WATER RESOURCES RESEARCH, VOL. 21, NO. 10, PAGES 1483-1493, OCTOBER
1985
Water-Limited Equilibrium of SavannaVegetationSystems
PETER S. EAGLESON
Massachusetts
Institute of Technology,Cambridge
RAFAEL I. SEGARRA
Virginia PolytechnicInstitute, Blacksburg
The average
annualwaterbalanceof savannavegetation
systems
is modeledasan interactive
competition betweentreesand grassfor water and energy.Ecologicaloptimalityhypotheses
are introduced
whichallowspecification
of thewoodlandcanopydensity,thegrasscanopydensity,andtheplant(water
use)coefficients
of both tree and grassunderconditionsof naturalequilibrium.Only one tree-grass
equilibrium
stateis foundandis shownto agreewellwith the observed
stateof savannain Sudanand
the Transvaal.This stateis stablewith respectto perturbationsof vegetationcanopydensity,but is
metastable
with respect
to shiftsin climate.Two otherequilibrium
statesexistas vegetation
monocultures,i.e.,grassland
and forest,but both are shownto be unstablewith respectto perturbations
of
vegetationdensity.
INTRODUCTION
It is generallyagreedthat the key to understandinglargescale environmentalimpacts,such as those associatedwith
increasingatmosphericconcentrationsof carbon dioxide, lies
with global-scalenumericalmodels(GCMs) of the coupled
ocean-atmosphere
dynamicsand thermodynamics
along with
their interactivesurfaceboundaryconditions.It has been observed recently [Shuttleworth,1983], however, that further
progressin formulatingthe interactionof landsurfaceand atmospherein GCMs requiresestablishing
a quantitativebasis
for describingthe transitions between adjacent vegetation
If these equilibria can be expressedin terms of the state
variables and fluxes of the soil and atmosphereand can be
kept computationallysimple,the knowledgewill have direct
utility in solvingthe GCM boundaryconditionproblem.
We begin this attempt here with a model of the tree-grass
savannavegetationsystem.
BACKGROUND
SavannaEtiology
The term "savanna"has a wide range of meaningsamong
biogeographers,
sometimesreferringto flat and open landtypes.
scapes,and other times referring to the vegetation that
Early attemptsby biogeographers
to put order in their ob- characterizes
that landscape[Sarmiento,1984]. We useit here
servationsof naturewereperforceempiricaland hencefocused in the senseof Dansereau[1957] to describea mixed formaupon thosefeaturescommonto differentobservations.
Where tion of grassesand woodyplantsin any geographicalarea.As
earth's natural plant cover was concerned,this empiricism such,it comprisesone of the four groups(along with forest,
took the form of simplecorrelationsof vegetationtype with grassland,and desert)into which he classifies
all vegetation
climaticvariablessuchas averageair temperatureand precipi- types.For later modelingpurposes
it is well to keepin mind
tation [e.g.,K6ppen,1900; Holdridge,1947]. As physicalun- the idealized description of the savanna given by Walter
derstandingof vegetativewater use grew, these same corre- [1973]:"... ecologically
homogeneous
grasslands
uponwhich
lations were recast by climatologists[e.g., Grigor'yev, 1958; woody plants are more or lessevenlydistributed."This "even
Mather and Yoshioka,1968] in terms of water and heat baldistribution"of the woody plants is indicativeof control by
ance parameters.
moisture availability and is a key to the conceptualization
Increasingknowledgeof vegetationbiophysicsand bio- used here.
chemistrytogetherwith the power of computer-based
modThe coexistence,
in the sameregion,of thesetwo very differelinghas(asin otherfields)spawnedcontemporaryinterestin ent plant typeshas been attributedvariouslyto edaphic(i.e.,
the dynamicaspectsof systembehavior.This and currentap- soil), anthropogenic,and climatologicfactors.For example,
preciationfor the role that may be playedby nonvegetative Beard [1953], in an extensivesurvey of American savannas,
ecosystemcomponentssuch as fire, herbivore,and humans summarizedthe various theoriesregardingsavannaformation.
has focusedattention on the complexitiesand differencesin He concludedthat "Savanna is the natural vegetationof the
systembehaviorat the expenseof continuedsearchfor the highlymaturesoilsof senilelandforms(or, in somecases,of
physicalbasisof their simplecommonalities.
Perhapsat meso- very young soils on juvenile sites)which are subjectto un-
scale(say,10'• km2),spatially,andat climaticscale(say,30-50
years),temporally,the pests,predators,and disastersmay be
viewed as merely perturbationsto what is a fundamentally
resilient climate-soil-vegetation
system.This simplisticview
admits expressionof "first-order"physicalequilibria among
the three components.Suchapproximationsof reality would
fill an emptynichein the hierarchyof climatemodels.
Copyright1985by the AmericanGeophysical
Union.
favorabledrainageconditionsand have intermittent perched
water tables with alternating periods of waterlogging(with
stagnantwater) and dessication.Frequent fires occur but are
not a necessity
for the maintenanceof the savannawhichis an
edaphicclimax."
Monasterio and Sarmiento[1975] also studied savannasin
tropical America.They found savannasover a wide range of
soil conditionsand proposeda classificationschemebased
upon the seasonalityof the climate.They, too, concludedthat
fire and other human influencesare modifiers rather than pro-
Paper number5W0489.
0043-1397/85/005W-0489505.00
ducers of savannas.
1483
1484
EAGLESON
AND SEGARRA:
SAVANNAVEGETATION
SYSTEMS
PA
1
,
--M
w
1-Mg--
•
epw'kvw
TREES
I
epg'kvg
GRASS
IIIIIll]1111•
•A•U•l•••U•U•i•
Jßeps
BAREI so•L
%=0
Rg= o
NO DEEP PERCOLATION
I
available to carry them through the dry season.The authors
sought to test some hypothesesregarding the structureand
dynamics of a savanna. Among other things they demonstrated the existenceof dual equilibrium states in systems
wherethe soil infiltration capacitydeclinedwith reducedgrass
cover. One state is a tree-grass mixture and the other is a
thicket with virtually no grass.
In a second study using this same model, Walker et al.
[1981] introduced grazers to reduce the grasscover and exploredthe stabilityof the resultingdual equilibria.
The major shortcomingof this simulation approach is the
amount of detailed information required. Several of the parameters obviouslypose estimation difficulties.Among these
are the water uptake coefficients,which include the proportioning of water uptake by trees among four soil compartments, rainfall penetration by stemflow, and the amount
of water flow through so-called "quick flow channels,"the
existenceand influence of the latter being still a matter of
some speculation.Such detailed information on savannasis
not available.
Fig. 1. Annual water balanceof tree-grasssavanna.
Hopkins [1979] presents a physiognomicclassificationof
West African savannasbasedupon tree density.He found the
local preferencefor a particular type to be determined by
either edaphicor biotic factors,although on a large scalethe
transition from forest to grass(the two savannaextremes)appearsto follow a gradient of decreasedhumidity.
Many more recent studieshave begunto reach a consensus
that there is continuumof savannatypes,from the dry savannas on semidesertfringes to moist woodlands, with soil
moistureavailability the controllingvariable as modified by
rainfall, soil properties, and geomorphology.For example,
Tinley [1982], in his study of southernAfrican savannas,concluded that the soil moisture balance is the "overwhelming
important factor determiningthe spatialdistributionof forest,
savanna,and grassland."In the face of his observationsof soil
moisturebalanceas the controllingfactor, Tinley totally refutesthe notion that the savannasof Africa are anthropogenic
In the interestof utility, and we hope, of generatinginsight,
we reformulatedthe Walker and Noy-Meir [1982] water balance model in a much simplified form and have introduced
competitionfor energyas well as water.
•
THEORETICAL
Model
DEVELOPMENT
Formulation
The averageannual savannawater balancewill be modeled
as illustratedin Figure 1. We assumethat the trees,grass,and
bare soil actually are distributedover the area in homogeneousfashion,but in this schematicdiagramthey are lumped
to facilitate
visualization
of the water balance. The area of
bare soil is indicatedby M s. The area shadedby grass(with
the sundirectlyoverhead)
is Mg andthat by thetreesis M w.
This is often called the "projective foliage cover" or the
"canopydensity."Sarmiento[1984] has noted "... in savannas
the balance betweenpluvial inputs and evapotranspiration
lossesis so tight, that only in exceptionallywet yearsis there
systems.
an excessof water that is lost through surface runoff or
It thus appears that we can speak of a natural savanna throughdeepinfiltration."We thereforemake the simplifying
ecosystemas differentiated from an anthropogenicsavanna. assumptionof no runoff and we will examine the effectof this
The natural savanna appears to be the biotic responseto assumptionlater.
alternating wet and dry seasons;the amount of soil moisture
Savannatypicallyoccursin climateshavingmarkedseasonavailablecontrolsthe densitiesof the woodlandand the grass. ality of the moisturesupply [Walter, 1973, p. 68]. We will
expressthis in our model primarily through a wet seasonof
SavannaEquilibriumModels
mean length m, during which the entire annual precipitation
Most existingmodelsof savannashave taken the ecosystem PAfalls,and the dry remainderof the year, 1 - m,.
approachthat simulatesthe dynamicsof biomassproduction
The grassland-woodland
competitionfor water is expressed
and the mass flow rates betweenits live, standing-dead,and through two conditions:(1) the trees having accessonly to
litter components[e.g., Grunowet al., 1980]. Such modelsare that water whichis unusedin the root zone of the grassand
primarily diagnosticrather than prognostic,however,and re- whichpercolatesto the level of the tree rootsand (2) the trees
quire the field determinationof numerousrate coefficients.
shieldingthe grass from solar radiation and from turbulent
Recently,somesimulationmodelshave appearedwhich at- heat and moistureexchange,and henceexertingsomecontrol
tempt to addressthe issueof the stabilityof savannasin terms over the ability of the grassto usewater.
of moisturecontrol. The attemptshave focusedon the deterFor the first of thesecompetitionconditionswe assumethe
mination of the equilibriumstatesin termsof the quantity of trees do not draw moisturefrom the root zone of the grass
moisture-dependent
woody and grassbiomasscomponents. (i.e.,"upper"soillayer).Becauseof the zero runoffassumption
Outstandingamong theseis the model of Walker and Noy- they must useall of the water percolatedfrom the upperlayer
Meir [1982]; their idea is that grass roots confined to the and to emphasizethis their root systemis shownin Figure 1
upper soil layer will draw water from this layer only and will as extending beneath the entire surface. Of course, the real
exhaustthe supplyquickly due to their high rate of transpira- mechanismfor getting all the percolated water to the tree
tion. Trees will be out competedin the top layer, but they roots may involve a high water table and lateral movementof
alone have accessto the lower soil layers where moisture is moisturein the capillaryfringe.
EAGLESONAND SEGARRA' SAVANNA VEGETATION SYSTEMS
1485
I
2
Soil Control
--
Climate Control
J=eT___•s
eps
_//Y
,-_(c-3)nK(1)*('l)•e
(c+5)/2
•rmtbeps-- -
•/,,•••
•rmtbeps
I
lO-1
I
1
Fig. 2. Bare soil evaporationefficiency[from Eagleson,1978a].
We assumethat the trees draw moisture only from the
nitude decliningin climatesof increasingaridity. Variation of
lowersoilandduringa season
of averagelengthrn0through- predominant species with climate makes the long-term
out which the soil moisturethere is sufficientto supporttran- averagek•,,species
dependent.
spiration.
For rn0,muchlessthanunitythetreesWillbe"ever- For the bare soil Eagleson[1978a] has shownthat the longgreen" in order to make optimum use of the short season term averagescan be written
ILarchef, 1975,p. 68]. For rno--•! the treeswill be deciduous.
ers= Jeps
The value of rnoin a particularcasewill be determinedin large
part by the elevationof the water table.
where the bare soil evaporationefficiencyJ has the form
We further assumethat the upper soil supportingthe grass
J = 1- [1+ x//•E]e-e + (2E)'/2F[•,
E]
has a spatially uniform soil moistureconcentrationwhich has
a time averagevalue s = soduring the rainy seasonand which where
is zero during the remainderof the year. The roots of the grass
are assumedto exploit the entire volume of the upper soil
layer including that underlying bare soil. These assumptions
•mtbep
s2
(5)
(6)
E=(c--3)nK(1)W(1)½
eso(C+5)/2 (7)
restrict grasswater use and bare soil evaporation to the wet
seasonof the year.
For the secondcompetition condition we will, reflect the
shieldingthrough suppressionby the upper canopy of the
in which
c soil pore disconnectedness
index;
n soil porosity;
annualatmospheric
vaportransportcapacityev that is ef- K(1) soil hydraulicconductivityat saturation,cm s-t;
fectivefor the lower canopy.Accordingly,and usinga linear
approximatio,n,
we assumefor the shadingof grassby trees
%, = (! - M•,)et,,•
(1)
W(1) soil matric potential at effectivesaturation,
cm (suction);
½e dimensionlessexfiltration diffusivity;
mtb averagetime betweenrainstorms,s;
So wet seasonspacetime averagesoil moisture concentration in root zone of grass.
and for the shadingof bare soil by grass
e.,,s
= (1 - M•)e.,,,,
(2)
For simplicity we will work with the asymptotes of (6)
The actualannualevapotranspiration
rate er is expressed which are givenby Eagleson[1978a] as
as a fraction of the potential for each surfacetype. For the
grass
wedefine
a plantcoefficient
kvg
suchthat
er,,= k,.,,,e.,,,,
=
'-
(3)
<
J- 1 E>(2/=)
It is characteristicof grassesthat they transpireat their maximum rate (without stomatalcontrol) until the s0il moistureis
(8)
Equations(6) and (8) are comparedin Figure 2.
Accordingto Brutsaert[1982], observationsof soil properindex c in the
constant throughout the transpiring (i.e., wet) season with ties give values of the pore disconnectedness
magnitudecloseto unity and Equation (3) may be applied range •5. Choosing,for analytical convenience,
over any time period during the wet season.
c= 5
(9)
exhausted
[Walter,
1973,
p.68].Thevalue
ofk• willthusbe
For thetreeswedefinek•wsimilarlyas
erw= k,,ep,
allows us to write
(4)
Trees exerciseconsiderablecontrol over their transpiration
rate, however, reducing it as moisture becomesscarce,and
thereforeko,,is variablewith time. To use(4) for long-term
averageskv,,becomes
an effectiveaveragevalue,with mag-
j m •JoSO5/2
(10)
where [Eagleson,1978b]
t/2
jo__[O.O23an3/2K(1)
/(rnt•
Y1/2 /•1/2 )it/2
(11)
1486
EAGLESON
AND SEGARRA'
SAVANNA
VEGETATION
SYSTEMS
The numerator of $ is the annual potential (i.e., So--1)
in which
percolation
o?"percolatlvitY,"
while
thedenominator
isthe
a surfacetensionof porewater,dyn cm-•;
•
grasslandtranspirativity.Thus S measuresthe relative sense
dynamic viscosityof pore water, poise;
(i.e,,upordown)
ofthesoilmaisture
movement.
Larger
values
7 specific
weightof porewater,dyncm-3.
of S will terid to favor trees.
By usingtheseparameters,the governing(12) through (15)
We will now write the annual tree-grasswater balance in
'
two parts, one for each of the two limiting bare soil functions can be written
of (8). With each term written in terms of long-term average
Mw
annual values we have
+ M•(1- Mw)+ (! - Mg)RSo
•/2
PA= Mwm•epwkvw
+ Mom•ep•kv•
+ (1-- Mo)m•JoSo
•/2 (12)
when
underthecondition
er, < ep,whichcanbewritten
Joso
•/2< (1- MgX1- Mw)e•w
(13)
(22)
Rso
•/2<-(! -- M•X1- Mw)/ko,
(23)
and
or
Mw = KSso•
PA= Mwm•e•k•+ Mgm•e•k•,
+ (1 - M•)m•ep, (14)
(24)
Considering
k,•,andthoprod•uct
mokv•
to beknownparame-
whener, = e•,.
tersof the system,
therearqthreeunknowns,
So,Mw,andM,,
In the long-term average,and with our assumptionof no but only two equations.Under t!aehypothesisthat natural
runoff,the treestranspire
exactlythat waterperco!ated
from vegetationsystemswill attemptto minimizewater demand
the upper soil layer. By usingthe steadypercolationapproxi- stress[Eaqleson,
1978c,1982]wecanaddthe third equation
mation given by Eagleson[1978b] we thus have a second
relationship
Mwm•e•k•---m•K(1)So
s
•5)
Because
the plantcoefficient
ko•is alsodetermined
by the
(15) climate
and the soil, we may hypothesize[Eagleson,1982]
We now simplifythesegoverningequationsby definingthe
dimensionless
r3So/OM
• -- 0
variables
that the long-term evolution of the vegetationalso tends to
maximize
soilmoisture.
Thi•saddsa fourthequation
t3So/t3k•
-- 0
(26)
K = m•kv,/(m•k•w
)
(16)
O = PA/(m,e•kv•)
(17) andallowsusto predicttheproductmoko,.
R = Jo/(ep.ko•)
(18) EquilibriumStates
S-- f(1)/(ep•kv•)
(19)
The parameter K will be called the "water use ratio." It
representsthe ratio of the annual water use of grasslandor
Letting
s, = So•/2
(27)
Equations(22) and (24) reduceto the quadratic
"opensavanna"(i.e.,M• = 1, M w= 0) to that of forest(i.e.,
M• = O,Mw -- 1).
a0s,2 + bos,+ Co= 0
(28)
The parameter G will be called the "potential grassland havingthe solution
density"because
it indicates
the valuethat M• wouldtakeon
if all availablemoisturewent to supportgrass(i.e., Mw = 0
and thusko•--ko,).The actualgrassdensityin a tree-grass
systemmust thereforeobeythe constraint
Mg <_G
[(
s,: :-ao
+-
col"
aol
(29)
where
(20)
When the moisture supply exceedsthat needed for the
M• -- 1 andMw-- 0 stateof fullgrassland,
thewoodland
densitybecomes
Mw > 0. For the caseof M• = 1, and using(1),
either (12) or (14) will become
(30)
a0 = [(1 - Mo)R]/[(1- K Mg)S]
(31)
Co= [M,-
G]/[(1 - K M•)S]
(32)
and (24) becomes
G=I-Mw(1--•)
M•=I (21)
and we seethat G will equal 1, regardless
of Mw, if K -- 1, G
a0 = 1
Mw= KSs.2
(33)
Equations(29) through (33) givethe woodlandcanopydensity
will be less than 1 if K > 1, and G will be greater than 1 if
K<I.
Unfortunately,R and S are not so simplycharacterized.The
numerator J0 of R is the annual potential (i.e., So-- 1) evaporation from a unit of bare soil surface.We will call J0 the soil
"evaporativity."The denominator of R is the annual transpiration from a unit area of grassland(i.e., Mw--0 and hence in which
ep•= epw),
andwe will callit the "grassland
transpirativity."
The ratio R is a measureof the relative potential water use of
thesetwo surfacecomponents.
i--k-g'/J
} (34)
X = R/(2S•/2)
(35)
It should be noted that only the positive radical in (29) is
EAGLESON
ANDSEOARRA:
SAVANNA
VEGETATION
SYSTEMS
1487
0.5
retained
in (34)in orderto satisfy
thereqhir/•ml•nt
thatboth
MoandM wbeequalto or lessthanunity.Thisforinitiation
of
the tree-grassproblem does not therefore adrhit alternate
equilibrium woodland states for a given climate and soil
K=I
0.4-
-
unless
therearealterfiate
grassland
densities
Mo.
As waspointedout above,Mg is undeterniined
without
additionalinformationor assumptions.
M•
0.r
..,
IX'
LimitingSolutions
As we havealreadyseen,K = 1 impliesthat a closedforest
............ '"'""'-'
œ
_- --.T,
;'-.--
(Mw= 1,Mo= 0)willusethesame
amoufit
Ofwaterannually
aswillgrassland
(Mw= 0, Mg= i). Theadditional
requirembnt that G = 1 determinesthat PA is exactlysufficientfor
both of theseextremestates.This can be seenfrom (1), (16),
S = 250
and (17) which give
PA= maepwkvw
= m•ep,
kv,
G K= 1
(36)
Byletting
G= 1in(34)
and
M•--}
0and
then
K• 1(inthat
0.11R
=25
order)we indeedfind M w= 1 for arbitraryX. If, however,
MgandMw
PA> K(1)thiswill oCeur
Withstarface
ponding
of thewater.
Fig. 4. Suboptimalequilibria(R =
BylettingG = 1 in (34)andsetting
M0= 1 weseethatMw
doesvanish
forarbitraryK andx.
This is satisfiedby either
Sinceforgrassland,
ep,= epw,
(36)gives
k,,•= PA/(m•ep,)
(37)
and then
•5, $ = •0, and K = 1).
s, = c•
(41)
which is physicallyinfeasible,or
mako•
= PA/e
w
(38)
Os,/OMo= 0
(42)
Equations
(42)and(29)havea physically
feasible
(i.e.,So_<
or
k•/k•, = m,/m,•
(39)
1) solution only for
G= K = 1
Most savannasare found in seasonalclimates[Monastei,io
(43)
andSarmiento,
19•75]
where,
bydefinition,
m•< 1.Because
of whichreduces(34) to
possiblewater table influencehowever,ma< 1. Equation(39)
demonstratesthereforethat the treesand grassesof a savanna
may have very differentwater usecharacteristics.
M w= [-X
+ (X 2 + 1)'/232
(44)
with
M0being
arbitrary.
Therearethusthreeequilibrium
states
of thesavanna
Ecologically OptimumSolution
system at G = K- 1' closed forest, grassland,and an intermediatetree-grassmixture.
Because(25) can be satisfiedin the short-term due to the
Applicationof (25)to (27)givesthecondition
Os,
0.4s,-0.6
0M
ø- 0
t
annualgrowthor deathof grasses',
(43) and (44) will be saidto
(40)define
the short-term equilibrium state of tree-grasssystems,
Note that Mw is a single-valuedfunction of X which is contrary to the speculationof some investigators[e.g., Walker
and Noy-Meir, 1982] that tree-grasssystemsmay have multiple (inhomogenous)equilibria.
X=CONSTANT
G=K=
1
/
/
/
dMw
Satisfaction
of thesecond
optimization
condition
(26),using
\
/
I
\
//
',
,,
\
dt
(27)..t.
hfough(32),leadsto M0= G. SincethisOptimiZation
conditionrequiresspeciesselectionwhich is presumablya
long-termphenomenon,
the statessatisfying
(26) shouldalso
satisfy(43). Under this restriction,the secondoptimization
',
/
\
EQUILIBRI•
condition
selec
tsMo= 1 andthelong-term
equilibrium
con-
(• ••/i EQUILIBRIA
•X7
O\
ditionsfor tree-grasssystemsare
/
,,
,,
',
I
1
\
1
/
G = K = M• = 1
(45)
In thelimit,as G, K, andM ogo to unity(in that order),(34)
again reducesto (44).
Stability
We begin by consideringperturbationsto the equilibrium
vegetationunderthe assumptionthat thesecauseno changein
the parameters of the climate and/or soil. To examine the
Fig. 3. Stability
ofsavanna
woodland
equilibria.
stability of the three equilibria identified above we write the
1488
EAGLESON
ANDSEGARRA'
SAVANNA
VEGETATION
SYSTEMS
TABLE 1. SampleClimatic Conditions
ep,•,
G
mm
PA,
m•
mm
(kv•= 1)
1113
544
1387
853
1036
1
0.58
1
0.88
1
548
986
986
1241
803
2.03
0.95
1.41
0.78
1.29
[Development
Analysis
Associates,
1978] . 115 0.42
i533
0.18
Locality
Reference
[Eagleson,1978c]
[Eagleson,1978c]
[Eaglesonand Tellers, 1982]
[Eaglesonand Tellers,1982]
[Eaglesonand Tellers,1982]
Clinton, Massachusetts
Santa Paula, California
West Point, Georgia
Neosho, Kentucky
Cartersville, Virginia
Riyadh, Saudi Arabia
unsteady
conservation
ofmass
equation
forthewater
in the
lower layer. By letting Z,• be the elevationof the water table
in the lower layer we have
constrained
by the hypothesized
conditionsof ecological
opti-
dZ,•
n• = m•K(1)So
5- Mwm•et,,•kv,
•
(46)
Differentiatingwith respectto time gives
dt
n
dt
mality. Some of these suboptimal solutionsare presentedin
Figure 4 for representativevaluesof R and $. In this figurethe
long-term optimum equilibriumstate of (45) is shownby the
plottedcircle.Theassociated
equilibrium
valueof M,• isgiven
rnaep,•kv,
• dM,•- 5rn,
K(1)So
½dso d2Z•,
n
The savannhsystem,however,is metastableregardingclimate change.EqUations(34) and (35) defineequilibriumsolutions to the .posed tree-grass configuration that are un-
dt 2
(47)
The sign of dM,•/dt is crucial to the stability issueas we can
seefrom the phaseplane diagram of Figure 3. In this illustration the three equilibrium statesfor M,• at G = K = 1 are
shownschematicallyby the plotted circleswheredM,ddt m O.
If the functiondM,•/dt connectingthesepointshas the general
shape of curve "a" (solid curve), the intermediate tree-grass
equilibriumwill be unconditionallystableto perturbationsin
M,•. If fire shoulddecreaseM,• from this equilibriumvalue,it
would be in a regionof positivedM,•/dt (i.e.,curvea), and the
systemwould return to equilibrium. Such restoration would
not follow perturbationsof the M,• = 1 and M,• = 0 equilibria
and they are thereforeunstableif curve a governs.Should the
other possibility,curve "la," govern, on the other hand, the
stabilitiesare reversedand a tree-grassmixture would be an
unstable state.
by the intersectionof the dashedM,•(So)curve with the solid
Ma(so;G= 1)'curve.
In Figure 4.the parameterK is held constantat its optimum
value of one. The dashedcurve representsM,•(So)as given by
(24)andisindependent
ofG.Thesolidcurves
represent
M•(so;
G) as given by (24) and (34). For G = 1, Sois independentof
Mg, and thereforethe short-termconditionOSo/OM•
= 0 is
satisfiedidentically everywhere.It is the long-term condition
OSo/Ok•w
= 0 thatfixestheoptimalMa = 1.
For G > 1 we seethat Ma increases
with SountilSoreaches
the value (So= 0.33 for G = 1.2) at which M,• becomesunity.
At thispointMa !s discontinuous,
sincethere,dueto (1),Mg
must drop to zero. The point to be noted is that as SoincreasesundernonoptimalG > 1, the equilibriumtree density
is driven to unity and the equilibi'iumgrassdensityto zero.
For G < 1 weseethat Ma decreases
withincreasing
So.At
Ma = 0, Soreaches
its maximumvaluewhichis lessthanthat
for optimum G = 1; the associatedequilibrium M,• is lessthan
its optimum also.
of the first term on the right-hand side exceedsthat Of the
Therefore if we accept the ecological optimality criteria of
secondterm on the right-hand side, then dMddt carries the (25) and (26) as the operative ones in determiningtree-grass
sign of dso/dt.We will make this assumption.Shouldwe be at equilibria, climates with G > or < 1 cannot have stable treetheM,• = 1, Mg = 0 equilibriumanddecrease
M,• somehow,
a grasssavannas.For G > 1 pressureto develop toward increasgrasscover will then grow, and Sowill decline.Should we be ing Sowill lead eventuallyto closedforest while for G < 1 this
at the M,• = 0, Mg = 1 equilibriumand decrease
M•, we will pressurewill lead to low densitytreesand an absenceof grass.
then get a rise in Sowhich will produce the percolationneeded It is worth noting that this latter.state correspondsto the
to begin a woodland component.
secondequilibriumreportedby Walker and Noy-Meir [1982].
This reasoningindicatesthe tree-grassequilibrium of (43) Theflatness
of thedriving
gradient
OSo/OMa
in thesmallM,•
and (44) to be stableto perturbationsin either vegetalcompo- range of most savannasindicates,however,that thereis ample
nent such as might occur by fire, pest, herbivore, or humans, opportunityfor other stress-producing
factors(suchas nutriprovidedthat the perturbationdoesnot bringaboui a change tion, pests,fire, etc. that have been neglectedin this waterin the parametersof the system.For example,we might expect basedanalysis)to play a stabilizingrole.
the tree-grassequilibrium to return to its former value of M,•
In Figure 4 we seehow very sensitivethe vegetationdensity
following tree removal, provided the deforestation did not is to smallchangesin the averagesoil moistureconcentration.
With these behaviors in mind we conclude that the treelead to changes in the soil properties and hence in X. Although the tree-grassequilibrium value of M,• is independent grasssavannaequilibriumis metastableto changingdimate.
of Ma and henceis stableto grazingand burningof the grass It is interestingto note that paleoclimaticstudiesof ice cores
component,it is known [e.g., Walter, 1973, p. 71] that this [Dansgaard,1981] have shownvery rapid shiftsbetweenforest
often leads to natural replacementof the grass by thorny, and grasslandduringglacialtime.
woody shrub that is resistantto theseinsults.
Finally, we shouldnote that the openingassumptionof this
We also conclude that under these savanna conditions
section(that climateremainsconstantin the faceof changesin
(G = K = 1) the foreststate(and the grasslandstateas well) is vegetalcover)may not alwayshold. Climate changecan result
in unstableequilibrium.In this regard it is worth noting that from vegetationlossin a moist convectiveclimate where much
slashand burn agriculturehasconvertedsome40% of African precipitationis derivedfrom locally evaporatedwater providequatorial foreststo savanna[Phillips, 1974].
ed that the removal of vegetationleads to a sustainedincrease
Returning to (47), if we can assumethat the absolutevalue
EAGLESONAND SEGARRA'SAVANNAVEGETATIONSYSTEMS
0
in runoff and henceto a sustainedreduction of local evaporation. Increased potential evaporation can also follow vegetation removal due to the increasein air temperatureaccom-
I
I
I
I
200-
panyinga highersurfacealbedo.Both of thesechangeswould
be naturally reversibledue to regrowthof the coverunlessthe
denudation caused permanent structural and/or chemical
changesin the surfacesoil that inhibitedthe process.
Tundra
400-
[ 4e_e,,owo,ne
I Conif
.....
f....
t(Canada)
SagqbrushI
Hardwood
forests
(Canada)]
Oak chestnut
Effect of Runoff
Our solution has been unduly restrictivein that it has for
convenience
I
1489
assumed
zero
runoff.
In
the savanna-covered
catchmentstributary to the Bahr el Ghazal swampsin Sudan,
for example, the long-term average annual yield is on the
order of one tenth the long-term annual precipitation ]Chart
and Eagleson,1980]. We will not derive an "exact" solution
for this case but will instead look at the qualitative effect of
allowingfirst surfaceand then groundwaterrunoff.
Supposewe allow only surfacerunoff and assumeit to be a
small fraction v of the annual precipitation. Our governing
equationswill be the same as beforeif we substituteG' for G
everywhere.By definition
G' = (1 - v)G
(48)
Annual
epw
So
L IBunch•l
"•
8001 I grassI I
-hickory
rests
(ram)
•
• •:
pine
I
Tropical rainforest
1800--100
I
-60
I
-20
+20
I
+60
I
+ 100
I
+ 140
+ 180
The optimum solution now gives G'= 1 and we see that
Thornthwaite
Moisture Index. Im = 100
L•Pw
G' = 1/(1 - v), which is slightlylargerthan one.
Let us now allow only groundwaterrunoff. If the ground0
.4
.7
1
water flow is positive (i.e., an outflow), it will have no effect
mr: Relative
Length
of RainySeason
upon the soil moisturebut will lead to a smaller M,•. If the
Fig. 5. Observed climatic bounds of common vegetation types
groundwaterflow is negative(i.e., an inflow), such as might
occurin low-lying,high water tablelands,both M,• and so will (G- K = 1 on auxiliary scale)(adapted and reprinted from Mather
[1978]).
be higherwhich correspondsto
G' = (1 + v)G
(49)
have, as in (37),
and to an optimum G = 1/(1 + v), which is slightly smaller
than one.
PA/e•,w-m•kvg
/
This crude argument demonstratesthat the optimum savanna conditions may vary somewhateither way from G = 1.
However, the small runoff percentagein most savanna climates should make the zero runoff solution a good approximation of reality. We will now test this solutionagainstavailable observations.
(51)
From a survey of the literature describingsavannasin South
Africa, West Africa, Sudan, and South America, Segarra
[1983]foundrn•from0.46to 0.67.Usingk, = 1 (51)givesa
range of Im from -33 to -54 and allows comparisonwith
Mather's [1978] observedrange of 30-60 through the auxilIary abscissaon Figure 5.
WoodlandDensity
OBSERVATIONS
A savanna in the Nylsvley (nails'-fiey) Provincial Nature
Reservein the northern Transvaal province of South Africa
To begin with, it is helpful to understandthe natural range (24.7øS-28.7øE)is described qualitatively in a recent comof the parameter G in order to appreciatethe utility of the pendiumon tropical savannas[Huntley and Walker, 1982].
condition G-- 1 as a discriminator
of savanna conditions. A
The trees are primarily Burkea africanawith a canopy denrange of conditionsgatheredfrom the literature are presented sity Mw = 0.275, and the grassesare tussock-formingperenin Table1. In calculating
G it is assumed
thatkvg= 1. Of the nials [Huntley and Morris, 1982]. The soils are describedas
six climates shown only Santa Paula, California has G • 1. "fine-grainedsands"having 0-6% clay in the A horizon and
While not all biomesnear Santa Paula are currently savanna 5-15% clay in the B horizon [Huntley and Morris, 1982].
[e.g., Eagleson, 1978d], the presenceof Mediterranean-type
For climate data we must go to other sources.Rutherford
savannasin this region is well known [e.g.,Eyre, 1968].
[1979] definesthe mean annual precipitation to be 630 mm
coming mainly in the mid-October through March rainy
Boundaryof SavannaZonobiome
season.The mean annual temperature is 18.6øC.
Mather and Yoshioka[1973] definedthe climatic bounds of
Pitman [1976] describes the rainstorms at Pretoria as
thecommon
zonobiomes
in termsof e•,wandthe Thornthwaite having durations normally less than 24 hours. In 9 years of
[1948] moistureindex with the latter written
record there were 525 rainy days. By usingthis as the number
of storms,and neglectingtheir duration, we can estimate the
Natural Range of G
Im:100[e••1]
meantimebetweenstormsasmtb= 2.93days.
(50)
Their correlation is presented in Figure 5 as taken from
Mather [1978]. By using our savanna criterion G = 1, we
Middleton et al. [1981], presenta map of long-term average
annual Symonspan evaporationover South Africa that shows
a value of 1800 mm/year for the Nylsvley region. Midgley et
1490
EAGLESON
AND SEGARRA:SAVANNAVEGETATIONSYSTEMS
.2O
JONGLEI
SUDAN
--
I
h
OBSERVED
RANGE
.16
.12
Mw
-
I
-
I
,I ./' /
I
!/ECOLOGICAL
r
OPTIMUM
i /
.08
_
-.04
1
/
/
- •
i .PARAMETERS
"GENERALIZE•"
ECOLOG,CAL
OPTIMUM
V --•:-RA'G•-OF
II I OBSERVED
RANGE
I
0
USING
I BAHR
EL•G
H_A_Z_A
L
I
I
I
I
10-5
I
I
I
I III
I
I
I
I
I
II
I
I
I
I I I
10-4
10-3
K(I), cm/sec
Fig. 6. Sensitivityof woodlandcanopydensityto soilhydraulicconductivity,Jonglei,Sudan.
al. [1983] give the long-term averagemonthly Symonspan
evaporation as a percentageof the annual total areally
averaged over the drainage region (Limpopo-Olifants) that
includes the Nylsvley Reserve.(The Symons pan is 6 feet
square and 2 feet deep, and buried to ground level (1
foot = 30.48 cm)) (D.C. Midgley, personal communication,
1984).Pitman [1973] givesa seasonalalgorithm for reducing
the pan evaporation to water surfacevalues.Applying these
findingsto the Symonsaverageof 1800 mm gives an annual
the value used here. At 26øC this translates to a saturated
intrinsicpermeability
k(1)= 2.80 x 10-9 cm2.
It is interestingto note the resultsof S. Andreou(personal
communication,1983),who appliedEaglesoh's[1982] ecological optimality theory to the catchmentsof three Bahr el
Ghazal tributaries(Jur, Loll, and Tonj) on the "ironstoneplateau" westof the Bahr el Jebel.The theory yieldsthe saturated
intrinsic permeability at which the vegetation cover is maximum. The average result for the three catchmentsis k(1)=
watersurface
evaporation
of 1680mm.Because
of thehigher' 2.21x 10-9 cm2 or K(1)= 2.5 x 10-4 cm s-x, whichis realbedoof vegetation,
e•,wwill be still lower.We will usethe markablycloseto the averageof the observedrangealongthe
factor of 0.94 estimated by Chan and Eagleson[1980] for the
reduction of water surface evaporation to that for wet bare
Jonglei Canal to the east of the Bahr el Jebel.This, of course,
may be fortuitous.
soil;thisgivesourfinalestimate
e•,w
= 1579mm.
The sensitivityof Mw to the estimateof K(1) as given by
(44) is illustratedin Figure 6. Also shownon this figure is a
semiempirical"generalized"optimum hydraulicconductivity
intrinsicpermeability10-8 < k(1)< 10-7 cm:. The geometric as given by Eaglesonand Tellers [1982, equation 26]. This
There are no quantitative observationsof the soil properties. Accordingly,we estimate the porosity n = 0.35 and the
averagepermeabilityis thus k(1)= 3.16x 10-8 cm•. These
propertiesare summarizedin Table 2.
A secondsavanna is that along the path of the Jonglei
Canal in Sudanto the eastof the Bahr el Jebel(7.5øN-31.3øE).
Aerial photographstaken by the seniorauthor along the path
of the canaleastof a point betweenShambeand Fangak yield
an estimateof tree canopy densityMw = 0.13. F. A. Rahim
(personal communication, 1980) reports these to be "Arabic
glue trees." Meteorological data for Shambe and Fangak
[Chan and Eagleson,1980] averageto give a 7-month rainy
seasonyielding an averageseasonalprecipitationof 815 mm.
The nearestmeteorologicalstationis at Rumbek,whereapplication of Van Bavel's[1966] combinationform of the Penman
TABLE
Nylsvley
Observed
815
630
e•,w,mm
1606
1579
m•
TA,øC
7/12
26
mt•,days
3.04
M,•
Species
0.13
?
monthly in the manner describedby Eaglesonand Tellers
k,,,
tivity in the range 10-4< K(1)< 10-3 cm s-x, the geometricalaverageof whichis K(1) = 3.16 x 10-4 cm s-x andis
Parameters
PA, mm
n
days.
An Egyptian report [Academy of ScientificResearchand
Technology, 1978] prepared for the design of the Jonglei
Canal describesthe soilsof the region as well-drained,moderately permeableloamy soilswith saturatedhydraulicconduc-
Savanna
Jonglei
equationgivesep•= 1606mm.Thiscomputation
is performed
[1982].The meanannualtemperature
is 26øCandmt•= 3.04
2.
k(1),cm•'
Estimated
0.35
1
2.21 x 10-9
5.5/12
18.6
2.93
0.28
Burkeaafricana
0.35
1
3.16 x 10-"
Calculated
0.92
0.87
61.9
591
19.5
32.6
EAGLESONAND SEGARRA:SAVANNAVEGETATIONSYSTEMS
approximatesemiempiricalrelationshipis designedto bypass
the extensivecomputationsassociatedwith evaluation of the
"exact" optimum but is based upon results from only nine
catchmentsin the United States.Clearly, the approximationis
not very good in this caseas it underpredictsMw by 30%.
Finally, the porosity is estimatedas for Nylsvley at n -- 0.35
and the full range of parametersfor the Jonglei savanna is
summarizedin Table 2. The parametersof Table 1 are now
used to compare the observedtree canopy densitieswith the
value predictedby (44) as is shownin Figure 7. To understand
the physicalsignificance
of the abscissa
R/(2S•/2) we first note
that the percolativity K(1)can also be interpreted through (15)
as the potential (i.e., so = 1) woodland transpiration rate. We
1491
TABLE 3. Savanna Plant Coefficients(G = K = 1)
Grasses
Location
= PA/(m,ep•,)
California
Transvaal
Sudan
0.95
0.92
0.87
Trees
m,•k,•,•
= PA/ep•,
0.55
0.40
0.51
Should (44) withstand further testing, its relationship of observableM• to "hidden" k(1) will provide a usefuladdition to
the arsenal of techniquesfor parameterizing the hydraulic
propertiesof soilsover large areas.
will call Mwrnoep•kv•/rn,
the "woodlandtranspirativity."
Now
Plant Coefficients
we have
Instead
ofassuming
kv,-- 1,aswedidto explore
thenatural
R
2X-
sl/2
soil evaporativity
I-(woodland
transpirativity)(grassland
transpirativity)-I
•n
rangeof G for a wide range of climatesin Table 1, we will now
estimate the actual plant coefficientsfor the three identified
savannasusing (37) and (38). The results are presented in
Table 3 and seemquite reasonableand very consistentfor the
grasses
at least.For the treeswe stillneedto knowmg.In the
(52)
The denominatorof (52) is the geometricalaveragetranspirativity; thus 2X is the ratio of potentialsfor return of moisture to the atmosphere,i.e., through the soil as opposed to
through the vegetation.M• is thus inverselyrelated to X.
Although the estimatedvalues of G for the two observed
savannasare both lessthan the theoreticalvalue of unity for
stable equilibrium, the decrementis well within the error of
estimate,and the agreementof the plotted points with the
theory of (44) in Figure 7 is remarkable.
It is important to gather data for other savannasto see if
this agreementis sustainedover a wider range of conditions.
THEORY
k(I)= 10-7 cm2
(I) = 10-8 crn
2
high water table case we expect m0--• 1 and the trees to be
deciduous.
In thiscasekv•is about0.5.With rno--•rn•--•0.5we
find evergreentreesand k,•--• 1. Suchvaluesare consistent
with the relative water use rates of thesetwo vegetationtypes
[Larcher, 1975].
SUMMARY AND CONCLUSIONS
The tree-grasssavannahas been modeledin its equilibrium
state as a competition for water and solar energy. The grass
allows the trees only that water which the grass cannot use,
and the trees control the grass'sability to use this water by
shieldingthe grassfrom solar radiation.
When it is assumedthat the grassseeks(through its canopy
density)to minimize water demand stress,the necessaryconditions for existenceof equilibrium savannasare G--K-1.
There are three suchequilibria: closedforest,grassland,and a
tree-grassmixture. Only the last of these is stable to perturbationsin the vegetationcomponentsbut it is metastablewith
respectto climatechange.
To assurethat the tree roots have accessto all the percolated water it appearsnecessarythat the water table be "high
enough."
Soil moisture variability below the grassroot zone likely
governsthe deciduousversusevergreennature of the woody
componentwith low water table tending to produce an evergreen savanna.
Mw
Observationsof two tree-grasssavannasprovide a limited
10-1 --
verification
_
8-
of the theoretical
climax conditions.
_
NOTATION
6_
4-
A
a0
bo
c
Co
E
OBSERVATIONS
_
(•) NYLSVLEY, TRANSVAAL
(G = 0.92)
fix JONGLEI,SUDAN
(G= 0.87)
2-
climate-vegetationparameter.
coefficient.
coefficient.
soil pore disconnectedness
index.
coefficient.
exfiltration parameter.
ep, average
annualpotentialevapotranspiration
from
.2
.4
6
.8
1
2
3
4
5
grass, ram.
ees averageannualpotentialevaporation
frombare
R
on3/2
X=• =0.076
•mtb
k(i)l/:
epw
kvg
Fig. 7. Savanna woodland canopy density, comparisonof theory
and observation.
soil, mm.
e•,• averageannualpotentialevapotranspiration
from
woodland, mm.
er• average
annualevapotranspiration
fromgrass,
mm.
1492
EAGLESON AND SEGARRA.' SAVANNA VEGETATION SYSTEMS
eTs averageannualevaporationfrom bare soil,mm.
eT,• averageannualevapotranspiration
from trees,mm.
G
G'
Zw
J
potential grasslanddensity.
transformedpotential grasslanddensity.
water table elevation,m.
bare soil evaporation efficiency.
Jo evaporativityof soil,cm s-•.
K
water use ratio.
K(1) soilhydraulicconductivity
at saturation,cm s-•.
kvg plantwaterusecoefficient
forgrass.
kvw plantwaterusecoefficient
for trees.
k(1) soilpermeabilityat saturation,cm'-.
Mg grasscanopydensity.
M s bare soil fraction of surface.
M w woodland canopy density.
rntb averagetime betweenrainstorms,
s.
rno averagelength of tree transpirationseason,fraction
of year.
rn, averagewet season,fraction of year.
n soil porosity.
P A averageannual precipitation,min.
R relative soil-grasspotential water use.
S soil percolativity.
So wet seasonspacetime averagesoil moistureconcentration in root zone of grass,So- 1 at saturation.
s.
t
X
transformedsoilmoistureconcentration.
time, s.
soil moisture flux parameter.
/• reciprocalof averagetimebetweenrainstorms,
s-•.
7 specificweightof porewater,dyn cm-3.
# dynamic viscosityof pore water, poise.
v ratio of runoff to precipitation.
a surfacetensionof porewater,dyn cm-•.
Eagleson,P.S., Climate, soil, and vegetation,3, A simplifiedmodel of
soil moisture movement in the liquid phase, Water Resour.Res.,
14(5), 722-730, 1978b.
Eagleson,P.S., Climate, soil, and vegetation,6, Dynamics of the
annual water balance, Water ReSour.Res., 14(5), 749-764, 1978c.
Eagleson,P.S., Climate, soil, and vegetation,7, A deriveddistribution
of annual water yield, Water Resour.Res.,14(5),765-776, 1978d.
Eagleson, P.S., Ecological optimality in water-limited natural soilvegetationsystems,1, Theory and hypothesis,Water Resour.Res.,
•8(2), 325-340, 1982.
Eagleson,P.S., and T. E. Tellers, Ecological optimality in waterlimited natural soil-vegetationsystems,2, Tests and applications,
Water Resour.Res., •8(2), 341-354, 1982.
Eyre, S. R., Vegetation and Soils: A World Picture, Edward Arnold,
London, 1968.
Grigor'yev,A. Z., The heat and moistureregimeand geographiczonality, in ProceedingsThird Annual Congressof the Geographical
Society of the USSR, pp. 3-16, Akademy Nauk SSSR, Moscow,
1958.
Grunow, J. O., H. T. Groeneveld, and S. H. C. Du Toit, Above
ground dry matter and dynamicsof the grasslayer of a SouthAfrican tree savanna,J. Ecol., 68, 877-889, 1980.
Holdridge, L. R., Determination of world plant formations from
simpleclimatic data, Science,105, 367-368, 1947.
Hopkins, B., Forest and Savanna,2nd ed., Heinemann Educational
Books Ltd., London, 1979.
Huntley, B. J., and J. W. Morris, Structure of the Nylsvley Savanna,
in Ecologyof Tropical Savannas,edited by B. J. Huntley and B. H.
Walker, pp. 433-455, Springer-Verlag,New York, 1982.
Huntley, B. J., and B. H. Walker (Ed.), Ecologyof Tropical Savannas,
669 pp., Springer-Verlag,
New York, 1982.
Koppen, W., Versucheiner Klassifikation der Klimate, vorzugsweise
nach ihren Beziehungenzur Pflanzenwelt, Geogr. Z., 6, 593-611,
1900.
Larcher, W., PhysiologicalPlant Ecology,translatedfrom German by
M. A. Biederman-Thorson,252 pp., Springer-Verlag,New York,
1975.
Mather, J. R., The Climatic Water Budgetin EnvironmentalAnalysis,
239 pp., Lexington Books, Lexington, Mass., 1978.
Mather, J. R., and G. A. Yoshioka, The role of climate in the distribution of vegetation,Ann. Assoc.of Am. Geogr.,58(1),29-41, 1968.
•be dimensionless
exfiltrationdiffusivity.
Middleton, B. J., W. V. Pitman, D.C. Midgley, and R. M. Robertson,
W(1) soil matric potential at effectivesaturation,cm (suction). The Limpopo-Olifantssystem,SurfaceWater Resourcesof South
AcknowledgmentsThis material is basedupon work supportedby
the National ScienceFoundation under grant ATM-8114723 when
the junior author was a student at the MassachusettsInstitute of
Technology.The authorsthank B. H. Walker for his helpfulintroduction to this problemand O. T. Solbrigfor bringingrecentliterature to
our attention.
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(ReceivedDecember 14, 1984;
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