxes. We start by dis- ability controls the onset of mineral N-limitation, as discussed
cussingthecaseofrelativelyhighconcentrationoforganicNinthe below.
substrate,andthenmovetoprogressivelylowerNavailability(i.e., When organic N availability further decreases, F becomes
increasingðC=NÞ ,asschematicallydepictedfromtoptobottomof increasingly negative, and larger amounts of mineral N are
S
Fig.9). extractedfromthesoil.ThemineralNpool,however,isnotalways
WhenðC=NÞ isrelativelylow(organicNinexcess;topofFig.9), abletosupplyenoughN,inwhichcasemineralN-limitationmay
S
carbon limits decomposition, so that no C overflow is necessary occur(lowerpartofFig.9;shadedareasinFig.10).Thethresholdof
ðRO ¼ 0Þ.Onthecontrary,Nisinexcess,andnetNmineralization substrateC=NatwhichN-limitationoccursisdefinedasðC=NÞLIM
is needed to keep ðC=NÞ constant and let the microbes grow at (Fig.10b).Biogeochemicalmodelsaccountforsuchalimitationin
B
a rate controlled by the C decomposition, i.e., ð1(cid:2)rÞDEC. The N various ways. Commonly, a factor is defined to reduce potential
demandforthemicrobesisthusgivenbyð1(cid:2)rÞDECðC=NÞ(cid:2)1,which immobilization and decomposition, as a function of the available
B
issmallerthantheamountoforganicNassimilated,hDECðC=NÞ(cid:2)1. mineralN(Fig.11a).Alternatively,aCoverflowtoeliminateexcess
S
Asaresult,FispositiveandiscomputedfromEq.(6)tocompen- C is employed (Fig. 11b). These two modeling approaches are
satetheexcessN,thatis, describednext.
F ¼ DEC(cid:2) h (cid:2) 1 (cid:3); (7) C N inhibition hypothesis (INH). According to this scheme,
ðC=NÞS ðC=NÞCR immobilization is limited by mineral N availability, and
decomposition is reduced to the point that ensures the
whereðC=NÞ ¼ ðC=NÞ =ð1(cid:2)rÞisthecriticalC-to-Nratioofthe
CR B correct C=N in the flux feeding the biomass pool. To model
substrate(BosattaandStaaf,1982;ManzoniandPorporato,2007).
this mathematically, we follow Manzoni and Porporato
Fig.10aplotsthefunctionF=DECfromEq.(7).Clearly,whenthe
(2007)andfirstdefinetheactualgrossimmobilizationasthe
Ncontentof thesubstratedecreases,theexcessmineralizationis
productofthepotentialNdemandF(Eq.(7))andareduction
progressively reduced until the threshold ðC=NÞ ¼ hðC=NÞ
IMM CR coefficient ð4MNÞ that accounts for mineral N availability
is reached, corresponding to F ¼ 0. When the substrate C=N is
(Eq.(9)),
further increased beyond that threshold (central part of Fig. 9),
Fig.9. ConceptualviewoftheeffectsofincreasingsubstrateC=Nondecomposerstoichiometricmodels(seealsoFigs.10and11).ShadedarrowsrepresenttheCfluxes,blank
arrowstheNfluxes;thewidthofthearrowsillustratestherelativeimportanceofCandNfluxesinrelationtothedecomposerrequirementðC=NÞB.