BioScience

American Institute of Biological Sciences

P U B L I S H E R Richard T. O'GradyE D I T O R - I N - C H I E F Rebecca Chasan

M A N A G I N G EDITOR Christina T. MaguireASSISTANT EDITOR Jill M. Scott

To eat without being eaten isof paramount importancefor insect herbivores—in-

deed, for many animals. In the ar-ticle beginning on page 35, Eliza-beth A. Bernays discusses the manyfeeding modes of insect herbivoresand the constraints that have se-lected for particular feeding be-haviors. For example, predationcan select for specialization on ahost plant that provides camou-flage for the insect. Larvae of thesatyrid butterfly Taygetes androm-eda, for instance, feed exclusivelyon young leaves of the tropicalgrass Lasciasus. Young larval in-stars rest along the veins of theleaf, where they are barely visible(left photo), whereas larger latelarval instars are most effectivelycamouflaged in their resting posi-tions along the old leaf sheath (rightphoto). These resting behaviorsprovide an example of specialized behaviors associated with extreme host plant specificity thathelp to protect insect herbivores from predators. In the article beginning on page 25, Steven L.Lima argues that although the effects of predators are usually discussed in terms of their lethaleffects on prey populations, anti-predator behavior by potential prey can have profound effectson ecological systems.

—Rebecca ChasanEditor

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Reid's Paradox of Rapid

Plant Migration

Dispersal theory and interpretation of paleoecological records

James S. Clark, Chris Fastie, George Hurtt, Stephen T. Jackson, Carter Johnson,George A. King, Mark Lewis, Jason Lynch, Stephen Pacala, Colin Prentice,

Eugene W. Schupp, Thompson Webb HI, and Peter Wyckoff

The oak, to gain its present mostnortherly position in North Britainafter being driven out by the coldprobably had to travel fully six hun-dred miles, and this without exter-nal aid would take something like amillion years. (Reid 1899)

Biologists have long regardedthe natural dispersal of largeseeds as an impediment to plant

range expansion after glacial peri-ods. Global maps predicting biomedistributions under future climatechange scenarios (e.g., VEMAP1995)are now prompting ecologists tothink about the dispersal problem: Ifrapid climate change or habitat de-

James S. Clark, Jason Lynch, and PeterWyckoff are in the Department of Botany,Duke University, Durham, NC 27708;Chris Fastie and Stephen T. Jackson are inthe Department of Botany, University ofWyoming, Laramie, WY 82701; GeorgeHurtt and Stephen Pacala are in the De-partment of Ecology and Evolutionary Bi-ology, Princeton University, Princeton, NJ08544-1003; Carter Johnson is in the De-partment of Horticulture and Forestry,South Dakota State University, Brookings,SD 57007; George A. King is at DynamicCorporation, US EPA National Health andEnvironmental Effects Research Labora-tory, Corvallis, OR 97333; Mark Lewis isin the Math Department, University ofUtah, Salt Lake City, UT 84112; ColinPrentice is at the School of Ecology, LundUniversity, Lund, Sweden; Eugene W.Schupp is in the Department of RangelandResources, Utah State University, Logan,UT 84322; and Thompson Webb III is inthe Department of Geological Sciences,Brown University, Providence, RI 02912-1846. © 1998 American Institute of Bio-logical Sciences.

A plausible explanation

for how rapid migrations

were achieved can

guide forecasts for tree

populations in the

twenty-first century

struction remove sizable portions ofa plant's range, will seed dispersalallow colonization of distant areasthat may become favorable? Dis-persal and population spread is acontinuing theme of paleoecologicalresearch (e.g., Webb 1986, Davis andZabinski 1992, Melillo et al. 1996),and it has demonstrated empirical mi-gration rates that are three orders ofmagnitude faster than ClementReid's estimate.

Tree populations migrated rap-idly after the most recent glacial pe-riod, which suggests that they mightdo so again if climate were to changerapidly in the future. However, theleap from late-glacial warming togreenhouse warming of anthrogeniclandscapes is broad (King andHerstrom 1996). Estimates of futuremigration rates cannot be taken di-rectly from paleoecological records,but they might be sharpened by anunderstanding of how life history anddispersal affected past migrations. Aplausible explanation for how rapidmigrations were achieved in the pastcan guide forecasts for tree popula-tions in the twenty-first century.

Although the rapid rates impliedby past records bode well for plantpopulations themselves, the rates arefar too high to have been producedby the dispersal mechanisms that areembodied in traditional notions of treelife history and restricted dispersal.We call this dilemma of rapid migra-tion Reid's Paradox, as voiced by himin the opening quotation. Explain-ing rapid migration has challengedpaleoecologists for a century and re-mains a point of debate (Davis 1987,Johnson and Webb 1989, Prentice1992, Clark 1993, Huntley 1996).

New insights gained from dis-persal theory hold promise not onlyfor understanding some of the in-triguing events of the past, but alsofor anticipating changes to come. Arecent workshop held at the NationalCenter for Ecological Analysis andSynthesis in Santa Barbara, Califor-nia, concluded that elements of asolution to Reid's Paradox can begleaned from recent advances in themathematical modeling of dispersal.Model populations disperse seedsaccording to a broad class of dis-persal "kernels," which are func-tional shapes that describe the scat-ter of dispersed seed away from theparent tree (van den Bosch et al.1990,Neubertetal. 1995). Althoughdispersal theory began with Reid'sParadox (Skellam 1951), only re-cently has research in paleoecologyand dispersal theory begun to con-verge toward new interpretations ofhow dispersal can contribute to therate and variability of an expandingpopulation front. Reid's Paradoxserves as an instructive example, not

January 1998 13

14kai 12kai

I' "i

10ka

Ska 6ka 4ka

Pollen Percentage

CJ 40-20• 20 -10• 10-5d 5 -2 .5~l 2.5-0

36

Oka

Figure 1. Time-series of beech (Fagus grandifolia) pollen percentages from 14 ka to 0 ka(1 ka = 1000 radiocarbon years before present) in eastern North America. Circles indicatelocations of paleorecords, and circle size is scaled to the percentage of F. grandifoliapollen in sediment samples. Contours interpolated within the spatial grid of pollensamples are shaded to show abundance levels. The general migration pattern is one ofnorthward, then westward, spread. Patterns are irregular, reflecting the scatter of siteswith differing soils and topography and also, perhaps, a variable dispersal process.

only of advances beyond traditionalviews of diffusion, but also of thevalue of dialogue among disciplinessharing complementary goals.

Reid's Paradox

The dilemma of rapid migration aroseearly in the interpretation of plantpopulation spread from leaf and seedremains in marsh and bog deposits.Reid's Paradox, laid out in Reid's1899 book, The Origin of the BritishFlora, points out an apparent con-flict between past tree distributions,as indicated by fossil evidence, anddistributions predicted from somerather simple life history consider-ations. Sometime between the end ofthe last ice age (which, in Reid's day,was estimated to have been 20,000years ago but is now known to havebeen approximately 10,000 yearsago) and the Roman occupation ofBritain, temperate taxa such as oakexpanded 1000 km northward tooccupy regions that were last popu-lated before the last ice age. Theimplied rate of spread appeared im-possibly great in view of averageseed dispersal distances, which wereobserved to extend not much beyondthe edge of the tree. Reid speculated

that relatively rare seed dispersalevents by birds must have facilitatedpopulation expansion.

Evidence gained over a century ofpaleoecological research has broad-ened Reid's Paradox to include manytree taxa in North America andEurasia. Early efforts to map chang-ing geographic distributions of treesinclude those of Firbas (1949) inEurope and of Davis (1976,1981) inNorth America. The record of pasttree distributions now includes alarge, cooperatively produced data-base of fossil pollen collected fromthousands of lakes worldwide (Hunt-ley and Birks 1983, NAPD 1990,Wright et al. 1993). These data havebeen used to produce sequential mapsof fossil pollen distribution and abun-dance, which demonstrate that manytree populations spread rapidly fol-lowing ice retreat (Figure 1). Severalkey insights into the abilities of treespecies to migrate in response to cli-mate change have emerged throughstudies of fossil pollen distributions.

• Species were capable of changinggeographic distributions in responseto 1000-year climate changes (Kutz-bach and Guetter 1986). These cli-mate changes affected large regions

that were quickly occupied as popu-lation ranges shifted at rates exceed-ing 100 m/yr (Davis 1981, Ritchieand MacDonald 1986, Htmtley andWebb 1989). For example, recentestimates of the tree population ex-pansion rate across Europe and NorthAmerica range from 150 to 500 m/yr,with rates being highest in the earlyHolocene and then diminishing(Huntley and Birks 1983, Delcourtand Delcourt 1987, Birks 1989).• Rapid migrations occurred in re-sponse to superimposed short-term(10-100 yr) climate changes (Gearand Huntley 1991, Chapellaz et al.1993, Lamb et al. 1995).• Tree species moved at differentrates and in different directions as aresult of climate change. These dif-ferences may have arisen from spe-cies-specific tolerances to differentclimate variables (Johnson and Webb1989, Prentice et al. 1991).• Large potential "barriers" to seeddispersal, such as the North Sea,Baltic Sea, and Lake Michigan, ap-pear to have posed few obstacles tomigration (Reid 1899, Webb 1986,Huntley and Webb 1989, Woods andDavis 1989, Kullman 1996), butlarger barriers, such as the AtlanticOcean, did block seed dispersal.

The rapid shifts in the ranges ofmany tree taxa at the time of iceretreat, between 16,000 and 10,000years ago, suggest the possibility thathaphazard, long-distance establish-ment events might explain theseshifts. Reid's rough calculations ofpopulation spread expected from lifehistory considerations, by contrast,assume that a diffusion process in-volving relatively uniform, incremen-tal, step-by-step advance would un-derlie tree migration. The rate ofsuch an advance would be deter-mined by two basic elements: thesize of the steps (dispersal distance)and the rate of population growth.Reid understood the importance ofdispersal, and his use of generationtime demonstrated a partial recogni-tion of the importance of populationgrowth. The average dispersal dis-tance implied by his calculationsclearly does not support a diffusionview of tree spread, and neither doesthe fact that populations apparentlyleaped across large bodies of water.When Reid invoked birds to explain

14 BioScience Vol. 48 No. 1

the movements of seed well beyondthe main population, he was denyingdiffusion in a strict sense and appeal-ing to a far more uneven (and qualita-tively different) process. Such unevenprocesses are also a feature of recentdevelopments in dispersal theory.

Dispersal theory embraces the"fat tails"

The benchmark article inauguratingcurrent interest in dispersal theory(Skellam 1951) was also the first toformalize the problem that diffusioncould not explain the migration oftree populations. Skellam (1951) wassufficiently inspired by Reid's Para-dox to open his broad treatment ofpopulation movement with this ex-ample. Skellam showed that Reidhad defined a problem containingthe elements of a diffusion equation.A diffusing population initiallyspreads as a Gaussian distribution.The distribution tends to grow at theedges, due to reproduction, and toflatten in the center, due to interac-tions that limit population growth athigh density; eventually, the popula-tion frontier expands as a wave trav-eling at constant velocity. Analysisof the leading edge of the wave isstraightforward because its rate ofspread does not depend on the com-plex, nonlinear interactions that con-trol growth rates in the populationinterior (e.g., competition).

The dispersal distances that wouldhave been necessary to explain wavevelocity—that is, the distance thefront had moved—can be calculatedgiven elapsed time since retreat ofthe Pleistocene ice, an estimate oftree generation time, and an estimateof seed production. Skellam statedthe problem in this way: Suppose apopulation of oaks contains indi-viduals that produce an average ofjR0 = 107 seeds in a lifetime. Whatdispersal distance, D, is needed toexplain a movement of X(n) = 103

km in n = 300 generations since iceretreat? Simple diffusion predicts thatafter n generations, the populationfront moves:

X ( n ) = Dn VIrLR7

Consistent with Reid's conclusion,the mean dispersal distance of seedsimplied by the model is impossibly

Tmean displacement

for all kernels

"fat-tailed"(c = 1/2, kurtosis =

exponential(c = 1, kurtosis =

Figure 2. Three dis-persal kernels thathave the same averagedispersal distance butdiffer in kurtosis. (left)The high kurtosis of thefat-tailed kernel meansthat it is more peaked atthe source (i.e., where 'I ' <c = 1/2, kurtosis = 25.2)distance from parent = -g0) than the lower-kur- "gtosis curves. Differences uiin the tails (i.e., far fromthe parent) of the threedispersal kernels appearminor, but a log scale(inset at right) illus-trates how a fat-tailed Distancs from Parent

distribution allows forrare long-distance dispersal. The values of the parameter c and kurtosis values aredescribed in the box on page 16.

0 100200300Distance (m)

Gaussian(c = 2, kurtosis = 3)

100

large, approaching 1 km. Thus, theclassic article that sparked interestin diffusion models of populationspread begins with an applicationdemonstrating its failure. Skellam'sanalysis of Reid's Paradox ends byconcurring with Reid's speculationsthat seed dispersal must have beenassisted by birds.

Skellam's early demonstration thatdiffusive spread could not accountfor Holocene tree migration mayexplain why theoretical treatment ofHolocene tree migrations ceased assoon as it began. The failure of dis-persal theory to explain Holocenetree spread appeared to spell an earlydoom for its application to paleo-ecological records, and the disciplinesof paleoecology and dispersal theoryembarked on divergent paths. Overthe last few decades, paleoecologicalresearch has rarely included applica-tion of migration models (but seeDexter et al. 1987). Dispersal theorywas used to explore problems in dif-fusion, a logical application. Diffu-sion models successfully describe di-verse problems in population biology(Okubo 1980, Hengeveld 1994), andthey yield a rich spectrum of dy-namic behaviors for simple commu-nities of interacting populations.

Although diffusion models haveproven useful in many contexts, ex-amples of population spread havecontinued to arise that cannot bewell described by simple step-by-stepmovement. For example, whereasmost dispersing seeds remain nearthe parent plant, some fraction dis-

perses great distances. This smallminority of long-distance dispersalevents might disproportionately in-fluence some aspects of populationdynamics, especially rates of geo-graphic spread. Models that relaxedsome of the strict assumptions ofdiffusion (see below) were, there-fore, clearly needed.

Recent analyses of populationspread have begun to include modelsin which the underlying dispersal ker-nel (seed distribution) is leptokurtic(see box page 16). Leptokurtic ker-nels describe distributions of off-spring that are clustered near thesource and "fat" in the tails (Figure2). The fat-tailed kernel in Figure 2has large kurtosis, meaning that thecurve density is high near zero and atlarge distances (Figure 2 inset). Theincorporation of rare long-distancewanderers in population modelsmakes the tail of the kernel fat andits analysis somewhat complex. Therandom walk that underlies diffu-sion, by contrast, simplifies to local,step-wise motion of individuals. Thislocal motion, in turn, greatly simpli-fies analysis of migration to an as-ymptotically constant rate of spread,as demonstrated in Skellam's calcu-lation for Reid's Paradox. However,leptokurtic kernels describe a situa-tion in which step size is highly vari-able, with a few individuals takinggreat strides. The longer strides arerare events and, thus, infrequentlyobserved and hard to measure.

Recent models that accommodateleptokurtic seed dispersal (Mollison

January 1998 15

Dispersal kernels

Dispersal is described in mathematical models by a kernel indicatingthe distance seeds travel. A "dispersal kernel" is a probability densityfunction that describes seed arrival at a distance x from a parent plant.For simplicity, assume that movement occurs along one dimension(right or left), that seed arrivals are concentrated near the parent anddecline with distance x, and that the dispersal process is "unbiased,"in the sense that seeds have an equal probability of being dispersed tothe right or left of the parent plant. The probability that a seed willarrive in the interval (x, x + dx) to the right or left of the parent, wheredx a small increment, is approximately k(x)dx. The kernel k ( x ) alwayshas mode at x - 0, but the shape of k can vary considerably, dependingon the nature of the dispersal process (e.g., Neubert et al. 1995). Theconcept of a "fat-tailed" kernel can best be demonstrated by a densityfunction that is expressed in terms of two parameters, a "distanceparameter," a, and a "shape parameter," c, that determines highermoments (including kurtosis):

*(*) =2oT(l/c)

exp

where F() is the gamma function with argument lie. From the mo-ments of k ( x ) , we find the mean displacement of a seed moving to theright or left to be

„ _ oT(2/c)

and kurtosis to be

r(5/c)r(i/c)

r2(3/c)

This density is quite general and includes several familiar densityfunctions, including the exponential (c = 1) and Gaussian (c = 2). Fat-tailed distributions are ones with kurtosis larger than the exponentialdistribution or c parameters less than 1 (Figure 2). These fat-taileddistributions are characterized by accelerating spread (Kot et al. 1996).

1972, 1977, van den Bosch et al.1990, Kot etal. 1996) have begun toyield results that are relevant to theHolocene spread of trees. Surpris-ingly, the rarest (and, thus, least oftenobserved) long strides turn out tohave overwhelming effects on mi-gration, making dispersal highly vari-able. Colonists that become detachedfrom the main population acceleratespread well beyond that expectedfrom the average dispersal distance(Figure 3), an average that is con-trolled largely by the small-steppingmajority. They also make the rate ofspread highly variable. The conse-

quence is an ill-defined populationfront that is difficult to describe and,hence, to analyze. Spatially and tem-porally variable densities includesatellite populations that becomeestablished by rare, distant colonistsand subsequently grow and coalescewith the expanding main population(Figure 4).

One of the non-intuitive resultsemerging from analyses of fat-tailed(i.e., highly leptokurtic) kernel shapesis the possibility for acceleratingspread of the population (Figure 3a).Unlike diffusion, which yields a trav-eling wave of constant velocity, a

kernel that is sufficiently fat in thetail (i.e., fatter than an exponentialdensity) can spread by "great leapsforward that get increasingly out ofhand" (Mollison 1977). Dependingon the "fatness" of the kernel tail,population spread can be a polyno-mial or exponential function of time(Kot et al. 1996). Because an ob-served migration (e.g., the Holocenespread of a given taxon) is but onerealization of a stochastic processthat spans finite time, there may beno way to determine asymptotic ratesof spread or even to guess at theirexistence. That is, not only can theasymptotic rate of spread be great,but a maximum rate may not exist.

Elements of a solution: theoryand paleoecology converge

Results from model analyses of seeddispersal can aid interpretation ofthe paleoecological record. Some el-ements of an explanation for rapidHolocene tree migrations have arisendirectly from paleoecology, but oth-ers can be sharpened by theoreticalanalysis. The first and most funda-mental element revealed by modelsis the inescapable role of local popu-lation growth rate. This point wasmade by Skellam (1951), but inter-pretations of Holocene tree spread,with the exception of Dexter et al.(1987) have largely ignored it. Pa-leoecologists since Reid's day haverecognized that generation time pacesthe leaps forward of a populationfrontier; new recruits at a popula-tion boundary must achieve seed-bearing age before they can dispersetheir seed beyond the advancing front(Webb 1986, Birks 1989, Johnsonand Webb 1989). In difference equa-tion models, such as used by Skellam(1951) and Kot et al. (1996), genera-tion time sets the time step of themodel. But the rate of spread alsodepends on seed number—the moreseed, the higher the populationgrowth rate, and the more rapid themigration. The odds of great stridesforward are enhanced simply by in-creasing seed amount, which raisesthe odds of seed arrival at long aswell as short distances. All repro-duction and dispersal models (diffu-sion and others) incorporate repro-ductive capacity when determiningspread (van den Bosch etal. 1990, Kot

16 BioScience Vol. 48 No. I

(a)400 600 800 1000

(c)

Location(x)

Figure 3. Population spread resultingfrom a fat-tailed kernel (top) comparedwith traveling waves of constant veloc-ity (middle and bottom). In each case,average seed dispersal distance is thesame but the kernel shape is different.Each panel shows the front at successivetime steps. Each dispersal kernel corre-sponds to one of the kernels in Figure 2:in the top panel, c = Vj; in the middlepanel, c = 1; and in the lower panel, c =2. The fat-tailed kernel (c = '/?) generatesaccelerating spread over time.

et al. 1996). Reproduction is an ele-ment of population spread that can beoverlooked in the absence of models.

A second contribution of modelanalysis has been to clarify how long-distance dispersal affects spread. No-tions of rapid spread and large leapsrepresent a convergence of theoreti-cal results and speculation from thepaleoecological record. Reid ap-pealed to this possibility as a wayaround the strict confines of diffu-sion. Animal dispersal and rare es-tablishment events that might resultfrom storms have been invoked toexplain rapid migrations of manytemperate tree taxa (Davis 1981,Davis etal. 1986, Webb 1986). Davis(1986), Prentice (1992), and Clark(1993) noted specifically that epi-sodic, long-distance colonizationevents violate a diffusion model. Pre-dictions from models containingleptokurtic dispersal may not con-flict with fossil pollen evidence.

Models showing that a large aver-age dispersal distance is not requiredfor rapid spread should influence theinterpretation of paleoecologicalrecords. Although it is not possibleto establish the details of the dis-persal kernel that might have gov-erned any particular realization of astochastic migration, the conclusionthat rates of spread were too fast forsimple diffusion appears inescapable.

(a)8f

(b)

'C

-80 -60 -40 -20 0 20Distance(x)

40 60 -80 -60 -40 -20 0 20Distance(x)

40 60

(c)

fo1.20

8

(d)

-80 -60 -40 -20 0 20Distance(x)

40 60 -80 -60 -40 -20 0 20Distance(x)

40 60

Figure 4. Monte Carlo simulation of patchy population spread using a mixeddispersal kernel (see box page 20) with c = 1, p = 0.99, a, = 0.5, and a, = 10. Twentyindividuals were established from coordinate (0, 0) at time zero. At each generation,each individual produces a Poisson number of offspring, with a mean of 1.2. Dotsindicate offspring locations after 20 generations (a), 30 generations (b), 40 genera-tions (c), and 50 generations (d). Note the presence of new outlying populations,which spread and eventually coalesce.

If fat-tailed dispersal kernels areneeded to explain rapid spread, butrates of spread decrease over time(rather than accelerate; Birks 1989),then there may be reason to suspectthat Holocene spread was actuallylimited by a comparatively slow rateof climate change.

Beyond the focus on rapid spreadis an additional parallel betweentheory and paleoecology: the evolvingconcept of an expanding front. Asapplied mathematicians were discov-ering that leptokurtic dispersal makesfor temporally variable (Mollison1977) and patchy (Lewis in press)population spread, paleoecologistsbegan to view migration as a processwhereby outlying populations mightcolonize well ahead of an advancingfront (Davis 1986, Prentice 1992).This changing concept of fronts hasimmediate implications for interpre-tation of the paleoecological record.

Much attention and debate onpatterns of Holocene tree spread hasrelied on "finding the front" (e.g.,Bennett 1985, Davis et al. 1991,MacDonald 1993). In the past, esti-mates of population spread rates in-volved drawing lines on a maparound the inferred distribution of ataxon based on a low, "threshold"pollen abundance. Paleoecologistsattempted to average over much ofthe poor geographic coherency inthese low values to calculate how therange limit changed from one timeincrement to the next. There are manysources of variability in these lowvalues that recommend this type ofaveraging. For instance, if 2.5%beech pollen were used as a thresh-old to mean that beech trees grewnear a study site in Figure 1, thenhow would the front be drawn inareas where adjacent sites vary be-tween 0% and 5%? If a few beech

January 1998 17

pollen grains appear in the paleoeco-logical record followed by a longabsence, does this observation meanthat a population appeared and thenwent locally extinct? Could a fewpollen grains have, instead, blown infrom distant populations, or couldthe sample have been contaminatedin the laboratory? Averaging can tendto overcome the potential for erroror noise in the data.

Models containing leptokurtic dis-persal actually predict this type ofvariability, including irregular fronts,possibly with outlying populationsinitiated by rare dispersal events (Fig-ure 4) that eventually coalesce andare overrun by an expanding corepopulation. Model analysis makessuch "outliers" a requirement of fat-tailed dispersal kernels—that is, theybecome the expectation of the dis-persal process rather than a sourceof noise that obscures it. This is notto say that every errant pollen grainmust necessarily represent a nearbypopulation of trees (e.g., Jackson1990JacksonandWhiteheadl991).But theory alters the interpretationof outliers and, therefore, the assign-ment of the "front."

Candidates for dispersal

Theory shows that rapid tree migra-tion is plausible given the properdispersal kernel, but it does not iden-tify what factors determine the ker-nel. Theory further identifies the fattail, which cannot be characterizeddirectly, as the most important partof the kernel. Fortunately, lessonsfrom mathematical models do notdepend on a precise characterizationof the tail. Different kernel formsand broad ranges of parameter val-ues predict that migration can be fastand variable. Rather than exact de-scriptions of the tail, therefore, onlya sense of the existence of long-dis-tance arrivals and what might causethem are required. As more is learnedabout the agents of dispersal, thebetter will be our ability to predictmigration potential in the face ofalternative scenarios of environmen-tal change.

Plausible candidates exist for rare,long-distance transport of seed. Birdscan transport fleshy fruits 1 km ormore. Frugivorous mammals, suchas foxes and bears, can travel more

than 10 km/d (Storm and Montgom-ery 1975, Willson 1993), making themimportant, albeit underappreciated,agents of long-distance seed dis-persal. Blue jays (Cyanocitta cristata)cache acorns and beech nuts as far as4 km from the parent tree (Johnsonand Adkisson 1985), an observationanticipated by Reid (1899). Clark'snutcrackers (Nucifraga columbiana)disperse pine seeds up to 22 km(Vanderwal l and Balda 1977).Whereas animal dispersal is confinedto taxonomically restricted groupsof trees, abiotic factors are, of course,more general. Tornadoes and otherstorms can transport seeds many ki-lometers (Snow et al. 1995). Someseeds (e.g., of birches; Betula spp.)are released in winter and second-arily dispersed by wind across snow(Matlack 1989). Water can trans-port seeds long distances along ri-parian corridors.

It is easy to point to potentialdispersal vectors and declare thattheir existence reconciles the con-flict between theory and data, butthe process of seed dispersal is com-plex and often ineffective (Schupp1993). The behavior of animal dis-persers may be conducive to neitherlong-distance directional movementof seeds nor successful establishment.Although migrating passenger pi-geons (Ectopistes migratorius, nowextinct) are thought to have movedfagaceous nuts long distances, it isoften overlooked that the nuts ma-ture when flocks would have beenmigrating south, in the opposite di-rection of Holocene tree spread inthe northern hemisphere (e.g., Birks1989). Moreover, passenger pigeonswould have killed most of the seedwith their muscular gizzards, andthey did not cache nuts in the ground(Johnson and Webb 1989). Largeflocks might still have enhancedspread by increasing the overall rateof seed dispersal (Webb 1986), butbird migrations were not a simplepipeline for poleward spread of treepopulations at the end of the last iceage. The same situation applies toautumn-ripening fleshy-fruited spe-cies dispersed by autumn migrantsmoving south. Summer-ripening spe-cies whose fruits mature before birdsbegin their autumn migrations facethe potential of limited seed move-ment. Moreover, much animal-in-

duced seed dispersal is not random,but rather directed toward otherfruiting individuals, often of the samespecies. Consequently, long-distancemovement from an individual mayonly infrequently result in long-dis-tance dispersal from a population(Hoppes 1987, Schupp 1993).

Even when a seed is carried a largedistance away from the current popu-lation range, dispersal may not leadto successful establishment. Animalspecies differ in the probability thata seed that they disperse will pro-duce a new adult (Schupp 1993).Compared with birds, temperate ter-restrial mammals often damage moreseeds in gut transit, deposit seeds inlarger clumps that may be more sus-ceptible to density-dependent mor-tality, and deposit seeds in micro-habitats that are poorly suited forrecruitment or survival (Johnson andWebb 1989, Schupp 1993, Willson1993). Moreover, deep burial of seedin burrows by small mammals oftenprecludes successful germination(Vanderwall 1990). However, mam-mals are more likely than most birdsto disperse seeds the requisite longdistances. Therefore, both mammalsand birds may contribute to rapidmigrations of fleshy-fruited species,with mammals providing the occa-sional successful long-distance dis-persal necessary for large leaps andbirds providing the effective localdispersal necessary for rapid popu-lation growth. A similar "division oflabor" may exist for nut-bearingtrees, with blue jays providing theleap forward and rodents providingeffective local dispersal.

Our incomplete identification ofplausible long-distance dispersersmakes the link between dispersaltheory and Holocene tree migrationstentative. Hickory (Carya spp.),whose nuts cannot be opened bymost jays, has no obvious long-dis-tance disperser, yet it appears to havespread northward almost as fast assome species with plausible long-dispersers (e.g., oaks; Quercus spp.).Were hickory dispersers present inthe past that are not recognized to-day? Or has some other explanationfor Holocene distributions of hickory(e.g., the presence of pre-existing iso-lated populations close to the icefront) been overlooked? It is alsopossible that extinct Pleistocene ani-

18 BioScience Vol. 48 No. 1

mals and humans that exploited nutsas a food source contributed to treepopulation spread. Although mod-ern studies of dispersal and migra-tions cannot tell us which animalstransported the seeds of which treespecies, they can shed light on pastand future seed movements and ontheir potential contributions to popu-lation spread.

Long-term implications ofshort-term analysis

Because tree populations migratedrapidly in the past, will they be ableto repeat this performance in theface of future climate changes?Whether rapid spread is a basic po-tential of tree populations or, in-stead, the outcome of special cir-cumstances following glaciations canbe determined only by analyzing thefactors that control migration. Be-cause past migrations cannot be ob-served directly, it is necessary to relyon indirect methods, combining ob-servational data and mathematicalmodels, to assess the efficacy of pos-tulated dispersal mechanisms to pro-duce rapid migration.

Two types of modeling efforts arenow being used to analyze how lifehistory and dispersal control popu-lation spread. The "forward" ap-proach starts with assumptions aboutlife history and dispersal and thenuses these to predict how popula-tions will migrate. The assumptionsconsist of functions (e.g., a diffusionequation) that are parameterizedwith data (e.g., average dispersal dis-tance). This analysis might yield suchquantities as rates and patterns ofspread. These predictions are thenjudged against the paleoecologicalevidence (e.g., Figure 1). The "in-verse" approach, by contrast, startswith an observed pattern of spreadand then asks whether the life histo-ries and dispersal kernels requiredby a model of that pattern are com-patible with observation. It was in-verse reasoning that allowed Reid(1899) and Skellam (1951) to con-clude that average seed dispersal dis-tances were not compatible with adiffusion model of the high migra-tion rates of the early Holocene.Obviously, forward and inverse meth-ods complement one another; Reid'scomparison compelled him to ponder

L a) Dispersal kernel fitted to Acer rubrumseed rain

Figure 5. Comparisonof spread rates simu-lated with a tree popu-lation model (JamesS. Clark, unpublisheddata) using three dif-ferent dispersal ker-nels showing the front20-year intervals, (a)Slow spread predictedfrom the dispersal ker-nel fitted to seed datafor Acer rubrum in thesouthern Appalachians(Clark etal. 1997). (b)Mixed model, describedin box page 20, with afraction p = 0.95 in the kernel &,(«) fitted to A. rubrum data and (1 - p) = 0.05 ina thin-tailed (c = 2) kernel k2(x) having a large dispersion parameter az = 100. (c)Mixed model similar to the middle panel, but the tail k.(x) is fat (c = 1/2). The fitteddata produce the slow, constant spread of diffusion, whereas allocating 5% of seedproduction to the fat-tailed kernel results in rates of spread comparable to thoseobserved in the early Holocene (d). The thin tail (b) produces constant spread at a higherrate than a kernel with no tail (a), but spread does not accelerate (d).

b) Acer rubrum kernel with 5% thin tail

c) Acer rubrum kernel with 5% fat tail

Year

the feasibility of rare, long-distancedispersal events by birds and water.

Fastie's (1995) investigation ofcolonization around Glacier Bay,Alaska, following rapid retreat ofglaciers there between 1750 and 1840provides basic elements for a for-ward analysis of tree spread. Re-cruitment, measured up to 4 km awayfrom refugial populations, yieldsrough estimates of reproductive ratesand dispersal. Detailed tree-ringanalyses of establishment withinthese stands provide the demographicdata needed to calculate populationgrowth. The broad dispersal observedat Glacier Bay resulted in migrationrates across 50 km of deglaciatedterrain of 300-400 m/yr, rates thatare comparable to early Holocenemigration rates for many tree taxa,including spruce (Piceaspp.) in west-ern North America (Ritchie andMacDonald 1986, Cwynar 1990).The observed pattern of seedling re-cruitment is consistent with a highlyleptokurtic dispersal kernel. Thus,reproduction and dispersal evidencefrom existing seedlings are elementsof a forward analysis that can beused to predict migration rates thatare consistent with tree-ring evidenceat Glacier Bay and fossil pollen datafor the early to mid-Holocene.

The complementarity of forwardand inverse modeling is especiallyvaluable when evidence is incom-plete. For example, a combined for-

ward and inverse approach is beingused to ask whether seed dispersalkernels of temperate hardwood spe-cies are compatible with rapid spread(James S. Clark, unpublished data).The approach involves data model-ing methods that make it possible toestimate local dispersal patterns un-der closed canopies with overlap-ping conspecific crowns (Ribbens etal. 1994, Clark et al. in press). Spa-tial distributions of parent trees andof seed accumulation in the under-story uniquely define dispersal ker-nels for "locally" dispersed seed atscales of 1-100 m. When these fitteddispersal kernels are embeddedwithin tree population models, theypredict rates of spread (1-30 m/yr)that are substantially slower thanmaximum rates in the pollen record(Figure 5a). The inability to measurethe noisy tails of dispersal kernels di-rectly frustrates further progress witha forward model. Additional headwayrequires an inverse approach.

Inverting the problem, therefore,one must search for a dispersal ker-nel that satisfies two criteria: it iscapable of producing rapid spread,(i.e., it has a fat tail), and it agreeswith seed production rates and dis-persal distances that can be observedin nature. Rates of spread dependlargely on the shape of this tail (" howfar") and on the fraction of seedallocated to the tail ("how much").The tail cannot be fitted directly to

January 1998 19

A mixed model to assess implications of

long-distance dispersal

How would the tail of the dispersal kernel have to look to explain thehigh migration rates observed in the early Holocene, but constrainedto be consistent with actual data on seed dispersal? One modelingstrategy uses a mixed dispersal kernel, which consists of two prob-ability density function components: k^x) and &,(*), with a fractionp allocated to one component and (1 - p) to the other:

The two components, k^x) and &,(#), have shapes determined by dif-ferent parameter values av cl and a,, c2, respectively.

The mixed model k(x) can be used to append a long, adjustable tailwithout changing the general shape of a kernel &,(x) that has beenfitted to locally dispersed seed in the forest understory. This mixedmodel thus allows large changes in the shape (c2) or length (a,) of the tail,described by k2(x), and to adjust how much seed is allocated to thetail — that is, (1 -p). Simulations of tree population dynamics provideestimates of the values of p and «2 needed to produce high migrationrates. A likelihood ratio test comparing the mixed model with thatfitted to data shows whether the ranges of p and a^ extend to thoserequired to produce rapid migration. Figure 5d compares the spreadobtained by a mixed kernel having 5% of the seed allocated to a tailwith mean dispersal of 200 m with the spread obtained by a kernelwithout this tail.

Acerrubrum Quercus15001000-

^ 100

Figure 6. Probabilitysurfaces from likeli-hood ratio tests of seeddata under the assump-tion that a fraction (1 -p) of seed comes fromlong-distance dispersaldescribed by a fat-tailedkernel k2(x) with dis-persal parameter a, (seebox this page). The pa-rameters represent theamount of dispersion(i.e., the "length" of thetail, a,) and fraction in the tail (1 -p). The contours express the probability that a modelwith a long, fat tail differs from purely local seed dispersal. High probabilities (thoseapproaching unity), in the lower left of contour plots, mean that a mixed model thatincludes a small tail (both (1-p) and a, are small) explains the data as well as purely localdispersal. Low probabilities, in the upper right of each plot, mean tails that are too longor contain too much seed to be compatible with the data. Plots show that observed seeddispersal for Quercus are compatible with long (more than 200 m) dispersion parametersin the tails, because the probability surface shows values well above those that might causeus to reject the hypothesis of local seed rain (e.g., probabilities less than 0.05). A. rubrumprobabilities for similar parameter values are lower, indicating that data are less likely tobe consistent with long-distance dispersal.

Fraction in fat tail (1 - p)

seed data because long-distance dis-persal is sporadic. One can, how-ever, ask how much tail a dispersalkernel can have and still be consis-

tent with observed patterns of seedrain. Perhaps a dispersal kernel witha long, fat tail will fit the data as wellas one without such a tail (Kot et al.

1996). Rather than demonstrate theexistence of the tail, one can ask:How much tail will the data admit?If the data admit a tail that is suffi-ciently fat to produce rapid migra-tion in a population model, then theParadox evaporates.

The inverse method involves com-paring likelihoods of observing apattern of seed rain in a forest underalternative assumptions of differentkernel shapes. A kernel that assumesonly local dispersal (i.e., a kernelwithout a long tail) includes a pa-rameter a (see box page 16) that isestimated by maximum likelihoodanalysis of seed trap data (Clark etal. in press). The likelihood of thedata is then computed under the al-ternative hypothesis that the kernelincludes a small amount of seed thatis dispersed longer distances, de-scribed by a fat tail. This alternativekernel introduces two new param-eters, a second dispersal parameteroc2 describing "how far" the tail ex-tends, and a fraction (1-p), describ-ing "how much" goes into the tail(see box this page). We cannot esti-mate these parameters from data,but we can search for parameter val-ues that are compatible with the data,as judged by likelihood ratio tests. Ifwe place "too much" seed in a tailthat extends "too far," then the fitbecomes unacceptable. But there is arange of parameter values (not toomuch or not too far) that the modelsfit equally well (Figure 6). If thisparameter range includes values thatcan predict rapid migration, we haveat least one candidate for a solutionto Reid's Paradox, one that is basedon real data. Current research (JamesS. Clark, unpublished data) analyzeshow alternative kernels that fit seedrain data affect spread in populationmodels (Figure 5).

Combined approaches that inte-grate analysis and theory can im-prove our understanding of how dis-persal affects population spread. Forinstance, George Hurtt and StephenPacala (unpublished data) are usinginverse methods to calibrate the dis-persal kernels in forward models tospatial and temporal pattern in fossilpollen data on the subcontinentalscale, that is, the "long-distance"component of dispersal. Dispersalparameters estimated at this scalecan be compared with parameter es-

20 BioScience Vol. 48 No. 1

timates from local studies, such asthat by Clark et al. (in press). Con-sistency between estimates obtainedby different methods and coveringdifferent scales builds confidence inthe interpretation of range move-ment by dispersal patterns that areobserved today. Inconsistent esti-mates can support alternative hy-potheses to rapid spread from dis-tant, full-glacial distributions, suchas expansion from local refugia (Reid1899, McGlone 1988).

New ways to analyze and model apopulation front from pollen datacan also broaden our view of migra-tion pattern. King and Herstrom(1996) abandoned the traditional cal-culation of migration rates based onchanges in the perceived locations ofan expanding front. They estimatedrates of spread from a hexagonalgrid of "arrival dates" interpolatedfrom nearest fossil pollen sites in thenortheastern United States and adja-cent Canada, sidestepping entirelythe problem of defining a front loca-tion. Histograms of migration ratesestimated by this method (Figure 7)indicate that a few episodes of highmigration rates could account for the"rapid" range expansion of tree popu-lations after deglaciation; median ratesare substantially less than mean rates.

Complementary to this analysis isa spatially explicit model of treepopulation dynamics (including seeddispersal) applied to the modern land-scape of Michigan (George A, Kingand Alan Solomon, manuscript inpreparation). This more mechanisticforward approach illustrates bothan uneven migration "front" causedby episodic dispersal events and thepotential for rapid migration, evenwhen such events are rare (Figure 8).Together, the range of forward andinverse modeling studies shed lighton the pattern and rates of spreadthat may aid in interpreting past rapidmigrations and in evaluating thepotential for spread in the face offuture climate and land-use changes.

Sharpening the data

Whereas models help paleoecologiststo explore the efficacy of postulatedmechanisms that produce rapidspread of tree populations, futureprogress will rely on improved pa-leoecological records. Analysis of mi-

<0IT

Figure 7. Histogram ofcalculated Picea migra-tion rates in easternNorth America duringthe Holocene. Rateswere calculated from a50-km grid of interpo-lated arrival dates. Vec-tors were drawn be-tween the grid point inwhich pollen first ap-pears and the nearestgrid point where pol-len was already present(King and Herstrom1996). The range ofeach rate class is 50 m/yr. The mean and median rates are 220 m/yr and 140 m/yr, respectively.

149 249 349 449 549 649 749 849 949 > 1000Migration rate class (m/yr)

gration requires accurate dating andspatially dense data. Interpreting pa-leoecological records depends onknowing how pollen data record treepopulations and how physical fac-tors influence seedling establishment.Characterizing population patternssuch as those in Figure 1 requiresspatial and temporal detail. The un-derpinning of migration analysis isradiocarbon dating, which is subjectto counting errors and to contami-nation from "dead" carbon in an-cient bedrock, from vertical move-ment of soluble carbon, and fromuncertain stratigraphy (Webb andWebb 1988, MacDonald et al. 1991,Pilcher 1993, Webb 1993). Manyfossil pollen sites were dated beforethese uncertainties were fully appre-ciated, so dating is often impreciseor inaccurate.

Spatial gaps in continental datanetworks (Huntley and Prentice1993, Webb et al. 1993, Jackson etal. 1997) also remain obstacles tocharacterization of pollen distribu-tion. Densities of fossil pollen sitesare low (approximately one site per40,000 km2) in all but a few regions(Webb 1993), so dispersal events orpopulation patterns cannot usuallybe resolved at scales of below 10-100 km. An exception is in the north-ern Great Lakes region, where a highdensity of fossil sites provides fine-resolution mapping of immigrations(Davis etal. 1986,1991,Davis 1987,Woods and Davis 1989).

Analysis of population spread alsorequires improved methods for datainterpretation. Migration studies relyon fossil pollen as a proxy of treepopulations. However, interpreting

populations from pollen grains isnot straightforward, because pollengrains of some taxa (e.g., pines andoaks) can move great distances(Prentice 1988, Jackson 1994). Plantmacrofossils constitute an alterna-tive source of data to test inferencesfrom pollen and to provide spatialand taxonomic precision that is notattainable from pollen data (Figure9; Dunwiddie 1987, Jackson andWhitehead 1991, Jackson et al.1997). Pollen and plant macrofossildata together can provide a basis forrobust inference of past plant distri-butions and population sizes at arange of spatial scales.

Interpreting past migrations re-quires not only high-quality paleo-records, but also knowledge of cli-mate trends and of fine-grainedpattern in environmental factors thataffect tree establishment. Althoughpast migrations were rapid at times,movement at many population fron-tiers was constrained by climate. Be-cause much of past climate change isinterpreted from fossil pollen data,there is a danger of circular reason-ing when attempting to interpret howpast climate change affected past mi-grations (Prentice et al. 1991). Pa-leoecologists will have to rely onevidence for past climate that is in-dependent of fossil pollen data ifthey are to interpret how climateconstrained population spread. Fine-grained environmental heterogene-ity, which actually controls the es-tablishment of seedlings in newlocations, is determined, in part, bydisturbances such as fire and windthrow. The evidence for past distur-bance is still rudimentary.

January 1998 21

Because deficiencies in the fossilrecord are subject to improvement, acontinuing dialogue between dataand theory will ensure continuedprogress. Accumulating radiocarbondates will refine chronologies, sharp-ening the record of population spreadand expanding the range of ques-tions that paleoecological data canbe used to test. The high site densi-ties available near the Great Lakescannot be attained in all regions, buthypothesis-driven data collection inselected regions can refine migrationestimates and be used to test as-sumptions and predictions of mod-els. Webb (1993) advocates con-s t ruc t ing data sets tha t allowresolution and coverage at severalspatial and temporal scales. Such hi-erarchically nested data sets will helpto surmount the limitation of notbeing able to obtain high-resolutiondata everywhere. Subcontinental-scale networks of plant macrofossildata are still in development, butthey have already added detail topopulation spread (Jackson et al.1997). Improved ways to identifyclimate changes (Wright et al. 1993)and disturbance (Bradshaw andZacknsson 1990, Clark et al. 1996)during invasions of new populationswill lead to better understanding ofthe factors that control migrationsat broad spatial scales.

Anticipating spread

Past conditions do not tell us what toexpect in the future. Future rates ofclimate change could exceed those ofthe late Quaternary, and future ratesof spread cannot be inferred fromHolocene rates, particularly becauseHolocene rates were far from con-stant. Migrations in coming decadesmay result in changing communitystructure, as spreading populationsoutrun their slower-migrating com-petitors, pollinators, seed dispers-ers, or natural enemies and as theyencounter new ones as the popula-tion frontier extends to new areas.To anticipate the responses of treepopulations to climate change, mod-els must cope with the specific condi-tions of today, including the factthat spread must occur through alandscape whose spatial structureand potential for plant establishmenthave been modified substantially

(Pitelka et al. 1997). The role ofpaleoecological data in developingpredictive models may be subtle. Thepaleoecological data reveal certainkey, qualitative features of the pro-cess—above all, that migration ratescan be rapid and are governed by thetail of the dispersal kernel, not by themean dispersal distance. In addition,these data may provide a test formore mechanistic models of the mi-gration process.

So far, global, grid-based dynamicvegetation models have treated mi-gration implicitly as an all-or-noth-ing phenomenon—that is, dispersalis assumed either to be not a barrier(so that abundant propagules of "ap-propriate" plant functional types arealways assumed to be present) or tobe confined to locations (model gridcells) where a given plant type al-ready exists. A full description of thetransient response of vegetation toclimate change, however, will alsorequire spatially explicit treatmentof propagule spread between loca-tions as a process modifying vegeta-tion dynamics, as well as some wayto represent the effects of landscapefragmentation on the effectivenessof spread. Possible approaches tothis problem were developed at aworkshop on climate change andplant migrations in Bateman's Bay,Australia. The workshop participantsagreed to represent vegetation dy-namics in large-scale models basedon insights derived from finer-scale,species-specific models of spread,with contemporary invasions andHolocene migrations serving to con-strain model parameters that may bedifficult to estimate directly frommodern observations (Pitelka et al.1997). Such a combination of model-and observation-based studies islikely to provide a dual benefit: ad-vancing the development of modelsthat can assess the range of out-comes of present and future globalchange, and illuminating the naturalprocesses by which plant species havesurvived the vagaries of natural en-vironmental change.

Acknowledgments

This article originated in a work-shop on "Dispersal and the HoloceneMigration of Trees" that was held atthe National Center for Ecological

Analysis and Synthesis in Santa Bar-bara, California, 1-3 July 1996. Thearticle benefited from further discus-sion at the workshop "Plant Dis-persal and Migration in Response toClimate Change," which was held atBateman's Bay, Australia, 15-22October 1996. For helpful commentson the manuscript, we thank BrianHuntley and four anonymous review-ers. Katherine H. Anderson preparedFigure 9.

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McGlone MS. 1988. Glacial and Holocenevegetation history, New Zealand. Pages557-599 in Huntley B, Webb T III, eds.Vegetation History. Dordrecht (The Neth-erlands): Kluwer.

Melillo J, Prentice 1C, Schuize E-D, FarqhharG, Sala O. 1996. Terrestrial biotic re-sponses to environmental change and feed-backs to climate. Pages 445-482 inHoughton JT, Meira Filho LG, CallanderBA, Harris N, Kattenberg A, Maskell K,eds. Climate Change 1995: The Science ofClimate Change. Cambridge (UK): Cam-bridge University Press.

Mollison D. 1972. The rate of spatial propa-gation of simple epidemics. Proceedingsof the Sixth Berkeley Symposium on Math-ematics, Statistics, and Probability 3: 579-614.

. 1977. Spatial contact models for eco-logical and epidemic spread. Journal ofthe Royal Statistical Society Series B 39:283-326.

Neubert MG, Kot M, Lewis MA. 1995. Dis-persal and pattern formation in a discrete-time predator-prey model. TheoreticalPopulation Biology 48: 7-43.

[NAPD] North American Pollen Database.1990. National Geophysical Data Center,Boulder (CO).

Okubo A. 1980. Diffusion and EcologicalProblems: Mathematical Models. NewYork: Springer-Verlag.

Pilcher JR. 1993. Radiocarbon dating andthe palynologist: A realistic approach toprecision and accuracy. Pages 23-32 inChambers FM, ed. Climate Change andHuman Impact on the Landscape. Lon-don: Chapman & Hall.

Pitelka LF, et al. 1997. Plant migration andclimate change. American Scientist 85:464-473.

Prentice 1C. 1988. Records of vegetation inspace and time: The principles of pollenanalysis. Pages 17-42 in Huntley B, WebbT III, eds. Vegetation History. Dordrecht:(The Netherlands): Kluwer.

. 1992. Climate change and long-termvegetation dynamics. Pages 293-339 inGlenn-Lewin DC, Peet RA, Veblen T, eds.Plant Succession: Theory and Prediction.London: Chapman & Hall.

Prentice 1C, Webb T III. 1986. Pollen per-centages, tree a b u n d a n c e s , and theFagerlind effect. Journal of QuaternarySciences 1: 35-43.

Prentice 1C, Bartlein PJ, Webb T III. 1991.Vegetation and climate change in easternNorth America since the last glacial maxi-mum. Ecology 72: 2038-2056.

Reid C. 1899. The Origin of the British Flora.London: Dulau.

Ribbens E, Silander JA, Pacala SW. 1994.Seedling recruitment in forests: Calibrat-ing models to predict patterns of tree seed-

ling dispersion. Ecology 75: 1/94-1806.Ritchie JC, MacDonald GM. 1986. The pat-

terns of post-glacial spread of white spruce.Journal of Biogeography 13: 527-540.

Schupp EW. 1993. Quantity, quality, and theeffectiveness of seed dispersal by animals.Vegetatio 107/108: 15-29.

Skellam JG. 1951. Random dispersal in theo-retical populations. Biometrika 38: 196-218.

Snow JT, Wyatt AL, McCarthy AK, BishopEK. 1995. Fallout of debris from tornadicthunderstorms: A historical perspectiveand two examples from VORTEX. Bulle-tin of the American Meteorological Soci-ety 76: 1777-1790.

Storm GL, Montgomery GG. 1975. Dispersaland social contact among red foxes: Re-sults from telemetry and computer simu-lation. Pages 237-246 in Fox MW, ed.The Wild Canids: Their Systematics, Be-havioral Ecology, and Evolution. NewYork: Van Nostrand Reinhold.

van den Bosch F, Metz JAJ, Diekmann O.1990. The velocity of population expan-sion. Journal of Mathematical Biology28: 529-565.

Vanderwall SB. 1990. Food Hoarding inAnimals. Chicago: The University of Chi-cago Press.

Vanderwall SB, Balda RB. 1977. Coadapta-tions of the Clark's nutcracker and thepinon pine for efficient seed harvest anddispersal. Ecological Monographs 47: 89-111.

fVEMAP]. Vegetation/Ecosystem Modellingand Analysis Project. 1995. A comparisonof biogeography and biogeochemistrymodels in the context of global climatechange. Global Biogeochemistry Cycles 9:407-437.

Webb RS, Webb T HI. 1988. Rates of sedi-ment accumulation in pollen cores fromsmall lakes and mires of eastern NorthAmerica. Quaternary Research 30: 284-297.

Webb SL. 1986. Potential role of passengerpigeons and other vertebrates in the rapidHolocene migrations of nut trees. Quater-nary Research 26: 367-375.

Webb Till. 1993. Constructing the past fromlate-Quaternary pollen data: Temporalresolution and a zoom lens space-timeperspective. Pages 79-101 in Kidwell SM,Behrensmeyer AK, eds. Taphonomic Ap-proaches to Time Resolution in FossilAssemblages. Knoxville (TN): Paleonto-logical Society.

Webb T III, Bartlein PJ, Harrison SP, Ander-son KH. 1993. Vegetation, lake levels,and climate in western Canada during theHolocene. Pages 415-467 in Wright HEJr, Kutzbach JE, Webb T III, RuddimanWF, Street-Perrott FA, Bartlein PJ, eds.Global Climates Since the Last GlacialMaximum. Minneapolis (MN): Univer-sity of Minnesota Press.

Willson MF. 1993. Mammals as seed-dis-persal mutualists in North America. Oikos67: 159-176.

24 BioScience Vol. 48 No. 1

caused by changes in the earth's orbit.Journal of Biogeography 16: 5—19.

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. 1994. Pollen and spores in Quater-nary lake sediments as sensors of vegeta-tion composition: Theoretical models andempirical evidence. Pages 253-286 inTraverse A, ed. Sedimentation of OrganicParticles. Cambridge (UK): CambridgeUniversity Press.

Jackson ST, Whitehead DR. 1991. Holocenevegetation patterns in the AdirondackMountains. Ecology 72: 641-653.

Jackson ST, Overpeck JT, Webb T III, KeattchSE, Anderson KH. 1997. Mapped plant-macrofossil and pollen records of LateQuaternary vegetation change in easternNorth America. Quaternary Science Re-views 16: 1-70.

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Johnson WC, Webb T III. 1989. The role ofblue jays in the postglacial dispersal offagaceous trees in eastern North America.Journal of Biogeography 16: 561-571.

King GA, Herstrom AA. 1996. Holocene treemigration rates objectively determinedfrom fossil pollen data. Pages 91—101 inHuntley B, Cramer W, Morgan AV, PrenticeHC, Allen JRM, eds. Past and Future RapidEnvironmental Changes: The Spatial andEvolutionary Responses of TerrestrialBiota. New York: Springer-Verlag.

Kot M, Lewis MA, van den Driessche P.1996. Dispersal data and the spread ofinvading organisms. Ecology 77: 2027-2042.

Kullman L. 1996. Norway spruce present inthe Scandes Mountains, Sweden at 8000BP: New light on Holocene tree spread.Global Ecology and Biogeography Letters5: 94-101.

Kutzbach JE, Guetter PJ. 1986. The influenceof changing orbital parameters and sur-face boundary conditions on climate simu-lations for the past 18000 years. Journalof Atmospheric Sciences 43: 1726-1759.

Lamb HF, Gasse F, Benkaddour A, El HamoutiN, van der Kaars S, Perkins WT, PearceNJ, Roberts CN. 1995. Relation betweencentury-scale Holocene arid intervals intropical and temperate zones. Nature 373:134-137.

Lewis MA. In press. Variability, patchinessand jump dispersal in the spread of aninvading population. In Tilman D, KareivaP, eds. Spatial Ecology. Princeton (NJ):Princeton University Press.

MacDonald GM. 1993. Fossil pollen analysisand the reconstruction of plant invasions.Advances in Ecological Research 24: 67-110.

MacDonald GM, Beukens RP, Kieser WE.1991. Radiocarbon dating of limnic sedi-ments: A comparative analysis and dis-

cussion. Ecology 72: 1150-1155.Matlack GR. 1989. Secondary dispersal of

seed across snow in Betula lenta, a gap-colonizing tree species. Journal of Ecol-ogy 77: 853-869.

McGlone MS. 1988. Glacial and Holocenevegetation history, New Zealand. Pages557-599 in Huntley B, Webb T III, eds.Vegetation History. Dordrecht (The Neth-erlands): Kluwer.

Melillo J, Prentice 1C, Schuize E-D, FarqhharG, Sala O. 1996. Terrestrial biotic re-sponses to environmental change and feed-backs to climate. Pages 445-482 inHoughton JT, Meira Filho LG, CallanderBA, Harris N, Kattenberg A, Maskell K,eds. Climate Change 1995: The Science ofClimate Change. Cambridge (UK): Cam-bridge University Press.

Mollison D. 1972. The rate of spatial propa-gation of simple epidemics. Proceedingsof the Sixth Berkeley Symposium on Math-ematics, Statistics, and Probability 3:579-614.

. 1977. Spatial contact models for eco-logical and epidemic spread. Journal ofthe Royal Statistical Society Series B 39:283-326.

Neubert MG, Kot M, Lewis MA. 1995. Dis-persal and pattern formation in a discrete-time predator-prey model. TheoreticalPopulation Biology 48: 7-43.

[NAPD] North American Pollen Database.1990. National Geophysical Data Center,Boulder (CO).

Okubo A. 1980. Diffusion and EcologicalProblems: Mathematical Models. NewYork: Springer-Verlag.

Pilcher JR. 1993. Radiocarbon dating andthe palynologist: A realistic approach toprecision and accuracy. Pages 23-32 inChambers FM, ed. Climate Change andHuman Impact on the Landscape. Lon-don: Chapman &: Hall.

Pitelka LF, et al. 1997. Plant migration andclimate change. American Scientist 85:464-473.

Prentice 1C. 1988. Records of vegetation inspace and time: The principles of pollenanalysis. Pages 17-42 in Huntley B, WebbTill, eds. Vegetation History. Dordrecht:(The Netherlands): Kluwer.

. 1992. Climate change and long-termvegetation dynamics. Pages 293-339 inGlenn-Lewin DC, Peet RA, Veblen T, eds.Plant Succession: Theory and Prediction.London: Chapman &c Hall.

Prentice 1C, Webb T III. 1986. Pollen per-centages, tree abundances, and theFagerlind effect. Journal of QuaternarySciences 1: 35-43.

Prentice 1C, Bartlein PJ, Webb T III. 1991.Vegetation and climate change in easternNorth America since the last glacial maxi-mum. Ecology 72: 2038-2056.

Reid C. 1899. The Origin of the British Flora.London: Dulau.

Ribbens E, Silander JA, Pacala SW. 1994.Seedling recruitment in forests: Calibrat-ing models to predict patterns of tree seed-

ling dispersion. Ecology 75: 1/94-1806".Ritchie JC, MacDonald GM. 1986. The pat-

terns of post-glacial spread of white spruce.Journal of Biogeography 13: 527-540.

Schupp EW. 1993. Quantity, quality, and theeffectiveness of seed dispersal by animals.Vegetatio 107/108: 15-29.

SkellamJG. 1951. Random dispersal in theo-retical populations. Biometrika 38: 196-218.

Snow JT, Wyatt AL, McCarthy AK, BishopEK. 1995. Fallout of debris from tornadicthunderstorms: A historical perspectiveand two examples from VORTEX. Bulle-tin of the American Meteorological Soci-ety 76: 1777-1790.

Storm GL, Montgomery GG. 1975. Dispersaland social contact among red foxes: Re-sults from telemetry and computer simu-lation. Pages 237-246 in Fox MW, ed.The Wild Canids: Their Systematics, Be-havioral Ecology, and Evolution. NewYork: Van Nostrand Reinhold.

van den Bosch F, Metz JAJ, Diekmann O.1990. The velocity of population expan-sion. Journal of Mathematical Biology28: 529-565.

Vanderwall SB. 1990. Food Hoarding inAnimals. Chicago: The University of Chi-cago Press.

Vanderwall SB, Balda RB. 1977. Coadapta-tions of the Clark's nutcracker and thepinon pine for efficient seed harvest anddispersal. Ecological Monographs 47:89-111.

[VEMAP]. Vegetation/Ecosystem Modellingand Analysis Project. 1995. A comparisonof biogeography and biogeochemistrymodels in the context of global climatechange. Global Biogeochemistry Cycles 9:407-437.

Webb RS, Webb T III. 1988. Rates of sedi-ment accumulation in pollen cores fromsmall lakes and mires of eastern NorthAmerica. Quaternary Research 30: 284-297.

Webb SL. 1986. Potential role of passengerpigeons and other vertebrates in the rapidHolocene migrations of nut trees. Quater-nary Research 26: 367-375.

Webb T III. 1993. Constructing the past fromlate-Quaternary pollen data: Temporalresolution and a zoom lens space-timeperspective. Pages 79-101 in Kidwell SM,Behrensmeyer AK, eds. Taphonomic Ap-proaches to Time Resolution in FossilAssemblages. Knoxville (TN): Paleonto-logical Society.

Webb T III, Bartlein PJ, Harrison SP, Ander-son KH. 1993. Vegetation, lake levels,and climate in western Canada during theHolocene. Pages 415-467 in Wright HEJr, Kutzbach JE, Webb T III, RuddimanWF, Street-Perrott FA, Bartlein PJ, eds.Global Climates Since the Last GlacialMaximum. Minneapolis (MN): Univer-sity of Minnesota Press.

Willson MF. 1993. Mammals as seed-dis-persal mutualists in North America. Oikos67: 159-176.

Clark, James S.; Fastie, Chris; Hurtt, George; Jackson, Stephen T.; Johnson, Carter; King, George A.;Lewis, Mark; Lynch, Jason; Pacala, Stephen; Prentice, Colin; Schupp, Eugene W.; Webb, Thompson, III;Wyckoff, Peter. 1998. Reid's paradox of rapid plant migration: dispersal theory and interpretation of paleoecologicalrecords. BioScience. 48(1): 13-24.

24 BioScience Vol. 48 No. 1