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e c o l o g i c a l m o d e l l i n g 2 1 3 ( 2 0 0 8 ) 293–307

avai lab le at www.sc iencedi rec t .com

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

rummholz and grassland coexistence above the forest-linen the Krkonose Mountains: Grid-based model of shrubynamics

an Wilda,∗, Eckart Winklerb

Institute of Botany, Academy of Science of the Czech Republic, Zamek 1, CZ-252 43 Pruhonice, Czech RepublicHelmholtz Centre for Environmental Research-UFZ, Department of Ecological Modelling, D-04301 Leipzig, Germany

r t i c l e i n f o

rticle history:

eceived 19 February 2007

eceived in revised form

9 November 2007

ccepted 27 December 2007

ublished on line 4 March 2008

eywords:

ompetition-colonization trade-off

ndividual-based model

attern oriented modelling

inus mugo

patial explicit

a b s t r a c t

The coexistence of the contrasting life forms krummholz and grass on the summit plateaux

of the Krkonose Mts., Czech Republic, was studied using a simulation model. A spatially

explicit, combined individual- and grid-based simulation model of krummholz dynam-

ics and its interaction with grassland was developed. The field-of-neighbourhood (FON)

approach was used to manage the krummholz individual interactions. The model was

parameterized using two types of field data: (i) direct measurements of parameter values

and (ii) data on the present spatial pattern of the krummholz–grassland mosaic derived from

aerial photographs. The latter was compared with simulated spatial patterns and several

parameters were estimated based on the patterns that gave the best fit. Two scenarios were

explored using the model: without and with disturbance affecting krummholz abundance.

The sensitivity of different response variables to most of the model’s parameters was esti-

mated under both scenarios. In general the results indicate that krummholz and grassland

can coexist as a mosaic in both scenarios for long time under a broad range of parameters.

Disturbance may shift the proportion of krummholz and grassland patches, but does not

necessarily affect the long-term persistence of the mosaic. The model also predicts irregular

fluctuations in the proportions of krummholz and grassland in the long-term equilibrium

and a high resilience of this system. A conceptual analytical model that summarizes the

dynamics of the system was used to discuss the mechanism of coexistence. It showed that

the obvious superiority of krummholz is modified by the ability of grass to reduce sexual

reproduction of krummholz shrubs, thus weakening their strict dominance. It was con-

cluded that the competition-colonization (CC) trade-off is the general mechanism allowing

the coexistence even of such different contrasting life forms as shrubs and grasses.

tion. The situation in the temperate and boreal zones is quite

. Introduction

wo different and highly competitive plant life forms,

raminoids (particularly grasses and sedges) and woody plantstrees or shrubs), successfully coexist and co-dominate in largearts of the world from tropical to boreal zones. Most intrigu-

∗ Corresponding author. Fax: +420 267750031.E-mail address: wild@ibot.cas.cz (J. Wild).

304-3800/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2007.12.013

© 2008 Elsevier B.V. All rights reserved.

ing is the tree–grass coexistence in tropical and subtropicalsavannas, where both life forms have favourable life condi-

different. Here trees and grass coexist mainly on ecotoneswhere woody life forms reach their growth limits (e.g., alpinetreeline, or forest–alpine tundra ecotone). For most European

i n g

294 e c o l o g i c a l m o d e l l

mountains that reach an altitude above treeline, a transitionzone of krummholz develops in the tree–grass ecotone. Inthe Eastern Alps, Carpathian and Hercynian Mountains, thiskrummholz zone is often formed by clonal dwarf (shrubby)forms of woody species like Pinus mugo (Nagy et al., 2003).Similarly to trees cover of krummholz decreases along an alti-tudinal gradient, and a shrub–grassland mosaic is formed inthe krummholz–grass ecotone. Although the factors limitingthe altitudinal distribution woody plants are not fully under-stood, the hypothesis that temperature limits their growth(Korner, 1998; Korner and Paulsen, 2004) is broadly accepted.Thus, this so-called treeline theory seems to be a sufficientexplanation for the existence of the krummholz–grasslandmosaic in many mountains.

But such an evident explanation may overshadow otherfactors affecting the dynamics of the krummholz–grass sys-tem. Previous studies carried out in the Krkonose Mts. (CzechRepublic) resulted in two mechanisms being proposed thecoexistence of krummholz and grassland: disturbance andlife history trade-offs. A study based on historical aerial pho-tographs (Kyncl and Wild, 2004; Wild, in preparation) showsthe occurrence of large disturbances (i.e., insect outbreaks)in the dwarf pine stands and indicates that the mosaicresults from an event-driven regime. Detailed differentiationof the krummholz belt vegetation into various habitat types(Wild and Wildova, 2002) indicates the role of gaps (originallyreferred to as succession sites created by dwarf pine dieback)as the only places suitable for dwarf pine regeneration, whichis almost prevented in grassland. But these gaps are gradu-ally colonized by grass. The presence of a recruitment-limitedsuperior competitor (krummholz) and an inferior competi-tor (grasses) having the ability to spread into gaps evokemechanism for species coexistence via life history trade-offs(Amarasekare, 2003).

In this paper, the possibility of natural krummholz–grassland coexistence and the mechanisms or key factorspromoting this coexistence are addressed using a simula-tion model. For this purpose a spatially explicit, combinedindividual- and grid-based simulation model was developedof krummholz and grassland dynamics and of their interac-tions. The field-of-neighbourhood (hereafter FON) approachdeveloped by Berger and Hildenbrandt (2000) was used tomanage the individual interactions. The FON method aug-mented the zone-of-influence (ZOI) or tessellation approaches(e.g., Czaran, 1998) by varying competition depending on thedistance between the individuals. This method has been suc-cessfully applied in both practical (Berger and Hildenbrandt,2000) and theoretical studies (Bauer et al., 2002, 2004). Thisapproach was modified by combining FON with a grid-basedmodel and taking into account the irregular shape of clonalshrubs.

Using this model, the values of those parameters that wereunknown from field studies, but that could determine coexis-tence, were determined first. For this purpose the present statewas projected by using long-term simulations and comparedwith the spatial patterns of the krummholz–grassland mosaic

with observed ones. Second, the importance of particularfactors influencing the dynamics of krummholz–grasslandmosaics was estimated by testing the sensitivity of the modeloutcome to variation in most of the parameters. To examine

2 1 3 ( 2 0 0 8 ) 293–307

how the disturbance regime contributes to species coexis-tence almost all simulations were run with and withoutdisturbance. Disturbance events were given by dieback of indi-viduals, variable in frequency and in the proportion of thekrummholz population affected. Finally, a conceptual modelof krummholz–grassland system is presented and the simu-lation results are discussed in the light of trade-off theory onspecies coexistence.

2. Empirical background

2.1. Study area

The Krkonose Mts. (Giant Mts., Riesengebirge, Karkonosze) liealong the boundary between the Czech Republic and Polandand belong to the Sudetes, a chain of non-calcareous mid-dle mountains shared by the Czech Republic, Poland andGermany. A part of the mountain range, 40 km in lengthand 20 km in width reaches a maximum altitude of 1600 mand highest part are in the alpine zone. Central parts ofthe mountains are formed of eruptive and metamorphicrocks (granite, gneiss, mica-shist). Only on the peripheryof the mountains do other rocks such as limestone, erlanand quartzite occur. Climate at the highest altitudes isharsh. Annual temperatures at the summits vary between0 and +1 ◦C, snow covers the summits for more than 180days per year and annual precipitation varies between 1200and 1600 mm. Only a few summits of the Krkonose Mts.reach the alpine zone. Most of these areas belong to thekrummholz belt, formed by a mosaic of prostrate dwarfpines (P. mugo TURRA) and grassland throughout its altitu-dinal range, and not only close to the upper dwarf pinegrowth limit. In contrast to other European mountains, thekrummholz zone in the Krkonose Mts. is not adjacent toalpine grasslands. At the highest altitudes the mountains arecovered by other types of vegetation, characterized by a dif-ferent species composition and lower occurrence of grasses(Soukupova et al., 1995). The krummholz belt is thus optimumfor both shrubs and grasses. Therefore, the coexistence ofkrummholz–grassland in the Krkonose Mts. may be a specialcase of grassland–shrub dynamics in boreal zones and repre-sent an excellent example for studying interaction of thesetwo life forms.

2.2. Species and community types

Krummholz stands occur between 1200 and 1450 m a.s.l. Dom-inant dwarf pine (P. mugo) is a procumbent shrub of variousgrowth forms, which differ in height (up to approximately 3 m)and branching structure (Stursa, 1966). Procumbent branchesof dwarf pine often produce adventitious roots, especiallywhere the bark is damaged by either physical or biotic injury(Stursa and Kyncl, 1995). This results in effective horizontalspreading and account for the prostrate form of the shrub.Occasionally, the rooted branches give rise to new inde-

pendent ramets and the clonal spread of dwarf pine. It isa long-lived species; the oldest individuals are more than250 years old (Kyncl, unpubl. data). Krummholz understoryvegetation is usually dominated by Vaccinium myrtillus and

g 2 1 3 ( 2 0 0 8 ) 293–307 295

bo

NairggiWl

3

3

FwgObaotals

0esmelgocicwh

iwoiBbore

Tfcgle

Fig. 1 – Scheme of the subjects modelled (ellipses) and

e c o l o g i c a l m o d e l l i n

ryophytes, but also the grass species Deschampsia flexuosaften occurs.

Grassland patches within krummholz are dominated byardus stricta accompanied by D. flexuosa, Anthoxanthumlpinum and Carex bigelowii. This community, phytosociolog-cally ascribed to the alliance Nardo-Caricion rigidae, possiblyepresents a relict type of tundra vegetation analogous torassland in Scandinavia and Scotland (Krahulec, 1985). Therassland vegetation in the krummholz–grassland mosaics described in detail in Soukupova (2001) and Wild and

ildova (2002). Krahulec et al. (1996) present a phytosocio-ogical overview of grassland in the Krkonose Mts.

. Model structure and implementation

.1. Basic features of the model

or simulation modelling an individual-based approachas used, which produced spatial patterns of krummholz–

rassland mosaic that could be compared with those observed.nly krummholz shrubs were modelled on an individualasis because of the large size differences between grassnd krummholz shrubs, and the physiognomic dominancef krummholz shrubs in this system. Thus, complication ofhe model was avoided by neglecting complicated interactionsnd the (mostly unknown) life histories of the dominant grass-and species. But the possibility of addressing questions onhrub–grass coexistence fully remained.

The model was discrete in time and space. An area of.25 ha was modelled, consisting in a grid of 200 × 200 cells,ach of 0.0625 m2 with toroidal boundaries. One step in theimulation represented 1 year. The following “objects” wereodelled: (i) individual shrubs generated either by seedling

stablishment or by the development a ramet from an estab-ished plant, (ii) gaps originating after shrub dieback, and (iii)rassland. Each cell of the grid was exclusively assigned to onef these objects. One shrub was represented by a number ofells (at least one cell) according to a horizontal projection ofts size. One of these cells represented the place of rooting andould either be the place of original seedling establishment orhere a new ramet rooted. A height parameter denoted theeight of the shrub in each cell.

In the field, the shrubs grow horizontally by gradual layer-ng with occasional rooting of lateral branches. This processas simulated by assigning new adjacent cells to the groupf cells representing a given shrub. Vertical growth was real-

zed by changing the height of the shrub in the relevant cells.oth types of growth – horizontal and vertical – were limitedy competition with other shrubs. Shrubs could not overlap, asne cell could be assigned only to one individual. Thus, one cellepresented the minimal space occupied by one individual,.g., a seedling.

After dieback of a shrub, all the cells were assigned to a gap.he gap could either be recolonized by a shrub (by spreading

rom an adjacent cell and/or by establishment of a seedling) or

olonized by grassland after some time. The colonization byrassland is modelled as abrupt transition from gap to grass-and. This simplification reflects changes in gap suitability forstablishment of shrub seedlings due to the naturally grad-

processes included in the model (square boxes). The arrowsillustrate the direction of transition between subjects.

ual process of regeneration of grasses surviving under shrubdense canopy or the colonization from regions already fullycovered by grass.

Fig. 1 illustrates the main features of our simulation model.The model is written in C++ (Borland C++ Builder ver. 6.0)and uses graphical components MmVisTools developed by M.Muller (MmVisTools@toomai.de).

3.2. Shrubs

3.2.1. Neighbourhood interaction between shrubsThe incorporation of neighbourhood interactions influencesmost of the modelled processes and is a key feature of theIBM. Following Berger and Hildenbrandt (2000) it was assumedthat the competition exerted by one focal individual decreasedwith increase in distance from that individual. Basically, theshape of the field of competition (FON) curve was based onthe stem diameter of modelled trees. However, because clonalshrubs have several stems other size-related parameters wereused. Average height of shrub, h, was chosen and used to cal-culate FON at distance d from the closest edge of the shrub. Azone-of-influence of each individual was defined based on arelative threshold value of FON, which was set to 0.05:

FON(d) = h for d < 0

FON(d)=h e−c(d2/h) for 0 < d < dmax, where FON(dmax) = 0.05

FON(d) = 0 for d > dmax

(1)

Parameter c influences the shape of the FON and consequentlythe intensity of competition between individuals Ci (withCi = 1/c). It was assumed that the FONs of individual shrubscan overlap. Hence, in a given grid cell (x, y) the competitionstrength F(x, y) is equal to the sum of the FON values exertedby N individuals in the neighbourhood which overlap this cell

by their ZOI:

F(x, y) =∑

N

FONn(x, y) (2)

296 e c o l o g i c a l m o d e l l i n g 2 1 3 ( 2 0 0 8 ) 293–307

enta

Fig. 2 – Scheme of implem

Since an individual was represented by a number of cells cre-ating an irregular shape, the distance d was measured fromthe closest edge cell of the individual. Hence, the shape ofZOI was not circular but copies the irregular shape of thehorizontal projection of an individual shrub (Fig. 2). Strongcompetition was assumed to occur close to a shrub, whichis reflected in observed striking changes in species compo-sition of grassland (Wild and Wildova, 2002). Therefore aGaussian-like shape curve was used to model FON (Eq. (1)),as it better described the competition-strength patterns inthe neighbourhood of shrubs than simple exponential curves.The competition strength F(x, y) affected both horizontal andvertical growth and also the mortality of shrubs (see below).

3.2.2. Seed production, dispersal and establishmentThe number of seeds per individual was given by the sum ofseeds produced in the area of a shrub. Production was givenper area unit (m2) by drawing a random number from a Gammadistribution with two parameters, mean Spa and a coefficientof variation (CV), multiplied by the horizontal projection size(hereafter size) of the individual. If mast years were included,the number of seeds produced by one shrub was summed overthe period between mast years My.

Seeds were individually dispersed from a randomlyselected edge cell of the producing individual. The distancedispersed by each seed was derived from a negative expo-nential distribution with mean Dda. Direction of dispersal wasgenerated randomly (0–360◦). Seed establishment was allowedonly in cells without shrubs. All seeds falling into cells occu-pied by a shrub or a seedling were lost. Establishment successwas modelled with different predefined probabilities: Ep forgap cells and Eg for grassland cells, with Ep � Eg. Neighbour-hood competition, expressed by FON, did not influence theestablishment of seeds in the first year, but did influence sur-vival in subsequent years by decreasing the maximum growthrate Gmax (see below).

3.2.3. Horizontal and vertical growthThe annual horizontal growth occured only in edge cellsand was calculated separately for each cell as a maximum

tion of the FON approach.

potential growth rate Gmax (constant over the whole lifespan)reduced by the effect of the FON acting on the cell. The reduc-tion factor was expressed as the ratio of the height of shrubhy to the competition strength in the given cell F(x, y). If theedge cell was not influenced by the FON of another individualthe F(x, y) in a given cell was equal to the height of a shrubwith a ratio of 1 (see Eq. (1)). Otherwise the ratio decreasedwith increasing value of FON exerted by other individuals. Asa result the horizontal growth Gact in a particular cell and yeary was calculated as

Gact = Gmaxhy

F(x, y)(3)

The growth Gact achieved in a particular edge cell was accu-mulated over years. If the accumulated distance Gtot exceededthe size of a cell (0.25 m), the shrub could spread into grass-land or gap cells in the immediate neighbourhood, defined bythe eight nearest cells (Queen or Moore neighbours; Czaran,1998). This approach limited the influence of FONs on neigh-bouring shrubs to only those particular edge cells of the focalshrub, and neglected their influence on the whole ZOI of thefocal shrub. This allowed, however, the modelling of the vari-able influence of field strength on different parts (edge cell)of focal individuals, which is important for prostrate growingshrubs.

Further, an ad hoc rule producing circular-like growth ofshrubs in a rectangular grid was implemented. If the distancecondition (Gtot > size of cell) was satisfied, the spreading wasadditionally limited by probability ps, related to the numberof cells occupied nc, by the same individual in the near neigh-bourhood of the focal edge cell. The maximum probability wasassumed to be equal to 1 for cases of 7 and 0 occupied cells.Seven cells was the maximum number of neighbours of thesame individual for an edge cell, which had to have at least

one empty cell in the neighbourhood to be an edge cell. Zero nc

occurred only when the individual is represented by only onecell (seedlings or saplings), and then it could spread wheneverthe distance condition was satisfied. Otherwise the probability

g 2 1 3 ( 2 0 0 8 ) 293–307 297

d

Vc

wwpawgbsiouw

h

Ti

3Oiwmttfrwdateasvs

3Tm

Fig. 3 – Clonal growth model. First a rooting site of theramet (black squares) is generated at distance Rda and arandomly chosen direction ˛ from the centre of the parent.

e c o l o g i c a l m o d e l l i n

ecreases with decrease in nc:

ps =(

nc

7

)2for nc < 7

ps = 1 for nc = 7 and 0(4)

isually assessed, the quadratic form in Eq. (4) yielded betterircular shapes than a linear relationship.

Vertical growth – increasing the height of an individual –as modelled similarly to horizontal growth, but one valueas calculated for the whole shrub. To obtain this value for aarticular year, the reduction factor h/F(x, y) (see Eq. (3)) wasveraged over all cells of the focal shrub and then this factoras used to reduce the Gmax value. In addition, the vertical

rowth rate rapidly decreased above a certain shrub heightecause the branches were too heavy to remain upright andag and touch the ground. Therefore, vertical growth was lim-ted by one minus the ratio of the actual and maximum heightf a shrub Hmax. As a result, the vertical growth h in a partic-lar cell and year y for a shrub of height hy occupying N cellsas calculated as

= Gmax1N

∑N

(hy

F(x, y)yn

)(1 −(

hy

Hmax

)2)

(5)

he quadratic form of the height ratio reflected the decreasen vertical growth close to the maximum height of the shrub.

.2.4. Clonal growthnly those parts of individuals controlled by ramet probabil-

ty Rp produced ramets. The number of ramets per shrub Rna

as derived from a Poisson distribution using the observedean. Two other parameters, the minimal size a shrub needs

o reach to start producing ramets Rcmin, and the average dis-ance of the ramets from the shrub centre Rda, were necessaryor mimicking clonal growth of a shrub. After selection of aooting site, at distance Rda (drawn from Gamma distributionith observed mean and coefficient of variation) and a ran-omly chosen direction from the centre of the parent shrub,new ramet was produced. A new ramet is only produced if

he selected cell was fully enclosed by others cells of the par-nt shrub. The size of a new ramet was defined by branchingngle Ba and the distance between the new and the parenthrub (see Fig. 3): the new ramet was treated as a new indi-idual competing with other individuals including the parenthrub.

.2.5. Mortalityhree components of mortality were incorporated in theodel:

(i) Dieback resulting from competition. The stress caused bycompetition between shrubs was expressed as compe-

tition rate Cr: the ratio of sum of competition-strengthvalues FON(x, y) exerted by the focal shrub in the Ncells of its entire ZOI and total competition strengthF(x, y) exerted by focal and other shrubs in these N

Then a part of parent shrub given by branching angle ˇ anddistance from centre is assigned to the new ramet.

cells:

Cr =∑

NFONn(x, y)∑

N

Fn(x, y)(6)

It was assumed that exposure of a particular shrubto unfavourable condition must last for some time inorder to cause dieback: if the competition rate Cr wassmaller than Crmin in each of the last 5 years, the par-ticular individual dies. This value was assumed to bestrongly correlated with the competition intensity andhence the parameter Crmin was set arbitrarily to a valueof 0.1.Dieback caused by competition with grass influencesonly shrub seedlings and its rate is incorporated inprobability of establishment in grass Eg.

(ii) Mortality based on age. The probability of survival, ps, of afocal individual decreased exponentially when the actualage L was larger than an age limit Lmax (Eq. (7)):

ps = e−(L−Lmax) (7)

(iii) Dieback caused by disturbance. This type of mortalityoccured only in simulations that focused on the influ-ence of disturbances on krummholz–grass coexistence.No data exists on disturbance; therefore, it was modelledas unspecified events causing dieback of part of the livingshrubs with a variable intensity, period, and regularityof occurrence. The number of randomly selected indi-viduals dying during one disturbance event (1 year in themodel) was derived from a binomial distribution with

given proportion Di. The period between disturbanceevents was drawn from a Gamma distribution with givenmean Dpa and a coefficient of variation Dv. The Gammadistribution allowed for continual variation in the fre-

i n g

298 e c o l o g i c a l m o d e l l

quency and regularity or irregularity of a disturbanceevent.

After dieback of a shrub a gap was created which remainedopen for a number of years given by parameter Gd.

3.3. Grass

Growth and spreading of grasses were modelled as changesof a gap cell after given time Gd into grassland in one step,reflecting the gradual changes in gap-cell quality with respectto seedling establishment Gd (see also Section 3.2). Becausethe process of changes of gap includes regeneration of grassesalready present in the gap, the grass spreading is modelledas spatially unconstrained process; i.e. it does not depend onpresence of grass in the neighbourhood of gap. Dieback ofgrass in a given cell is caused by horizontal growth of shrubsinto this cell or by successful establishment of shrub seedlingin this grassland cell. Grass cannot spread into shrub cell.

3.4. Simulation

Each simulation started with 100 established seedlings ran-domly positioned in the simulation grid. It was assumed thatgrassland occurs in all other cells. Then the following pro-cesses were simulated sequentially each year: competitionstrength, horizontal and vertical growth of shrubs, coloniza-tion of gaps by grasses, shrub mortality, seed production anddispersal, seedling establishment, production of shrub ramets,and disturbance if applied.

Two scenarios were simulated: (i) one without disturbancerunning over 3000 years and (ii) one with disturbance runningover 4000 years. Disturbance was introduced into the simula-tion after 1500 years, when the system had reached a stablestate and the influence of unrealistic initial conditions wasreduced. The longer simulation period for the scenario withdisturbance was necessary in order to reveal the influence ofdisturbance.

4. Model parameterization and analysis

The model was parameterized using field data, see below,or data obtained from other authors (Kyncl, in preparation;Soucek et al., 2001; Soucek, unpubl. data). Two types of fielddata were used: direct measurement of parameter values,and data on spatial pattern of present krummholz–grasslandmosaics (Wild, in preparation). The former gave direct esti-mates of parameter values or at least their order of magnitude.The latter were compared with simulated spatial patterns,and several parameters were estimated based on the best fit.Parameters and their symbols are listed in Table 1.

4.1. Field data

Most of the field data were collected from study plots estab-

lished in the eastern part of Krkonose Mts. near the peak ofSmogornia (50◦45′18′′N; 15◦40′39′′E) during 2001–2004. Basedon both aerial historical photographs and written recordsthese are thought to be little affected by man.

2 1 3 ( 2 0 0 8 ) 293–307

In four plots (20 m × 20 m) all krummholz individuals,except seedlings (total 134) were precisely mapped includingthe centre (place of primary rooting) of both parent plants andramets using a laser rangefinder (Impulse 200) with an angle-measuring module (Angle Encoder) and Field-Map technology(http://www.fieldmap.com). This yielded data on maximumheight of individuals (Hmax) and their clonal growth: theaverage number of ramets per individual (Rna) and averagedistance of ramet from parent plant (Rda). The minimumsize a shrub needs to reach before producing ramets (Rcmin)was estimated as minimal distance between centre of parentshrub and ramet plus half of the size of the smallest ramet.The number of layered branches in a circle of radius (Rda)around the centre of the parent shrub was counted, and thebranching angle (Ba) estimated by dividing 360 by the numberof layered branches. However far it was attempted to modelthe growth of individuals including ramet formation the gridapproach was not able to reproduce the exact size and shape ofa shrub, especially those competing with other shrubs. There-fore, the predicted and observed sizes (horizontal projection)of shrubs were different and were not used for parameteri-zation.

Twenty-six mature shrubs were selected, and the averageannual increment of the shoots over more than 100 years wasmeasured using dendrochronological techniques (Wild et al.,2004). Only free growing branches were measured. A decreasein apical growth of branches occured mainly on contact withother shrubs. Thus we assumed that the average annual shootincrement recorded was close to the maximum value and usedit as the maximum growth rate (Gmax) in the model. The sameshrubs were used to estimate the age structure. However, thiswas complicated because it was difficult to identify individu-als due to their clonal growth, reduction or cessation in radialgrowth of branches after rooting, and decomposition of theoldest branches (Kyncl and Wild, 2004). Thus, it was impossi-ble to estimate of the age of many individuals. The age of theoldest branches (individuals) recorded was approximately 250years and used as an estimate of the maximum lifespan of theshrubs (Lmax).

In the same area, three other plots (50 m × 50 m) were estab-lished in 2002, and the age of all saplings up to height of 0.5 m(total 85) was measured in terms of the number of internodeson the stem. The distance of a sapling from a mature shrubat the time of seedling establishment was estimated usingthe average annual shoot increment of mature shrubs (Kyncl,unpubl. data), which is a good indicator of horizontal growth.The current distance was increased by multiplying the aver-age annual shoot increment by the age of the sapling. Thesevalues were used to calculate the average dispersal distance(Dda). These data also yielded the proportion of seedlings col-onizing gaps and grassland (81–4), which was used to calculatethe relationship between the probabilities of establishment ina gap Ep or grassland Eg cell.

Number of germinative seeds produced by a shrub was esti-mated based on 20 years’ of observation of 24 shrubs (Souceket al., 2001; Soucek, unpubl. data). Seed production was highly

variable both between years and individuals; nevertheless theaverage number of seeds per square meter (Spa) of a shrub waswell approximated by the Gamma distribution. Further, thecoincidence of a year of high seed production and the age of

e c o l o g i c a l m o d e l l i n g 2 1 3 ( 2 0 0 8 ) 293–307 299

Table 1 – List of the parameters used in the simulation

Symbol and name Basic value Range tested Unit Comment

Growth of shrubsGmax Maximum growth rate 0.028 0.001–0.07 m Maximum potential growth

(horizontal and vertical) per yearHmax Maximum height 2.5 1.0–3.0 m Maximum height of shrub

Mortality and competitionLmax* Lifespan 300 50–450 year From this age the probability of dying

increases exponentiallyCi* Competition intensity 0.3 (0.1) 0.0–2.0 – Influence of the size of the ZOI of a

particular individual (in Eq. (2), c = 1/Ci)Gd* Gap duration 20 0–150 year Age of gap after which it is covered by

grassland (unsuitable for krummholzestablishment)

Seed production, dispersal and establishmentFs Minimum age for seeds production 30 0–70 year At this age individuals start to

produce seedsSpa Average seed production 25.67 0.0–50.0 – Average number of seeds per square

meter of an individual shrubDda Average dispersal distance 0.92 0.1–5.0 m Average distance over which seed is

dispersedMy Mast year 7 1–25 year Period between mast yearsEp* Probability of seed establishment in

gaps0.011 (0.001) 0.0–0.3 prop. –

Eg* Probability of seed establishment ingrassland

0.000543 0.0–0.04 prop. –

Clonal growthRp* Probability of producing ramet 0.25 (0.5) 0.0–1.0 prop. Probability of an individual shrub

becoming clonal (start to produceramets)

Rcmin Minimal radius for clonal growth 1.42 1.0–3.0 m Distance from centre of individualshrub to its farthest cell; reaching thissize is necessary condition forproducing ramets

Rna Average number of ramets perindividual

1.74 0.5–4.0 – –

Rda Average ramet distance 2.2 1.0–5.0 m Average distance from centre ofparent plant to point where a rootingsite of ramet is produced

Ba Branching angle 35 15–90 ◦ Average branching angle betweenramets; determines the size of newramet

DisturbancesDi* Disturbance intensity (0.1) 0.0–0.9 prop. Proportion of the shrubs dying during

a disturbanceDpa* Average period between disturbances (140) 20–140 year –Dv* Disturbance variability (0.1) 0.1, 0.8 – Coefficient of variance controlling

obse. In pa

typcsF

4

Spo

Basic values are those that generate a response that best matches theAsterisks in superscript indicate parameters estimated by simulation

he saplings indicate the likelihood of the existence of a mastear (My), but the time series is too short to be certain. Theeriod between mast years adopted was 7 years (Stursa, pers.omm., 2000). Based on author’s field observations of plantedtand the age at which shrubs start to produce viable seed iss = 30.

.2. Data estimated by simulation

everal parameters crucial for mosaic dynamics namely, lifes-an Lmax, probability of seedling establishment in a gap Epr grassland Eg, competition intensity Ci, probability of ramet

Gamma distribution of Dpa

rved data. Range is that used for testing the sensitivity of the model.rentheses are the basic values used in the scenario with disturbance.

production Rp, and duration of gap Gd, could not be obtainedby observation. This is also true for three additional parame-ters describing the disturbance regime (intensity Di, averageperiod Dpa and variability Dv).

These parameters were estimated by comparing the modelresults with the observed patterns of several response vari-ables. For each parameter, values were selected within theadmissible range (Table 1), and simulations were performed

with combinations of these parameter values (altogether 3360and 11,340 combinations for without- and with-disturbancesscenarios, respectively). Three response variables were usedto compare a simulated and observed pattern: (i) extent

i n g

300 e c o l o g i c a l m o d e l l

of gap and grassland together (hereafter openings) withinkrummholz, (ii) cover of grassland and (iii) core area index(CAI) of openings. The core area represents the interior area ofpatch (in this case opening) unaffected by the user-specifiededge (McGarigal and Marks, 1995). Edge size was set at 0.5 mbased on a previous study on the interaction between shruband grassland (Wild and Wildova, 2002). Then the CAI was cal-culated as a ratio of core area to the total area of openings.Comparable field data was derived from five plots of the samesize as the simulation grid, randomly positioned on classifiedaerial photographs from the study area (Wild, in preparation).

From the last 500 years of simulation runs, the averagevalues and standard deviation of the response variables werecalculated, to cope with the stochasticity of the system. Thenthe ranges of modelled data (mean ± standard deviation) werecompared with observed minimum and maximum values.These cases that gave values that overlapped to some extentwith observed ranges were accepted as corresponding to thefield data. The set of parameter values which gave the smallestsum of squared differences between observed and simulatedresponse values was used as the basic value for the sensitivityanalysis (listed in Table 1).

Unfortunately, it was not possible to estimate the extent ofthe gaps from the aerial photographs. This value, however, iscomplementary to the grassland cover and one of these valuesis necessary for deciding whether shrub and grassland coexistunder particular simulated conditions. Therefore, only resultswith a gap/grassland ratio lower than the arbitrarily set 60%were considered as corresponding to field data.

Finally, the parametric spaces of the estimated parametersin the two scenarios were compared using standard Euclidiandistances between parameters to reveal whether disturbanceincreases parametric space and thus coexistence.

4.3. Sensitivity

First, the sensitivity of three response variables (see above)

to most parameters, without and with disturbances, was esti-mated. One parameter was changed within a predefined range(see Table 1), and five simulations per one parameter changewere run. The average value and standard deviation of the

Fig. 4 – Example of 50 m × 50 m area of the spatial pattern of kruphotographs (A) and generated by simulation using the basic paphotographs, therefore the white colour represent open areas co

2 1 3 ( 2 0 0 8 ) 293–307

response value over the last 500 steps averaged over 5 simu-lation runs were then plotted.

Additionally the effects of small changes (±10%) in the val-ues of parameters on response variables were explored. Thesensitivity of the response variables to different disturbanceregimes given by a combination of three parameters (intensityof disturbance, average period between events and length ofa period) was also tested. Both intensity and period were var-ied using several values from within the ecological meaningfulrange of values (see Table 1), but only two values (0.1 and 0.8)of the last parameter were used, which represent low and highvariability during the average period between disturbances.

Finally the stability (resilience) of the modelled systemparameterized with a basic set of parameters was explored.One disturbance event was introduced after 2000 years, whichkilled a different proportion of shrubs, and whether the sys-tem returns to the previous state and how long it took wasobserved. The return to previous state was stated when allthe response variables returned to the interval defined by themean ± standard deviation recorded over previous 500 yearsin the simulation without disturbance.

5. Results

5.1. Parameterization

In general, the model was able to produce a spatial mosaicof krummholz and grassland similar to that observed in thefield (Figs. 4 and 5). Both scenarios, without and with distur-bance, could be parameterized to generate a long-term stablecoexistence of krummholz–grassland mosaic. In such cases,the model reached a stable state after 1500–2000 years. Themodel predicted coexistence under relatively broad ranges ofparameters values (see Table 2). Forty combinations of param-eters from 3360 for the without-disturbance scenario and 106from 11,340 for the with-disturbance scenario matched theobserved field pattern. The “without-disturbance” scenario

indicated that a long lifespan of the krummholz (“lifes-pan” ≥ 300 years), a short period when gaps are suitablefor the establishment of krummholz seedlings (“gap dura-tion” ≤ 20 years), and a relatively low vegetative reproduction

mmholz–grassland mosaic observed in the aerialrameter set and no disturbance (B). Gaps are not visible innsisting of gaps and grassland in picture A.

e c o l o g i c a l m o d e l l i n g 2 1

Fig. 5 – Photographs of krummholz–grassland mosaic takenf

(tdswtmogirit1d

nbt(

(see Fig. 6). However, the inclusion of disturbance decreased

rom the model of an airplane. Area of ca 1 ha is displayed.

“ramet probability” = 0.25) might be necessary conditions forhe long-term stability of this mosaic (see Table 2). Inclusion ofisturbance shifted the emphasis to lower competition inten-ity (0.1). Most combinations of disturbance parameters testedere able to generate the observed pattern, but mainly when

he disturbance occurred at long interval (140 years) and atedium or high intensity with both low and high variability

f periodicity. The simulations also showed the importance ofenerative reproduction for system dynamics. In both scenar-os the model predicted that there should be high incidenceegeneration from seed. The ratio of individuals that orig-nated from seed compared to vegetative reproduction forhe simulations matching the observed field pattern was.54 ± 1.01 in the absence of disturbance and 4.87 ± 6.22 whenisturbances were included.

The parametric space of the without-disturbance sce-ario measured in terms of the maximum Euclidian distance

etween estimated parameters combination was smaller (5.8)han the parametric space of the scenario with disturbance7.0). Both parametric spaces fully overlapped: that of with

Table 2 – Range and frequency of values of estimated paramete

Disturbance Parameter

No (n = 40) Gap durationProbability of seed establishment in gapsCompetition intensityProbability of producing rametLifespan

Yes (n = 106) Gap durationProbability of seed establishment in gapsCompetition intensityProbability of producing rametLifespanDisturbance intensityAverage period of disturbanceDisturbance variabilitya

a Only these two boundary values were tested.

3 ( 2 0 0 8 ) 293–307 301

disturbance included that of without disturbance. All param-eterized simulations exhibited some degree of fluctuation inthe area occupied by grassland and krummholz, often overperiods of more than a 100 years. This fluctuation changed inintensity and period, but never entirely ceased during the testperiod of 10,000 years.

5.2. Sensitivity

Fig. 6 shows the sensitivity of areas of grassland to changes indifferent parameters in both scenarios. Not surprisingly, theseresponses were most sensitive to parameters connected withthe growth potential of the krummholz: competition intensity,maximum growth rate, lifespan, mortality (minimum compe-tition rate) and average seed production. The influence of theprobability of seedling establishment (both in gaps and grass-land habitats) on the response variables was highest whenthe probability is small and sharply decreases with increasein probability. The parameters connected with architectureand production of new ramets and the height of shrubs onlyslightly influenced the response variables. Further, the modeldid not appear to be sensitive to the age at which the shrubsstart producing seed and the frequency of mast years (notdisplayed). Both openings and grassland areas showed thesame trends in sensitivity with variation in almost all parame-ters. Only grassland areas strongly decreased with increase ingap duration. The decrease resulted mainly from the mutualexclusivity of grassland and gap; only one can be present inthe same place. The shape of openings (core area index) wasalso very sensitive to competition intensity and mortality ofkrummholz (Table 3). However, compared to the size vari-ables, core area index showed less sensitivity to the maximumgrowth rate and lifespan of krummholz. Further, the increas-ing probability of producing ramets seemed to decrease thecore area index more than the total size of the openings.

Both simulations, with and without disturbances, revealedsimilar sensitivity curves for most of the parameters tested

the sensitivity of the model to most of the parameters (Table 3).Further, small changes (±10%) to the basic values of theparameters highlighted only a few key parameters. In the

rs that match the field data

Minimum Maximum Most frequent

0 60 20 (32.5%)0.001 0.031 0.006 (20%)0.1 0.9 0.3 (32.5%)0.25 0.75 0.25 (52.5%)300 400 300 (56.1%)

0 100 20 (30.1%)0.001 0.031 0.016 (46.2%)0.1 0.9 0.1 (49.1%)0.25 0.75 0.25 (44.3%)200 400 300 (53.2%)0.01 0.9 0.1 (19.8%)20 140 140 (54.7%)0.1 0.8 0.8 (61.3%)

302 e c o l o g i c a l m o d e l l i n g 2 1 3 ( 2 0 0 8 ) 293–307

Fig. 6 – The sensitivity of the area of grassland to a wide range of selected parameters with disturbance (open circles) andwithout disturbance (solid circles). The point and bars represent the means and standard deviations of the values predictedover the last 500 years of the simulation, calculated as the average of the values from five repetitions of each simulation.The insertions show the main effects of changes of abscissa parameters on components of the conceptual model equations(8) and (9).

Table 3 – Sensitivity of response variables (opening area, grassland area and core area index (CAI)) to small changes(±10%) in the values of the basic parameters

Name of parameter Without disturbance Disturbance

Opening Grass CAI Opening Grass CAI

Maximum growth rate −15.4 −21.5 −2.4 −8.0 −12.4 1.28Maximum height 11.9 17.5 9.0 10.2 16.2 6.3Lifespan −11.9 −12.1 0.6 −14.0 −16.1 −1.0Competition intensity 14.0 19.9 8.36 4.8 7.6 2.7Minimum age for seeds production 1.2 0.7 −0.1 1.0 2.4 1.2Average seed production −9.4 −15.4 −8.5 −4.3 −6.3 −4.1Average dispersal distance −7.5 −13.9 −8.3 −14.6 −21.9 −9.4Mast year −2.6 −3.8 −1.8 0.7 0.0 −1.4Probability of seed establishment probability in gaps −0.5 −1.7 −1.5 −4.2 −6.5 −2.2Probability of seed establishment probability in grassland −8.2 −12.0 −5.7 −6.9 −10.6 −3.9Gap duration −10.8 −23.5 −10.5 −4.5 −13.0 −2.2Probability of producing ramet 0.6 −0.3 −0.4 −10.1 −16.9 −8.2Minimal radius for clonal growth 0.8 0.4 1.5 0.6 1.7 1.9Average number of ramets per individual −3.6 −3.8 −1.8 −7.2 −11.5 −4.5Average ramet distance 0.4 0.4 0.4 −0.8 −1.7 −1.1Branching angle − 6.6 −8.6 −2.9 −3.7 −5.1 −2.9Disturbance intensity 2.0 3.8 2.6Average period of disturbance −4.5 −7.2 −1.8Disturbance variability 2.4 1.8 0.3

The changes in response variables are expressed as the percentage of values obtained from the simulation with a basic set of parameters. Thethree highest values of each response variable in each scenario are italicised.

g 2 1 3 ( 2 0 0 8 ) 293–307 303

wagaaepmi

tspdged

5

Tsptdtd

Fibvdn

Fig. 8 – The time it takes for the simulation (transition time)to return to the previous state after a disturbance(measured in terms of the proportion of individuals killed)

e c o l o g i c a l m o d e l l i n

ithout-disturbance simulation the openings and grasslandreas were very susceptible to small changes in maximumrowth rate, intensity of competition and age of a gap. The corerea index was at least affected by growth rate and stronglyffected by the height of krummholz. Disturbance shiftedmphasis to lifespan, dispersal distance and probability ofroducing ramet for all response variables and the maxi-um height of a shrub (only for grassland area and core area

ndex).Because disturbance was associated with three parame-

ers, intensity, periodicity and variability in periodicity, theensitivity of the response variables to combinations of thesearameters was tested. The results indicated a linear depen-ency between the intensity and period of disturbance, whichenerated the same value in the response variable (Fig. 7). Theffect of the period for which the disturbance lasted, however,id not differ between low and high periods.

.3. Stability

he model with the basic parameter set predicted a stableystem, which was revealed by the system returning to itsrevious state after disturbance. The transition time from dis-

urbance to the initial state increased as the percentage ofead shrub increased (disturbance intensity; Fig. 8). However,he increase is slow up to 80% and the average transition timeoes not exceed the lifespan of a krummholz. Even when 95%

ig. 7 – Relation of the area of openings (a) and core areandex (b) to intensity of disturbance and average periodetween disturbances. The size of circles indicates thealue of the response variable. The cases whenisturbances resulted in the extinction of krummholz areot displayed.

occur. The average transition time ± standard deviation isbased on 10 repetitions of each disturbance intensity.

of the shrubs died the system, in most cases, was able to returnto the previous state. Some simulations that were run usingthis percentage and all runs with higher percentage led toextinction of krummholz.

6. Discussion

6.1. The parameters predicted by the model

The model indicates that under the present conditions themosaic of krummholz and grassland is stable and likelyto coexist for a long time without any intervention. Thelimitation on krummholz propagation mainly to seedlingestablishment in gaps seems to be a sufficient but not theexclusive factor promoting coexistence. The simulations indi-cate that other conditions limiting krummholz regenerationshould also need to be satisfied, such as the gaps should existonly for a short period and that the shrubs should have amoderate ability of clonal growth by producing ramets. Themodel predicts also a long lifespan for krummholz individu-als. Finally, higher percentage of individuals originated fromseed than from vegetative reproduction was predicted.

These predictions mainly correspond with field observa-tions. The long lifespan of krummholz was already knownfrom a dendrochronological study (Stursa and Kyncl, 1995;Kyncl, in preparation). Further, preliminary isozyme analy-sis (Wild et al., 2004) indicates that a higher percentage ofindividuals are generated from seed than from vegetativereproduction, although vegetative reproduction was supposedto predominate (Stursa, pers. comm., 2000). Also the modellingof clonal growth based on a time series of aerial photographsrevealed that a remarkably high percentage of individuals orig-

inated from seed, although the percentage was much smallerthan that predicted by the model (Wild, in preparation). Onlythe short existence of the gap seems to be contradicted bythe field observations, which indicate that visually distin-

i n g

304 e c o l o g i c a l m o d e l l

guishable gaps persisted for more than 100 years (Wild andWildova, 2002). However, the persistence of gaps in the modelis the length of time they are suitable for krummholz seedlingsestablishment, not the period for which the gaps are visuallydistinguishable in the grassland. Two phases in successionwere observed in gaps: (i) an early stage characterized bythe remains of krummholz and a high cover of lichens andbryophytes and (ii) a late stage in which there is a high coverof grasses (Wild and Wildova, 2002). The increase in grasses inthe later stage probably inhibits krummholz seedling estab-lishment. Thus the predicted short period of existence of gapsgives only the early stage observed in the field.

6.2. Sensitivity and stability of the model

The sensitivity analysis, not surprisingly, shows that param-eters influencing krummholz propagation mainly affect theproportion of the area covered by grassland and openingsin the mosaic. Some of the observed relations, however,could have consequences for predicting the real mosaics.The observed growth rate (annual increment of shoots)increases inversely with altitude (Soucek et al., 2001; Kyncl, inpreparation). In the model the openings and grassland areasdecrease markedly with increase in krummholz growth rate.However, they do not disappear even at the highest valuestested (Fig. 6). Thus the predicted area of openings basedon the growth rate decreases with decreasing altitude, butsuch openings are still present at the lowest border of thekrummholz belt. The next parameter strongly affecting thearea of grassland is gap duration. It could be interpreted asa simple result of the implemented rules, which allow thespread of grass in a gap after the gap duration has elapsed.However, gap duration also strongly influences the spreadingof krummholz by favouring krummholz seedling recruitmentduring the existence of gap; hence, both factors interact andaugment the effect of gaps on the mosaic structure.

The model is very sensitive to the intensity of intraspecificcompetition between krummholz and the derived parameterssuch as, minimal competition rate (see Table 3 and Fig. 6).As the FON approach is phenomenological, the parameterscould not be related to some resource used by individuals(Berger and Hildenbrandt, 2000; Bauer et al., 2004). This couldbe assessed by comparing the model output with an observedpattern (Grimm et al., 1996). It should be stressed that despitethe fact that interspecific competition determines the propor-tion of grassland in the mosaic, coexistence is predicted overa wide range of values.

Analysis also showed cases when grass and shrubs do notcoexist; long gap duration caused extinction of grasses andlifespan shorter than 100 years or high competition betweenshrubs result in extinction of shrubs. It indicated that speed ofchanges of gaps (colonization or regeneration of grasses) andsurviving of both, seedlings and mature shrubs are of highimportance for krummholz–grassland coexistence.

The low sensitivity of the model in the absence of dis-turbance to all parameters associated with clonal growth,

especially the probability of producing ramets, is surprising.The low probability of producing ramets may be counterbal-anced by krummholz seedling establishment in gaps. Highincidence of ramet production, however, leads to smaller gaps

2 1 3 ( 2 0 0 8 ) 293–307

that are partly occupied by new ramets and high competitionwithin the gaps that suppresses seedlings recruitment. Vege-tative reproduction and seed production by krummholz seemto compensate each other.

The model predicts irregular fluctuations in the propor-tions of krummholz and grassland. The presence of naturallong period fluctuations might be important for the man-agement of the system, which is often based on short-termstudies of limited informative value. However, the fluctuationcould be an artefact of unrealistic initial conditions of the sim-ulation which started only with seeds randomly positioned inthe grassland. The strong tendency of the system to return toa similar state after a disturbance makes this unlikely. In addi-tion, the high resilience and relatively short time to recovery(short in relation to lifespan of krummholz) indicate the sys-tem’s ability to withstand the occasional severe disturbance.

Further, the pronounced resilience of the predicted equi-librium decreases the probability of the existence of multiplestable states being confounded using only one set of unreal-istic initial conditions (Connell and Sousa, 1983; Schroder etal., 2005). The initial conditions used correspond well with thesituation in new krummholz plantations. Thus, the transitionof a plantation to natural stable dynamics is likely to be veryslow.

6.3. Influence of disturbance

The inclusion of disturbance in the model led to an increasein the range of parameters enabling coexistence. In addition,the system reacts linearly to an increase in the magnitude ofthe disturbance (frequency and/or intensity) by a decrease inkrummholz cover in the mosaic. Thus, disturbance is a sur-rogate for other factors limiting the krummholz propagationand stabilizes proportions of composition of the mosaic.

Disturbances of low or middle intensity and/or of a longduration resulted in simulation patterns comparable to thoseobserved. Field data on the frequency of disturbance are notavailable, but a previous study indicates that periods of morethan 70 years (determined from historical photographs) arelikely (Kyncl and Wild, 2004). This does not contradict themodel’s predictions.

In general, the inclusion of disturbance decreases the sen-sitivity of the model to most parameters. If it is assumedthat krummholz propagation is the most important factordetermining the mosaic pattern, then reducing krummholzabundance by disturbance could reduce the influence ofother parameters. Further, Bolker et al. (2003) suspect thatexogenous disturbances increase the variation in populationdensities of coexisting species above that generated by popu-lation dynamics. Such an increase could enhance the effectsof the mechanism allowing coexistence of two species basedon differences in their life histories.

Furthermore, disturbance shifts the highest sensitivityfrom growth-related parameters and gap duration to dis-persal distance, vegetative reproduction and the lifespan ofkrummholz. This shift can be explained by the necessity to

disperse greater distances in krummholz stands, thinned bydisturbances and to counterbalance the abundance decreaseby a longer lifespan and an increase in vegetative reproduc-tion.

g 2 1

spo

6m

IpcTtavsmugsaodoKgtpc(

K

G

w

g

k

f

f

p

Mtifspcs

e c o l o g i c a l m o d e l l i n

Inclusion of disturbance also increases the area of gapsuitable for seedling establishment, which resulted in a higherroportion of reproduction by seed than in the scenario with-ut disturbance.

.4. Mechanism of long-term coexistence: a conceptualodel

n the introduction a life history trade-off was evoked as aossible global mechanism to explain krummholz–grasslandoexistence and to elucidate possible coexistence limitations.he consequences of the classical competition-colonization

rade-off theory between species, with a modification (Levinsnd Culver, 1971; see Amarasekare, 2003 for review), can beisualised for our case by a conceptual difference-equationystem of the krummholz–grass system. This pseudo-spatialodel assumes a finite number N of sites (cells) as in the sim-

lation model. To cope with the intricate spatial aspects ofrowth, seed dispersal, and clonal reproduction of krummholzhrubs K we introduce a “shrub size” y = y(K), denoting the aver-ge number of cells occupied by a shrub, which summarizesver all the consequences of these aspects and which is onlyepending on the number of krummholz shrubs K, but notn the number of cells occupied by grass G (see Winkler andlotz, 1997, for the application of this method for describingrowth and division of grass tufts). As ramets are placed nearhe parent shrub their formation is not considered as a birthrocess but as an enlargement of the parent shrub. The con-eptual model based on the number of birth and death eventsB and M) for both species per time step reads as:

t+1 = Kt − MK + BK = Kt − mKKt + bKKt(p1 f + p2g)

= Kt − mKKt + ayKt (p1 f + p2g) (8a)

t+1 = Gt − MG + BG = Gt − BKyg

g + f+ bG(Gt − MG)f ∗ (8b)

ith

= G

N(9a)

= Ky

N(9b)

= 1 − g − (1 − mK)k (9c)

∗ < f (9d)

1 � p2 (9e)

ortality mK of shrubs comprises death by ageing and also dueo disturbance. Sexual reproduction of krummholz shrubs BK

s given by seed production (fecundity) bK = ay (where a theecundity per shrub cell), modified by the ability to establish

eedlings on gaps (probability p1) or grass cells (probability

2). The fraction f of gap cells also comprises the krummholzells just set free by shrub mortality. After establishment ofeedlings it is assumed that the new shrub immediately fills

3 ( 2 0 0 8 ) 293–307 305

its average number of cells, y, thus replacing grass G accord-ing to the proportion of grass cells to the sum of grass plusgap cells g/(g + f). (Note that intraspecific interactions duringgrowth are included in the definition of shrub size y.) Thisreplacement gives the mortality of grass cells. “Birth” of grassis due to spreading of the remaining grass after shrub mor-tality into empty cells. Here the fraction f* of available emptycells is smaller than that of shrub-seedling establishment f asgrowing shrubs are not only displacing grass but also coveringcells being empty before, and shrub growth is superior to grassexpansion in the simulation model (a detailed formulation off* is omitted). Grass “fecundity” bG is inversely proportional togap duration.

In this model, summarizing the main features of the simu-lation model, krummholz K is competitively superior to grassG: shrub growth, giving rise to y(K), is not affected by G, butin its consequence grass G is replaced. Also seedling estab-lishment in gaps is assumed in the model to have a temporaladvantage over grass spread. Grass must counterbalance itsreplacement by a sufficiently large spread rate, i.e., gaps mustbe filled by grass as rapid as possible. By a rapid filling of gapsgrass G can prevent, to a large extent, seedling establishmentof shrubs K. Because of this model feature dynamics of K isalso affected by G as long as p2 < p1: as compared to emptycells, grass restricts establishment of K seedlings.

The consequences of changes in simulation model param-eters and in components of the conceptual model can besimultaneously studied by Fig. 6. Increasing growth rate ofshrubs or increasing ramet probability positively affects aver-age shrub size y(K) whereas stronger intraspecific competitionwithin K reduces y(K): both changes inversely affect replace-ment of G. Very high intraspecific competition between shrubswill lead to such a low birth rate bK of shrubs and relativespread advantage of grass that K cannot persist at all. Higherdispersal of krummholz seeds gives larger K shrubs and lowergrass cover. An increase in krummholz “birth” by larger seedproduction or higher establishment rates negatively affectsgrass cover, but when probability of establishment of shrubs ingrass is very low grass will dominate the community. Longergap duration means lower birth rates of grass and also lowergrass cover. The same is given by longer average lifespan ofkrummholz shrubs.

To summarize, krummholz is the superior competitor butis recruitment-limited by low seed-establishment success anddispersal ability, and grass is the inferior competitor but read-ily colonizing the gaps. However, not all the assumptions of theCC trade-off model hold. The colonization ability of grass maybe less than that assumed, shifting the dominance towardskrummholz. The dominant species of grass, N. stricta, has avery limited seed production above the forest-line (Stursova,1985; but see Hejcman et al., 2005 for contradiction) and itsclonal growth is very slow (Klimes and Klimesova, 1999). Onthe other hand, this is counterbalanced by the temporal limi-tation for krummholz in colonizing gaps. A gap, once createdand not colonized by krummholz in the early successionalstages by seedlings, slowly changes to grassland and remains

covered with grass until it is overtaken by krummholz spread-ing horizontally from patches in the neighbourhood.

The condition of complete asymmetric interspecific com-petition is not fully met in our model. The CC trade-off theory

i n g

r

306 e c o l o g i c a l m o d e l l

was originally proposed to account for coexistence when thesuperior species will always displaces the inferior species—so-called “displacement competition” (Yu and Wilson, 2001).However, Kisdi and Geritz (2003) showed that this trade-offcould also maintain the coexistence of perennial plants if so-called “replacement competition” occurs in which the speciescompete only for unoccupied places created by the death ofadult plants. For already occupied places a “pre-emption rule”holds, even for the weaker species. Recently Calgano et al.(2006) showed that relaxing the assumptions of full asymmet-ric competition and introducing a pre-emptive effect did notpreclude coexistence.

In the scenarios studied here displacement competitionoccurs at the edges between patches, where krummholzclones overgrow the grass. Seedlings of krummholz competewith grass mainly for empty gaps and are the superior com-petitor. But at sites already occupied by grass krummholzseedlings can only establish very rarely—a partial pre-emptiveeffect in the opposite direction, which weakens full asymme-try of competition. As a result, there is a mix of displacementand replacement competition, which depends on location inthe mosaic and life history stage of krummholz. The interplayof these mechanisms, which cannot be expressed by the con-ceptual model but only by the full simulation model, can resultin fluctuations in the spatial structure of the mosaic predictedby the model. Furthermore the poor ability of krummholzseedlings to colonize grassland, their clonal growth, and poordispersal ability leads to aggregations of krummholz individu-als and generates a particular spatial pattern. Although recenttheoretical studies tend to devalue the importance of space asa factor promoting coexistence (Neuhauser and Pacala, 1999;Chesson and Neuhauser, 2002) the biologically generated spa-tial segregation (Pacala and Levin, 1997) seems to be importantfor the coexistence of contrasting life forms. Indeed the CCtrade-off also includes spatial segregation or intraspecific clus-tering, which is achieved by the poor dispersal ability of thesuperior competitor and higher colonization ability of the infe-rior competitor (Amarasekare, 2003).

Despite deviations from the theoretical model, whichweaken the coexistence conditions to the disadvantage ofboth species it is concluded that the competition-colonizationtrade-off is the general mechanism allowing the coexistenceeven of such different contrasting life forms as shrubs andgrass.

7. Conclusion

Our model indicates that a field-of-neighbourhood approachcan be successfully applied to prostrate clonal shrubs, eventhough the model was originally designed for trees having asingle trunk.

The results of the simulations indicate that different lifeforms can coexist krummholz and grasses, mediated by spa-tial segregation and life history trade-offs. The temporallimitation on the ability of krummholz seedlings in coloniz-

ing gaps is identified as one of the important life history traitsgoverning coexistence. Disturbance affects the proportion ofkrummholz and grassland patches within the mosaic, but isnot necessary for long-term persistence of the mosaic. How-

2 1 3 ( 2 0 0 8 ) 293–307

ever, this simulation study does not exclude the possibilitythat coexistence could be mediated by repeated disturbances.

The simulations also provide hints on how to managethe system. The predicted fluctuations in the proportions ofkrummholz and grass are useful for interpretation of currentobserved changes. The high resilience of the modelled systemindicates its ability to withstand a severe disturbance affectingkrummholz. The long transition time from initial conditionto equilibrium, however, indicate it will take a long time forkrummholz plantations to achieve natural dynamics.

Acknowledgements

We are grateful to T. Kyncl for providing the results ofdendrochronological analysis. We thank T. Herben and Z.Munzbergova, for their useful comments on the manuscript.We appreciate the assistance of Krkonose National Park per-sonnel, particularly J. Stursa and J. Vanek. We are finallyindebted to A.F.G. Dixon for the linguistic revision of themanuscript. The research was funded by the project of Min-istry of the Environments of the Czech Republic no. VaV610/3/00, by the researcher intention AV0Z60050516 and byBiodiversity Research Centre, grant no. LC 06073.

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