Ecological Applications, 5(4), 1995, pp. 1040-1055
? 1995 by theEcological Societyof America
TRANSITION AND GAP MODELS OF FOREST DYNAMICS1
M. F. ACEVEDO
Instituteof Applied Sciences and Departmentof Geography,Universityof NorthTexas, Denton, Texas 76203 USA
and CESIMO, Universidadde Los Andes, Merida, Venezuela
D. L.
URBAN2
Fort Collins, Colorado 80523 USA
Departmentof Forest Sciences, Colorado State University,
M.
ABLAN
Instituteof Applied Sciences and Departmentof Geography,Universityof NorthTexas, Denton, Texas 76203 USA
and CESIMO, Universidadde Los Andes, Merida, Venezuela
Abstract. We describe and apply a correspondencebetween two major modeling approaches to forestdynamics:transitionmarkovianmodels and gap models or JABOWAFORET type simulators.A transitionmodel can be derivedfroma gap model by defining
or otherconvenientcover
stateson the basis of species, functionalroles, verticalstructure,
types.A gap-size plot can be assigned to one stateaccordingto dominanceof one of these
cover types. A semi-Markovframeworkis used for the transitionmodel by considering
not only the transitionprobabilitiesamong the states,but also the holding times in each
transition.The holding times are consideredto be a combinationof distributedand fixed
time delays. Spatial extensionsare possible by consideringcollections of gap-size plots
and the proportionsof theseplots occupied by each state.The advantagesof this approach
include: reducingsimulationtime,analytica-lguidance to the simulations,directanalytical
explorationof hypothesisand thepossibilityof fastcomputationfromclosed-formsolutions
and formulae.These advantagescan be usefulin the simulationof landscape dynamicsand
of species-richforests,as well as in designingmanagementstrategies.A preliminaryapplicationto theH. J.Andrewsforestin theOregonCascades is presentedfordemonstration.
roles; gap; GIS; H. J.Andrewsforest;landscape modeling;
Key words: forestdynamics;functional
Markov; mosaic; semi-Markov;shade tolerance; succession; transition.
INTRODUCTION
We describe a linkage between the two approaches
most commonlyused formodelingforestsuccessional
dynamics: markovian or transitionmodels and JABOWA-FORET simulatorsor gap models. This linkage provides analytical guidance to numerical simulations and a consistentmethodologyto scale-up from
the gap level to the stand level. This methodologyis
needed for applications requiringnumerous simulations, as in landscape dynamicsand analysis of management strategies. Further simplificationscan be
achieved by groupingtree species according to their
functionalrole (e.g., regenerationand mortality),or
theirstructuralrole (e.g., verticalposition in the canopy), or a combinationof both. These simplifications
are convenientforapplicationsto species-richforests.
Markovianmodels have been used to analyze forest
dynamicsby simplifyingthe effectof environmental
conditions on demographyand growth (e.g., Horn
1975, Runkle 1981, and others reviewed in Usher
1992). Simulators of forest dynamics based on gap
models have linkedenvironmentalparametersto demographics and growthusing the JABOWA approach
I Manuscriptreceived 29 July 1994; accepted 12 October
1994; finalversion received 2 December 1994.
2Present address; School of the Environment,Duke University,Durham,NorthCarolina 27708 USA
(Botkinet al. 1972) in cases wherefloristicsand natural
historyof the species involved are available (e.g., the
FORET simulatorof Shugartand West 1977, and variants reviewed in Urban and Shugart 1992). Both approaches have faced limitsforapplicationsto speciesrich forestsand landscape-scale analysis due to mathematical and computationaldifficultiesinduced by the
increased numberof species and of model plots.
The adequacy of using Markov processes and differentialequationsto model plantsuccessionhave been
extensively discussed (e.g., Weinstein and Shugart
1983, Leps 1988, Usher 1992). The limitationsinclude:
vegetationhistorycannot be described by the current
state (e.g., Hulst 1979b, 1980), time-inhomogeneities
of the transitionmatrixare difficultto model (e.g.,
Hulst 1979b, Usher 1979, Hobbs and Legg 1983, Dorp
et al. 1985), and definitionof discretestatesin a continuumis arbitrary
(Usher 1979, 1981, 1992). However,
capabilities of
the predictiveand hypothesis-exploring
Markov models are advantageous (e.g., Runkle 1981,
Usher 1981, Hobbs and Legg 1983, Scanlan and Archer
1991, Pastor et al. 1993), the potentialfor expanding
and other
thestatedescriptionto includeenvironmental
biotic factorscan lead to ecosystem dynamics (e.g.,
Hulst 1980) and the theoreticalapparatus of Markov
processes forcalculatingstochasticdynamicscould be
betterexploited (Usher 1992).
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November1995
TRANSITION AND GAP FOREST MODELS
We will concentrateon thepossibilitiesof including
some of thevegetationhistoryby using a semi-Markov
model, which makes the transitionsdependenton the
time spent in a given state (Howard 1971). This type
of formulationcould, forexample, account forthe dependencyon the age structuresuggestedas necessary
by Hulst (1980) and Hobbs and Legg (1983). The semiMarkov model has been relativelyless used in ecological modelingresearch;Marcus et al. (1979) proposed
its use forcompartment
models,Acevedo (198 1a) used
its correspondenceto electricalnetworksformodeling
forestsuccessional dynamics,Moore (1990) used its
discreteformulationto relatedisturbanceand management strategiesof vegetationdynamics,and Matis et
al. (1992) applied it to shrimpmigration.
Gap models have been modifiedand adapted to sitespecificapplicationsbut have lost genericcapabilities
(Urban et al. 1991, Urban and Shugart 1992). While
thistrendhas helped us to understandthe dynamicsof
several forests,it is still necessaryto infergeneralbehavior and formulategeneral hypotheses. Deriving
transitionmodels fromgap models can help to extract
basic featuresand closed-formsolutionsforgap models.
Modeling species-richforests and landscape-scale
forestpatternscan be simplifiedby aggregatingspecies
accordingto convenientcriteria.For example, species
can be classified according to theirlight-dependence
or gap-requirement
characteristics,such as "pioneer"
or "gap-requiring" species and "climax" or "shadetolerant"species (Acevedo 1980, 1981a, b, Swaine and
Whitmore1988, Whitmore1989). These classes have
been the subjectof muchresearchand discussion (Brokaw 1982, 1985, 1987, Hubbell and Foster 1986, Brokaw and Scheiner 1989), and permita dynamicinterpretationof the forestas an ever-changingmosaic of
patches cycling throughgap, building, and mature
phases (Watt1947, Oldeman 1978, Whitmore1989) or
similar phases (Whittakerand Levin 1976, Bormann
and Likens 1979). Even thoughthisclassificationmay
be simplistic(Barton 1984, Hubbell and Foster 1986,
Denslow 1987, Brokaw and Scheiner 1989, Canham
1989, Lieberman et al. 1989, Smith et al. 1992), it
allows for a practical modeling methodologyable to
answer questions relatedto the coarse-scale dynamics
of the forestmosaic.
Other examples include: definitionof cover-states
according to successional status for simulatinglarge
regions (Shugart et al. 1973); definitionof types accordingto shade and droughttolerance(Smithand Huston 1989); definitionof functionalgroups based on
physiological,reproductive,and life historycharacteristics (Moore and Noble 1990); and identificationof
several tree species roles based on patternsobserved
in FORET-type simulations(Shugartet al. 1981, Shugart1984, 1987). This last approachcombinesthegapcreatingpropertiesof trees(derivedfromthemortality
process) with the gap-requiringproperties (derived
1041
fromthe regenerationprocess), to obtain four main
groups of tree species thatplay functionalroles in the
dynamics of the forest.This scheme associates size
with gap creationand thereforeis similarto the classificationproposed by Swaine and Whitmore(1988)
for tropical forests.Shugart and Urban (1989) have
used simulatedforestswithsingle species representing
thefourroles to infertypicaldynamicalpatterns.Transition models based on these fourroles and theircorrespondenceto gap simulatorshave been explored in
detail by Acevedo et al. (1995a).
In thefollowingpages, transitionmodels using semimarkoviantransitionsamongcovertypesare developed
and parameterized.First,the conceptual basis of the
approach is presented,emphasizingthe semi-Markov
transitionmodel. Next, an example of state definition
bothspecies functionalroles and canopy
incorporating
layersis developed. Third,the methodto parameterize
the transitionmodel fromthe gap model is described
using four functionalroles as an example. Next, the
basis forsimulatinglandscape dynamicsusingthetransition model is described. Lastly, the ideas presented
in theprevioussectionsare applied to theH. J.Andrews
forest in the Western Cascades of Central Oregon;
states are definedby species and vertical position in
the canopy, a transitionmodel is parameterizedfrom
a gap model,and thetransition
model is used to explore
landscape dynamics.
MARKOVAND SEMI-MARKOVMODELS
A Markovchain describingthetransitionsof a forest
plot among states is used to establish a semi-Markov
model thatalso considersthe different
longevitiesand
growthratesof the treespecies. A gap-size forestplot
is assumed to make transitionsamong several states
definedon thebasis of dominanceof one of theseveral
cover types. These types can be species, functional
roles, structuralroles, or a combinationof both. Explicitconsiderationof theunoccupiedgaps can be done
very simplyby definingthemas anothertype.At time
t, the total coverage in a collection of n gap-size plots
will be distributedamong the N types according to
proportionsXi(t),whichshouldbe approximatelyequal
to the probabilitiespi(t) thata plot is covered by type
i at time t.
The Markovchainis givenby a matrixP oftransition
probabilities.An entrypI of this matrixis associated
withthe transitionfromstatej to state i. These probabilities are definedfromknown relationshipsamong
the cover types or fromthe outputof gap simulators.
Examples of thesemethods,using fourfunctionalroles
to define the states, are described in Acevedo et al.
(1 995a).
The steady-statederived from a Markov chain to
inferthe long-termforestcompositionfails to account
for the time spent in a given state before making a
lontransition,which is importantgiven the different
gevitiesand growthratesof thetreespecies. Therefore,
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1042
M. F. ACEVEDO ET AL.
a semi-Markov process is a more realistic model.
Steady-statecover values can be analyzed under differentscenarios; for example, conditionsof high disturbance,conditions favoringgrowthof small trees,
and those favoring growth of large trees (Shugart
1987). A special case of conditionsof highlyaccentuatedroles, i.e., extremesof shade-toleranceand gapcreation characteristicsis analyzed in Acevedo et al.
(1995a).
The holding time densities he's, thatis, the probabilitydensitiesforthe time spentin makingthe transitionfromtypej to i, can be convenientlydefinedby
a gamma density (Lewis 1977, Hennesey 1980, Acevedo 1981a) h,(t), whichhas two parameters:dij,and
rate,and thesecond,
ki0.The firstone, dij,is a first-order
the orderof the function.
kij,is an integerrepresenting
For ki0> 1 the probabilityof makingan instantaneous
transitionis zero. The mean and varianceof thisdensity
are equal to
mi0= kly/did
vij = kijl(d
j)2
(1)
Statisticsof interest,e.g., occupancyprobabilities,entranceprobabilities,transittime,etc.,can be calculated
fromsemi-Markovtheory(e.g., Howard 1971). In thi-s
paper, only the formulaefor the limitingbehavior of
the occupancyprobabilitiesare used. These stationary
probabilitiesrepresenttheprobabilitiesof findinga gap
occupied by any one of the cover types.
The waitingtimedensity,thatis, theprobabilitydensity of the time spentin statej, beforemakinga transitionto any one of theotherstates,can be calculatedby
N
Wj(t) = , pjhi(t)
=1=
j = 1, 2, ... N,
(2)
as thesum of all theholdingtimedensitiescorresponding to transitionsout of statej. The mean waitingtime
in statei is then the mean of Wj(t), and denoted as
MJ,
N
MJ=plJmij
=1=
j = 1, 2,... N,
(3)
Ecological Applications
Vol. 5, No. 4
elapsed. Note thatX** is differentfromX* obtained
fromthe embedded chain, demonstratingthe importanceof theholdingtimeforthestationarydistribution.
A large value forXj** comparedto the othertypescan
be due to a highervalue for the mean of the holding
time in transitionsout of statej, as well to a high
probabilityof transitioninto statej.
Note thatforeach transition,threeparametervalues
are needed: the rate and orderfor each time lag, and
one transitionprobability.Instead of specifyingthe
rates,it is possible to specifythe ordersand themeans
of the lags, and then use Eq. 1 to compute the rates.
The dynamicscan be determinedby computersimulation of the mean behaviorof a collection of plots. A
convenientapproach forthe simulationis to integrate
numericallythe set of first-order
differential
equations
that emulate the gamma function(Acevedo 1981a).
This equivalence has been referred
to as thelinearchain
trick(McDonald 1978), or the catenarysystem(Van
Hulst 1979a), or pseudo-compartments
(Matis et al.
1992). The transitionfrom state j to state i can be
considered as a sequence of transitionsamong intermediate states,at the rate dj. The transitionfromthe
statej to the firstintermediatestateis affectedby the
probabilityp,. All the proportionsin the intermediate
states of this transitionare assumed to add up to the
totalproportionof thestatej undergoingthattransition.
Therefore,the sum over all the destinationstatesi will
give the total proportionin statej.
For long holding times withlow variance it is convenient to add a fixed time delay fJto the gammadistributedtime delay to obtain the holdingtime density (Blythe et al. 1984, Acevedo et al. 1995a). The
computersimulationis more complicated because it
involves the numericalintegrationof a set of delaydifferentialequations. The mean holding time is the
sum of the mean of the distributeddelay and the fixed
delay. Examples are given in Acevedo et al. (1995a).
FUNCTIONAL
ROLES
AND CANOPY
VERTICAL
STRUCTURE
The definitionof cover type is sufficiently
general
or the sum of all productsof theprobabilitypj and the to allow theinclusionof functionalroles,verticalstrucmean kIjldljof the hij(t)density.In turn,the mean time ture,or othercharacteristics.For example, structural
betweentransitionsM is calculated as a weightedsum aspects can be used to characterizesuccessional states
of the mean waitingtimes in each state
in tropical forests (e.g., Lescure 1978, Tomlinson
N
1987). In this section we will combine the fourfuncM = E Xj*Mj,
(4) tional roles discussed in the Introductionand a twoJ=1
layer canopy.
wheretheweightsused in theaverage are thestationary
The fourfunctionalroles are definedin thefollowing
proportionsXJ*of the embedded chain. The steady- manner(Shugart1984, 1987). Role 1: gap-creatingand
state occupancy probabilities,and thereforethe sta- gap-requiring,shade-intolerant
trees thatcan grow to
tionaryproportionsXj** for all the types,can be cal- a large size. Role 2: gap-creatingand non-gap-requirculated as
ing, shade-toleranttreesthatcan grow to a large size.
Role
3: non-gapcreatingand gap-requiring,shade-inXj*MjMI
j = 1, 2, ... . N,
X**=
(5)
toleranttrees thatgrow to relativelysmall size. Role
whereXj** representsthefractionof theplots thatwill 4: non-gap-creatingand non-gap-requiring,
shade-tolbe occupied by typej aftera sufficiently
long timehas eranttreesthatgrowto relativelysmall size. Transition
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November1995
TRANSITION AND GAP FOREST MODELS
1. Definitionof feasible statesin the four-role,twolayer transitionmodel.
TABLE
State
Overstory
1
2
3
4
5
6
7
8
9
10
11
Gap
Gap
Gap
Gap
Gap
Role
Role
Role
Role
Role
Role
Understory
Gap
Role
Role
Role
Role
Gap
Gap
Role
Role
Role
Role
1
2
1
1
2
2
1
2
3
4
0/1
2l
0/2
0
0
0
0
0
0 P62 ?
0
0
0
P51
P
0
0 P38 0 P310 0
0 0 0 0 0
O p73 0 0
0
0
0
0
0 0 00 ?
P86 0
0 0 0 0 0 P96 0
0
0
00 0 ?
0 0 0 00
0
0
0
0
0
P1070
0 PI170
p59
0
0
0
00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
P5I1
(6)
Thereare 24 transitions
withnon-zeroprobabilitiesand
only 13 of these probabilities need to be specified.
However, all 24 sets of delay parametersneed to be
estimated.The transitiongraphof this chain is shown
in Fig. 1.
Parametervalues are obtainedby groupingthe transitionsinto: mortality,
growthand establishment.Mortalitytransitionsare those to stateshavinggap in both
2/0
93
P16
0/3
P19
12
P107 2)2
P12
/P1
P14
0/
P211
P3-
2
4
2
4
1/4
P96
P628
0 P12 P13 1 1 P16 P17 P1s P19 PIIo Pill
P21 0 0 0 0 0 0 0 0 0 0
0 0 0 0
0 0 0 0
1/0
68
models based on these fourroles and theircorrespondence to gap simulatorshave been explored in detail
by Acevedo et al. (1995a).
Since roles 3 and 4 are consideredto be low stature
relativeto roles 1 and 2, it is convenientto assume a
so thatstatesof the model explictwo-layerstructure,
itlyconsidertherelativeproportionsof mixed canopy.
In fact,the presenceof role 4 is underestimated
when
consideringonly one layer,because these small trees
can never dominatean intactforest.
The statesare composed of role i in theuppercanopy
and rolej in the lower canopy. By combiningthe four
roles, emptygaps, and two canopy layers we obtain a
totalof 11 statesas definedin Table 1. The valid transitions are derivedassumingthattreescan only be establishedin theunderstoryand in emptypositionscreated by mortality.The fall of large trees can create
openings in both layers (fromstates 8, 9, 10, and 11
to state 1), butcan also createopeningsin theoverstory
leaving already establishedshade-toleranttrees in the
understory(fromstates8 and 10 to state3; fromstates
9 and 11 to state 5). Accordingto these assumptions
the followingtransitionmatrixis derived:
P31 0
P41 0
6
1043
t
P07/110P1
1
P41
v
P59
P51
P15
0/4
P 17
O = state i
x/x= overstory/understory
FIG. 1. A Markov model of forestdynamics based on
aggregationof species in fourroles and a two-layercanopy.
State numbers,as defined in Table 1, are enclosed in the
circles; the transitionsare shown in the transitionmatrix,Eq.
6. The two-layercanopy structureis shown on the side of
in understory.
each circle as: role in overstory/role
layers, i.e., to state 1, PI, i = 2, 3, . . ., 11; or those
to stateshavinggaps in theoverstory,i.e., states3 and
5, P3s, P310, P59' P51,i Growthtransitionsare fromthose
states having role i in the understoryto those states
having the same role in the overstory,i.e., fromstates
2 and 3 to states 6 and 7, respectively,P62, p73 Establishmenttransitionsare those fromstateshaving gaps
in the understory,i.e., states 1, 6, and 7, P21, P31, P41,
P51, P86, P96' P107, P117. For the sake of illustrationwe
select low-mortality
probabilitiesp1 i = 0.1 for i = 2,
3, 6, 7, 8, 10,as wellas forp59, P51I- The growth
prob-
abilities will thenbe forcedto P62 = p73= 0.9. A high
value of 0.9 is selected forplg,PI I' P38, P310o All establishmentprobabilitiesfromstateshavingroles 1 and 2
in overstory(states 6 and 7) will be assumed to take
the same value; therefore
P86, P96 = 0.45 and P107, P117
= 0.45. Establishmentprobabilitiesin gaps spanning
both canopy layers (thatis fromstate 1) are assumed
to be high for shade-intoleranttrees (roles 1 and 3),
trees(roles
P21 = P41 = 0.45, and low forshade-tolerant
2 and 4), P31 = Ps5 = 0.05.
Takinginto account characteristicsof the functional
roles, we assume the values listed in Table 2 for the
means, orders,and fixed latencies of the delays correspondingto these threeprocesses. The rates are obtained using Eq. 1. The parametervalues for all transitionsare summarizedin Table 3. A 500-yrsimulation
with these parametervalues, for an initial condition
correspondingto all plots in state 1, yield resultsfor
the dynamicsof all 11 states.
As an example of these results,Fig. 2 (top) shows
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1044
M. F. ACEVEDO
2. Delay densityparametersfor the four-role,twolayermodel (m = meanin years,k = order,f= fixedlatency
in years).
Role
Role
Role
Role
1
2
3
4
Ecological Applications
Vol. 5, No. 4
Dynamics witha two layer canopy
Example ofsome states
TABLE
Establishment
ET AL.
0.5
Growth
Mortality
m
k
f
m
k
f
m
k
f
3
2
3
2
2
2
2
2
2
1
2
1
30
30
30
30
2
2
2
2
10
10
10
10
10
10
10
10
4
4
4
4
100
100
10
100
0.4
00.
0.2
Lm
U.
0.1
theresponseof states4, 9, and 11. The earlypeak (state
4 = Gap/Role 3) correspondsto initialcolonizationof
gaps by small shade-intoleranttrees (role 3), which
subsequentlydecreases withtime.The othertwo traces
correspond to those states (9 and I 1) having small
shade-tolerant
trees(role 4) in theunderstory.
Note that
these states are well representedin the canopy, even
thoughrole 4 treesare of low stature.This resultdemonstratesthe advantage of includingcanopy layers in
the statedefinition.It will be shownin thenextsection
that excluding the vertical structurein the state definitionwould lead to misleadinglow values of role 4.
As anotherexample of theresults,in Fig. 2 (bottom)
we show three aggregatedvariables: the sum of all
states withgap in the overstory,of all stateswithrole
1 in the overstory,and of all states withrole 2 in the
overstory.Early phases of the simulationshow a decrease of the proportionof gaps, followed by an in3. Summaryof parametervalues for all transitions
of the four-role,two-layermodel. p,, k, are a-dimensional,
mUand fJ in years.
TABLE
i,j
2,1
3,1
4,1
5,1
8,6
9,6
10,7
11,7
PyJ
0.45
0.05
0.45
0.05
0.45
0.45
0.45
0.45
mlJ
Establishment
3
2
3
2
2
2
2
2
ki
f
2
2
2
2
2
2
2
2
2
1
2
1
1
1
1
1
6,2
7,3
0.9
0.9
Growth
30
30
2
2
10
10
1,2
1,3
1,4
1,5
1,6
1,7
1,8
1,9
1,10
1,11
3,8
3,10
5,9
5,11
0.1
0.1
1
1
0.1
0.1
0.1
0.9
0.1
0.9
0.9
0.9
0.1
0.1
Mortality
10
10
10
10
10
10
10
10
10
10
10
10
10
10
4
4
4
4
4
4
4
4
4
4
4
4
4
4
100
100
10
100
100
100
100
100
100
100
100
100
100
100
0
10
Gap/Role3
20
30
Time(decades)
40
50
Rolel/Role 4 - Role2/Role4
Dynamics witha two layer canopy
Aggregatedstates
1
GapI-- -RolelI--
0.8
0
Role2/--
~0.6
0.4
0.2
0:
10
20
30
Time(decades)
40
50
FIG. 2. Simulationresultsforthe mean of the stateprobabilities accordingto the semi-Markovmodel. States defined
on the basis of fourroles and two canopy layers (Table 1).
A complete dominance of state 1 (gap) is assumed for the
initialcondition.Top panel: dynamicsof states4, 9, and 11.
Bottompanel: aggregatedstatesaccordingto overstorydominance: gaps/anyrole, role 1/anyrole, role 2/anyrole.
crease of theproportionof stateshavingrole 1 in overstoryand subsequent increase of those states having
role 2 in overstory.
PARAMETERIZING
TRANSITIONMODELS FROM
GAP MODELS
In this section,we demonstratehow to estimatethe
parametersof the semi-Markovmodel fromruns of a
gap model simulatorusing prototypesof species that
correspondto the cover types.Countingthe frequency
of occurrence of each transitionpair and its timing
allows an estimationof the transitionprobabilitiesand
the holdingtimedensities.To illustratethisprocedure
we summarizeheretheexample developed in Acevedo
et al. (1995a). Four species wereconstructedin thegap
model ZELIG (Urban and Shugart 1992) to represent
thefourcontrastingfunctionalroles definedin theprevious section. The basal area dynamicsexhibita suc-
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1045
TRANSITION AND GAP FOREST MODELS
November1995
Holding time densities
TalliedfromZELIG
0.03
2 04
?0.02
h21+h42
h1 3
|
30
20
Time (decades)
40
50
X
P24
forthe modelas extracted U. 0.01
structure
FIG. 3. Transition
in thisstruc- co
Cyclesarecontained
fromZELIG simulations.
are shown.
ture.Onlythemajortransitions
cessional patternover several hundredyears. Role 3
asserts an early importancebut quickly declines; role
1 reaches its greatestdominancein the first100 yr,and
role 2 increases steadilyinto the next centuries.Role
4 is the most commoncomponentof the understoryin
older stands,but does nothave the statureto dominate
can be removedby including
thecanopy.This difficulty
a two-layercanopyas discussedin theprevioussection.
States were assigned to each plot by the role with
the greatestbasal area. A fifthstate, open gap, was
defined as plots with < 1 m2/habasal area. Better
schemes can be devised using, for example, cluster
analysis as demonstratedby Usher (1981, 1992) and
Leps (1987). Using this simple classification,the relative abundance of the states shows a trendsimilarto
the relativebasal area per role. The notable exception
to this is thatrole 4 rarelyoccurs because these trees,
althoughcommon,are nearly always dominatedby a
role 1 or 2 overstory.This resultindicates the convenience of a classificationcombiningthe roles and the
vertical position in the canopy, as was done in the
previous section.
Plot transitionsare tallied in the ZELIG runs by
countingthe cases when the plot changes fromstate i
matrixforeach
to statej, to generatea transition-count
time step. Results for400 plots over 500 yr of simulation with ZELIG were produced in this case. The
transitiongraphcorrespondingto this structurecan be
seen in Fig. 3, where only the most importanttransitions are shown.
The time course of the frequencyof occurrenceof
a transitionfromstatei to statej, allows estimationof
the holding time densityforthe transitionfromi to j.
Fig. 4 illustratessome of these results.Fixed latencies
can be extractedfromthese time traces and the estimated means of the gamma densities are obtained by
subtractingthe latency fromthe observed mean. For
example, the holding time densityfor the transition
fromrole 3 to 1 can be estimated,using the corresponding time trace in Fig. 4, to have the following
values for the parameters:orderk13= 3, mean Mi3 =
50 yr,and fixeddelayf3 = 30 yr.Following this procedure for all feasible transitions,the matricesof orders, means, and lags are obtained.
10
0
FIG. 4. Time tracesof theholdingtimeforthe transitions
fromrole 1 to role 2, fromrole 2 to role 4, and fromrole 3
to role 1, as tallied fromthe ZELIG simulations. A 10-yr
average is shown to display the main featuresof the density.
The transitionmatrixis adjusted togetherwith the
matrixof themeans to fitthe semi-Markovmodel simulation resultsto the ZELIG simulationresults.Using
this procedure,the following matricesof parameters
are obtained:
P
0.000 0.200 0.200 0.000 0.000
0.000 0.000 0.100 0.875 0.000
0.000 0.700 0.000 0.125 1.000
1.000
0.000
0.000
0.100
0.000
(7)
0.000 0.000 0.700 0.000 0.000
k=
m
0
0
2
0
3
3
0
1
2
0
0
0
0
5
0
0 260 400
0
80 50
0
0 40
0
80
0
0
1 40
0
0 300
0
[0 90
f=
2
0
0
2
0
0
40
0
0
0
90
0 30
0 60
0
0
0
80
0
0
30
0
0
yr
01
0
0
0
0
yr.
(8)
can be calculateddirectly
The stationarydistribution
fromthe closed formsteady-statesolution (Eq. 5) as
[0.0029, 0.2082, 0.6256, 0.0886, 0.0746]. Calculation
by simulationexhibittheresultsshown
of thetransients
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All use subject to JSTOR Terms and Conditions
1046
M. F. ACEVEDO
ET AL.
EcologicalApplications
Vol. 5, No. 4
ZELIG
-
300
400
MOSAIC
0.8
0.6
0
0
&_ 0.4
U.
0.2
0
0
100
200
Years
500
FIG. 5. Plot- or patch-typedynamics of the four differentroles (1, 2, 3, 4). Comparison of semi-Markovsimulations
results (smooth lines) to ZELIG simulationresults (fluctuatinglines) illustratingthe adequacy of fittingthe parametersof
the semi-Markovmodel fromthe ZELIG simulation,performedfor400 plots and 500 yr.Typical successional dynamicsare
exhibitedby this simulation.G = gap.
in Fig. 5 togetherwith the ZELIG simulationresults. MOSAIC prototypethatrunsin theUNIX environment
This exampledemonstrates
thefeasibilityof calibrating and is linked to a raster-basedGIS, such as GRASS
the transitionmodel fromthe ZELIG output.
(U.S. ArmyCorps of Engineers 1991) and the GRID
module of ARC/INFO (ESRI 1992). Model outputfiles
APPLICATION TO LANDSCAPE DYNAMICS
in ASCII are convertedand importedto the GIS. ReThis section illustratesthe use of transitionmodels sultsare shownin Fig. 6 foryear400 of one simulation
to analyze forestdynamicsat the landscape scale fol- run withrandominitialconditions.
lowing Acevedo et al. (1995b). The transitionsemiComputerresourceutilizationdepends on the nummarkovianmodel, hereafterreferredto as MOSAIC,
ber of states,the orderof the delays, and the number
has probabilities,distributedlags, and discrete time of non-zerotransitionprobabilities.For illustrationof
lags estimatedfromsimulationruns of a gap model model performance,MOSAIC runs in -20 min, oc(ZELIG) at the plot scale as describedin the previous cupying -30 Mb of memory,when executed in a
section. This parameterestimationprocedure assures SPARC 2 workstationwith32 Mb memoryand 38 Mb
consistencyin thechange of scale. Environmentalfac- swap, fora 500-yrsimulationof a landscape of 10 000
torsare storedas GeographicInformation
System(GIS)
ha, (1000 cells of 10 ha each), fortheparametervalues
files and transferred
to MOSAIC to adjust the param- used in theprevioussection.
By contrast,a 500-yrsimetersforsimulation;values forthe statesat each landulationrun will take -3 h fora small watershed('50
scape cell are generatedby MOSAIC and transferred
ha) using thecurrentversionof ZELIG. Large numbers
to the GIS fordisplay and analysis.
of cells would degradeMOSAIC's performancedue to
The landscape is composed of a collectionof a large
increased swapping.
numberof cells (units, ecotopes, or tesserae; Naveh
The parametricdependence of successional models
and Lieberman 1984). Each one of these cells is modon environmentalconditions are importantfor landeled as a mosaic of smaller,gap-scale plots, and its
scape applications (e.g., Weinsteinand Shugart 1983,
stateis given as the proportionof its totalarea in each
Shugart
1989; D. L. Urban and T. M. Smith,unpubof several cover types. Landscape dynamicsare simlished
data).
The interactionof these variables include
ulated as changing proportionsof within-cellcover
slope/aspect
effects
on radiation;elevation effectson
types. The transitionmodel is used to simulate landscape dynamics,by repetitivecalculationsforeach cell temperatureand precipitation;and terraineffectson
hydrologyvia topographicconvergence.Terraineffects
at each time step.
limitingfactorsare incorporatedby
Due to the Monte Carlo natureof gap-typesimula- and environmental
assuming
GIS
layers
containingthese factors,for extors,theiruse forlandscape applicationsrequiresmultiple runs for every plot constitutinga cell, whereas ample: thermaleffect(representingelevation and temthetransitionmodel only requiresone runper cell. For perature),soil moisture(combiningprecipitationand
illustration,
an area of 10 ha per cell would requireruns topographicalconvergence), and soil fertility(repremethof a gap model for 100 plots of 0.1 ha each, which sentingnutrientavailability).There are different
implies a reductionby a factorof 100 when using ods to extractthe transitionmodel parametersfrom
the semi-Markov simulation. We have developed a these environmentalconditions.In this section we il-
This content downloaded from 129.120.92.148 on Fri, 6 Jun 2014 13:14:57 PM
All use subject to JSTOR Terms and Conditions
November1995
TRANSMON
Cover Maps (%)
......
N.
Year 400
Role 1
M
.....
. ... jb'Ffd&
...
.........
- -
-
----------
M,
Year 400
Cover Maps
M
M
.. ..... ......N
Himmiiiiii-.110
- ----MM111
...
Role 2
..............
M
...
.........
..............
W '
M.",W
1047
AND GAP FOREST MODELS
U
...
M.:
.
............
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.. . .......
N
N
U.
M.
M
:
MN
.....
M.,
E
in: .... ...
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M
.....
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---
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Z;
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V=W
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N
:.:-
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N N
M.-
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..... .. ...........................
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UIN
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H.H.:
.
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2
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.
.
HE
......................
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.................
M=
Role 4
...........
...t-!.:
...........
mw
--M :....
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.....
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M
........
Year 400
Cover Maps (%)
"M
...............
:,,,, Cover Maps
2-
...
. . ..
....
NZ
...........
.
............
. .....
......
%.-:1
......
EM.1
ME
..... U-M
......... ......
... ........
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in: .......
......-0 n.Z.
......
M
FIG. 6. Homogeneous landscape dynamics simulationresults for year 400 of one simulationrun with random initial
conditions.Four relativecover values are shown fora hypotheticallandscape of 50 x 50 = 2500 cells. Darkercells indicate
largerproportionof the correspondingrole.
lustrateone possiblemethod;in thenextsectionwe
will providean exampleof a different
method.
Foreverycell k,jwe use threevariables(whichtake
relativevaluesbetween0 and 1) storedas maps,Tk,,
soilmoisture,
Sk, andFk,representing
temperature,
and
soil fertility,
respectively.
For each covertypei and
everycell kjI theenvironmental
limiting
factors(relativevaluesbetween0 and 1) are calculated,taking
intoaccounttheresponseofthecovertypeto thelimA compound
itingfactor.
environmental
limiting
factor
is thencalculated foreach role i and everycell k,I. For
example, a simple assumptionis that the composite
factoris theproductof thesethreefactors.The resulting
maps foreach type i are then used to multiplyevery
nominalor optimalratedji of the transitionfromstate
i to all otherstatesj. This is done forevery transition
witha non-zeroprobabilityand foreach cell everyyear.
As examples of implementing
thesefactors,we can use
expressionssimilarto those employed in gap models:
a parabolic expressionfor the thermallimitingfactor
This content downloaded from 129.120.92.148 on Fri, 6 Jun 2014 13:14:57 PM
All use subject to JSTOR Terms and Conditions
1048
M. F. ACEVEDO ET AL.
Ecological Applications
Vol. 5. No. 4
4. Maximum and minimumtemperaturetolerances
for the hypotheticalparabolic thermalresponse of four
roles.
TABLE
Role
Role
Role
Role
I
2
3
4
Maximum
(0C)
Minimum
(OC)
10
15
00
20
30
30
20
30
controllingthe thermalenvironmental
factor.Soil
moisture
and fertility,
as well as neighbor
effects,
are
ignoredin thisexample.Fig. 7 illustrates
thehypothermal
factor
mapwhichcorresponds
toa Digthermal
FIG.7. A hypothetical
factor
established
tocor- thetical
ranging
from
toa DEM of50 X 50 = 2500cellsof 10haeach, italElevationModel(DEM), withaltitude
respond
withaltitude
valuesranging
from
0 to5 kin,andtemperature
0 to 5 km,covering
25 000 ha in 50 x 50 = 2500cells
values from
toO,C.
of 10 ha each.Assuming
a linearlapserateof5?C/km,
hypothetical
temperature
valuesrangefrom250 at the
lowestsitesto 00 at highestsitesin stepsof 5?C. All
with maximum and minimum tolerances for each role
thesevaluesare hypothetical
and used onlyforillusfor the soil moisture
i; a square-root
expression
limiting
Five
cover
tration.
are
fourfunctional
types
assumed,
factor
tolerable
parameterized
bythemaximum
dryness
roles
and
gap.
The
semi-Markov
parameter
valuesare
fortypei; anda quadraticexpression
forthesoil ferfactor
withthreeparameters
foreachtype thesameas thosederivedin theprevioussection.The
tility
limiting
andminimum
thermal
tolerances
forthepari. A totalof six parameters
are thenrequiredforeach maximum
abolic
thermal
inTable
response
of
each
role
are
given
effects.
typei to computetheterrain
4.
2
Roles
I
and
are
assumed
to
have
optimum
growth
cell
interactions
should
alsobe included
Neighboring
whereasroles3 and4 areassumed
locations,
to developspatialpatterns
ofthedynamicmosaicdue atwarmer
The simulation
results
tocontagious
disturbance
(SprugelandBormann1981, to requirelowertemperatures.
after
all
500
for
roles
are
in
ilyr
shown
8
and
Fig.
Knight1987)andseeddispersalorotherneighborhood
lustrate
the
distribution
of
the
roles
according
to
their
effects
reP*j A similar
Na
,
. has been
(10)
(Turner1987).
approach
response.In thenextsectionwe will applya
centlyreported
by Wissel(1992) to studytheforest thermal
similar
method
to a morerealisticexample.
mosaiccycle applyingthecellularautomatamethod
and considering
onlythedominant
species.
APPLICATION TO H. J. ANDREWS FOREST
of neighbor
effectsis important
to
Implementation
In
this
sectionwe illustrate
theapplicationof the
accountfordispersal.
The layersofcovermapsXi,one
in
methodology
developed
the
previous
sectionsto the
foreachtypei, areavailableat everytimestep.Therein
H.
J.
coniferous
forest
Andrews
intheOregon
forest,
fore,foreverycell k, therearerelativecovervariables
This area is locatedat about440
rel- Cascade Mountains.
(valuesbetween0 and 1) foreach typei. Another
threeimportant
species
ativevariable(valuesbetween0 and 1) yki.is obtained N, 1220W.Forthisapplication,
are
considered:
Douglas-fir
(Pseudotsuga
menziesii),
theXik.,of all cells suffounding
kLI
by averaging
western
hemlock(Tsugaheterophylla)
andPacificsilq +a r+a
X
ver firor truefir(Abies amabilis).Cover typesare
verticalpositionin thecangenerated
by considering
+ 1)2'(9
qar=a(2a
and
the
of
the
is a disopy
stand.
Standstructure
age
wherea is an offset.For exampleif a = 1, we have
tinctive
feature
of
different
and
classes
of
stages
age
(2a + 1)2 = 9 cells. The resulting
mapsy, (one for theregion(CohenandSpies 1992).The following
six
eachtypei) represent
theabundance
ofthattypearound
statesaredefined:
I = gap,2 = youngeven-aged
Douga cell. ThenMOSAIC multiplies
everynominalorop- las-fir
(<30 mheight),
3 = mature,
even-aged
Douglastimaltransition
factor fir
probability
pijbya correction
in
(>30 m height),4 = "old-growth"
Douglas-fir
theupperstoryandwestern
hemlockintheunderstory,
5 = youngtrue(Pacificsilver)fir(<30 m tall),6 =
yipij
maturetrue(Pacificsilver)fir(>30 m tall withan
Thesetypescorrespond
to a classification
understory).
foreverypairi,j of statesand at each cell. This cor- basedon remotesensing(Cohenand Spies 1992),exrectionfactorassuresthatthesumof thenewp*i is ceptthatwe addedtheage class forDouglas-fir
and
to 1.
thatwe excludedmountain
hemlock.
equal
As an illustration
of theapplicationof theseideas
TheZELIG modelwas modified
totakeintoaccount
we construct
a hypothetical
used for
landscapewithaltitude aspect,slope, and altitude.The parameters
250
*!
P*
yii
)
This content downloaded from 129.120.92.148 on Fri, 6 Jun 2014 13:14:57 PM
All use subject to JSTOR Terms and Conditions
November
1995
t
TRANSITION AND GAP FOREST MODELS
Cover Maps (%): Year 500 1
Cover 4r4
Mops (%): Year 500
Ro I e I
u
Ro I e 3
___l
O5le!!!
|
RoIe
-
3.43
N.ur!In
J t-;=
r.
1
-i
1049
FIG. 8. Resultsofheterogeneous
landscapedynamics
simulation
takingintoaccounttheenvironmental
factor
established
in Fig. 7. The responseof therolesis illustrated
foryear500. Darkercells indicatelargerproportion
of thecorresponding
role.
each speciesaregivenin Table5. Runsweremadeon
a transect
from500 to 1600 m of altitudeat 100-m
I + exp[(-alt + u,)/s,]'
(11)
intervals.Slope was set at 50% and aspectat 1800
prob(south-facing
ofthesimulation
runs areusedto accountforthechangesin transition
slopes).Theresults
abilities
and
when
a
of
delays
moving
along
gradient
of thegap modelat thesesiteswerethenused to eshereI = 1, 2 to accountfortwoecotones,alt
timatethestructure
and parameters
of thetransitionaltitude,
model.The effectof temperature
and precipitation
is = altitude,u, = threshold,s, = sensitivity,all in m.
in studiesof
builtintotheresultsof thegap simulation
modeland An ecotoneat 1100-1200m is reported
this
area
and
(Fiorella
Ripple
1993).
summarized
as effects
of altitude.
Some fixedlatenciesfortransitions
to statei are
ThegraphofFig.9 wasderivedon thebasisofthese
with
the
value
of
changed
function
linearly
b,as
runs.A transient
pseudo-state,
labeled0, was included
in thegapphaseto separatethosegapspreviously
colfj = fij + aib,
(12)
onizedfromthosegaps withbaregroundat thestart
of thesimulation.
of altitude,
Sigmoidfunctions
wherefj is thebaselinefixedlatencyand a, is a coTABLE
Spp.
ABam
PSme
TShe
5. ZELIG parameters
valuesforthespeciesconsidered
in HJA.*
Am
Dm
Hm
G
600
1100
500
200
300
225
62.5
84.1
63.6
1250
1700
1080
F
1
1
1
Tmin
Tmax
L
118
441
311
1815
2411
2480
5
2
5
M
3
4
3
N
2
2
2
Sds
13.3
13.3
13.3
Si
0
0
0
* Spp. = species, ABam = Abies amabilis, PSme = Pseudoisuga menziesii,TShe = Tsuga heterophylla,Am = maximum
dbh(cm),Hm = maximum
age (years),Dm = maximum
rate(cm3wood/M2
height(m), G = growth
leaf),F = lifeform
(generictypes),Tmin= minimum
temperature
tolerance(growingdegree-days),
Tmax = maximum
temperature
tolerance
(growingdegree-days),
L = shadetolerance
(rankI to 5, 1 = veryintolerant),
M = maximum
drought
tolerance
(rank1 to
N = nutrient
5, 1 = veryintolerant),
response(rankI to 3, 1 = responsive),
Sds = seeds,SI = sprouts.
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1050
M. F. ACEVEDO
2 = Young Douglas Fir
4 = Mature Douglas Fir/
Hemlock
5 = Young True Fir
6 = Mature True Fir
p61 =bX6
pl4
P51
1b1
P6
P66
P20
P41
P41 =1-b2(1-bX6)-bx6
Gap
P16
P51 =b2 (1 -bx6)
FIG. 9. Transition
graphforHJAforestcovertypesas
derivedfromZELIG simulation
andrates
runs.Probabilities
are a function
of altitudeby usinga sigmoidfunction.
ET AL.
Ecological Applications
Vol. 5, No. 4
6. Summaryof MOSAIC parametervalues estimated
fromZELIG output.pt,,kl,are a-dimensional,mi,and f, in
years. Some values depend on b, and b,6functions.
TABLE
i,Aj
2,0
1,2
4,2
4,1
5,1
6,1
1,4
4,4
6,5
1,6
6,6
1
bi
1 - b1
1 - b2(1 - bX6)- bx6
b2(1- bX6)
bX6
0.25
0.75
1
0.35
0.65
ml,
k,
fi,
2
3
10
10
8
10
200
10
2
400
5
2
1
4
1
1
1
1
1
2
1
1
10 + a2b,
59 + a1b
77
10 + a4b,
20
0
0
0
50
0
0
thewhole area of theDEM thatcontainstheHJAforest
efficient,both expressed in years. Abundance effects (Fig. 11) by aggregatingcells of 30 X 30 m into cells
are accounted forby functionsof cover type,e.g., by of largersize. The elevationsof n neighboringcells are
a sigmoid functionof state 6 (maturetruefir),
combinedusing thenearestneighboroptionin the RESAMPLE command of GRID (ESRI 1992) to obtain
the
altitudeof one aggregatedcell. For demonstration
(13)
bx6(t)= 1 + exp[(-X6(t) + U6)/S6]'
we used n = 3 and 5. In thefirstcase (n = 3), the cells
where X6(t) is the value of state 6 at time t, U6 is a are 90 X 90 m (8100 m2or 0.81 ha), and the DEM of
Fig. 11 is convertedinto an aggregatedDEM having
threshold,and S6 a sensitivity,all a-dimensional.
371/3
123 rows and 539/3 - 179 columns, for a
MOSAIC parametervalues estimatedfromZELIG
outputare summarizedin Table 6. Note thatsome of total of -22000 cells, or a 10-fold reductionin the
these parametersdepend on elevation and cover dy- numberof cells, withaltituderangingfrom400 to 1612
namics accordingto Eqs. 11, 12, and 13. In turn,the m. In the second case (n = 5), the cells are 150 X 150
parametervalues forthe sigmoid and linear functions m (22 500 m2or 2.25 ha), and the DEM of Fig. 11 is
given by these equations are listed in Table 7. For il- convertedinto the aggregatedDEM of Fig. 14, having
lustration,the temporaldynamicsat 500 and 1200 m 371/5 74 rows and 539/3 - 108 columns,fora total
of altitudeare displayed in Fig. 10. The resultsof the of -8000 cells. Of course, some loss of resolutionis
calibratedMOSAIC model are superimposedon these observed.Altitudeof aggregatedcells now rangesfrom
400 to 1600 m.
graphsforcomparison.
We now use the aggregatedDEM fordemonstration
We use the DEM that includes the HJA forestas
cell sizes.
illustratedin Fig. 11 withaltituderangingfrom400 to runsof MOSAIC forall 18 000 ha at different
A
500-yr
simulation
run
on
a
SPARC
2,
with
48 Mb
1617 m. This map correspondsto an area of -18 000
RAM and 1 Gb SWAP, takes -36 h forcells of 90 X
ha, from 44011'20.41
N to 44017'15.911
N and from
1220511.511
W to 122017'15.411
W. The DEM has 371 X 90 m and 12 h forcells of 150 X 150 m. As an example
539
200 000 cells of 30 X 30 m. For a preliminary of this last cell size, Fig. l5a and b show MOSAIC
demonstration
of MOSAIC operatingat plot-sizescale, simulationresultsat theend of the500-yrrunforcover
types4 and 6, matureDouglas-fir/hemlock
and mature
we selected a small sector of Fig. 11, from -44012l N
to 44013' N and 12208' W to 12209' W (Fig. 12), with true fir,respectively.Distributionof these types acaltituderangingfrom846 to 1573 m. A total of 56 X cordingto altitudeis noticeable.
In order to furtherreduce simulationtime, we can
59 = 3304 landscape cells of 900 m2 (0.09 ha) cover
an area of -300 ha or 1.6% of total area in Fig. 11. eitherselect a smaller sector of the aggregatedDEM
This relativelysmall landscape size could also be sim- and runMOSAIC onlyin thissector,or we can maintain
ulated using ZELIG, but we use MOSAIC to illustrate a large area and proceed withfurtheraggregation.Of
the firststep of the scaling-upprocedure.Even at this course, the desired approach will depend on the resosmall landscape size, MOSAIC offersthe advantageof lutionrequiredto answerthequestionsto be asked from
closed form solutions. The results at the end of the the model. In the second case, we would be interested
simulationrun are shown in Fig. 13 forcover type 6,
correspondingto maturetrue fir,which illustratesthe TABLE 7. Parametervalues forsigmoidfunctionsand linear
functions.
effectof altitudeon the distributionof thiscover type.
A noticeable ecotone is indeed displayed at -1100S2
U6
S6
a2
a4
Ul
U2
SI
a,
1200 m altitudedue to the value of 1050 m selected
(m) (m) (m) (m) (adim.) (adim.) (yr)
(yr) (yr)
for u2.
550 25 1050 20 0.1 0.001
-0.5
-5.5
1.5
To illustratethescaling-upprocedurewe now modify
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All use subject to JSTOR Terms and Conditions
November1995
TRANSITION AND GAP FOREST MODELS
1=Gap, 2=DFir
4 = DFir/Hemlock
1051
Dynamicsat 500 m
ZELIG
-
MOSAIC
-
I~~~~~~~
(0.
0
0.6
0.2 0
-
I
100
0
"'
'
I
200
1" " '
'
300
-
lS
lDI
400
Years
- .11
I
500
1=Gap,2=DFir
4=DFir/Hemlock
5=YTFir,6=MTfir
Dynamicsat 1200 m
ZELIG
-
MOSAIC
0.8
0
L0.4
0.2
0
100
200
Years
300
400
500
FIG. 10. Calibrationof MOSAIC model using resultsof ZELIG runs at different
altitudesat HJA. The dynamicsat (top
panel) 500 m and (bottompanel) 1200 m are shown forillustration.
only in coarse-scale dynamics;forexample, using an
aggregatedcell made up of 49 neighboringcells of 210
X 210 m = 44100 m2 or 4.41 ha, a modifiedDEM
(Fig. 16) of 371/7 53 rows and 539/7 77 columns,
or =4000 cells, is obtained. Altituderanges from402
to 1601 m. The simulationtime requiredfora 500-yr
run is now reduced to - 1.67 h on the same machine.
Fig. 17 shows theresultsforcover types4 and 6. Broad
featuresof distribution
withaltitudeare stillnoticeable,
but not so much resolutionis obtained in the ecotone
areas and the valleys.
This applicationis presentedfordemonstration
purposes only. Calibrationusing ZELIG outputwas performedonly at south-facingand 50% slopes for only
one soil type.Improvementsin model performanceare
expected by calibrating for differentenvironmental
conditions.
DISCUSSION
AND CONCLUSIONS
The linkagedeveloped in thispaper betweenthetwo
most widely used modeling approaches to forestdynamics, transitionmarkovianmodels and JABOWAtypesimulators,has several advantages.In additionto
simplifying
thesimulations,theexistingtheoreticalapparatusforMarkov and semi-Markovprocesses can be
veryhelpfulto provideanalyticalguidance to the simulations and provide fast, direct exploration of hypotheses fromclosed-formsolutions and formulae.A
semi-Markovtransitionmodel offersflexibility
in scale
and simplicityof analysis.
There are manyways of definingthe statesor cover
types, and this definitionis an importantpart of designinga transitionmodel. Functionalroles can be definedaccordingto the requirementof canopy gaps for
regenerationand the capacityforopeningcanopy gaps
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1052
M. F. ACEVEDO ET AL.
Ecological Applications
Vol. 5, No. 4
U
U0
FIG. 11. Digital Elevation Model (DEM) of HJA forest
covering -18000 ha, from4411'20.4' N to 44017'15.9 N
and from122?'51.5" W to 122017'15.4" W. The DEM has 371
FIG. 13. MOSAIC results
atendofsimulation
runof500
x 539
200 000 cells of 30 x 30 m. Altituderanges from yr,atcell sizeof30 X 30 m,formature
truefirintheselected
400 to 1617 m.
sectorofHJAforest
ofthis
displayedinFig. 12. Distribution
covertypeaccordingto altitudeis illustrated.
upon death. For considerationof structure,states can
be definedas a combinationof layerand role, yielding
a more detailed and realisticmodel.
Other criteriato group the tree species into types
could be used, such as those derivedfromarchitecture
or morphologyfollowingtheHalle prototypes,
or those
relatedto climaticconditions(e.g., Box 1981, Woodward 1987). However,the typesanalyzed in thispaper
are easily characterizedfromthe well-studiedJABOWA-typesimulators,and have the advantageof being
relatedto the dynamicconcept of the forestcycle.
Long-termtrendscannotbe investigatedby thetransitionprobabilitymatrixalone; thetimedelays or holdin determining
are important
ing timesin thetransitions
as well as thetransientsfolthestationarydistribution,
lowing a disturbance.Definingthedelays as a chain of
first-order
processes has the advantage of facilitating
computersimulation,and of introducingintermediate
FIG.
12. DigitalElevationModel(DEM) ofa smallsector
of HJA forest,from -44?12' N to 44?13' N and 122?8' W
runof56 x 59 cells
to 12209'W,selectedfordemonstration
rangesfrom846 to 1573m.
of size 30 x 30 m. Altitude
states that could potentiallybe used as size classes.
These age classes could be used to inferdbh structure.
The establishmentand mortalitypatternsof the JABOWA and FORET typeof simulatorsare capturedin
thetransitionmodel in termsof transitionprobabilities
among different
species roles. Growthand aging rates
are accountedforby distributedand fixedtimedelays
associated withthe transitions.
Once the parametersof the transitionmodel are obtained fromdetailed simulationsusing a JABOWAtypesimulator,the transitionmode' can be used to run
a simplifiedsimulationof the dynamicsin those cases
when a large numberof simulationruns are required,
e.g., landscape applications,or when a large number
of parametersare required,e.g., forspecies-richforests.
The potential of the transitionmodel to simulate
landscape dynamics is enhanced by linkingit with a
GIS. The parameters,probabilities,and delays are
cells of
DEM forHJAbyaggregating
FIG. 14. Modified
30 x 30 m intocells of 150 x 150 m. The altitudesof 25
cells arecombinedto obtainthealtitudeof one
neighboring
cell. Altitude
rangesfrom400 to 1600m.
aggregated
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November
1995
TRANSITION AND GAP FOREST MODELS
1053
demonstration
FIG. 15. Resultsat theend of simulation
demonstration
FIG. 17. Resultsat theend of simulation
runof MOSAIC, at a cell size of 150 x 150 m, forthe
DEM ofFig. 14.Toppanel:covertype4. Bottom runof MOSAIC, at a cell size of 210 X 210 m, forthe
aggregated
DEM ofFig. 14.Toppanel:covertype4. Bottom
to aggregated
of thesetypesaccording
panel:covertype6. Distribution
of thesetypesaccordingto
cover
type6. Distribution
panel:
altitudeis illustrated.
altitudeis illustrated.
and
made functionsof soil moisture,slope, fertility,
conditionsof each landscape cell.
otherenvironmental
Cell interactionsare includedto develop spatial patterns of the dynamic mosaic (Bormann and Likens
1979, Knight 1987), as for example wind-induced
waves (Sprugel and Bormann 1981). A similar approach has been recentlyreportedby Wissel (1992) to
studythe forestmosaic cycle applyingthe cellular automatamethodand consideringonly thedominantspecies.
The approachwe have presentedherecan play a role
ecosystemsdynamicsat thelandscape
in understanding
scale. Questionsat thislarge scale can be addressedby
the transitionmodel while maintaininga consistent
conceptual and empirical basis with finerscale ecological detailthroughitscorrespondencewithgap models. Consistencyacross scales is much needed in landscape ecology (Meentemeyerand Box 1987, Risser
1987).
This frameworkprovides new capabilities for land
use and managementstrategiesby makingit possible
to obtain fast simulationresultsfor the interactionof
of land management,nattimeand space heterogeneity
ural disturbances,and successional dynamics.
ACKNOWLEDGMENTS
cells of
DEM forHJAbyaggregating
FIG. 16. Modified
30 x 30 m into cells of 210 x 210 m. The altitudesof 49
neighboringcells are combined to obtain the altitudeof one
aggregatedcell. Altituderange from402 to 1601 m.
to H. H. Shugart,UniWe wantto expressourgratitude
totheconcept
forthestimulating
impetus
versity
ofVirginia,
models.We
rolesandtheirlinkagetotransition
offunctional
of Minnesota,D.
also wish to thankJ. Pastor,University
of Colorado,and an anonSchimel,UCAR and University
on a preceding
forveryvaluablecomments
ymousreviewer
to B. Hunter,
versionof thispaper.We arealso verygrateful
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All use subject to JSTOR Terms and Conditions
M. F. ACEVEDO
1054
Universityof NorthTexas, forprovidinggreathelp withthe
GIS work.
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