Short‐Range Dispersal Maintains a Volatile

2 Received Date: 09-Jul-2016

3 Revised Date: 19-Dec-2016

4 Accepted Date: 16-Feb-2017

5 Article Type: Articles

6 Short-range dispersal maintains a volatile marine metapopulation: the brown alga

7 Postelsia palmaeformis

9 Robert T. Paine

10 Department of Biology

11 University of Washington

12 Seattle, WA 98195-1800

14 Eric R. Buhle

15 Northwest Fisheries Science Center

16 2725 Montlake Blvd. East

17 Seattle, WA 98112-2097

19 Simon A. Levin

20 Department of Ecology and Evolutionary Biology

21 Princeton University

22 Princeton, NJ 08544Author Manuscript

This is the author manuscript accepted for publication and has undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/ecy.1798

24 Peter Kareiva

25 Institute of the Environment and Sustainability

26 UCLA

27 Los Angeles, CA 90095

28 [email protected]

29 (corresponding author)

33 Running Head: Local dispersal allows persistence

34 Abstract. The annual brown alga Postelsia palmaeformis is dependent for its survival on

35 short-distance dispersal (SDD) where it is already established, as well as occasional long-

36 distance colonization of novel sites. To quantify SDD, we transplanted Postelsia to sites

37 lacking established plants within ≥10 m. The spatial distribution of the first naturally

38 produced sporophyte generation was used to fit dispersal kernels in a hierarchical Bayesian

39 framework. Mean dispersal distance within a year ranged from 0.16 to 0.50 m across sites;

40 95% of the recruits were within 0.38 to 1.32 m of the transplant. The fat-tailed exponential

41 square root kernel was the best among the candidate models at describing offspring density

42 and dispersal. Independent measurements of patch size over 2-5 generations permitted an

43 evaluation of whether models parameterized by individual-level data could adequately

44 predict longer-termAuthor Manuscript persistence and spread at the patch scale. The observed spread rates

45 generally fell within the 95% predictive intervals. Finally, Postelsia was eliminated from 14

46 occupied sites that were then followed for ≥27 years. The probability of invasion when

48 from the nearest propagule source. Sites >10 m from a source were rarely invaded, and one

49 initially densely populated site isolated by 39 m has remained Postelsia-free since 1981. In

50 spite of dispersal that is almost entirely within 2 m of the parent, the ability of our models to

51 capture the observed dynamics of Postelsia indicates that short-range dispersal adequately

52 explains the persistent and thriving Postelsia metapopulation on Tatoosh Island. However,

53 the presence of Postelsia over a 2000-km coastal range with many gaps >1 km makes it clear

54 that rare long-distance dispersal must be required to explain the geographic range of the

55 species.

57 Key words: Postelsia palmaeformis, dispersal, metapopulation, colonization, ephemeral

58 populations, persistence

60 Running head: Local dispersal allows persistence

61 INTRODUCTION

63 Postelsia palmaeformis Ruprecht, or sea palm, is a widely distributed and locally

64 abundant brown alga found in the upper rocky intertidal zone from southern California to

65 Vancouver Island, and is a characteristic species of wave-exposed, highly disturbed sites

66 (Dayton 1973, Paine 1988). It is an annual that must recolonize each year without any sort of

67 seed bank as well as a fugitive species that depends upon wave-mediated gap formation to

68 remove the competitivelyAuthor Manuscript dominant mussel Mytilus californianus and release space, the

69 limiting resource (Paine 1979). The persistence of vigorous Postelsia populations in a highly

70 capricious and severe environment is due to dispersal and recolonization, but unlike annual

72 entirely on spatial dispersal for persistence. The absence of resting stages or seed banks and

73 the inevitability of local competitive exclusion by Mytilus makes Postelsia the quintessential

74 dispersal-dependent metapopulation. Yet unlike many “classic” fugitive species, Postelsia’s

75 dispersal ability appears to be quite limited (Dayton 1973, Paine and Levin 1981). The key

76 question is whether or not local, short-distance dispersal is adequate for this species’

77 persistence, or whether long-distance rescue is necessary.

78 To answer this question we combine experiments and models to ask whether local

79 dispersal processes can account for the year-to-year pattern of persistence and spread of

80 Postelsia on Tatoosh Island, WA. While several researchers have asked whether local

81 observations of dispersal can explain broader patterns of spread (Andow et al 1990,

82 Shigesada and Kawasaki 1997, Clark 1998), none have done so with direct annual

83 observations spanning >20 years and frequent extinction and colonization events. Similarly,

84 models of patch occupancy and recolonization in metapopulations (Hanski 1994) have not

85 been based on detailed dispersal characterizations of individual organisms. Thus this study is

86 unique in building an empirically based microscale (i.e., at the scale of individual dispersal

87 events) model of population turnover and colonization in a highly volatile population

88 experiencing frequent disturbance.

89 Our approach builds upon a long history of theoretical and empirical examinations of

90 the dynamical consequences of space and movement in population theory, tracing back to

91 Haldane, Fisher and Wright, and later Skellam (1951). Levins (1969) coined the term

92 “metapopulation”Author Manuscript to describe populations hierarchically structured in space, and the

93 discipline of spatial ecology has developed rapidly since (Levin 1976, Tilman and Kareiva

94 1997, Turchin 1998, Hanski and Gaggiotti 2004). The theme uniting these approaches is that

96 even though local populations are ephemeral.

97 A number of modeling frameworks have been used to relate the dispersal kernel (i.e.,

98 the distribution of individual dispersal endpoints about a source) to various aspects of spatial

99 population dynamics. Classical reaction-diffusion (RD) models (Skellam 1951, Okubo and

100 Levin 2001) implicitly assume that dispersal distances are normally distributed and show

101 population spread converging to a constant-speed traveling wave, provided Allee effects are

102 absent. Integrodifference equations (IDEs; Kot et al. 1996) are the discrete-time analog of

103 RD models, but more importantly they allow the shape of the dispersal kernel to be specified

104 explicitly. For exponentially bounded kernels such as the normal distribution, IDEs also

105 predict asymptotically constant spread rates; however, leptokurtic or “fat-tailed” kernels

106 (which contain a greater frequency of short distances near the source and long distances in

107 the tails compared to a normal distribution) can produce accelerating, asymptotically

108 unbounded population spread (Kot et al. 1996, 2004). IDEs have seen increasing use among

109 ecologists in response to growing recognition that many organisms have leptokurtic

110 distributions of dispersal distance (Clark et al. 1999) and that rare long-distance dispersal

111 events, often mediated by different mechanisms than local dispersal, can drive patterns of

112 population spread (Kot et al. 1996, Clark 1998, Nathan and Muller-Landau 2000). However,

113 by their very nature such events are difficult to observe and quantify, and thus pose a

114 challenge for parameterizing fat-tailed kernels empirically (Nathan and Muller-Landau

115 2000). Moving from the spread of a single continuous population to the dynamics of patchy

116 population aggregatesAuthor Manuscript can involve stochastic patch occupancy models (SPOMs; Ovaskainen

117 and Hanski 2004), which describe metapopulation dynamics driven by local extinction and

118 migration between patches in a network. Both the colonization and extinction rates in these

120 directly related to the dispersal kernel.

121 Here we describe a series of field experiments used to parameterize empirical models

122 spanning three scales in a hierarchy of spatial population dynamics. In the first section

123 (Quantifying local dispersal at the individual scale), we examine local, short distance

124 dispersal over a single generation in which the distribution of offspring can be attributed to

125 known source populations within a few meters. These data allow us to parameterize simple

126 models which are then employed in the second section (Predicting multi-generation spread

127 at the patch scale) to model the growth of discrete patches over multiple generations, and to

128 compare these predictions against independent observations. In the third section (Dispersal

129 limitation and patch turnover dynamics), we use long-term (> 27 years) monitoring of the

130 reinvasion and extinction probabilities of sites from which Postelsia had been eliminated to

131 ask whether we can successfully scale up from centimeters to tens of meters, and from

132 individual dispersal to metapopulation dynamics. Thus we meld field measures with an

133 integrative mathematical treatment of the dispersal process. We conclude by discussing the

134 implications of our results for Postelsia’s seemingly paradoxical strategy of persistence as a

135 gap-dependent fugitive with limited long-range dispersal ability.

136 Author Manuscript

138

139 All experiments and observations occurred on or in the immediate vicinity of Tatoosh

140 Island, Washington State (48° 23’ N, 124° 43W). Details of this wave-swept cluster of islets,

141 including maps and descriptions of the sites referenced below, are given in Paine and Levin

142 (1981), Leigh et al. (1987) and Paine et al. (2010).

143 P. palmaeformis occupies a broad geographic range, 34° to 52° N. latitude, along the

144 Pacific shores of North America (Abbott and Hollenberg 1976). It is a biological indicator of

145 wave-swept rocky shores and inhabits gaps or patches in intertidal beds of the mussel Mytilus

146 californianus (Dayton 1973, Paine 1979, Blanchette 1996). Postelsia is an annual whose life

147 cycle includes a microscopic and ecologically poorly known gametophyte and an easily

148 identifiable macroscopic sporophyte. Spores are apparently released during low tide and drip

149 onto the adjacent rock surface where they immediately adhere (Dayton 1973, Paine 1979,

150 1988). As in other members of the Laminariales, the female gametophyte is sessile (Luning

151 and Muller 1978, Van Der Meer and Todd 1980) and thus does not represent a secondary

152 dispersal phase. However, sporophytes may disperse by rafting (Paine 1988) as seen in

153 subtidal kelps (Reed et al. 2004); rafting individuals cast on the beach can retain fertile

154 spores, but the contribution of this potentially long-distance dispersal process to colonization

155 and establishment has not been quantified.

156 Postelsia is a dominant competitor at small spatial scales (~100 cm2, which

157 corresponds to the area of an uncrowded mature holdfast), but loses space during the

158 gametophyte or earlyAuthor Manuscript sporophyte stage to encroachment by mussels. At the latitude of

159 Tatoosh, the species requires wave disturbance to remove mussels and create open space

160 (Paine 1979, 1988). Because disturbances to mussel beds vary intra- and interannually in

162 between local extinction and dispersal to new patches. Postelsia is thus “doomed locally, but

163 survives globally by a balance between dispersal and competitive escape ability” (Levin

164 1976: 294).

165 Fig. 1 captures these dynamics, showing spatially and temporally varying patterns,

166 huge swings in local abundance suggesting a high potential rate of increase, and an inability

167 to colonize or persist on mussel shells.

168

169 QUANTIFYING LOCAL DISPERSAL AT THE INDIVIDUAL SCALE

170

171 Field methods

172

173 Postelsia transplants (Paine 1979, 1988, Barner et al. 2011) were initiated in 1975,

174 maintained consistently until 1984, and continued intermittently until 2001. Transplants were

175 made to sites isolated from any resident Postelsia by at least 10 m and usually >100 m,

176 effectively eliminating propagule sources other than the transplant. To establish a transplant,

177 the distal portions of sporogenic plants were cut off, enclosed in a mesh basket and anchored

178 to the rock surface in a 30-40 cm square clearing bounded by mussels. Released spores

179 dribbled onto the surface and developed into gametophytes that reproduced to form the next

180 generation (F1). In most cases, natural dispersal was measured from the F1 to F2 generation

181 by recording the locations of all F2 progeny (immature sporophytes in the spring) in suitable

182 habitat (gaps in theAuthor Manuscript mussel bed) adjacent to the transplant site. Altogether, 16 transplant

183 datasets were collected. In some cases offspring x- and y-coordinates were recorded to the

184 nearest cm. Sometimes groups of grid cells were binned together, for example when high

186 locations. In a few cases only the maximum distance of offspring from the center of the

187 transplant region was recorded, providing a low-resolution “map” comprising two annuli

188 defined by the maximum offspring distance and the maximum radius of suitable habitat.

189 These data are summarized in Table 1, and Appendix S1 gives detailed descriptions of each

190 site and the experimental protocols.

191

192 Fitting dispersal kernels

193

194 Each of the 16 datasets provides a spatially explicit sample of offspring produced by a

195 known number of parent sporophytes located within a bounded region of suitable habitat.

196 These maps contain information on the rate of increase and spread of Postelsia, and we draw

197 on developments in the terrestrial plant dispersal literature to model these processes (Ribbens

198 et al. 1994, Clark et al. 1998, Nathan and Muller-Landau 2000, Canham and Uriarte 2006).

199 This approach is based on the dispersal kernel, a two-dimensional PDF f(x, y) which gives

200 the probability density of offspring dispersing to a point (x, y) from a parent located at (xp ,

-2 201 yp). The expected density of offspring n(x, y), which has units of m , is the product of the

202 dispersal kernel and the number of offspring produced per parent, λ. If there are multiple

203 parents, the expected total offspring density is the sum of their overlapping contributions:

204

m 205 nxy,, f x x y y . (1) () =λ∑ ( −−pp) p=1 Author Manuscript 206

207 The number of offspring NR in a grid cell R= [x1, x2] × [y1, y2] is the integral of the total

209

yx22 210 N n(, x y ) dxdy . (2) R = ∫∫ yx11

211

212 Each Postelsia transplant dataset consists of the number of offspring counted in

213 discrete grid cells. Assuming a Poisson distribution, the likelihood for the observed count

obs 214 NR is

215

216 NNobs ~ Pois . (3) R ()R

217

218 The total likelihood for the entire vector of observations Nobs is the product of Eq. 3 over all

219 grid cells at all sites. In practice, the size of the area constituting a single observation differed

220 among and sometimes within sites (Table 1).

221 We considered four candidate functional forms for the dispersal kernel: Gaussian,

222 exponential, exponential square-root, and 2Dt (Table 2). The first three are one-parameter

223 special cases of the more general power-exponential PDF; however, like other investigators

224 (Ribbens et al., Clark et al. 1999), we found that simultaneously fitting the shape and scale

225 parameters led to unstable estimates. Instead we estimated the scale for three fixed values of

226 the shape, two of which (Gaussian and exponential) produce thin (i.e., exponentially

227 bounded) tails whileAuthor Manuscript the third (exponential square-root) produces a fat tail. The 2Dt kernel is

228 a fat-tailed distribution derived by considering the scale parameter of the Gaussian kernel as

229 an inverse-χ2 random variable, and has been shown to fit leptokurtic dispersal data better than

231 we fixed the 2Dt shape parameter at 1 and estimated the scale. Thus each candidate model

232 has two free parameters, namely λ and a parameter a that scales the width of the kernel. We

233 assumed directional isotropy in all cases. Finally, we considered a “global” model of

234 recruitment as a baseline against which to compare the dispersal kernel models. In this model

235 offspring density is constant across space and independent of local parent sporophyte

236 abundance, as would be expected if propagules rained down from a well-mixed bath.

237 We used a hierarchical Bayesian approach (Clark 2005, Cressie et al. 2009) to fit each

238 candidate model to the data. To accommodate spatial variation in the population dynamics,

239 we modeled the site-specific per capita growth rate λ j as a random effect drawn from a

240 lognormal hyperdistribution with hyper-mean μλ and standard deviation σλ. Similarly, the

241 constant spore density u in the global model was given a lognormal hyperdistribution.

242 Although in principle both a and λ might vary among sites, the data were not informative

243 enough to estimate random effects on both parameters, as evidenced by failure of the MCMC

244 sampler to converge. Preliminary analyses indicated that variation in λ contributed more to

245 the overall fit, so we made a identical at all sites (i.e., ai = a).

246 An additional source of uncertainty is the fact that parent sporophytes were not

247 mapped explicitly; only the number of parents and the boundaries of the parent stand are

248 known. We therefore treat the parent coordinates as latent states to be estimated. For site j,

249 the prior probability of the parent locations is

250 Author Manuscript mmjj 11   251 p xyjj, =   , (4) () xx  yy  21jj−−  21jj 

253 where xj and yj are the vectors of x- and y-coordinates for the mj parents within a rectangular

254 region bounded by [x1j, x2j] and [y1j, y2j].

255 We used noninformative uniform priors for μλ, σλ, and a. The posterior is

256 proportional to the product of the hyper-distribution of the random effects λ (Eq. 4), the prior

257 probability of the parent coordinates (x, y) for all sites (Eq. 5), and the Poisson likelihood for

258 the counts of offspring (Eq. 3):

259

260 P(µσλλ, ,,,,λ a xy | N)()()()∝ h λ |µσλλ , pL xy , N |,,, λ a xy . (5)

261

262 We used Markov chain Monte Carlo (MCMC) simulation with alternating Gibbs and

263 Metropolis updates to draw a sample of 2000 parameter vectors from the posterior

264 distribution for each candidate model and assessed convergence by monitoring

265 autocorrelation, Geweke statistics, and the potential scale reduction factor (Gelman et al.

266 2013). The MCMC sampler was implemented in Matlab R2008b (The MathWorks Inc.); see

267 Data S1 for code and data files.

268 Model selection for hierarchical Bayesian models is an active area of research; one

269 widely used statistic is the deviance information criterion (DIC; Spiegelhalter et al. 2002),

270 which penalizes model fit by a measure of complexity. Calculating DIC, however, requires

271 the posterior mean, which does not make sense for the estimated parent coordinates x and y.

272 Instead, since our candidate dispersal kernels all have the same number of fixed parameters, Author Manuscript 273 we simply compared the posterior mean of the deviance D(θ), defined as -2 times the log of

274 the likelihood (Eq. 3). The model with the lowest posterior mean deviance, D()θ , represents

276

277 Results

278

279 Progeny produced naturally by transplanted Postelsia were always tightly clustered in

280 the neighborhood where parent sporophytes had stood the previous year. Offspring density

281 declined monotonically with distance from the parental transplant area (Fig. 2). Across the

282 five datasets where locations were recorded with sufficient resolution to calculate empirical

283 moments and quantiles, offspring were found at an average distance of 0.16-0.50 m from the

284 center of the transplant region, and 95% fell within 0.38-1.32 m from the center (Table 1).

285 This distribution of offspring is consistent with our assumption that the transplant sites were

286 effectively isolated from outside sources of Postelsia spores.

287 The data strongly favored an exponential square-root dispersal kernel over the three

288 other local dispersal models (Table 2). All four kernel models were unequivocally favored

289 over the global dispersal model (Table 2), providing further evidence that dispersal is

290 distance-limited and that transplant sites were effectively closed populations. The

291 exponential square-root kernel provided a good fit to the observed offspring densities,

292 capturing the leptokurtic pattern of a sharp peak at the source and a relatively long tail (Fig.

293 2). Posterior distributions of λ j showed wide variation in local population growth rates, with

294 up to 30-fold annual increase at some sites and 80-fold decline at others (Fig. 3). This

295 variation in population growth rates is consistent with the heterogeneity observed in earlier

296 studies of PostelsiaAuthor Manuscript population persistence (Paine 1988). The hyper-mean of log(λ) indicates

297 rapid population increase on average (μλ = 0.50, 95% credible interval [0.26, 2.19]; Table 2).

299 PREDICTING MULTI-GENERATION SPREAD AT THE PATCH SCALE

300

301 Field methods

302

303 The Postelsia transplants described above often persisted beyond the F2 generation,

304 providing information on spatial spread of discrete, isolated patches. These longer-term

305 observations generally could not be used for dispersal kernel fitting because the number of

306 individuals and their exact locations were not recorded. Likewise, in some cases the number

307 of F1 sporophytes was unknown and offspring were never mapped explicitly. Instead, these

308 time series consist of patch extent, i.e., the maximum distance of any sporophyte from the

309 center of the original transplant region in each generation. As a patch spreads, it becomes less

310 dense and the probability of local extinction increases (e.g. Table 5 in Paine 1988). For these

311 reasons measures of spread were discontinued after the F5 generation. Because these

312 observations are independent of the single-generation data used to fit the dispersal kernels,

313 they can serve as a test of our ability to scale up from dispersal process at the individual level

314 to dynamics at the patch scale.

315

316 Models of patch spread

317

318 The repeated measurements of patch size allowed us to ask whether the individual-

319 level processes thatAuthor Manuscript govern local dispersal (summarized by the fitted dispersal kernel) can

320 predict spread rates at larger spatiotemporal scales. We used integrodifference equations

321 (IDEs; Kot et al. 1996) as a theoretical framework to bridge individual- and patch-scale

323 spatial domain, so they are well suited to an annual species such as Postelsia. Each year

324 individuals produce offspring according to a production function g(·) which are then

325 redistributed via the kernel f(x,y). The density in the next year at a point (x,y) is found by

326 integrating over all possible source locations:

327

∞∞ 328 n(, xy ) gn (, xy ) f ( x xy  ,ydxdy  ) . (6) tt+1 =−−∫∫ [] −∞ −∞

329

330 We used a hockey-stick production function to represent density-dependence due to space

331 limitation,

332

λntt(, xy ), n (,) xy≤ K 333 gn[]t (, xy )=  , (7) λK, nt ( xy , ) > K

334

335 where K is the maximum local density of adult sporophytes. For the dispersal kernel, we

336 compared predictions based on the exponential and exponential square-root kernels, the latter

337 of which had the strongest support from the dispersal data (Table 2).

338 Spatially restricted Postelsia patches are ephemeral and a patch may not approach the

339 asymptotic spread rate during its lifetime, so rather than relying on asymptotic theory we

340 generated predictions of spread by stochastic simulation. We iterated the IDE on a two-

341 dimensional grid (30×30Author Manuscript m, cell size 0.06×0.06 m). At each annual time step we drew the

342 number of individuals in each cell from a Poisson distribution given the expected density

343 (Eq. 7) multiplied by cell size. In addition to the process error inherent in this stochastic

345 each parameter vector (μλ, σλ, a) in the posterior sample obtained by MCMC (see Fitting

346 dispersal kernels, above). For a given value of μλ and σλ we drew a value of the population

347 growth rate λ from its lognormal distribution, so that each simulated trajectory represented a

348 possible realization of spatial population dynamics at an arbitrary site. The individual-level

349 dispersal data did not provide enough contrast in density to estimate K, so we set it to 400 m-2

350 based on observed maximum densities and the typical size of mature Postelsia holdfasts in

351 crowded stands (Dayton 1973, Paine 1979). Ignoring uncertainty in K will underestimate the

352 uncertainty in spread predictions, but should not significantly affect the comparison between

353 candidate models. In the multi-generation spread data, the parent sporophytes at t = 0 were

354 located within a 0.2×0.2-m region; the number of parents was known, but their exact

355 positions were not. We therefore began each simulated trajectory by sampling the initial

356 number of individuals at random from all the observed values and assigning them random x-

357 and y-coordinates on the interval [-0.1, 0.1]. Simulated trajectories ran for five years and

358 patch radius in each year was measured as the distance from the origin to the furthest

359 individual, matching the definition used in the field. Simulations were done in Matlab

360 R2008b (The MathWorks 2008); code is provided in Data S2.

361

362 Results

363

364 Isolated patches of Postelsia spread at an observed average rate of 0.47 m/yr (range: -

365 1.62 to 2.67). SpreadAuthor Manuscript rates predicted by the stochastic IDE model with an exponential square-

366 root kernel were consistent with these observations, but the predictions were slightly biased

367 upward (Fig. 4A, B). The predicted mean annual spread rate across all five years (0.77 m/yr)

369 95% of the observed annual spread rates, 74% of the observed rates fell below the median

370 prediction. In contrast, predictions based on the fitted exponential kernel (Fig. 4C, D) had

371 lower bias and greater precision, with a mean spread rate of 0.42 m/yr, narrower 95%

372 posterior predictive intervals that still included 91% of the observed annual spread rates, and

373 49% of the data falling below the predicted median. Neither the empirical spread rates nor

374 the model predictions for either kernel showed a clear tendency to accelerate or decelerate

375 (Fig. 4B, D). The variance in predicted spread rates increased for the first two years of the

376 simulation and then declined slightly, suggesting a weakening influence of initial conditions

377 as the spread began to converge to its asymptotic speed.

378

379 DISPERSAL LIMITATION AND PATCH-TURNOVER DYNAMICS

380

381 Field methods

382

383 We removed Postelsia from physically well-defined and previously occupied sites to

384 quantify reinvasion frequency into suitable habitat, with other sites left as unmanipulated

385 controls. Resident sporophytes (most pre-reproductive) at removal sites were hacked off at

386 the holdfast, counted and discarded into the water. Sites were considered locally extinct only

387 when no Postelsia were observed the following year. After the initial eradication, sites were

388 not perturbed further, but were monitored annually and the number (or in some cases

389 presence/absence)Author Manuscript of Postelsia was recorded. Together with the unmanipulated controls, this

390 provides time series of patch occupancy at a total of 16 sites over a minimum of 27 and a

391 maximum of 30 years (Table 3).

393 density and hence spore production at the source, target site area and habitat suitability. For

394 each removal site we estimated a single, static distance to the nearest source for each removal

395 site. These distances are known with certainty when the removal site was contiguous or

396 immediately adjacent to a consistently occupied control site. For the sites that were separated

397 by some distance from source populations that themselves were variably occupied through

398 time, we employed an average or the most likely distance. The combination of using only

399 nearest neighbor distances (as opposed to all possible sources) as well as average nearest

400 neighbor when the sources were variably occupied attributes all reinvasion to one and only

401 one distance (or source), when in fact the reinvasion could arise from several sources. This

402 simplification can either under- or overestimate dispersal, depending on the size of the

403 neglected source pools and their distances from the target. In the absence of more detailed

404 data we did the best with what was available.

405

406 Models of colonization and extinction probability

407

408 To understand how the dispersal ability of Postelsia contributes to turnover dynamics

409 in a network of local patches, we modeled the probabilities of colonization and extinction as

410 a function of the distance of each patch from its nearest occupied neighbor. This simple

411 approach ignores several factors that likely influence the dynamics, such as habitat suitability

412 of the target site, density and per capita fecundity in the source patch (i.e., source strength),

413 and the possibilityAuthor Manuscript of spores arriving from multiple sources. Nonetheless, we believe this

414 approach can provide an overall estimate of the degree of dispersal limitation at spatial scales

415 up to two orders of magnitude greater than an individual’s mean spore dispersal distance and

417 Given the resolution of the data, we did not attempt to fit process-based models (e.g.,

418 SPOMs; Hanski and Gaggiotti 2004) to the patch turnover time series but rather used

419 descriptive regression models. Here each observation is a single colonization (an occupied

420 patch in year t that was vacant in year t - 1) or extinction (a vacant patch in year t that was

421 occupied in year t - 1) event. Repeated observations from a given site are likely correlated

422 (e.g., due to differences in habitat quality and disturbance regime), as are observations across

423 sites in a given year (e.g., due to fluctuations in climate or wave action that affect all sites;

424 Paine and Levin 1981). To account for spatial and temporal nonindependence, we used

425 generalized linear mixed-effects models (GLMMs; Bolker et al. 2009) that included random

426 effects of site and year on the intercept and a fixed effect of distance from source. For site s

427 in year t, the full model describes the annual probability of either colonization or extinction

428 (pst) as

429

logit ()pst =ββ0 +++bbs t dist ds

430 bNs ~() 0,σ site , (8) bN~ 0, t ()σ year

431

432 where β0 and βdist are the intercept and slope respectively, ds is the distance of site s from its

433 nearest occupied neighbor, and bs and bt are the normally distributed random effects with

434 standard deviations σsite and σyear. We also considered reduced models that excluded either

435 the distance effect,Author Manuscript one of the two random effects, or both. For each dataset (colonization or

436 extinction), models were ranked in order of their support from the data using Akaike’s

437 information criterion (AIC; Burnham and Anderson 2002). We used Monte Carlo simulation

439 involved randomly drawing a new outcome for each observation in the real dataset, using the

440 fitted probabilities under the AIC-selected best model. The same model was then fitted to the

441 pseudo-data, and predicted probabilities were calculated for an arbitrary site and year by

442 combining the point estimates of the fixed parameters with a random draw from the

443 estimated random effects distributions. All GLMM analyses were done in R 3.2.5 (R Core

444 Team 2016); see Data S3 for details.

445

446 Results

447

448 The four most consistently occupied removal sites were vacant in only 30/111 (27%)

449 total site-years despite their relatively small areas. There were 11 invasions of these sites

450 when unoccupied (11/30 site-years), an invasion probability of 0.37 yr-1. Such sites retain

451 Postelsia, albeit in a spatially varying pattern (e.g., Fig. 1). On the other hand, eight removal

452 sites remained unoccupied in 209/226 (92%) of site-years, and were invaded only 10 times in

453 those 209 site-years (0.05 yr-1). Such poor sites attest to the continuing presence of suitable

454 and locally invasible but generally vacant habitat.

455 Binomial GLMMs indicated that probabilities of both colonization and extinction

456 were related to the minimum distance from a source of Postelsia spores, but the strength of

457 evidence for the two relationships differed. For colonization probability, the estimated best

458 model included a negative effect of distance along with random effects of both site and year,

459 but this model’s AICAuthor Manuscript score was only 0.51 units lower than that of a model without the

460 distance effect (Table 4). In contrast, the top three models for extinction probability all

461 included positive effects of distance and received much stronger support from the data than

463 showed strong evidence for both spatial and temporal variation, the data could not distinguish

464 between random effects of site and year in the extinction models, and inclusion of both did

465 not improve explanatory power (Table 4).

466 On average, the annual probability of colonization for a vacant site declined and the

467 probability of extinction of an occupied site increased with distance from the nearest

468 potential spore source (Fig. 5). Considerable uncertainty surrounded these average

469 relationships, however, particularly for colonization. The greatest uncertainty in both fitted

470 and observed colonization rates occurred near the origin, that is, for sites with nearby

471 neighboring Postelsia. Invasion rates of some of these sites were fairly high while others

472 were near zero despite presumably high propagule pressure (Fig. 5A). Empirically, habitable

473 sites >10 m from the nearest source population were colonized in only 4% of site-years

474 (5/136), while the average colonization rate for more proximal sites was 14% (22/155), with

475 some sites colonized nearly three times more frequently than this (Fig. 5). Likewise, sites >

476 10 m from a source went extinct 57% of the time when occupied (5/7 site-years), while sites

477 with closer neighbors had an extinction rate of only 13% (20/149 site-years).

478

479 DISCUSSION

480

481 Our three independent data sets document spatially restricted spread from a fixed

482 propagule source. The estimated dispersal kernels indicate that the next generation of

483 Postelsia individualsAuthor Manuscript establish themselves within a very limited radius of the parental

484 generation, generally consistent with the earlier qualitative estimates of 3.0 m (Dayton 1973)

485 and 1.5 m (Paine 1979). Modeling this multi-generation patch spread using IDEs, we

487 model employed, in very close agreement with the empirical estimates (Fig. 4).

488 Our final technique to quantify the ability of Postelsia to invade previously occupied

489 sites involved eliminating the species from physically well-defined sites of known area and

490 proximity to a source population (Table 3). Long-term observations revealed that some sites

491 were consistently “poor” whereas others were consistently “good”, and despite their small

492 areas hosted persistent populations that were rapidly recolonized whenever local extinction

493 occurred. These persistent spatial differences (i.e., site random effects) may reflect variation

494 in community structure and disturbance regime that affect habitat suitability for Postelsia

495 (Paine 1979, 1988) as well as the effects of site area, local abundance, and source strength on

496 extinction risk (Paine 1988). Nevertheless, the overall average relationships for colonization

497 and extinction probabilities (Table 4, Fig. 5) reflect well-established biogeographic

498 phenomena: extinction risk increased and colonization frequency declined with isolation

499 from potential sources. Reinvasion rarely occurred at sites >10 m from sources, even three

500 decades after removal, again emphasizing the importance of dispersal limitation to the

501 dynamics of Postelsia.

502 Our study system permitted us to evaluate the dispersal of an annual kelp at scales up

503 to tens of meters. However, local populations may be separated by inhospitable gaps of many

504 kilometers, which raises the issue of how fat the tails of the dispersal kernel might be. Theory

505 based on IDEs has highlighted the sensitivity of spread rates to the tail of the dispersal

506 distribution in continuous habitats (Kot et al. 1996, Clark 1998), although the persistence of a

507 metapopulation networkAuthor Manuscript may be relatively insensitive to the precise kernel shape (Lockwood

508 et al. 2002). The best-supported models for the Postelsia dispersal data are fat-tailed

509 (exponential square-root and 2Dt), suggesting these data might be consistent with rare long-

511 fixed source. For example, waves or currents might carry spores far from the parent (Gaylord

512 et al. 2006). However, the likelihood of successful reproduction at such long distances is

513 diminished because male and female gametophytes must land in close proximity to one

514 another (Dayton 1973). Moreover, SDD alone is sufficient to predict local patch dynamics

515 over multiple generations even without invoking a fat-tailed kernel; such patches grew by no

516 more than 3 m/yr.

517 These considerations suggest that a different mechanism is needed to account for the

518 long-distance (~20 m) colonization events that we observed, albeit rarely. Rafting of

519 sporogenic plants is the most likely explanation (Dayton 1973, Paine 1988). Such events

520 clearly do occur, although their rarity makes them inherently difficult to quantify and predict

521 (Nathan and Muller-Landau 2000). Clark (1998) shows that rare LDD events that are

522 effectively undetectable in data, and thus cannot be captured by a fitted kernel, may

523 nevertheless be frequent enough to account for the rapid post-glacial migration of temperate

524 trees. We hypothesize that LDD by rafting likewise maintains Postelsia’s coastwide

525 distribution over geological and evolutionary time.

526 However, the contribution of LDD to metapopulation dynamics over shorter

527 ecological time scales appears to be minimal. We observed only five colonization events >10

528 m from a source in 136 site-years, and fitted distance-dependent models suggest that such

529 events are vanishingly rare beyond 20 m. Thus at the scale of Tatoosh, where Postelsia

530 patches occur over kilometers of coastline and are often isolated from one another by

531 hundreds of metersAuthor Manuscript (Paine 1988), we conclude that most dynamics are local. At this spatial

532 scale and over ecological time (e.g., tens to hundreds of generations), Postelsia’s persistence

533 in the face of high local extinction risk requires a disturbance regime that opens habitat at a

535 found that the species occurs only at sites with high disturbance rates (average proportion of

536 mussel bed cleared annually), but even among such sites, it occurs more frequently where

537 disturbance is regular and moderately intense than where it is occasional and severe. The

538 picture that emerges is one of local patchy populations that persist in a highly dynamic

539 spatial mosaic with mussels, largely decoupled from other such local patches.

540 It is perhaps surprising that we were able to detect an isolation effect for colonization

541 and extinction in spite of the vagaries of dispersal and a highly heterogeneous environment.

542 Even more surprising is the finding that although at least 95% of the propagules fell within

543 130 cm (and most within 20 cm) of their parent plant, this short-range dispersal was adequate

544 to maintain the metapopulation. This result could only be uncovered by combining explicit

545 spatial models with long-term field data on spread and persistence. The broader lesson for

546 metapopulations is that dispersal kernels can reflect extremely local dispersal processes and

547 still be adequate for persistence in highly capricious environments. Remembering that

548 Postelsia lacks a seed bank or overlapping generations, short range dispersal would likely be

549 an even more viable strategy when combined with the buffering mechanism of a temporal

550 rescue effect.

551 Although Postelsia is in many ways unique in its natural history, its commitment to

552 short-distance dispersal is common in many marine species that have sessile adults and a

553 minimally dispersing, crawl-away larval stage. Examples include some crustose (red)

554 coralline algae, anemones, solitary corals, starfish, numerous gastropods and tunicates. For

555 such species, extenAuthor Manuscript sive geographic distribution must be explained by effective rafting (e.g.,

556 Highsmith 1985), or simply gradual diffusion over millennia. While long-distance dispersal

557 is likely crucial for range expansion and broad geographic patterns, local dispersal may be

559 environments.

560

561 ACKNOWLEDGMENTS

562 All work-intensive studies of long temporal duration require adequate financial support and

563 assistance in the field. RTP acknowledges decades of support from The National Science

564 Foundation, and in the later years, by the Andrew W. Mellon and Betty and Gordon Moore

565 Foundations. J. HilleRisLambers and M. Kot have contributed their expertise to our

566 considerations of dispersal kernels. We express deep gratitude to the Makah Tribal Council

567 for permission to study on tribal lands. The manuscript was improved by the comments of

568 two anonymous reviewers.

569

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670

671

672 Author Manuscript

673 Table 1. Summary of Postelsia transplant data used to fit dispersal kernels. Transplants

674 established an F1 sporophyte generation, and natural dispersal was measured by recording

676 year in which offspring locations were observed. Datasets vary in the resolution of the parent

677 and offspring location maps, resulting in different numbers and configurations of grid cells. Author Manuscript

Transplant Sampling Area area Grid cell No. No. No. distance

Site date date (m2) (m2) size (m2) cells parents offspring (m) †

S-78/79-7 1978 1980 4.00 0.09 1.72-2.28 2 1 21 0.74

S-78/79-8 1978 1980 4.00 0.09 0.84-3.16 2 1 1 1.00

S-78/79-9 1978 1980 1.67 0.04 0.01-0.13 23 6 52 0.16

(0.44)

S-78/79-14 1978 1980 4.00 0.09 0.84-3.16 2 15 2 1.00

S-80-8 1980 1982 4.00 0.09 0.01 400 1 0 —

S-80-9 1980 1982 4.00 0.09 0.01 400 1 0 —

S-80-12 1980 1982 1.83 0.09 0.01-0.16 183 27 337 0.30

(1.32)

S-80-13 1980 1982 4.00 0.09 0.04-0.52 14 1 1 1.00

S-80-16 1980 1982 4.00 0.09 1.54-2.56 2 2 11 0.70

S-83-2 1983 1985 4.00 0.09 0.5-3.5 2 7 77 0.40

S-83-3 1983 1985 4.00 0.09 0.84-3.16 2 7 10 1.00

S-83-4 1983 1985 4.00 0.09 0.84-3.16 2 1 5 1.00

G-83-4 1983 1991 5.40 0.31 0.09-1.26 14 20 145 0.50

(1.21)

G-99-2 1999 2001 4.00 0.09 0.04-0.52 14 1 1 0.23

G-99-3 1999 2001 0.89 0.06 0.0001 8856 5 106 0.18 Author Manuscript (0.38)

G-99-4 1999 2001 1.17 0.02 0.0001 11700 2 55 0.22

678 †Where a single value is given, it is the maximum observed distance of an offspring

679 sporophyte from the center of the region that contained the parents. In these datasets, either a

680 single offspring was present or offspring positions were not recorded individually.

681 Otherwise, the values are the mean and 95th percentile (in parentheses) of dispersal distance.

682 Table 2. Dispersal kernel models fitted to data from 16 Postelsia transplants using

683 hierarchical Bayesian methods. Equations give the probability density of offspring at radial

22 684 distance r xx yy from a parent sporophyte. Posterior means and 95% =()() −pp +−

685 credible intervals are shown for the deviance D(θ), scaled so the best model has a mean

686 deviance of zero, the hyper-mean μλ and hyper-variance σλ of the log population growth rate

687 λ, and the dispersal scale a. Author Manuscript

Model Density D()θ μλ σλ a

† Global nr() = u 2494.59 — — — (2485.92,

2507.38)

Gaussian 2 766.06 1.32 1.91 44.87 λ r nr=exp − () a2 a π  (742.69, (0.25, (1.27, (42.74,

791.50) 2.29) 2.91) 47.08)

λ r Exponential nr exp 254.44 1.30 1.79 15.71 () =2  − 2πaa (235.90, (0.26, (1.18, (14.62,

275.30) 2.23) 2.75) 16.81)

Exponential 12 0.00 0.50 0.28 27.20 λ r nr=− exp () 4 (4)a2 a square-root πΓ  (-18.30, (0.17, (0.11, (24.2, 30.6)

21.50) 0.84) 0.61)

2Dt λ 163.09 5.97 0.44 26.09 nr= () 2 2 r (143.90, (5.42, (0.17, (23.02, πa 1+ a 185.10) 6.52) 1.01) 29.22)

688 †In the global model, offspring density is independent of location and of the number of

689 parents.Table 3. Postelsia removal manipulations and associated unmanipulated control sites.

690 Research began in 1978 on sites 1-8 and 1981 for 9-14 and the two controls, and all time

691 series were collected through 2008. Author Manuscript

removed (m2) to source observed invasions extinctions occupied

(m)

Removal 1 6 1.5 9.6 28 6 5 39

Removal 2 7 3.0 4.0 29 4 3 66

Removal 3 4 0.8 16.8 29 1 1 3

Removal 4 574 1.2 17.4 30 3 2 17

Removal 5 72 1.0 20.6 30 1 1 3

Removal 6 185 2.4 2.3 28 4 4 61

Removal 7 489 4.5 7.0 29 4 4 31

Removal 8 4000 40.0 39.4 27 0 0 0

Removal 9 8 1.0 10.5 27 0 0 0

Removal 10 34 2.5 1.0 27 2 1 78

Removal 11 8 1.0 1.0 27 1 1 4

Removal 12 128 2.0 2.0 27 0 0 0

Removal 13 13 1.5 3.5 27 0 0 0

Removal 14 86 4.0 0.0 27 1 1 85

Control 2c 0 1.0 1.0 28 0 1 75

Control 10c 0 3.0 1.0 28 0 0 100

692

693 Author Manuscript 694 Table 4. Logistic GLMMs fitted to data on patch colonization (n = 279) and extinction (n =

695 153), where each observation is an individual colonization or extinction event. Probabilities

696 were modeled as a function of the distance of each site from its nearest occupied neighbor.

698 for parameter definitions.

699

700

Fixed Random

† ‡ § ¶ effects effects k β0 βdist σsite σyear Deviance ∆AIC wAIC

Probability of colonization

distance site, year 4 -2.79 -0.11 1.83 1.93 142.71 0.00 0.562

— site, year 3 -3.82 — 1.83 2.18 145.23 0.51 0.435

distance site 3 -1.82 -0.08 — 1.33 156.52 11.81 0.002

— site 2 -2.59 — — 1.53 159.02 12.31 0.001

distance year 3 -2.23 -0.06 1.14 — 164.11 19.39 0.000

— year 2 -2.76 — 1.12 — 169.70 22.98 0.000

Probability of extinction

distance site 3 -2.66 0.25 — 0.13 107.88 0.00 0.422

distance year 3 -2.66 0.25 0.00 — 107.88 0.00 0.421

distance site, year 4 -2.66 0.25 0.00 0.13 107.88 2.00 0.155

— site 2 -1.38 — — 1.46 120.96 11.07 0.002

— site, year 3 -1.38 — 0.00 1.46 120.96 13.07 0.001

— year 2 -1.68 — 0.00 — 132.94 23.05 0.000

701 †All models include a fixed intercept and may or may not include distance from nearest Author Manuscript 702 neighbor (dist).

703 ‡Models may include random effects of site and year on the intercept.

704 §Number of estimated parameters.

706

707 Figure 1. Photographs of the eastward face of the 49m2 site 20 (Paine and Levin 1981) on the

708 extreme northwest side of Tatoosh. The spatial patterning of Postelsia varies greatly through

709 time (1975 vs. 1981 vs. 1989 vs. 2006) indicative of a high λ. Note that shells of Mytilus

710 californianus tend not to be colonized successfully, at least to the point of harboring adult

711 Postelsia sporophytes (1975, 1981, 1989, 1991, 2004, 2008). The entire face of site 20

712 appears to provide suitable habitat for sea palms; note the essentially reciprocal occupancy

713 patterns, 2004 vs. 2008. Photographs of many intervening years have been omitted because

714 of poor quality. Some indication of scale is given by noting that Postelsia typically grows to

715 60-75 cm in height (http://eol.org/pages/902813/details). Author Manuscript

717 Figure 2. Dispersal kernel of Postelsia. In (A) - (E), points show mean (±1 SE) density of

718 offspring (immature sporophytes) in radial bands at the five plots where the highest

719 resolution data were available. Distances are measured from the center of the region where

720 the transplanted parent sporophytes were located. The smooth curves show predicted values

721 for the exponential square-root kernel fitted to the dataset in a hierarchical Bayesian

722 framework. (F) Cumulative distribution of radial displacement, showing empirical

723 cumulative distribution function (CDF) for all 16 sites used in kernel-fitting (thin gray lines)

724 and fitted CDF. In all panels, thick black lines show posterior means and thin black lines

725 show 95% credible intervals.

726

727 Figure 3. Estimates of annual population growth rate (λ; note log scale) for the exponential

728 square-root kernel fitted to Postelsia dispersal data. Boxplots show posterior median,

729 quartiles, and 95% CI. The hierarchical analysis assumes that plot-specific values of λ (open

730 boxes) are drawn from a lognormal hyper-distribution with hyper-mean μλ (shaded box).

731

732 Figure 4. Observed and predicted spread of Postelsia patches. Observations (black points and

733 lines) are from 55 unmanipulated patches. Predictions were generated from a stochastic

734 simulation of an integrodifference equation model that tracks the extreme disperser in each

735 generation. Posterior means (solid gray lines) and 95% credible intervals (dashed gray lines)

736 of predicted patch radius (A, C) and annual spread rate (B, D) are shown, based on the

737 exponential squareAuthor Manuscript -root or exponential kernel fitted to individual dispersal data.

738

740 patches as a function of distance from the nearest occupied site. Each point represents a site,

741 and exact 95% CIs around the observed proportions are shown. Thick lines show mean fitted

742 probabilities from logistic GLMMs that include random effects of site and year (for

743 colonization) or site (for extinction), and thin lines show 95% CIs around the fitted values.

744 Means and CIs were generated by Monte Carlo simulation from the fitted model.

745 Author Manuscript

750

751

752

753

754

755

756

757

758

759

760

761

762

763

764

765 Figure 2.

766 Author Manuscript

768 S-83-3

769 S-83-2

770 S-78-14

771 S-78-9 S-78-8 772 S-78-7 773 S-80-16 774

Estimate S-80-13 775 S-80-12

776 S-80-9

777 S-80-8

778 G-83-4

779 G-99-4 G-99-3 780 G-99-2 781 -4 -3 -2 -1 0 1 2 3 4 782 ln(λ) 783

784

785

786

787

788 Author Manuscript

789

790

792

793 Figure 3

. 794 .

795 Exponential square root kernel Exponential kernel 12 12 796 A C 10 10 797 8 8 798 6 6 799 4 4

800 (m) radius Patch 2 2 801 0 0 0 1 2 3 4 5 0 1 2 3 4 5 802 4 4 803 B D 3 3 804 2 2 805 1 1 806 0 0 807 Spread rate (m/yr) rate Spread -1 -1

808 -2 -2 1 2 3 4 5 1 2 3 4 5 809 Time (yr) 810

811

812 Author Manuscript

813

814

816

817

818

819 Figure 4.

820

821

822 Author Manuscript

824

825 Figure 5.