MARINE ECOLOGY PROGRESS SERIESMar Ecol Prog Ser

Vol. 313: 27–41, 2006 Published May 11

INTRODUCTION

Many sedentary marine species are dispersed bycurrents that transport planktonic larval stages. Conse-quently, the duration of planktonic larval stages is amajor determinant of dispersal distances for thesespecies (Strathmann 1980, 1985, Shanks et al. 2003).Thus, simple models of marine dispersal commonly usemean current speed and direction to predict bio-geographic boundaries, sources, and sinks of marinepopulations, and other important spatial patterns inbiology. However, if larvae were moved only by themean currents they would only be transported down-stream (downstream is defined here as the direction ofthe mean current, and upstream is defined as opposite

the direction of the mean current, i.e. analogous tostream systems). For an established species to avoidbeing swept downstream, it must develop a strategy toallow its larvae to be retained against the mean cur-rent. In the absence of some mechanism returning lar-vae upstream against the mean current, the edge ofthe species range upstream of the mean current will beeroded away as older generations die, and eventuallythe entire population will be swept downstream andthus become locally extinct. The observed persistenceof sedentary adult populations against mean currentsis known as the ‘drift paradox’ (Müller 1982). Becauselarvae of most benthic marine invertebrates are unableto swim significant horizontal distances, the resolutionof this paradox must depend on the upstream dispersal

© Inter-Research 2006 · www.int-res.com*Email: jebyers@unh.edu

Going against the flow: retention, range limits andinvasions in advective environments

James E. Byers1,*, James M. Pringle2

1Department of Zoology, University of New Hampshire, 46 College Road, Durham, New Hampshire 03824, USA2Department of Earth Sciences, and the Institute of Earth, Oceans, and Space, University of New Hampshire, Durham,

New Hampshire 03824, USA

ABSTRACT: Increasing globalization has spread invasive marine organisms, but it is not well under-stood why some species invade more readily than others. It is also poorly understood how species’range limits are set generally, let alone how anthropogenic climate change may disrupt existingspecies boundaries. We find a quantitative relationship that determines if a coastal species with abenthic adult stage and planktonic larvae can be retained within its range and invade in the direc-tion opposite that of the mean current experienced by the larvae (i.e. upstream). The derivation of theretention criterion extends prior riparian results to the coastal ocean by formulating the criterion as afunction of observable oceanic parameters, focusing on species with obligate benthic adults andplanktonic larvae, and quantifying the effects of iteroparity and longevity. By placing the solutions ina coastal context, the retention criterion isolates the role of 3 interacting factors that counteract down-stream drift and set or advance the upstream edge of an oceanic species’ distribution. First, spawningover several seasons or years enhances retention by increasing the variation in the currents encoun-tered by the larvae. Second, for a given population growth rate, species with a shorter pelagic periodare better retained and more able to spread upstream. And third, prodigious larval productionimproves retention. Long distance downstream dispersal may thus be a byproduct of the manypropagules often necessary to ensure local recruitment and persistence of a population in anadvective environment.

KEY WORDS: Advection · Biogeographic boundaries · Diffusion · Dispersal · Drift paradox · Hemigrapsus sanguineus · Physical–biological coupling · Planktonic larvae · Recruitment

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Mar Ecol Prog Ser 313: 27–41, 2006

of the larvae by stochastic fluctuations of the currentsaround the mean current.

The balance of mean currents, fluctuating currents,and larval production which allows a population to beretained and to invade upstream is most clearly seen atthe upstream edge of the species range. The existingliterature has explained the location of species bound-aries using 2 seemingly distinct theories. Speciesboundaries have traditionally been described as thelocation where the domain becomes uninhabitable,where the species is no longer able to increase itspopulation from low abundance (Caughley et al. 1988).Often, this limit is associated with a temperaturedependent increase in mortality of a species (Bird &Hodkinson 1999, Stachowicz et al. 2002, Crozier 2003).More recently, Gaylord & Gaines (2000) have demon-strated that many marine species boundaries arelocated where currents reverse, and propagules arethus constrained by an oceanographic boundary. Thetheory we derive below unifies these 2 existing theo-ries by demonstrating that both biological and physicalfactors and their interaction determine where a speciescan be retained, and thus its range. The 2 existingtheories are shown to be endpoints on a continuumof possible population growth and oceanographic con-ditions that will cause a species boundary to form.Specifically, in a species whose range is not expand-ing, the upstream edge will often occur at the pointwhere the tendency of mean currents to move thespecies downstream is exactly balanced by the abilityof the species to exploit fluctuating currents to produceand return larvae upstream against the mean current.

Several recent papers have quantified the popula-tion growth needed to allow an essentially planktonicspecies to persist and spread upstream in idealizedmodels of river and stream circulation. Speirs & Gur-ney (2001) used a continuous space/time model to finda criterion for the retention of an organism in a streamor estuary. Pachepsky et al. (2005) extend the model ofSpeirs & Gurney (2001) by explicitly including benthicand planktonic life phases in a continuous space/timemodel, and they noted the link between the ability of apopulation to invade upstream and to be retained. ThePachepsky et al. (2005) model of the interaction of thebenthic and planktonic stages is not appropriate formany oceanic species, for in their model benthic adultsdirectly produce benthic offspring, while at the sametime some fraction of the adults detach from thebenthos and enter the plankton for a well defined time.Neither the Speirs & Gurney (2001) nor the Pachepskyet al. (2005) models explicitly analyze the retention ofiteroparous organisms which can reproduce overmultiple generations, and they do not quantitativelyaddress the effects of extended adult lifespan. Spiers &Gurney (2001) and Pachepsky et al. (2005) also find

that the ability of an organism to be retained dependson, among other parameters, the variation of the dis-persal distance of offspring from their parents. Theyrepresent this variation in terms of a single eddy diffu-sivity for each location in a stream. It is shown belowthat this is not always appropriate, that retention in theocean depends on many forms of variability in thecirculation, and that this variability must be calculatedon time and space scales relevant to the life historyof the organism.

We find criteria for the retention of a population forapplication to a coastal marine environment similar tothat modeled by Largier (2003) and Gaines et al. (2003)using an integro-differential model similar to theanalysis of retention by Lutscher et al. (2005). Weadapt and extend the Lutscher et al. (2005) model tothe coastal ocean using a formalism that easily andnaturally incorporates observable physical oceano-graphic data, including the interannual variability ofthe currents and multiple spawning events in differentseasons. Further, with a simple numerical model wequantify the importance of iteroparity on retention.

Our model suggests adaptations that enable coastalspecies to be retained and to spread upstream, andclarifies how organisms can modify their larval andspawning behavior to adapt to local oceanographicconditions. The model further suggests how severallongstanding life history observations of marine spe-cies (e.g. prodigious larval production and multiple orprolonged spawning pulses) can be interpreted asmethods to aid retention and upstream dispersal.

We find support for the predictions of our model in ananalysis of the timing of planktonic larval release in thenortheast Pacific; the majority of larval release times areconsistent with a strategy which optimizes the likelihoodof species retention. Indeed, recent studies suggestthat in general, self-recruitment or local retention maybe more common than previously thought in marinepopulations (Swearer et al. 1999, 2002, Strathmann et al.2002). We also discuss how the interplay of growth,mean currents, and retention may affect ecologicalprocesses with important conservation ramifications,such as the spread of nonindigenous marine species andspecies’ responses to climate change.

MODEL AND RESULTS

Semelparous species

We proceed by combining discrete time models oflarval dispersal and population growth to predict para-meter values that promote upstream dispersal andlocal retention of larvae in a coastal ocean. In orderto make these results more applicable in an oceanic

28

Byers & Pringle: How marine species spread upstream

environment, we use a formalism that can easily incor-porate observations of the coastal currents and larvalbehavior into the predictions of species invasion andretention in the coastal ocean.

The environment we consider is a coastal habitatof limited cross-shore extent, so the domain can beconsidered 1-dimensional. The coastline need not bestraight, just describable by a single alongshore dis-tance. Our model organisms are assumed to follow thecommon marine invertebrate life history of having aplanktonic larval stage at which most or all of the dis-persal occurs. This larval stage remains in the planktonfor a time Tm before settling. On average, the meancurrents cause the larvae released by a mother over anentire generation to recruit a distance Ladv downstreamfrom their mother, though all larvae do not settle at thesame distance downstream. Variability in the currentsdisperse the larvae so that when they settle they aredistributed with a standard deviation (SD) of Ldiff

around the point Ladv downstream of their mother. Weemphasize that we do not explicitly specify the spatialdistribution of the larvae, though it is assumed that thedispersal kernel is not leptokurtic, which is reasonablein the coastal ocean (Davis 1985, Siegel et al. 2003).

If larvae are released into a coastal ocean whoseflow statistics are statistically stationary over the timethe larvae are in plankton (Tm), and over the distancethey are likely to move (Ladv), then Ladv and Ldiff can beestimated from observable oceanic properties with asimple model of Siegel et al. (2003). Siegel et al. (2003)show that larvae released in a single spawning eventin such an ocean move on average a distance:

Ladv = UcurrentTm (1)

with a random spread around this mean distancespecified by:

Ldiff = (σ2τLTm)0.5 (2)

where Ucurrent is the mean alongshore flow experiencedby the larvae, σ is the SD of the alongshore currents, andτL is the Lagrangian timescale of the fluctuations of thealongshore currents. The τL is the timescale of the fluctu-ations of the currents as experienced by a larva movingwith the currents, i.e. along the path a larva takes. Allof these parameters are defined at the depth(s) thatthe larvae reside and only over the time period thatlarvae are planktonic (Table 1). Ucurrent, σ and τL can beobserved with standard physical observing tools (Davis1985) (if the statistics of the ocean currents change sig-nificantly over the distance or the time the larvae are inthe plankton, then it is necessary to use more complextools to characterize their Ldiff and Ladv, perhaps obser-vations made with Lagrangian drifters or floats [e.g.Davis 1985], or particle tracking simulations made innumerical models [e.g. Tilburg et al. 2005]).

A population can be retained in an area, and evenmove upstream, if the random fluctuations of thecurrents return enough larvae to the habitat of theadult residents to offset the mortality of the residents.The propagule production is quantified by the para-meter Nfec, the number of successfully recruitinglarvae produced in the single spawning event of asemelparous organism, in a population where densitydependent effects are not significant. In this definition,a larva is ‘successful’ if it recruits and reaches repro-ductive competency. Nfec is thus net of all mortality inthe larval pool and mortality after settlement butbefore reproductive competence. In the text below,Nfec is referred to as the number of potentially success-ful larvae, to emphasize that it is the number of larvaethat would be successful in the absence of densitydependence. In a small, growing population therewill be in each generation Nfec times more individualsthan the generation before. Thus after n generations,there will be N n

fec individuals. For a semelparousspecies, Nfec is essentially the same as the discreteintrinsic population growth rate, R; however, wechoose this notation because it makes the analysis ofiteroparous populations considerably more straight-forward (see next subsection).

29

Parameter Explanation

Nfec Number of larvae per generation per adult which would successfully recruit and growto reproductive competency in a sparselypopulated environment (i.e. with no density dependence). This variable is analogous to thevariable R, the intrinsic rate of population growth, used in population growth models

Ngen The number of generations an organismreproduces

Nfec × Ngen Number of successfully recruiting larvae produced over an individual’s lifetime, in the absence of density dependence

Ladv Mean distance a larva recruits downstream ofits mother

Ldiff SD of distance a larva recruits downstreamof its mother

Ldiffeffect Effective value of Ldiff to use in Eqs. (5) & (6)when Ladv undergoes interannual variability; itis the SD of larval dispersal distance for all lar-vae released over the lifetime of an individual

n Time in units of generations

σ SD of Lagrangian alongshore currents experi-enced by planktonic larvae

σLadvSD of Ladv when Ladv varies interannually

τL Lagrangian timescale of alongshore currents experienced by planktonic larvae

Tm Time larvae spend in plankton

Ucurrent Mean alongshore current speed experienced by larvae in plankton

Table 1. Defined parameters

Mar Ecol Prog Ser 313: 27–41, 2006

In an advective environment our definition of Nfec

may differ subtly from the way R is typically measuredby ecologists (e.g. Grosholz 1996). Ecologists estimat-ing R have typically measured the change in popula-tion at a location in space relative to the population atthat place, an Eulerian definition of population growth.Our Nfec is defined for an adult and its progeny, even asthe progeny move (on average) downstream of thelocation of the mother and thus is a Lagrangian defini-tion. In a non-advective environment, or an environ-ment which does not change in the alongshore direc-tion, these definitions are equivalent. In an advectiveenvironment in which the population varies along theshore (e.g. at an invasion front), births and deaths at apoint in space can differ dramatically from measure-ments that track a cohort of larvae as they move onaverage downstream.

To determine the values of Ladv, Ldiff, and Nfec whichallow retention of a semelparous species, consider asingle individual introduced into an empty uniformdomain at y = 0. If the species is retained, the popula-tion will grow at the point of introduction, and if thespecies is not retained, the population will decrease atthat point. Each generation after the introduction, thelarvae will be displaced on average Ladv ± Ldiff down-stream of its parent, so the distribution of individuals attime n will consist of N n

fec instances of a sum of nrandom steps of mean length Ladv and SD Ldiff. For afinite Ldiff, the central limit theorem states that fornon-leptokurtic dispersal kernels, the adults after ngenerations will tend to be normally distributedaround a location nLadv downstream, with a SD ofthe location of the individuals of n1/2Ldiff. Thus, as nbecomes large, the density of the individuals per unitlength along the shore P(y) will become close to theGaussian:

(3)

This equation describes a normal distribution whoseintegral over the entire domain is N n

fec, which is thetotal population. On average, the majority of larvaemove downstream, regardless of the choice of parame-ters, as long as there is a mean current (Fig. 1). How-ever, the population is retained at y = 0 if the pop-ulation there remains constant or grows, instead ofdecreasing to zero. The equation above can be writtenat y = 0 as (using the identity ab = eb lna):

(4)

If the term in the square bracket is >0, the populationgrows at y = 0 and upstream, and if <0, the populationat y = 0 and upstream goes rapidly to 0. Thus, thespecies will be retained if

(5)

Thus the population will be retained if the meantransport of larvae (Ladv) is small, if the spread of thelarvae (Ldiff) is large, or if the production of potentiallysuccessful larvae Nfec is large (Fig. 2).This derivation isvalid for any kernel where the location of the conver-gence of the population distribution to a Gaussianadvances upstream by a distance greater than Ladv

each generation. It will fail when the dispersal kernelhas a finite upstream extent, so that no offspring arereturned to the location of their parents. In a uniformdomain, the ability to be retained and to invade

ln( )NLLfecadv

diff

>2

22

PL n

n NL

L( ) exp ln( )0

1

2 2

2

2= −

difffec

adv

diffπ⎡⎡⎣⎢

⎤⎦⎥{ }

P yN

L ny nL

L n

n

( ) exp= − −( ){ }fec

diff

adv

diff2 2

2

2π

30

Generation 3Generation 2Generation 1

Ucurrent

Ucurrent

Alongshore distance

Alongshore distance

Alongshore distance

Pop

ulat

ion

den

sity

Pop

ulat

ion

den

sity

Pop

ulat

ion

den

sity

a

b

c

Fig. 1. Spread of a semelparous organism with a planktoniclarval stage in an advective/diffusive environment. (a) Spreadof the organism introduced in the middle of the domain (ver-tical dashed line) in an environment with no mean current.(b) Spread and downstream transport of an organism intro-duced at the same location, but in an ocean where Ladv islarge enough that Eq. (5) is not satisfied. The total populationis still growing, but at the introduction site the populationdensity is falling as the population is swept downstream.(c) Growth of the population in an ocean whose Ladv issmall enough that Eq. (5) is satisfied. The total population isincreasing at the same rate as in the middle plot, but becauseLadv is less, fewer larvae are swept downstream and the pop-ulation increases at the point of introduction and upstream.

See Table 1 for parameter explanations

upstream is linked. As can be seen inFigs. 1 & 2, when Nfec is large enough toallow a species to be retained, the pop-ulation grows not only where the spe-cies has been introduced, but upstreamof that point as well, allowing the popu-lation to expand its range upstream.Pachepsky et al. (2005) relate this resultto the invasion speed of Fisher (1937)and Skellam (1951).

The derivation above for Eq. (5) isonly formally correct in an infinite do-main, in the absence of density depen-dent growth, and for non-leptokurtickernels. Lutscher et al. (2005) showsthat Eq. (5) remains valid even if thesuccess of larvae depends on the popu-lation density (cf. Kot et al. 1996). Theyalso show that Eq. (5) is valid for finitedomains much larger than Ladv and Ldiff,but for smaller domains Nfec must beincreased to allow retention. Lutscheret al. (2005) shows that for leptokurtic(‘fat-tailed’) kernels, a smaller Nfec thanpredicted by Eq. (5) is needed to allowretention; however, observations andtheory suggest that oceanic dispersalkernels for Lagrangian particles in theocean are not leptokurtic (Davis 1985,Siegel et al. 2003).

Iteroparous species

The results derived above are forsemelparous species, and must be mod-ified for a species that lives and spawnsover multiple generations. Unfortunately, the simplederivation above becomes significantly more complexfor iteroparous organisms because of the need toaccount for the age structure of the population. Todescribe the impact of iteroparity, we analyze a simpli-fied numerical model in which an adult reproduces forNgen generations, and then dies. In each generation,each adult releases Nfec larvae which would recruitand survive to adulthood in the absence of densitydependence. Thus the total maximum number of off-spring an adult will leave in a lifetime in the absence ofdensity dependence is Ngen × Nfec (the intrinsic capac-ity for increase over a lifetime). As above, the larvaereleased in each generation are dispersed on average adistance Ladv with a SD of Ldiff around that average.The numerical model incorporates a logistic densitydependence. Further details of the numerical modelare presented in Appendix 1.

An analytic solution to this non-linear model is com-plex. However, the number of parameters involved aresmall (Ladv, Ldiff, Ngen, and Nfec), and dimensional ana-lysis finds that any property of the system can bedescribed by 2 non-dimensional parameters (Kundu1990). The non-dimensional parameter 1⁄2 × L2

adv × L–2diff

was found to be useful in Eq. (5), so the other appropri-ate parameter must be Nfec × Ngen. To determine howthese 2 parameters control retention, the numericalmodel described above was initialized with a singleindividual in the center of the domain and was rununtil the population either filled the domain, or wentextinct in the domain. The numerical model was thenused to find the maximum value of exp(1⁄2 × L2

adv × L–2diff)

that allowed the population to be retained and to per-sist in the domain. This was done for a large number ofpermutations of Nfec and Ngen (Fig. 3), and it was foundthat the population is retained when:

Incr

easi

ng L

diff

Pop

ulat

ion

den

sity

0 500 1000 Alongshore distance (km) 0 500 1000

Increasing Ladv

generation 3 generation 2 generation 1

Byers & Pringle: How marine species spread upstream 31

ln N NLLfec genadv

diff

×( ) >2

22ln N N

LLfec genadv

diff

×( ) >2

22

ln N NLLfec genadv

diff

×( ) >2

22ln N N

LLfec genadv

diff

×( ) =2

22

ln N NLLfec genadv

diff

×( ) =2

22ln N N

LLfec genadv

diff

×( ) <2

22

Fig. 2. Spread of a population introduced at y = 0 (dashed line) for 3 generations.The current moves to the right, and Ladv and Ldiff are varied from 100 to 300 km.In all plots, Nfec = exp(0.5) = 1.6. Retention occurs when the population at y = 0increases. This growth rate is sufficient to allow retention at y = 0 and upstreaminvasion in the upper left panels, where Ldiff is relatively large and Ladv is rela-tively small. In contrast, in the lower right hand panels, where Ldiff is relativelysmall and Ladv relatively large, the population growth rate is insufficient to al-low retention. Along the diagonal, where Ladv = Ldiff, the growth rate is verynearly large enough to allow retention. See Table 1 for parameter explanations

ln N NLLfec genadv

diff

×( ) =2

22

ln N NLLfec genadv

diff

×( ) <2

22

ln N NLLfec genadv

diff

×( ) <2

22

Mar Ecol Prog Ser 313: 27–41, 2006

(6)

This is equivalent to Eq. (5) when Ngen = 1 and theorganism is semelparous.

Nfec × Ngen is simply the number of surviving off-spring that a species would leave after a lifetime in anenvironment so sparsely populated that density depen-dence does not affect recruitment success. Additionalnumerical experiments confirm that when an organismhas a different Nfec for each age, retention is still con-trolled by the total number of offspring produced overa lifetime that would be successful in the absence ofdensity dependence. Thus, longevity increases reten-tion by allowing an individual to leave more offspringover its entire lifespan. The importance of lifetimesurviving offspring as the crucial parameter contrastswith earlier results which did not explicitly includelongevity, in which retention depended most on theinstantaneous growth rate of the population (e.g.Speirs & Gurney 2001, Pachepsky et al. 2005). Theircriteria would overstate the growth rate needed toallow retention by a factor of Ngen. In the followingsections, the impact of temporally variable Ladv and Ldiff

on retention are considered, and it is shown that itero-parity has a further benefit on retention.

Physical oceanographic controls on retention

By substituting the definitions of both Ladv (Eq. 1) andLdiff (Eq. 2) into the criterion for retention at a given site(Eq. 6), it is possible to see how larval planktonic dura-tion and population growth interact with the meancurrents and the strength and duration of the currentvariations to determine retention. Retention and inva-sion will occur when

(7)

Note that while Ucurrent, σ and τL are physical para-meters, they will be influenced by larval behavior (e.g.the depth of the larvae). The variables Nfec × Ngen andTm are determined solely by the biology of the speciesand interactions with water properties, e.g. tempera-ture. Ultimately, more Nfec × Ngen and less Tm areequivalent in their effects on population retention, soin an evolutionary sense, an individual can increase itsability to be retained by either increasing its fecundityor decreasing the time its larvae spend in the plankton(Fig. 4).

Impacts of multiple spawning events within a generation

For an individual to increase its ability to beretained at a given fecundity Nfec and Tm, it musteither reduce the mean advection of its larvae, orincrease the variability of the advection experiencedby larvae. It can do both by releasing its larvae duringdifferent seasons with different mean currents. Wecan quantitatively define how fluctuations in themean current between spawning events will allow aniteroparous species to be retained with a lower levelof fecundity. The simplest case to examine is when aspecies can spawn in 2 different seasons within asingle generation. If the larvae from each spawningevent live to the same time as part of a single cohort,and if (somewhat artificially) the offspring from thefirst spawn do not adversely affect the ability of thesecond spawning group to find available habitat, wecan treat these 2 spawning events as a single event,with a composite Ldiff and Ladv. If the larvae arereleased in multiple seasons with different oceano-graphic conditions, then Ladv is the mean distancefrom the mother to the settling location of all of thelarvae released in a generation, and Ldiff is the SD ofthe recruitment locations of all those larvae. Let usassume that the larvae from each spawning event aredispersed with a Gaussian kernel. For 2 spawningevents in different seasons, the first of which releasesc1 larvae that successfully recruit Ladv1 ± Ldiff1 down-

ln( )N NU Tm

gen feccurrent

L

>2

22σ τ

ln( )N NL

Lfec genadv

diff

>2

22

32

Fig. 3. For iteroparous and semelparous species the maximumvalue of exp[L2

adv/(2L2diff)] that allows retention in the numeri-

cal model (x-axis) and Nfec × Ngen in the numerical model(y-axis). Model runs were made for Nfec = 2, 3, 4, 5, 10 and 20larvae per adult per generation for adults that lived 1, 2, 4 and6 generations. Solid line is the 1:1 line. Model results do not lieexactly along the line due to small numerical errors described

in Appendix 1. See Table 1 for parameter explanations

Byers & Pringle: How marine species spread upstream

stream, and the second of which re-leases c2 larvae that successfullyrecruit Ladv2 ± Ldiff2 downstream (sothat c1 + c2 = Nfec), the resulting larvaldispersal parameters are:

(8)

(9)

from the direct computation of thevariance and mean of 2 Gaussian ker-nels. These composite Ldiff and Ladv canbe used in Eq. (6) to judge if a speciesis retained.

In order to understand the impact ofmultiple spawning events on reten-tion, it is instructive to consider thecase in which the dispersal and thefraction of larvae released that are suc-cessful are equal between spawning(Ldiff1 = Ldiff2 and c1 = c2). In this case,

and

From this we can see that retention ismade easier by a second spawningevent if the second spawn is transporteda lesser distance or in an opposite direc-tion than the first spawn both becausethe composite Ladv is decreased, and be-cause the composite Ldiff is increased. Thus, multiplespawning events will enhance retention if they increasethe variability of the currents larvae encounter, andreduce the mean distance the larvae move.

Longevity and multiple spawning events across generations

In the prior section it was shown that multiplespawning events in a single year could enhance reten-tion by increasing the SD of the larval dispersal dis-tance Ldiff. If an organism lives for multiple years, inter-annual variation in the currents would similarly tend toincrease its ability to be retained by the mean currents.

However, the discussion in the previous section isnot formally applicable to iteroparous species thatspawn in separate years, for that derivation did not for-

mally account for changes in population level and dis-tribution between spawning events. Nevertheless, it isreasonable to apply Eq. (9) to the case of inter-annuallyvarying mean flows, and to use the numerical modeldescribed in the appendix to judge its accuracy. Inorder to simplify the discussion, Ldiff will be kept con-stant between years.

Eq. (9) is written for just 2 spawning events. Toinclude the case in which Ladv varies between severalyears, we note that Eq. (9) is the quadratic mean of Ldiff

and the square root of the sample variance of Ladv,defined as:

(10)

where Ladv�� is the mean value of Ladv calculated over thelifetime of a single individual. This estimate of the vari-ance of Ladv is biased (Roberts & Riccardo 1999) from

1

1

2

NL L

ii

N

genadv adv

gen

−( )=

∑

L L L Ldiff diff2

adv1 adv2( )= + −( )14

2

12

L L Ladv adv1 adv2( )= +12

L

cc c L L L

cdiff

diff1 adv adv( ( ) )=

+ + −[ ] +1

1 2

21

2

2cc c L L L

1 2

2 2+ + −[ ]( ( ) )diff2 adv2 adv

Lc L c L

c cadvadv adv= +

+1 1 2 2

1 2

33

Incr

easi

ng N

fec

Pop

ulat

ion

den

sity

Alongshore distance (km)

Increasing Tm

generation 3 generation 2 generation 1

25

20

15

10

5

0

15

10

5

0

6

4

2

0

25

20

15

10

5

0

15

10

5

0

6

4

2

0

25

20

15

10

5

0

15

10

5

0

6

4

2

0

Fig. 4. Spread of a population introduced at y = 0 (dashed line) over 3 genera-tions for a variety of growth and larval planktonic durations. The currentmoves to the right and Nfec increases from 2 to 4 per generation in the vertical.Tm was chosen so that ln(Nfec × Ngen) = (U 2

currentTm)�(2 σ2τL) along the diagonal,and thus populations are retained in the upper left of the plot where Nfec islarge and Tm is small, and populations are swept downstream in the lowerright, where Nfec is small and Tm large (Eq. 7). As Tm increases across thecurves, larvae spend more time on average exposed to the mean advectionand are spread farther downstream. See Table 1 for parameter explanations

ln N NLLfec genadv

diff

×( ) >2

22ln N N

LLfec genadv

diff

×( ) >2

22

ln N NLLfec genadv

diff

×( ) >2

22

ln N NLLfec genadv

diff

×( ) =2

22

ln N NLLfec genadv

diff

×( ) =2

22

ln N NLLfec genadv

diff

×( ) =2

22

ln N NLLfec genadv

diff

×( ) <2

22

ln N NLLfec genadv

diff

×( ) <2

22ln N N

LLfec genadv

diff

×( ) <2

22

Mar Ecol Prog Ser 313: 27–41, 2006

the true variance of Ladv by a factor of (1 – N –1gen). Thus

Eq. (9) can be extended to include interannual vari-ability of Ladv by writing it as:

(11)

where σLadv is the true interannual variance of Ladv. Ldiff

is the SD of larval dispersal within a single year orspawning event, whereas Ldiffeffect is the SD of larvaldispersal distance for all larvae released over the life-time of an adult. The variable σLadv differs from σ usedin Eq. (2) in that σ is the SD of the instantaneous cur-rents experienced by larvae from a single spawningevent, while σLadv is the SD of the mean distance movedby larvae from different spawning events. In order totest if Eq. (11) can be used to determine the retentionof a species, we ran the numerical model described inAppendix 1, changing Ladv for the entire domain eachgeneration, and allowing this variable Ladv to have aSD of σLadv. The model was then run with many differ-ent values of σLadv, and for each, the minimum value ofNfec was found which allows retention. In Fig. 5, theresults of these model runs are shown and compared toEq. (6) as calculated with the interannual mean Ladv

and Ldiffeffect from Eq. (11) in place of Ldiff. The changein the minimum Nfec needed to allow retention is con-sistent with the predictions of Eqs. (6) & (11), showing

that they capture the effects of interannual variation inthe mean flow on the retention of a species. Thus yearto year variability in the mean currents can signifi-cantly aid in the retention of organisms that spawn formultiple years by increasing variability in the dispersalof larvae.

DISCUSSION

The results presented here illustrate the importanceof variability in the currents of the ocean to the reten-tion of a species and its ability to spread upstream.However, a single value, for example for eddy diffu-sion (e.g. Speirs & Gurney 2001 or Pachepsky et al.2005) or the σ of Eqs. (2) & (7), will not capture theimpact of flow variability on all species. The variabilityin the dispersal of the larvae that allows retentiondepends sensitively on the life history of the organism.If the species is semelparous, a single eddy diffusivityor a single number describing the strength of the cur-rent fluctuations during the release of a larvae candescribe the random component of larval dispersal. Butif an organism spawns in 2 seasons, it is necessary toinclude both effect of the variability in the currentsexperienced by the larvae in each spawning event,and the variation in the mean currents betweenspawning events (Eq. 9). If an organism lives for multi-ple years, the interannual variability of the mean dis-tance the larvae are moved each year, σLadv, becomesimportant as well (Eq. 11). What matters in all of thesecases is the spatial spread of the larvae released by asingle organism over its entire lifetime, and in mostcases it will be advantageous for retention (all otherthings being equal) to release larvae at multiple timesinto different currents.

In the following section we discuss how the majority oflarval release times for benthic organisms with plank-tonic larvae in the North East Pacific are consistent witha strategy to enhance retention by taking advantage ofoceanic variability, suggesting that enhancing retentionis an important life history goal. This is followed by adiscussion of larval retention and vertical behavior; theevolutionary tradeoffs between retention, fecundity, andthe time in plankton; and the ecological implications ofretention on species range limits and how these mightchange with a changing climate.

Larval release timing in the northeast Pacific:optimized for retention?

Reitzel et al. (2004) compiled data from Strathmann(1987) on larval release times of 142 species in thenortheast Pacific. They found that brooding species

L L N Ldiffeffect diff2

gen–1

adv( )= + −( )1 2

12σ

34

σLadv

Ngen = 1

Ngen = 2

Ngen = 4

min

imum

Nfe

c

2.5

2

1.5

1

0.5

0 0 10 20 30 40 50 60 70 80 90 100

Fig. 5. Minimum value of Nfec required to allow retention as afunction of the interannual variability of the mean larval trans-port, σLadv. Mean value of Ladv is 100 km, and Ldiff is 75 km.Lines are from Eq. (6) calculated using an effective diffusivityfrom Eq. (11), and the individual points are from the numericalmodel described in Appendix 1. Curves are shown for organ-isms which live for 1, 2 and 4 generations. The numericalmodel growth minimum values of Nfec were slightly largerthan expected from theory due to approximations made in thenumerical model and discussed in Appendix 1. See Table 1 for

parameter explanations

Byers & Pringle: How marine species spread upstream

release progeny evenly over the whole year, but for the89 benthic species with planktonic larvae, 62% releaselarvae in April (Fig. 6). Both lecithotrophic and plank-totrophic larvae peaked in this same period. Thesepatterns held even for phylogenetically controlledcomparisons. Thus, the confluence of many plankton-producing species (but not brooding species) acrossmany phyla on this single reproductive peak suggestsstrong selective force(s) at work on the timing of larvalrelease into the plankton.

Because lecithotrophic larvae are provisioned withfood, it seems unlikely that food supply (e.g. the springphytoplankton bloom) can fully explain the springpeak in larval release. Parental food supply also seemsunlikely to explain this pattern since brooding speciesdo not share a similar spawning peak in spring. Reitzelet al. (2004) invoked 2 primary explanations based ontemperature. Both essentially revolve around the fact

that because larvae typically develop faster in warmerwater, the spring release may be an adaptation toreduce development time. However, because thegreatest proportion of species spawn in April, not thetime of peak water temperatures in the NE PacificOcean, Reitzel et al. (2004) had to invoke secondaryassumptions to explain the discrepancy. We believeadaptation for retention may more parsimoniouslyexplain the peak in spring spawning. Although fasterdevelopment time would decrease Tm and thus aidretention, a more important factor may be the currentpatterns themselves.

Retention of larvae released in a single month will beoptimized if larvae are released in the month whenmean currents are minimal, when variability in cur-rents is maximal, and, for iteroparous organisms, whenthere is the largest interannual variability in the meancurrents. In the northeast Pacific these conditions con-verge in April. Many studies have found that upperwater column, mid-shelf currents from Northern Cali-fornia to the Canadian border change from mean pole-ward to equatorward flow in March/April, and back topoleward flow in late-summer (Strub et al. 1987, Huyeret al. 1979, Lentz 1987). There are fewer current obser-vations that are long enough to evaluate interannualvariability; however, in 10 yr of data from the 95 misobath near Coos Bay, Oregon (43.15°N, 124.56°W)(1981 to 1991 data courtesy of OSU Bouy group and2001 to 2004 from Barbara Hickey/GLOBEC), theintra-seasonal current variability σ (which leads toLdiff), is large all winter and begins to decline sharplyin May to reach a minimum in July. Furthermore,the inter-annual variability in the monthly mean along-shore currents, and thus σLadv, is greatest in April,somewhat weaker in winter, and weakest in summer(Fig. 6).

Thus, the month that Reitzel et al. (2004) found 62%of species releasing larvae, April, is a good time forthe retention of species with planktonic larvae be-cause mean currents (Ladv) around this time reverseand the inter-annual and intra-seasonal variability ofcurrents (σLadv

and σ, respectively) are large. Further-more, an additional 13% of species with planktoniclarvae release them in multiple months in which thealongshore currents are flowing in opposite directions(currents from Strub et al. [1987] Fig. 8, midshelf at43°N). This larval release pattern would also tend toenhance retention by spreading larvae both up anddown the coast. Thus 75% of species are found torelease their larvae in the month in which the along-shelf currents are very favorable for retention (April),or in multiple months in which the mean alongshorecurrents disperse larvae both up and down the coast,or both. We thus believe the most parsimonious expla-nation may be that the larvae are being released to

35

J F M A M J J A S O N D month

J F M A M J J A S O N D month

J F M A M J J A S O N D month

60

40

20

0

15

10

5

0

22

20

18

16

14

12

10

per

cent

sp

awni

ngS

D o

f mon

thly

mea

n

flow

, cm

s–1

σ, c

m s

–1

c

a

b

Fig. 6. (a) Fraction of northeast Pacific benthic species withplanktonic larvae releasing larvae in each month (fromTable 1 in Reitzel et al. [2004]). Included are 89 benthic spe-cies from 12 phyla. (b) SD of the monthly mean alongshoreflow at 35 m from 10 years of data on the mid-shelf near CoosBay, Oregon. Variability in the monthly mean flows willdirectly lead to variability in the mean transport of larvaeeach year, increasing σLadv. (c) Mean SD of the detrendedalong-shore velocity from the same mid-shelf location. See

Table 1 for parameter explanations

Mar Ecol Prog Ser 313: 27–41, 2006

maximize retention, and that the observed patternstrongly supports the importance of adapting larvalrelease to increase retention.

Adaptations to allow invasion and retention atlower growth rates: effects of larval depth

In the bulk of the analysis presented here, retentionis described as a function of the circulation, the timingof spawning, and the fecundity of the species. How-ever, individuals can also reduce the ratio Ladv/Ldiff,and thus the propagule production (Nfec × Ngen) neededto allow retention and encourage invasion, throughlarval behavior. Here we examine one possibility—thevertical positioning behavior of larvae. Larvae deeperin the water column often (but not always) experienceweaker currents (e.g. in Central California: Winant etal. [1987]; in the Mid-Atlantic Bight: Beardsley et al.[1985]). However, these weaker currents do not them-selves aid upstream retention since to reduce thegrowth rate needed to allow retention, the ratio of themean current (Ucurrent) to its fluctuations (σ) must bereduced (Eq. 7). It is only advantageous for an individ-ual trying to have its larvae retained to stay deep in thewater column if the mean flow is reduced with depthwhile the fluctuations in the flow are not, since it is therelative strength of fluctuations that determines up-stream retention and invasion. These conditions dooccur in some oceans. Shearman & Lentz (2003) ana-lyzed the flow south of Cape Cod in the approaches tothe Mid-Atlantic Bight and found that the mean cur-rents decreased approximately linearly towards thebottom, while the strength of the fluctuating currentswas relatively independent of depth outside of the sur-face and bottom boundary layers. Thus in this ocean,the ratio of Ucurrent to the fluctuating component of thecurrents, σ, decreases with depth (Fig. 7). A larvaresiding deeper in this ocean would be more easilyretained, and would invade upstream more easily, allelse being equal.

Evolutionary trade-offs between fecundity, planktonic duration, iteroparity, and retention

As can be seen in Eqs. (1) & (2), the mean distancemoved by larvae Ladv increases linearly with time, butthe SD of the distance Ldiff only increases as the squareroot of time because the effects of random motionswill tend to average out with time. Thus, as seen inEq. (7), a longer time in plankton reduces the likeli-hood that a semelparous species will be retained. Thelonger the larvae are in the plankton, the more likelyit is that the sum of the random transport by fluctuat-

ing currents will be less than the transport by themean current (Fig. 4). Thus a larva can defy the oddsof being transported downstream in the mean currentand move upstream by fortuitously catching a fluctu-ating current over a short time. However, its luck willrun out if it stays in the plankton too long, since theodds ultimately favor movement in the predominantdownstream direction. In the face of this result, whydo organisms produce larvae which spend a long timein the plankton? Two possibilities are given below.The first is that reproductive output and planktonicduration are, in an evolutionary sense, correlated. Thesecond is that for iteroparous species, increased timein plankton can have less of an impact on retentionthan for a semelparous species due to the interannualvariation of the currents.

Reductions in Tm over evolutionary time are unlikelyto be achieved independently of Nfec. This is becausehighly fecund adults typically produce many relatively

36

dep

th, m

Subinertialalongshore current U2/σ2 ln(Nfec × Ngen)

a b c

U2/σ2 ln(Nfec × Ngen)speed, cm–1

0

–10

–20

–30

–40

–50

–60

–70

0

–10

–20

–30

–40

–50

–60

–70

0

–10

–20

–30

–40

–50

–60

–70–20 –10 0 0 1 2 0 5

STD (s)Mean (U)

Fig. 7. (a) Mean (Ucurrent) and negative of the SD (σ) of inertialalongshore currents on the 70 m isobath south of Cape Codfor the fall of 1996 (Shearman & Lentz 2003). A negative cur-rent is towards the west. (b) Square of the ratio of the mean tofluctuating sub-inertial Eulerian currents (U2/σ2) on the 70 misobath. The amount of growth needed for a population to beretained in its habitat along this coast scales as (U2/σ2), and atthis location would reach a minimum near the bottom. (c) Log-arithm of the recruit production, Nfec × Ngen, as a function ofdepth, that would be needed to retain the population of anorganism with a larval duration of 2 wk in the presence of thismean current (Eq. 7). This growth rate is smallest near thebottom with a secondary minimum near the surface wherethe mean wind-driven Ekman flows particular to this locationtend to reduce the mean alongshore surface currents. Dataare from Coastal Mixing and Optics experiment, courtesyof R. K. Shearman and S. J. Lentz (Shearman & Lentz 2003).

See Table 1 for parameter explanations

Byers & Pringle: How marine species spread upstream

unprovisioned larvae that require a longer time in theplankton to feed and develop before achieving compe-tency to settle and assume a benthic lifestyle (e.g.Thorson 1950, Vance 1973, Strathmann 1980, 1990,Pechenik 1999). Thus reducing Tm necessitates moreinvestment in each offspring, limiting the total numberof young that can be produced by a mother. Since low-ering Tm and raising Nfec have identical effects on up-stream retention, evolutionarily which strategy shouldprevail? From the perspective of upstream retention,the answer depends on whether Nfec increases fastenough to offset the coupled increase in Tm. If the log-arithm of the production of larvae that can successfullyrecruit, ln(Nfec), increases faster than the time spentin the plankton, it is advantageous for a species toincrease Tm in order to increase Nfec (Eq. 7, Fig. 4).Strategies that decrease Tm without affecting Nfec

should be heavily favored. For example, because lar-vae usually develop faster in warmer water, all elsebeing equal, summer spawning could be due in part toselection to minimize Tm without otherwise reducingNfec.

For iteroparous species, the impact of a larger time inplankton Tm may be less than suggested by Eq. (7).Eq. (11) states that the SD of the distance dispersed byall the larvae an iteroparous adult releases over its life-time, Ldiffeffect, depends in part on the SD of Ladv fromyear to year, σLadv. Because Ladv for each year scaleslinearly with Tm, σLadv will scale linearly with Tm to theextent that the timescale of the interannual variations incurrents is longer than Tm. As can be seen from Eq. (11),as σLadv increases, it grows to dominate Ldiffeffect, and sowhen Tm grows large enough, Ldiffeffect will tend to scalelinearly with Tm. In this limit, the fecundity needed toallow retention (from Eq. 7) will only increase slowly asTm increases. Thus while there is always an advantageto a reduced Tm, it may be slight for iteroparous speciesif interannual variability of the currents is high.

Ecological implications and species range limits

The edge of a species’ distribution is often consid-ered an area where the environment has becomeuninhabitable, and thus the population at the edge isunable to grow (see reviews by Caughley et al. 1988,Brown & Lomolino 1998, Gaston 2003). The modelabove shows that in advective environments this isnot necessarily true; the upstream edge of the speciesrange limit can also occur where the species is able tosuccessfully reproduce, but the reproduction is insuf-ficient to balance the downstream loss of larvae dueto the mean currents. If a population at the upstreamedge of a domain has reproductive rate Nfec that doesnot satisfy Eqs. (5) or (6), the population there willdwindle rapidly, and that portion of the domain willbecome uninhabited. Thus regions that would behabitable in the absence of a mean current may beunoccupied (Gaylord & Gaines 2000, Gaines et al.2003, Largier 2003). The species range boundary willthen occur farther downstream, where growth is justsufficient to balance advection, and Eqs. (5) or (6) aresatisfied. This is illustrated in Fig. 8, in which Ladv isdoubled in the upstream fifth of the domain, prevent-ing Eq. (5) from being satisfied there. The populationonly becomes finite in the downstream portion ofthe domain, where the advection speed is slower.Prodigious output of dispersive propagules may haveevolved in part to ensure that enough larvae surviveand are moved upstream by variations in currentsto replenish upstream populations. Ironically, thisresult suggests that long dispersal downstreamcould largely be a byproduct of copious larval pro-duction by species trying to stay in place in an advec-tive environment.

It may seem un-intuitive that a theory built aroundthe potentially successful larvae (Nfec) in the absence ofdensity dependence will be useful in determining the

37

popu

latio

n de

nsity

alongshore distance (km)0 100 200 300 400 500 600 700 800 900 1000

1

0.8

0.6

0.4

0.2

0

Larger Ladv Smaller Ladv

Fig. 8. Population density in a domain of 1000 km in which the mean currents vary in strength along the domain. The meancurrent flows to the right. Ladv is 50% above the threshold for retention to the left of the dashed line, and is 50% below thethreshold to the right of the dashed line. The population can only be retained to the right of the line, and so there is nearly nopopulation to the left of the line. The very small population to the left consists of the occasional larvae which arrive left of the

line, but are unable to establish a population that persists because too many of their offspring are washed downstream

Mar Ecol Prog Ser 313: 27–41, 2006

range limit in a species whose population has reachedsteady state. However, when the population is at thethreshold of not being retained, where Eqs. (5) and (6)are only just satisfied, the population density willbecome small enough at the upstream edge thatdensity dependent effects no longer matter (Fig. 8).

If range boundaries are set by strong advection, oneway a species with a flexible life history could retainpopulations on the edge, and thus extend its rangeupstream of where it might not otherwise be able tomaintain its population, is by reducing Tm to near zeroat the upstream edge and thus more easily satisfyingEqs. (6) & (7). Such a strategy may help to explainwhy many species with mixed modes of reproduction(e.g. vegetative or sexual reproduction) primarilyreproduce vegetatively or asexually at their distri-butional limits (Dixon 1965, Whittick 1978, De Wreede& Klinger 1988, Eckert 2002, Billingham et al. 2003).A preponderance of marine algae and angiospermsuse vegetative propagation as the principal means ofreproduction at extremes of their distributions andin areas with strong advection (De Wreede & Klinger1988).

Any environmentally induced changes in larvalparameters can also limit or alter a species’ range bychanging its ability to be retained. Most notably, thelarvae of many species develop more slowly as temper-ature decreases (e.g. Pearse et al. 1991, Clarke 1992).As the obligate time in plankton Tm increases, thegrowth a population needs to retain and spread up-stream increases (Eq. 7). On the northeast coast ofNorth America, the mean flow is southwestward, themean temperature increases to the southwest, and sothe upstream northern edge of a species’ range couldbe limited to the point where Nfec is not large enoughto compensate for the increase in Tm in colder waters toallow retention. This may also limit the northwardspread of some invasive species. For example, theobligate minimum time in plankton for the Asian crabHemigrapsus sanguineus measured in lab experimentsincreased from 16 d at 25°C to 21 d at 20°C to 53 d at15°C, for oceanic salinities (Epifanio et al. 1998). Themean near-shore shelf flows along much of the north-eastern coast of North America (i.e. the Mid-AtlanticBight, Gulf of Maine, and Scotian Shelf) are about 9 cms–1 (Beardsley et al. 1985, Pettigrew et al. 1998, Smithet al. 2001, Hetland & Signell 2005), the strength of thesub-inertial alongshore current fluctuations (σ) inthe summer are roughly 9 cm s–1, and the Lagrangiandecorrelation timescales of the east/west currents (τL)is about 2 d (Brink et al. 2003, Beardsley et al. 1985).

The number of potentially successful larvae Hemi-grapsus sanguineus must produce to allow retention is,from Eq. (7), Nfec × Ngen = 50 for 25°C water, or about0.02% net larval survival in a single year based on an

upper estimate of per capita fecundity of 200000 to300 000 eggs yr–1 (McDermott 1991). For 20°C, Nfec ×Ngen = 200, or 0.08% survival. These temperatures thusproduce net larval recruitment rates that are easilyachieved given laboratory-based mortality rates for H.sanguineus (Epifanio et al. 1998), and decapod crus-tacean larvae in general (Morgan 1995). However, for15°C, Nfec × Ngen = 568 000, which even at 100% sur-vival exceeds the total number of larvae a female isexpected to release in 2 yr. (These values of Nfec × Ngen

are much larger than the values of R typically observedin situ, which are usually between 1 and 10. However,in situ estimates of R often include density dependenteffects and neglect advection.)

These calculations suggest how Hemigrapsus san-guineus has been able to expand rapidly hundreds ofkilometres upstream from its initial discovery in 1988in Cape May, New Jersey (McDermott 1991), wheresummer surface water temperatures are roughly 20°C(Loder et al. 1998). But they also suggest that popula-tions of H. sanguineus observed near the central coastof Maine (R. Seeley pers. comm.) where summer watertemperatures are ~15°C should be ephemeral. We cau-tion that all of these estimates are hedged by greatuncertainty, including the questionable applicability oflaboratory based estimates of larval duration and mor-tality. However, the increase in the minimum intrinsicgrowth needed for retention as temperatures fall is sogreat that it suggests that the H. sanguineus invasion isnear its northern limit in central Maine, and that limitis set by the decrease in temperature it experiences asit expands its range northward and the mean currentsmoving against the direction of its invasion. This con-clusion would be invalidated if the population on theopen shores is maintained by populations retained inestuaries where the mean alongshore current is small,or if the interannual variability in the currents is largeenough that there are sufficient years when the trans-port of the larvae is northeastward and thus againstthe climatological average current (cf. Eq. 11). Theformer seems unlikely, for H. sanguineus larvae areobserved to fail to develop at low estuarine salinities(Epifanio et al. 1998). However, we do not have suffi-ciently long time series of alongshore coastal currentsin the Gulf of Maine to rule out the latter scenario, andwill not until ocean observing systems in this area havebeen able to measure current variability for severalmore years.

More generally, decreased larval development timesin warmer water implies that any increase in meanocean temperatures due to global warming could allowequatorward species to extend their range polewardeven against mean currents. Even small increases inannual mean temp (0.5°C) can dramatically influencespecies distributions (Southward et al. 1995). These

38

Byers & Pringle: How marine species spread upstream

temperatures changes are usually suggested to alter aspecies’ distribution by directly affecting adult andlarval tolerances to temperature extremes (e.g. South-ward et al. 1995). In contrast, our model suggests thatupstream range expansion could occur in some oceanssimply because an increase in water temperaturedecreases development time (Tm) which facilitatesretention and expansion upstream. Therefore globalclimate change may not need to change temperatureenough to kill larvae and adults to change a speciesrange; its influence on their development time maybe enough to alter species ranges substantially.

CONCLUSION

In summary, currents can benefit spawning marinespecies by spreading larvae (often over large dis-tances) with little energetic cost to the organism itself.Superficially that is often the extent of our perceptionof currents’ influence on larval spread—that it is anenergetically efficient dispersal mechanism. However,these same currents present a very real challenge to anorganism, for it must avoid having too many of itslarvae swept downstream, and thus being unable topersist in a region. For an organism (and its progeny) tobe retained and to spread upstream, it must do one ormore of the following: (1) it can spawn in multiple sea-sons or multiple years, to increase the variability in itslarval dispersal by increasing the variability in the cur-rents its larvae encounter; (2) its larvae can spend lesstime in the plankton, or through their behavior reducethe distance which the mean currents move them(relative to Ldiff); or, (3) it can have a high reproductiverate, integrated over its entire life. In the coastalmarine environment advection is a dominant influenceon life. Life histories of organisms must include a com-bination of these traits that minimize downstreamadvection by the mean currents or maximize the vari-ability of this advection to retain essential upstreampopulations.

Acknowledgements. We thank C. Dibacco, B. Gaylord, M.Graham, R. Karlson, R. Strathmann, J. Wares, and 2 anony-mous reviewers for helpful reviews of the manuscript and D.Houseman for Eqs. (8) and (9), and for his insight on longevity.We thank F. Lutscher for pointing out the convergence prop-erties of the central limit theorem at the tails of a distributionand the implications this has for retention. We thank B.Hickey and S. Geier from the University of Washington forcurrent meter records from the North Pacific GLOBEC pro-gram and J. Fleischbein, A. Huyer and the Oregon State Uni-versity buoy group for their compilation of historical data usedin computing Fig. 6, and the scientists and technicians whotook these data. This work was financially supported by theNOAA-Cinemar program at the University of New Hampshireand NSF OCE-0219709. This is GLOBEC contribution No. 280.

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Byers & Pringle: How marine species spread upstream 41

Editorial responsibility: Ron Karlson (Contributing Editor),Newark, Delaware, USA

Submitted: April 12, 2005; Accepted: September 5, 2005Proofs received from author(s): March 4, 2006

Some of the results presented in this study for semel-parous species are found analytically, but the quantitativepredictions of the importance of iteroparity and interannualvariations of Ladv must be confirmed with a numericalmodel. The numerical model is a straightforward implemen-tation of the idealized organism described above, a benthicorganism with an obligate planktonic stage in an environ-ment with a finite carrying capacity. The habitat in the com-puter is a 1-dimensional array of settlement sites each ofwhich can hold 1 adult. Each adult produces Nfec larvaewhich move on average Ladv downstream from its mother,with a SD of this movement of Ldiff (both Ladv and Ldiff can befunctions of the mother’s location, and can change withtime). If a larva arrives at a site that is occupied by an adult,the larva dies. If several larvae arrive at the same site in thesame generation, 1 is randomly chosen as the survivor. In auniformly settled domain, this would lead to a logisticdensity dependent population growth rate. Larvae whichmove out of the domain die. The adults live for Ngen genera-tions, and then die. The larval dispersal kernel in most runswas Gaussian, but results did not change significantly whena top-hat or double-exponential dispersal kernel was usedwithin the limits of validity given in the ‘Methods’ section ofthe text. When Ladv changes between generations, thisvariation is drawn from a Gaussian distribution. The resultsdo not change if it is drawn from a uniform distributioninstead. The numerical model was written in MATLAB andFortran, and is available upon request from the authors.

The numerical model is not a perfect analogue of theidealized problem described in the model and resultssection. To have an operational simulation model weneeded to make a few simplifying changes to the underly-ing assumptions of the theory/equations. These changeswere to make the domain size and habitat availability finite.However, we show that both of these changes have verylittle effect on model outcomes. First, the model domain isfinite while the derivation of Eq. (5) above assumes aninfinite domain. However, the finite domain does not havea material effect on the results presented above—all runsshown were run with a domain of twice the size, and theminimum growth needed to allow retention changed by lessthan 10%. In all cases the model domain is much larger thanLadv and Ldiff.

A more subtle limitation in the model arises from the factthat the maximum population density of the larval habitat

must be much lower in the numerical model than in the realworld, due to our finite computer resources. Because of this,the number of successful larvae per lifetime at low popula-tion densities is not exactly Nfec × Ngen, for even when only asingle organism inhabits the domain, 2 of its larvae mighttry to settle in the same location, resulting in the failure ofone of them. The importance of this can be monitored byexamining the threshold for retention of semelparousorganisms, for which we have an analytic solution. For aspecies which produces 30 potentially successful larvae inits lifetime, the minimum Nfec × Ngen needed to allow reten-tion is 30% higher than predicted if there are 250 settlementsites in an stretch of coastline of length Ldiff. If this samestretch of coastline has 500 settlement sites, the minimumNfec × Ngen is 12% higher than predicted, and if it has 5000sites, the Nfec × Ngen is 5% higher than predicted. We usedthis latter density of settlement sites/Ldiff in the results pre-sented in the text and figures. Holding Ldiff constant, theerror increases roughly as square root of Nfec for a givenNgen. The ratio of settlement sites to Ldiff is inconsequentialfor most oceanic species; for example, for a barnacle whoseLdiff is only 10 km, the number of potential settling sites in adistance Ldiff along a rocky coast would be much greaterthan the 5000 in the model.

The numerical model has an additional shortcoming dueto its finite domain size when Ladv is allowed to vary ran-domly in time. A randomly varying Ladv has some probabil-ity of exceeding the value which allows retention for somefinite amount of time. In any finite domain with a fluctuatingLadv, there will eventually be a period of anomalously highLadv which drives the species extinct in the domain. Thecloser the species is to failing to satisfy Eq. 6, the morerapidly this will occur because the smaller the sustainedanomaly in Ladv must be to drive the species extinct. Like-wise, the smaller the numerical model domain, the morelikely it is that a species will go extinct in a finite time, for ananomalously high Ladv must be sustained for a smaller timeto drive the species to extinction. This can be seen clearly inthe semelparous species in Fig. 5. As the variabilityincreases, the growth rate needed to retain the species inthe finite domain increases slightly. We confirmed that thiswas an artifact due to the finite domain by re-running themodel in a domain half as large; the increase in growthneeded to allow retention at high levels of variability wasmore than doubled.

Appendix 1. Numerical model for formulation and errors