Variable processes that determine population growth and an invariant mean-variance relationship of intertidal barnacles KEIICHI FUKAYA,1, ,6 TAKEHIRO OKUDA,2 MASAKAZU HORI,3 TOMOKO YAMAMOTO,4 MASAHIRO NAKAOKA,5
AND
TAKASHI NODA1
1
Faculty of Environmental Science, Hokkaido University, N10W5, Kita-ku, Sapporo, Hokkaido 060-0810 Japan 2 National Research Institute of Far Seas Fisheries, Fisheries Research Agency, 2-12-4, Fukura, Kanazawa-ku, Yokohama 236-8648 Japan 3 National Research Institute of Fisheries and Environment of Inland Sea, Fisheries Research Agency, Maruishi 2-17-5, Hatsukaichi, Hiroshima 739-0452 Japan 4 Faculty of Fisheries, Kagoshima University, Shimoarata 4-50-20, Kagoshima, Kagoshima 890-0056 Japan 5 Akkeshi Marine Station, Field Science Centre for the Northern Biosphere, Hokkaido University, Aikappu, Akkeshi, Hokkaido 088-1113 Japan Citation: Fukaya, K., T. Okuda, M. Hori, T. Yamamoto, M. Nakaoka, and T. Noda. 2013. Variable processes that determine population growth and an invariant mean-variance relationship of intertidal barnacles. Ecosphere 4(4):48. http://dx.doi.org/10.1890/ES12-00272.1
Abstract. Although researchers recognize that population dynamics can vary in space and time as a result of differences in biotic and abiotic conditions, spatial and temporal variability in the patterns and processes of population dynamics have not been well documented on a seasonal time frame. We quantified seasonal changes in the coverage of intertidal barnacles, Chthamalus spp., with data collected for as many as 9 years at 88 plots in five regions located along more than 1800 km of the Pacific coastline of Japan from 318 N to 438 N. To examine how seasonal changes and the spatial heterogeneity of environments can interact to influence patterns and processes of population dynamics, we analyzed the data with two models of population variability: a population dynamics model, which provides knowledge about processes that determine population growth rates; and Taylor’s power law, which summarizes the relationship between the temporal mean and variance of the size of a population (temporal mean-variance relationship). We found that seasonal differences were prevalent in population growth rates, as well as in the strength and spatial scales of processes that determine population growth rates. In addition, the seasonality of these rates and processes varied between habitats at different spatial scales ranging from the scale of amongrocks within a shore to that of among-regions located in different latitudes, suggesting that the effects of seasonal environmental fluctuations on population growth can depend on the spatial heterogeneity of biotic and abiotic conditions that vary at multiple spatial scales. In contrast, the evidence for spatiotemporal differences in temporal mean-variance relationships was weak. Unlike theoretical expectations, spatiotemporal differences in the variability of population size were best explained by a unique power law, despite remarkable regional and seasonal differences in the processes that determine population growth rates. These results suggest that spatiotemporal environmental variability can affect population dynamics at multiple spatial scales but do not necessarily alter the scaling law of population size variability. Key words: Bayesian inference; Chthamalus; environmental variability; intertidal rocky shore; mean-variance relationship; population dynamics; population models; population regulation; population synchrony; seasonal variability; spatial scale; Taylor’s power law. Received 30 August 2012; revised 20 February 2013; accepted 25 February 2013; final version received 20 March 2013; published 17 April 2013. Corresponding Editor: S. Navarrete. Copyright: Ó 2013 Fukaya et al. This is an open-access article distributed under the terms of the Creative Commons
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FUKAYA ET AL. Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. http://creativecommons.org/licenses/by/3.0/ 6 Present address: The Institute of Statistical Mathematics, 10-3 Midoricho, Tachikawa, Tokyo 190-8562 Japan. E-mail:
[email protected]
INTRODUCTION
and the mean-variance relationship have rarely been examined concurrently in field studies of population dynamics. One effective approach to test this prediction would be to apply two different models of population variability, a population dynamics model and Taylor’s power law, to population time series data obtained at multiple sites characteristic of different environmental regimes. Fitting a population dynamics model to such data enables inferences to be made concerning the processes that drive wildlife population dynamics (Royama 1992, Turchin 2003). One of the most frequently used population models is the logistic growth model, which decomposes the population growth rate into several density-dependent and -independent components such as the intrinsic growth rate, strength of density dependence and process variance (Royama 1992, Turchin 2003, Clark and Bjørnstad 2004). The recent development of relevant statistical techniques (e.g., De Valpine and Hastings 2002, Clark and Bjørnstad 2004) has facilitated the widespread application of this approach in the analysis of ecological time series to elucidate spatial and temporal variations in the strength of density-dependent and density-independent processes on population growth (e.g., Saitoh et al. 2003, Stenseth et al. 2003, Fukaya et al. 2010). In contrast, Taylor’s power law describes a power-law relationship between the mean and variance of the size of a population (Taylor 1961) which has been shown to provide a good characterization of ecological time series (reviewed in Kendal 2004). The temporal meanvariance relationship can be characterized by the coefficients of temporal Taylor’s power law, which can be estimated from the coefficients in the regression of log variance versus log mean population size over time (Taylor 1961). Elucidating variations in coefficients of Taylor’s power law using empirical time series is a fundamental step to understand patterns of population variability in space and time (Taylor and Woiwod 1982, Thomas et al. 1994, Keeling and Grenfell 1999) which can also be useful to predict variability of populations (Mellin et al. 2010).
Density-dependent processes such as competition and density-independent processes such as climatic fluctuations both determine population size variability. For various organisms, researchers have recognized that the effects of these processes can vary in space and time because of differences in biotic and abiotic conditions that cause the patterns of population size and growth rate to vary (e.g., Saitoh et al. 2003, Stenseth et al. 2003, Fukaya et al. 2010). The season of the year is one of the most important factors that can clearly affect population dynamics. Because seasonal changes in biotic and abiotic conditions characterize ecosystems outside of the tropics, it is likely that seasonality in the patterns and processes of population dynamics is common for many organisms that inhabit higher-latitude ecosystems. It is also possible that seasonality in population dynamics varies spatially at different spatial scales because it may depend on both regional climates and/or local environmental factors (Stenseth et al. 2003, Helmuth et al. 2006, Gedan et al. 2011). Unfortunately, however, knowledge of the effects of environmental seasonality on population dynamics is limited to a few taxa, such as voles (Hansen et al. 1999a, b, Saitoh et al. 2003, Stenseth et al. 2003). Therefore fundamental questions remain about how patterns and processes of population dynamics vary seasonally and spatially across scales. Theoretical studies have suggested that differences in environmental conditions can affect processes that drive population dynamics, such as the strength of density dependence and magnitude of stochastic forces, which ultimately change the pattern of population variability known as the temporal mean-variance relationship of population size (Anderson et al. 1982, Hanski and Woiwod 1993, Perry 1994, Keeling 2000, Ballantyne and Kerkhoff 2007). Despite a large number of theoretical investigations, however, this prediction has not been tested empirically, because processes of population dynamics v www.esajournals.org
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The slope of Taylor’s power law, also known as the scaling exponent, can be of particular interest because it has been hypothesized that the temporal mean-variance relationship arises from population processes and a modification of them induced by environmental differences results in intraspecific variations in the slope of Taylor’s power law (demographic hypothesis : Anderson et al. 1982, Hanski and Woiwod 1993, Perry 1994, Keeling 2000, Ballantyne and Kerkhoff 2007). Application of these two models, the population dynamics model and Taylor’s power law, to the analysis of ecological time series will enable us to use comparisons of spatiotemporal patterns in the estimated parameters of population dynamics models and of Taylor’s power law to test the theoretical prediction that differences in environmental conditions can both affect processes that drive population dynamics and temporal meanvariance relationship of population size. Barnacles are sessile animals commonly found in intertidal rocky habitats of which population dynamics are well studied. Their life cycle consists of the pelagic larval stage and sessile adult stage, and therefore changes in coverage of their adult populations are determined by the recruitment of new individuals, somatic growth and mortality. Recruitment plays an important role in determining population abundance and biotic interactions including competition and predation (Gaines and Roughgarden 1985, Menge 2000, Connolly et al. 2001, Navarrete et al. 2005) while post-recruitment growth and mortality may also drive changes in adult population coverage (Bertness 1989, Menge 2000, Jenkins et al. 2001, Fukaya et al. 2010). A great variation may exist in their demographics which can be found at different spatial scales. For example, recruitment intensity may vary at regional (hundreds of kilometers) or mesoscales (tens of kilometers) because of differences in oceanographic conditions and processes (Connolly et al. 2001, Lagos et al. 2005, 2007, Navarrete et al. 2005, 2008, Broitman et al. 2008) while they may also vary at smaller spatial scales due to differences in small-scale hydrographic processes and other local conditions related to their settlement cues (Crisp 1961, Wethey 1984, 1986, Raimondi 1988, 1990, Larsson and Jonsson 2006). On the other hand, somatic growth rate and mortality rate may also vary at v www.esajournals.org
multiple spatial scales depending on regional climates (Bertness et al. 1999, Leonard 2000, Jenkins et al. 2001), food availability (Bertness et al. 1991, Sanford and Menge 2001), local hydrodynamic processes (Sanford et al. 1994, Larsson and Jonsson 2006), local thermal conditions (Lively and Raimondi 1987, Bertness 1989, Gedan et al. 2011), disturbance (Chabot and Bourget 1988), and species interactions (Connell 1961a, b, Bertness et al. 1999, Leonard 2000). These findings suggest significant spatial and temporal variations in long-term dynamics of adult populations that are likely to be found at different spatial scales. Although long-term fluctuations of adult population abundance of intertidal barnacles have been analyzed in previous studies (e.g., Southward 1991, Dye 1998, Burrows et al. 2002), little is known about how they can vary over geographic scales and among different seasons. The intertidal barnacle Chthamalus spp. is a dominant sessile animal in the mid- to highintertidal zone along the shores of the Pacific coast of Japan (Nakaoka et al. 2006 ). We previously reported that the population dynamics of C. challengeri are driven by seasonally variable density-dependent and density-independent processes in the intertidal zone of a region of the North Pacific coast of Japan (Fukaya et al. 2010). In this study, we focused on a larger geographic area compared to that considered by Fukaya et al. (2010) to examine the seasonality and spatial variability of population dynamics across multiple spatial scales. We quantified changes in the coverage of intertidal barnacles, Chthamalus spp., with data collected for as many as 9 years at 88 plots in five regions located along more than 1800 km of the Pacific coastline of Japan from 318 N to 438 N. Our analysis consisted of fitting population dynamics models to the data and examining temporal mean-variance relationships. We found that seasonality was prevalent but spatially variable for population growth rates and their associated regulatory processes, whereas mean-variance relationships were stable despite significant spatiotemporal differences in the processes that determine population dynamics.
MATERIALS
AND
METHODS
Census design Six regions (eastern Hokkaido, Oshima Penin3
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Fig. 1. Study shores established along the Pacific coast of Japan. Filled boxes are shores where data were not analyzed (see Materials and methods: Census design).
sula, Sanriku Coast, Boso Peninsula, Kii Peninsula, and Osumi Peninsula; Fig. 1) were chosen along the Pacific coast of Japan between 318 N and 438 N, with the intervals between neighboring regions ranging from 263 to 513 km (mean 6 SD: 404.9 6 107.3 km). Within each region, we chose five shores at intervals of 2.7 to 17 km (mean 6 SD: 8.2 6 4.3 km) along the coastline. Within each shore, we established five 5000-cm2 census plots on rock walls at semi-exposed locations at intervals of 3.1 to 378 m (mean 6 SD: 37.3 6 48.9 m). Each plot was 50 cm wide by 100 cm high, and the mean tidal level corresponded to the vertical midpoint of the plot. Detailed descriptions of the study sites and biogeographic features of the area can be found in previous works (Okuda et al. 2004, Nakaoka et al. 2006). We defined four spatial scales that reflected hierarchical levels of the census design: an among-regional scale, corresponding to the entire study area; a regional scale, corresponding to a single region; a shore scale, corresponding to a single shoreline site; and a rock scale, corresponding to a single census plot. Two species of Chthamalus were found in our census plots: a northern species, C. dalli Pilsbry, v www.esajournals.org
and a southern species, C. challengeri Hoek. The geographic ranges of these two species are thought to overlap in the region from northern Honshu Island to southern Hokkaido Island, although distinguishing these two species in the field is difficult. However, based on identifications by barnacle taxonomists, we concluded that the Chthamalus population in eastern Hokkaido consisted mainly of C. dalli, whereas C. challengeri dominated in other regions (Y. Hisatsune and H. Yamaguchi, personal communication). Details of our examination of the frequencies of the two species at the study shores are described in Appendix A. We measured the coverage of Chthamalus spp. at each plot twice per year, in April or May (spring) and in October or November (autumn), from autumn 2002 to spring 2011. We estimated the coverage of the entire plot by a point sampling method that involved recording the occurrence of Chthamalus spp. at 200 points on a grid with 5-cm intervals in both the vertical and horizontal directions. All measurements were made at low tide. We defined the periods between the spring and autumn censuses and the autumn and spring censuses as summer and 4
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FUKAYA ET AL. Table 1. Regions and shores where census plots were located. Region
Census period
Shore (Abbr.)
Latitude (N)
Longitude (E)
No. plots
Eastern Hokkaido
Autumn 2002 to spring 2011
Oshima Peninsula
Autumn 2002 to spring 2011
Mochirippu (MC) Mabiro (MB) Aikappu (AP) Monshizu (MZ) Nikomanai (NN) Iwato (IW) Usujiri (US) Shishibana (SS) Esan (ES) Hiura (HR) Myojin (MJ) Aragami (AG) Akahama (AK) Katagishi (KG) Tonda (TD) Hacchoiso (HC) Bansho (BN) Tenjinsaki (TN) Metsuura (MT) Hezuka (HZ) Takenoura (TK) Tajiri (TJ)
43801 0 0200 42859 0 1400 43801 0 0200 43802 0 5900 42856 0 2300 42800 0 1800 41856 0 3100 41852 0 1000 41847 0 2000 41844 0 1200 39828 0 4500 39824 0 3400 39821 0 0200 39820 0 2800 33838 0 0700 33839 0 1700 33841 0 4300 33843 0 3000 33845 0 5000 31805 0 3700 31802 0 4600 31800 0 1200
145801 0 3000 144853 0 2400 144850 0 0200 144846 0 4200 144840 0 2600 140852 0 4300 140856 0 0500 141806 0 5800 141809 0 4300 141803 0 3500 142800 0 1100 141858 0 5100 141856 0 0900 141854 0 2500 135823 0 2300 135821 0 0500 135820 0 0900 135821 0 1000 135817 0 4600 130849 0 5100 130843 0 3200 130840 0 0300
5 5 5 5 4 3 5 5 4 5 5 5 5 5 1 2 2 5 4 2 5 1
Sanriku Coast
Autumn 2002 to autumn 2010à
Kii Peninsula
Autumn 2002 to autumn 2007
Osumi Peninsula
Autumn 2002 to spring 2011
No census was conducted in 2008. No data were available for three plots in SS and two plots in ES after the autumn of 2006 or the spring of 2007 because they were accidentally destroyed. àNo census was conducted in the spring of 2008. No data were available for a plot in AK after the autumn of 2006 because it was accidentally destroyed.
winter, respectively. Seasonal variations are characterized by several factors that may affect the population dynamics of Chthamalus spp. First, recruitment rate varies within a year. Within the entire census region most of the recruitment of C. dalli and C. challengeri occurred in summer; recruitment was rarely observed during the winter (Maruyama 2005, Munroe and Noda 2010, Munroe et al. 2010). Second, heat and desiccation stress, which Chthamalus spp. experience during low tide, may be intense in summer but potentially weak in winter, because along the Pacific coast of Japan spring low tides occur during the day from late March to early September but during the night from late September to early March. Finally, the fact that phytoplankton blooms in the study region occur during the winter and peak between February and April (Furuya et al. 1993) suggests that food availability may be higher in the winter than summer. We excluded from the analysis several plots where Chthamalus spp. rarely occurred and where their average population size was quite small. We omitted all of the plots in one of the study regions (Boso Peninsula, 358 N) where Chthamalus spp. were rare. We also excluded the plots where Chthamalus spp. occurred fewer than v www.esajournals.org
four times during the census period or where their mean coverage was ,0.5%. Having excluded these plots, we then analyzed the population dynamics of Chthamalus spp. at 88 rocks within a total of 22 shores in five regions (eastern Hokkaido, Oshima Peninsula, Sanriku Coast, Kii Peninsula, and Osumi Peninsula; Table 1, Fig. 1). Such an exclusion of data was motivated because of a statistical reason that fitting the statistical model to data including these low abundance plots was difficult. Our study therefore reporting population dynamics that can be found in relatively abundant populations and results might not be generalized to populations where their mean abundances are regularly low. Note that there were some differences in the durations of census periods among regions and even among plots within a region because of census termination or accidental loss of plots (see Table 1 for details).
Data analysis We conducted a two-step analysis: we first used statistical procedures to fit a logistic population model to the data and then examined the temporal mean-variance relationship of population size (Taylor’s power law) by using parameters estimated from the fitted model. 5
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Statistical modeling of population dynamics.—We used a state-space model to analyze data because the model enables the observation error to be taken into account in the parameter inference, leading to a less-biased estimate of density dependence (De Valpine and Hastings 2002, Clark and Bjørnstad 2004, Freckleton et al. 2006). We applied a hierarchical Bayesian modeling framework to estimate model parameters. The observed number of sampling points occupied by Chthamalus spp. on plot i on census date number t, yti, was assumed to follow a binomial distribution (BIN). Given that the total numbers of dates (T ) and plots (I ) correspond to the number of temporal replicates (18) and number of examined rocks (88), respectively, that is:
ð2Þ
ð2CÞ ð3Þ
ð2DÞ ð4Þ
ð2EÞ ð1Þ et1 ,
ð1Þ
2ð1Þ
et1 ; Nð0; rs½t1 Þ for t ¼ 2; . . . ; T
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ð3Þ et1j ,
ð4Þ et1i
2ð1Þ
rs[t1]i þ (1 as[t1]i )ln At1i. The terms rs½t1 , 2ð2Þ
2ð3Þ
2ð4Þ
rs½t1k , rs½t1k 0 ½ j , and rs½t1k½i are the process ð1Þ
ð2Þ
ð3Þ
ð4Þ
variances that govern et1 , et1k , et1j , and et1i , respectively, in season s[t 1] in region k (k 0 [ j], k[i]). We assumed that the intrinsic growth rate and strength of density dependence have a multivariate normal distribution (MN): ! ðrÞ csj½i ð1Þ rsi ; MN ðaÞ ; R ; asi csj½i for s ¼ 1; 2; i ¼ 1; . . . ; I; ðrÞ csj
ð3Þ
ðaÞ csj
and denote the averages of rsi and where asi, respectively, in season s on shore j, and R(1) denotes the covariance matrix for rsi and asi. We assumed shore averages of intrinsic growth rate ðrÞ ðaÞ csj and strength of density dependence csj to have multivariate normal distributions: ! ! ðrÞ ðrÞ lsk 0 ½ j ð2Þ csj ; MN ðaÞ ðaÞ ; R csj lsk 0 ½ j for s ¼ 1; 2; j ¼ 1; . . . ; J ðrÞ lsk
ð4Þ
ðaÞ lsk
where and denote the regional average of intrinsic growth rate and strength of density dependence, respectively, in season s in region k, ðrÞ and R(2) denotes the covariance matrix of csj and ðaÞ csj . Seasonal differences in regional averages of intrinsic growth rate and strength of density
ð1Þ
ð4Þ
ð2Þ et1k ,
represent normally where distributed (N), stochastic fluctuations of growth rate between time t 1 and t at the amongregional, regional, shore, and rock scale, respectively, centered around the expectation value,
ln Ati ¼ rs½t1i þ ð1 as½t1i Þln At1i þ et1 ð3Þ
2ð4Þ
et1i ; Nð0; rs½t1k½i Þ for t ¼ 2; . . . ; T; i ¼ 1; . . . ; I
where pti is the fraction of the area of each plot covered by Chthamalus spp. In our census design pti equaled the area of each plot covered by Chthamalus spp. (Ati cm2) divided by the plot area (5000 cm2). The number of trials in the binomial distribution corresponds to the number of census points in a plot (200). We let s[t], j[i], k[i], and k 0 [j] be index variables such that s[t] is represented by a number s (s ¼ 1 and 2 indicate summer and winter, respectively) that corresponds to the period between sample date numbers t and t þ 1; j[i] is represented by a number of shore j where plot i is located; k[i] is represented by a number of region k where plot i is located; and k 0 [j] is represented by a number of region k where shore j is located. The numbers of examined shores (22) and regions (5) equal J and K, respectively. We then described changes in coverage of the Chthamalus population Ati by using the Gompertz population model, a loglinear form of the discrete logistic equation (Royama 1992, Turchin 2003), with seasonally varying intrinsic growth rate rsi and strength of density dependence asi:
ð2Þ
2ð3Þ
et1j ; Nð0; rs½t1k 0 ½ j Þ for t ¼ 2; . . . ; T; j ¼ 1; . . . ; J
yti ; BINð200; pti Þ for t ¼ 1; . . . ; T; i ¼ 1; . . . ; I; for t ¼ 1; . . . ; T; i ¼ 1; . . . ; I; ð1Þ
þ et1k½i þ et1j½i þ et1i
2ð2Þ
et1k ; Nð0; rs½t1k Þ for t ¼ 2; . . . ; T; k ¼ 1; . . . ; K
ð2AÞ ð2BÞ
6
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dependence in each region were modeled as follows: ðrÞ
ðrÞ
ðrÞ
ð5AÞ
ðaÞ
ðaÞ
ðaÞ
ð5BÞ
l2k ¼ l1k þ dk
l2k ¼ l1k þ dk ðrÞ dk
analysis random intercept and slopes, which varied at each shore and rock, and at each combination of season and shore. We fit all models with the lmer() function in the lme4 package in R (Bates et al. 2011).
RESULTS
ðaÞ dk
and represent the seasonal where differences in the regional average of intrinsic growth rate and strength of density dependence in region k, respectively. We obtained posterior distributions with a Markov chain Monte Carlo (MCMC) method run in WinBUGS (Lunn et al. 2000). Details for parameter inference are described in Appendix B. We used estimated model parameters to calculate the following derived parameters: average seasonal population growth rates and their seasonal differences at regional, shore, and rock scales; the sum total of process variance as a measure of the magnitude of stochastic forces on population growth rate and its seasonal variability; the relative strength of stochastic fluctuations, which measures the proportional impact of stochastic forces on population growth rates for each spatial scale (Fukaya et al. 2010); and the log of the temporal average and variance of the population size. Appendix C provides a detailed list of derived parameters and their definitions. Mean-variance relationship.—We examined the temporal mean-variance relationship by using the posterior median of the temporal mean and the variance of population size at each rock in each season, which we obtained as derived parameters of the fitted model (see Appendix C). We applied generalized linear mixed models (GLMM), in which the log of the variance, log10(s2), and log of the mean, log10(m), were treated as the response variable and predictor, respectively, a procedure that corresponds to estimating coefficients of temporal Taylor’s power law (Mellin et al. 2010). To examine spatiotemporal variations in the parameters of Taylor’s power law, we also considered region, season, and their interaction terms to be predictor variables, and we used the Akaike Information Criterion (AIC) to carry out variable selection by fitting all possible models that involved subsets of predictor variables. To take into account the dependence of response variables belonging to the same shore or rock, we included in the v www.esajournals.org
Population growth rate Regional average population growth rates were seasonally different for all of the studied regions, and seasonal differences varied across latitude (Fig. 2). Population growth rates were higher in the summer than winter in eastern Hokkaido, but were lower in summer than winter in other regions (Fig. 2A). The 95% credible interval of seasonal difference in regional average growth rate did not include zero in each region (Fig. 2B), suggesting that in these regions’ population growth rates are significantly different between seasons. Seasonal differences in population growth rates were smallest on the Oshima Peninsula and tended to increase with distance from the Oshima Peninsula (Fig. 2B). Seasonality in population growth was spatially variable even between shores and between rocks within a region (Fig. 3). The high-growth-rate season at some shores and rocks did not coincide with the regional average high-growth-rate season. For example, population growth rate in Hiura (HR) on the Oshima Peninsula was higher during the summer than winter (Fig. 3B) although population growth rates in Oshima Peninsula tended to decrease during the summer (Fig. 2). Seasonality in population growth rates was more variable between rocks: in eastern Hokkaido, Oshima Peninsula, and Sanriku Coast, population growth rates were higher during the summer at some rocks but higher during the winter at other rocks (Fig. 3F–H). On Kii Peninsula and Osumi Peninsula, population growth rates were consistently higher in the winter at all rocks, but some variability in seasonal differences in population growth rate among rocks still existed (Fig. 3I, J).
Processes determining population growth Seasonal patterns in regional averages of intrinsic growth rate and the strength of density dependence varied among regions (Fig. 4A–D). Intrinsic growth rates were higher in summer 7
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Fig. 2. (A) Regional average of population growth rate for each season. (B) Seasonal differences in regional average of population growth rate (a positive value indicates higher population growth rate in winter compared to summer). Definition of each variable is provided in Appendix C. Circles and bars represent medians and 95% credible intervals of posterior distribution, respectively.
compared to winter in eastern Hokkaido and the Oshima Peninsula, but were lower in summer compared to winter on the Sanriku Coast, Kii Peninsula, and Osumi Peninsula (Fig. 4A, B). Density dependence was stronger in summer compared to winter in eastern Hokkaido and the Oshima and Osumi Peninsulas, whereas it was weaker in summer compared to winter on the Sanriku Coast and Kii Peninsula (Fig. 4C, D). In contrast, total process variance, a measure of the magnitude of stochastic forces on population growth rate, was consistently higher in summer than winter in all of the regions (Fig. 4E, F). At the rock scale, average population growth rate, intrinsic growth rate, and the strength of density dependence were related to each other (Fig. 5). The average population growth rate for each rock in each season was positively correlated with the corresponding intrinsic growth rate and strength of density dependence. We also found a strong positive relationship between intrinsic growth rate and the strength of density dependence (posterior median of correlation coefficient was 0.96, Fig. 5F). The seasonal patterns of the relative strengths of stochastic fluctuations, which measure the v www.esajournals.org
proportional impact of stochastic forces on population growth rate at each spatial scale, also varied among regions (Fig. 6). In eastern Hokkaido, population growth rates were strongly affected by fluctuations at the scales of shores and rocks throughout the year (Fig. 6A). On the Oshima Peninsula, where seasonal differences in these processes were most noticeable, rock-scale fluctuations dominated during the summer, whereas regional- and shore-scale fluctuations dominated during the winter (Fig. 6B). On the Sanriku Coast, population growth was generally determined by rock-scale fluctuations, although regional-scale fluctuations tended to be stronger in summer (Fig. 6C). On the Kii Peninsula, fluctuations in population growth at the scales of rocks and shores were apparent in the summer, whereas shore- and regional-scale fluctuations were important in the winter (Fig. 6D). On the Osumi Peninsula, rock-scale fluctuations were most important throughout year, followed by the effect of shore- and regional-scale fluctuations (Fig. 6E). Overall, among-regional-scale fluctuations contributed little to the variability in population growth. 8
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Fig. 3. Seasonal differences in average of population growth rates at each shore (A–E) and at each rock (F–J). A positive value indicates higher population growth rate during the winter compared to the summer. Abbreviations for each shore and the definition of each variable are provided in Table 1 and in Appendix C, respectively. Shading indicates boundaries between shores. Circles and bars represent medians and 95% credible intervals of posterior distribution, respectively. Note that the 95% credible interval is smaller than the circle for some shores and rocks.
The result of model selection (Appendix D) showed that the logarithm of population size was the best single predictor variable (log10 s2 ¼ 0.825 þ 1.501[log10 m]). These results indicate that differences in population variability can be best explained by a unique Taylor’s power law,
Mean-variance relationship There was a linear relationship between the logarithms of population variance (s2) and mean population size (m), indicating that Taylor’s power law can characterize the temporal meanvariance relationship of population size (Fig. 7). v www.esajournals.org
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Fig. 4. Regional averages and seasonal differences of intrinsic growth rate (A–B), strength of density dependence (C–D) and sum total of process variance (E–F). Definition of intrinsic growth rate and strength of density dependence can be found in Eq. 5, and the definition of the sum of the total of process variance and seasonal differences can be found in Appendix C. A positive value of the seasonal difference indicates that the intrinsic growth rate, strength of density dependence, or total process variance is higher in the winter compared to the summer. Circles and bars represent medians and 95% credible intervals of the posterior distribution, respectively.
weight of the second best model (0.31) suggest that the second best model, which accounts for differences in intercept among regions, was also partially supported (Appendix D).
despite significant regional and seasonal differences in the processes that determine population growth. Note, however, that the small DAIC value between the best and the second best model (1.23) and the large value of the AIC v www.esajournals.org
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Fig. 5. A matrix of scatter plots of posterior medians of the average population growth rate (A and B), intrinsic growth rate (A and C), and strength of density dependence (B and C) estimated for each plot and in each season. Lower panels (D–F) show posterior distributions of the correlation coefficients among the parameters.
multiple spatial scales. To our knowledge, this is the first study to report that seasonal population dynamics can vary at multiple spatial scales. For intertidal barnacles, it has been reported that in temperate zones, thermal and desiccation stress during the summer can reduce somatic growth and survival of adult individuals (e.g., Bertness 1989, Sanford et al. 1994, Gedan et al. 2011). On the other hand in subarctic zones, low temperature or scouring by drifting ice during the winter can limit the distribution and abundance (Crisp 1964, Bergeron and Bourget 1986). These findings suggest geographical variations in seasonal changes in demographic rate of intertidal barnacles, which may underlie the latitudinal variation in population growth rate found in this study. The seasonal forces on demographics of intertidal barnacles may also vary at smaller spatial scales. For example, the existence of shades and algal canopies can have a positive effect on survival of barnacles during summer because it ameliorates
DISCUSSION In this study we evaluated seasonal and spatial variations in the population dynamics of the intertidal barnacle Chthamalus spp. at multiple spatial scales. Results showed that seasonal differences were prevalent in population growth rate (Figs. 2 and 3), and in the strength (Fig. 4) and influential spatial scales (Fig. 6) of processes that determine population growth rate. These results indicate that seasonal fluctuations in environmental conditions can have discernible impacts on changes in population abundance of intertidal barnacle populations. We also found that seasonality in population growth rate varied at different spatial scales ranging from between rocks to between regions. This result suggests that effects of seasonal environmental fluctuations on population growth can be dependent on spatial heterogeneity in biotic and abiotic conditions that occur at v www.esajournals.org
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Fig. 6. Relative strengths of stochastic fluctuations, which measure the proportional impact of stochastic forces on population growth rate at each spatial scale. The definition of variables is provided in Appendix C. Circles and bars represent medians and 95% credible intervals of posterior distribution, respectively.
Fig. 7. Scatter plots of posterior medians of the logarithm of the average population size (log10 msi ) and logarithm of the variance of the population size (log10 s2si ) estimated for each plot and in each season (these calculations can be found in Appendix C). Fitted lines are estimates of the temporal Taylor’s power law of the lowest AIC (log10 s2 ¼ 0.825 þ 1.501[log10 m]; Table D1 in Appendix D).
thermal conditions (Bertness et al. 1999, Leonard 2000). The size of substrates may also affect summer survival since it determines thermal capacity of substrates and hence affects rock v www.esajournals.org
surface temperature (Bertness 1989, Gedan et al. 2011). Our results imply that these features of rocky intertidal systems can result in complex spatial patterns of seasonal population dynamics. 12
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Future experimental studies are needed to elucidate details of the mechanisms that cause spatial differences in the seasonality of population dynamics. Because recruitment of individuals occurs during the summer in each studied region (Maruyama 2005, Munroe and Noda 2010, Munroe et al. 2010), spatial differences in postrecruitment processes such as body growth and mortality of sessile individuals presumably explain the observed spatial variations of seasonality in population growth rate. It might also be possible, however, that variations in recruitment intensity, which may occur at multiple spatial scales ranging from geographic or mesoscales (Connolly et al. 2001, Lagos et al. 2005, 2007, Navarrete et al. 2005, 2008, Broitman et al. 2008) to local scales (Crisp 1961, Wethey 1984, 1986, Raimondi 1988, 1990, Larsson and Jonsson 2006), were the underlying cause because recruitment density can strongly influence post-recruitment processes and dynamics of adult populations (Gaines and Roughgarden 1985, Menge 2000). Clarifying spatiotemporal differences in the strength of density dependence and its causal mechanisms are essential for understanding population regulation (Royama 1992, Turchin 2003). In our previous work (Fukaya et al. 2010), we proposed two mechanisms that could explain estimated seasonal variations in the strength of density dependence of C. challengeri populations on the Sanriku Coast. First, seasonal differences in the strength of density dependence may be caused by changes in intrinsic growth rate, because increases in recruitment, somatic growth, and survival rate can lead to direct contact between individuals and thus induce density-dependent mortality of sessile organisms (Bertness 1989, Sanford et al. 1994, Menge 2000). Second, decreases in the strength of density dependence during the summer may be caused by facilitative (inversely density-dependent) interactions, which can offset negative density dependence and are known to occur in intertidal habitats during the summer (Lively and Raimondi 1987, Bertness 1989, Kawai and Tokeshi 2006). The importance of these two mechanisms can partly be tested with the results of the work reported here, a large-scale survey of density dependence and intrinsic growth rate. If the first mechanism were important, a positive correlation would exist between intrinsic growth rate v www.esajournals.org
and the strength of density dependence. If the second mechanism were important, density dependence during the summer would be consistently weaker than during the winter. Our results are consistent with the first mechanism: there was a high degree of positive correlation between intrinsic growth rate and the strength of density dependence (Fig. 5), whereas density dependence was not consistently weaker in the summer for Chthamalus spp. populations (Fig. 4). These results imply that spatiotemporal differences in intrinsic growth rate, which will be high when recruitment, somatic growth, and survival rate are high, would be important determinants of the strength of regulation of Chthamalus spp. populations. Positive correlations of both intrinsic growth rate and the strength of density dependence with average population growth rate (Fig. 5) suggest that in Chthamalus spp. populations, the regulatory force of negative density dependence tended to operate where and when intrinsic growth rates, and hence average population growth rates, were high. Our analysis revealed a notable variation in the relative strength of stochastic fluctuations of Chthamalus spp. coverage (Fig. 6). The relative strength of processes that operated at different spatial scales varied among regions, the result being, for example, that populations were mainly driven by rock-scale processes on the Sanriku Coast (Fig. 6C), whereas they were strongly affected by shore-scale processes on the Kii Peninsula (Fig. 6D). The relative strength of processes can also differ temporally within a region. In Oshima Peninsula, for example, rockscale processes had a strong influence during the summer, whereas regional-scale processes dominated in the winter (Fig. 6B). This result suggests that the spatial scale of processes that strongly affect population dynamics can vary greatly as a function of season and region, a dependence that probably reflects spatiotemporal differences in the environmental factors that affect population growth. As the spatial scale of processes can determine that of population synchrony (Fox et al. 2011), our results imply that generalizations about population synchrony among intertidal sessile organisms (e.g., Dye 1998, Burrows et al. 2002, Gouhier et al. 2010) may require large-scale comparisons that explicitly incorporate seasonal variability. 13
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In contrast to the significant differences in patterns and processes of population growth, the evidence of spatiotemporal difference in temporal mean-variance relationships was weak (Fig.7, Appendix D). Our results suggest that although environmental differences can strongly influence processes that regulate population dynamics, they may not necessarily alter temporal meanvariance relationships. This result contradicts the theoretical expectation for the temporal meanvariance relationship: theoretical studies have demonstrated that environmental differences can induce intraspecific variations in temporal meanvariance relationships. These variations can manifest themselves as differences in the scaling exponent (i.e., slope) of Taylor’s power law by modifying processes that drive population dynamics, such as the strength of density dependence and magnitude of stochastic forces (Anderson et al. 1982, Hanski and Woiwod 1993, Perry 1994, Keeling 2000, Ballantyne and Kerkhoff 2007). The evidence of differences in the scaling exponent was, however, absent in our analysis (although intercept may potentially be different among regions; Appendix D). To our knowledge the present study is the first test of this theoretical prediction in which the meanvariance relationship and processes determining population growth were compared directly from a population time series collected in the field. Our results imply that the widely-accepted hypothesis for mean-variance relationship, that is, mean-variance relationship is determined by population processes (demographic hypothesis: Anderson et al. 1982), may not fully explain patterns of mean-variance relationships found in nature. We suggest that a combined application of the population dynamics model and Taylor’s power law in empirical studies will provide useful insights relevant to the patterns and causal mechanisms of the mean-variance relationship by revealing links between processes that control population dynamics and spatiotemporal differences in mean-variance relationships. Our analysis of the temporal mean-variance relationships has implications in terms of understanding of population ecology of barnacles as well. For example, our result indicates that the average population size can be regarded as a primary predictor for variability of barnacle populations because the Taylor’s power law links v www.esajournals.org
population variability and mean population size. The invariability of coefficients of the Taylor’s power law further suggests that a unique meanvariance relationship can simply be used to predict variability of the Chthamaloid populations over large geographic areas and different seasons. These insights should be the basic knowledge about the long-term population dynamics of these barnacles which would also be useful to explore general patterns of population variability. Two models of population variability, the population dynamics model and Taylor’s power law, independently provide useful insights relevant to population variability. The former offers knowledge about ecological processes that regulate population dynamics by decomposing population growth rates into several densitydependent and density-independent components (Royama 1992, Turchin 2003), whereas the latter provides a simple and robust framework for predicting population variability (Keeling and Grenfell 1999, Mellin et al. 2010). However, there has been limited application of these two models in studies of marine sessile organisms where size of a population is measured by coverage. In our previous work (Fukaya et al. 2010) and this study, we demonstrated that the logistic growth model can describe variation in coverage of the intertidal barnacle C. challengeri. In this study, we also showed that Taylor’s power law can provide a good characterization of spatiotemporal differences in the variability of Chthamalus spp. population coverage. These results suggest that some poorly understood issues for marine sessile populations, such as the relative influence of density-dependent and density-independent processes and spatiotemporal differences in meanvariance relationships, may be effectively addressed by using common analytical frameworks that have primarily been used in abundancebased studies (e.g., Saitoh et al. 2003, Stenseth et al. 2003, Mellin et al. 2010). We suggest that using these models to analyze cumulated time series data from intertidal rocky shores (e.g., Southward 1991, Dye 1998, Burrows et al. 2002) may provide new insights for understanding the population and community dynamics of these open and variable systems. In this study, we pooled two Chthamaloid species (C. dalli and C. challengeri ) because they 14
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FUKAYA ET AL. animal and plant species. Nature 296:245–248. Ballantyne, F., and A. J. Kerkhoff. 2007. The observed range for temporal mean-variance scaling exponents can be explained by reproductive correlation. Oikos 116:174–180. Bates, D., M. Maechler, and B. Bolker. 2011. lme4: linear mixed-effects models using S4 classes. R package version 0.999375-42. http://CRAN. R-project.org/package¼lme4 Bergeron, P., and E. Bourget. 1986. Shore topography and spatial partitioning of crevice refuges by sessile epibenthos in an ice disturbed environment. Marine Ecology Progress Series 28:129–145. Bertness, M. D. 1989. Intraspecific competition and facilitation in a northern acorn barnacle population. Ecology 70:257–268. Bertness, M. D., S. D. Gaines, D. Bermudez, and E. Sanford. 1991. Extreme spatial variation in the growth and reproductive output of the acorn barnacle Semibalanus balanoides. Marine Ecology Progress Series 75:91–100. Bertness, M. D., G. H. Leonard, J. M. Levine, and J. F. Bruno. 1999. Climate-driven interactions among rocky intertidal organisms caught between a rock and a hot place. Oecologia 120:446–450. Broitman, B. R., C. A. Blanchette, B. A. Menge, J. Lubchenco, C. Krenz, M. Foley, P. T. Raimondi, D. Lohse, and S. D. Gaines. 2008. Spatial and temporal patterns of invertebrate recruitment along the west coast of the United States. Ecological Monographs 78:403–421. Burrows, M. T., J. J. Moore, and B. James. 2002. Spatial synchrony of population changes in rocky shore communities in Shetland. Marine Ecology Progress Series 240:39–48. Chabot, R., and E. Bourget. 1988. Influence of substratum heterogeneity and settled barnacle density on the settlement of cypris larvae. Marine Biology 97:45–56. Clark, J. S. and O. N. Bjørnstad. 2004. Population time series: process variability, observation errors, missing values, lags, and hidden states. Ecology 85:3140–3150. Connell, J. H. 1961a. Effects of competition, predation by Thais lapillus, and other factors on natural populations of the barnacle Balanus balanoides. Ecological Monographs 31:61–104. Connell, J. H. 1961b. The influence of interspecific competition and other factors on the distribution of the barnacle Chthamalus stellatus. Ecology 42:710– 723. Connolly, S. R., B. A. Menge, and J. Roughgarden. 2001. A latitudinal gradient in recruitment of intertidal invertebrates in the northeast Pacific Ocean. Ecology 82:1799–1813. Crisp, D. J. 1961. Territorial behaviour in barnacle settlement. Journal of Experimental Biology
are difficult to distinguish in the field. In such cases, results should be interpreted carefully because if a local population consists of different species, estimated parameters of the population might not represent the species specific characteristics. Such artifacts, however, should not be prevalent in our results because in most of our census plots, only one Chthamaloid species were dominating at each plot (Appendix A) and hence parameters estimated for each population should be representing the characteristic of that species. At the largest spatial scale, our results involve estimates from the most northern region where C. dalli dominated and that from the four southern regions where C. challengeri was mostly found. We note that similar results would be obtained even if we excluded the most northern region from the analyses and focused solely on C. challengeri, and therefore the pooling of the two species should not be relevant to our main findings about spatiotemporal variations in population dynamics over geographic areas.
ACKNOWLEDGMENTS We are grateful to S. Higashi, T. Saitoh, I. Koizumi and J. A. Royle for their helpful comments on an earlier version of this manuscript. This manuscript also benefited by comments provided by S. A. Navarrete and two anonymous reviewers. We thank Y. Hisatsune and H. Yamaguchi for taxonomic identification of the intertidal barnacles. For providing access to laboratory facilities, we are grateful to the staffs and students of the Akkeshi and Usujiri Marine Stations of Hokkaido University, the International Coastal Research Center of the Atmosphere and Ocean Research Institute, the University of Tokyo, the Marine Biosystems Research Center of Chiba University, the Seto Marine Biological Laboratory of Kyoto University, and the Education and Research Center for Marine Environment and Resources of Kagoshima University. We also acknowledge the many researchers and students who helped our fieldwork. This research was partially supported by a Grant-in-Aid for Scientific Research by the Ministry of Education, Science, Sports and Culture in Japan to TN (no. 20570012), to MN (nos. 14340242, 18201043, 21241055) and to TY (no.17510197) and by a Grantin-Aid for a Research Fellow of the JSPS to KF (no. 235649).
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67:29–39. Helmuth, B., B. R. Broitman, C. A. Blanchette, S. Gilman, P. Halpin, C. D. G. Harley, M. J. O’Donnell, G. E. Hofmann, B. Menge, and D. Strickland. 2006. Mosaic patterns of thermal stress in the rocky intertidal zone: implications for climate change. Ecological Monographs 76:461–479. ˚ berg, G. Cervin, R. A. Coleman, J. Jenkins, S. R., P. A Delany, S. J. Hawkins, K. Hyder, A. A. Myers, J. Paula, A. M. Power, P. Range, and R. G. Hartnoll. 2001. Population dynamics of the intertidal barnacle Semibalanus balanoides at three European locations: spatial scales of variability. Marine Ecology Progress Series 217:207–217. Kawai, T. and M. Tokeshi. 2006. Asymmetric coexistence: bidirectional abiotic and biotic effects between goose barnacles and mussels. Journal of Animal Ecology 75:928–941. Keeling, M. J. 2000. Simple stochastic models and their power-law type behaviour. Theoretical Population Biology 58:21–31. Keeling, M. and B. Grenfell. 1999. Stochastic dynamics and a power law for measles variability. Philosophical Transactions of the Royal Society B 354:769–776. Kendal, W. S. 2004. Taylor’s ecological power law as a consequence of scale invariant exponential dispersion models. Ecological Complexity 1:193–209. Lagos, N. A., S. A. Navarrete, F. Ve´liz, A. Masuero, and J. C. Castilla. 2005. Meso-scale spatial variation in settlement and recruitment of intertidal barnacles along the coast of central Chile. Marine Ecology Progress Series 290:165–178. Lagos, N. A., F. J. Tapia, S. A. Navarrete, and J. C. Castilla. 2007. Spatial synchrony in the recruitment of intertidal invertebrates along the coast of central Chile. Marine Ecology Progress Series 350:29–39. Larsson, A. I., and P. R. Jonsson. 2006. Barnacle larvae actively select flow environments supporting postsettlement growth and survival. Ecology 87:1960– 1966. Leonard, G. H. 2000. Latitudinal variation in species interactions: a test in the New England rocky intertidal zone. Ecology 81:1015–1030. Lively, C. M. and P. T. Raimondi. 1987. Desiccation, predation, and mussel-barnacle interactions in the northern Gulf of California. Oecologia 74:304–309. Lunn, D. J., A. Thomas, N. Best, and D. Spiegelhalter. 2000. WinBUGS—a Bayesian modelling framework: concepts, structure, and extensibility. Statistics and Computing 10:325–337. Maruyama, T. 2005. Effects of recruitment intensity on population density of intertidal barnacle Chthamalus spp. Thesis. Chiba University, Chiba, Japan. [In Japanese.] Mellin, C., C. Huchery, M. J. Caley, M. G. Meekan, and C. J. A. Bradshaw. 2010. Reef size and isolation
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Chapman & Hall, London, UK. Saitoh, T., N. C. Stenseth, H. Viljugrein, and M. O. Kittilsen. 2003. Mechanisms of density dependence in fluctuating vole populations: deducing annual density dependence from seasonal processes. Population Ecology 45:165–173. Sanford, E., D. Bermudez, M. D. Bertness, and S. D. Gaines. 1994. Flow, food supply and acorn barnacle population dynamics. Marine Ecology Progress Series 104:49–62. Sanford, E., and B. A. Menge. 2001. Spatial and temporal variation in barnacle growth in a coastal upwelling system. Marine Ecology Progress Series 209:143–157. Southward, A. J. 1991. Forty years of changes in species composition and population density of barnacles on a rocky shore near Plymouth. Journal of the Marine Biological Association of the United Kingdom 71:495–513. Stenseth, N. C., H. Viljugrein, T. Saitoh, T. F. Hansen, M. O. Kittilsen, E. Bølviken, and F. Glo¨ckner. 2003. Seasonality, density dependence, and population cycles in Hokkaido voles. Proceedings of the National Academy of Sciences USA 100:11478– 11483. Taylor, L. R. 1961. Aggregation, variance and the mean. Nature 189:732–735. Taylor, L. R., and I. P. Woiwod. 1982. Comparative synoptic dynamics. I. Relationships between interand intra-specific spatial and temporal variance/ mean population parameters. Journal of Animal Ecology 51:879–906. Thomas, J. A., D. Moss, and E. Pollard. 1994. Increased fluctuations of butterfly populations towards the northern edges of species’ ranges. Ecography 17:215–220. Turchin, P. 2003. Complex population dynamics: a theoretical/empirical synthesis. Princeton University Press, Princeton, New Jersey, USA. Wethey, D. S. 1984. Spatial pattern in barnacle settlement: day to day changes during the settlement season. Journal of the Marine Biological Association of the United Kingdom 64:687–698. Wethey, D. S. 1986. Ranking of settlement cues by barnacle larvae: influence of surface contour. Bulletin of Marine Science 39:393–400.
SUPPLEMENTAL MATERIAL APPENDIX A
Pilsbry, and a southern species, C. challengeri Hoek. Geographic ranges of these two species are thought to overlap in the region from northern Honshu Island to southern Hokkaido Island. Therefore they potentially occur sympatrically within our northern region plots, although distinguishing these two species in the field is
Detailed description of the identification of Chthamalus species and examination of their distribution and frequency in studied shores Two species of Chthamalus exist in the census region: a northern species, Chthamalus dalli v www.esajournals.org
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difficult. To examine the distribution of these two species at the studied shores, we enlisted the help of two barnacle taxonomists, Y. Hisatsune and H. Yamaguchi, to assess the frequency of C. dalli and C. challengeri by random sampling of Chthamalus individuals at each shore in April 2004. In eastern Hokkaido, only C. dalli was found on four shores, and only three individuals (3.6%) of C. challengeri were found in 83 samples from Nikomanai (see Table 1 in main text for list of shore names). In Oshima Peninsula, only C. challengeri was found on all of the shores. In Sanriku Coast, only C. challengeri was found on three shores, whereas 30 individuals (23.1%) were included in 130 sampled individuals from Katagishi. A re-examination of the frequency of the two species in Katagishi in November 2004 yielded 23 individuals (11.3%) of C. dalli among 204 Chthamalus individuals. In Kii Peninsula and Osumi Peninsula, only C. challengeri were found on all the shores. These assessments suggest that although both Chthamalus species may be present in some barnacle populations on several shores in eastern Hokkaido and the Sanriku Coast, C. dalli accounts for most Chthamalus individuals in eastern Hokkaido, whereas C. challengeri dominates in other regions.
An inverse-Wishart distribution (IW), IW(R, 3), where R is a diagonal matrix in which all diagonal elements are 0.01, was specified as the priors for R(1) and R(2). A normal distribution with a mean of 0 and variance of 100 was specified as the priors for l and d. We obtained posterior distributions by using the Markov Chain Monte Carlo (MCMC) run in WinBUGS (Lunn et al. 2000). To confirm independence of the posterior probability on initial values, we executed three independent iterations. We obtained estimates from 1 3 106 iterations after a burn-in of 3 3 106 iterations, thinned at intervals of 200. We monitored the convergence of posterior distributions with the Gelman-Rubin diagnostic (Gelman et al. 2004). This statistic was less than 1.1, an acceptable level of convergence (Gelman et al. 2004), for each parameter of interest. We evaluated model goodness-of-fit by posterior predictive assessment with a v2 discrepancy quantity defined as Tðy; hÞ ¼
t
i
varðyti jhÞ
where y and h represent data and model parameters, respectively, and its posterior predictive P-value, defined as P ¼ Pr[T(yrep, h) T(y, h)jy], where yrep is the posterior predictive value of the data (Gelman et al. 2004). The obtained posterior predictive P-value was 0.014, an indication that our model tended to overfit the data and hence may have weak predictive power. The BUGS code of the model is provided as the Supplement.
APPENDIX B Details for parameter estimation and posterior assessment of model fit We specified vague priors for all estimated parameters. A uniform distribution (U), U(0, 1000) was specified as the priors for the square 2ð1Þ 2ð2Þ 2ð3Þ 2ð4Þ root of rs , rsk , rsk , and rsk , respectively.
v www.esajournals.org
X X ½yti Eðyti jhÞ2
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APPENDIX C Table C1. List of derived parameters and their definitions. Parameter Regional average growth rate ð1Þ (Dlog Ask ) Summer (s ¼ 1)
Definition
Result Fig. 2A
0
Ik X ½ðlog A3i log A2i Þ þ ðlog A5i log A4i Þ þ þ ðlog AT1i ;i log AT1i 1;i Þ i¼Ik
ðIk0 Ik þ 1Þ 3ðT1i 1Þ=2 0
Ik X ½ðlog A2i log A1i Þ þ ðlog A4i log A3i Þ þ ðlog AT2i ;i log AT2i 1;i Þ i¼Ik
Winter (s ¼ 2)
ðIk0 Ik þ 1Þ 3 T2i =2
Seasonal difference in regional growth rate Shore average growth rate ð2Þ (Dlog Asj ) Summer (s ¼ 1)
ð1Þ
ð1Þ
Dlog A2k Dlog A1k
Fig. 2B Not shown
0
Ij X ½ðlog A3i log A2i Þ þ ðlog A5i log A4i Þ þ þ ðlog AT1i ;i log AT1i 1;i Þ i¼Ij
ðIj0 Ij þ 1Þ 3ðT1i 1Þ=2 Ij0
X ½ðlog A2i log A1i Þ þ ðlog A4i log A3i Þ þ þ ðlog AT2i ;i log AT2i 1;i Þ i¼Ij
Winter (s ¼ 2)
ðIj0 Ij þ 1Þ 3 T2i =2 ð2Þ
ð2Þ
Dlog A2j Dlog A1j
Seasonal difference in shore average growth rate Rock average growth rate ð3Þ (Dlog Asi )
Fig. 3A–E Not shown
Summer (s ¼ 1)
ðlog A3i log A2i Þ þ ðlog A5i log A4i Þ þ þ ðlog AT1i ;i log AT1i 1;i Þ ðT1i 1Þ=2
Winter (s ¼ 2)
ðlog A2i log A1i Þ þ ðlog A4i log A3i Þ þ þ ðlog AT2i ;i log AT2i 1;i Þ T2i =2
Seasonal difference in rock average growth rate Sum total of process variance (r2sk )
ð3Þ
2ð1Þ rs
Seasonal difference in sum total of process variance Correlation coefficients among parameters Relative strength of stochastic fluctuation Among-regional level
2ð2Þ rsk
2ð3Þ þ þ rsk þ logðr22k =r21k Þ
corrðpsi ; psi0 Þ
2ð4Þ rsk
Fig. 3 F–J Fig. 4E Fig. 4F Fig. 5 Fig. 6
2ð1Þ rs =r2sk 2ð2Þ rsk =r2sk 2ð3Þ rsk =r2sk 2ð4Þ rsk =r2sk
Regional level Shore level Rock level
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ð3Þ
Dlog A2i Dlog A1i
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April 2013 v Volume 4(4) v Article 48
FUKAYA ET AL. Table C1. Continued. Parameter
Definition
Result
Log of average population size (log10 msi )
Fig. 7
Spring (s ¼ 1)
Autumn (s ¼ 2) Log of variance of population size (log10 s2si ) Spring (s ¼ 1)
log10
A2i þ A4i þ þ AT2i ;i T2i =2
log10
A1i þ A3i þ þ AT1i ;i ðT1i þ 1Þ=2
" # ðA2i m1i Þ2 þ ðA4i m1i Þ2 þ þ ðAT2i ;i m1i Þ2 log10 ðT2i =2Þ 1 where m1i ¼
Autumn (s ¼ 2)
Fig. 7
log10
A2i þ A4i þ þ AT2i ;i T2i =2
" # ðA1i m2i Þ2 þ ðA3i m2i Þ2 þ þ ðAT1i ;i m2i Þ2 fðT1i þ 1Þ=2g 1 where m2i ¼
A1i þ A3i þ þ AT1i ;i ðT1i þ 1Þ=2
Notes: Ik( j ): Beginning index number of plot in region k (or, in shore j ). Ikð0 jÞ : Last index number of plot in region k (or, in shore j ). T1i: Index number of time when the last autumn census was conducted at plot i (maximum 17). T2i: Index number of time when the last spring census was conducted at plot i (maximum 18). corr(x, y): Pearson’s correlation coefficient of x and y, where x ð3Þ and y are the posterior samples of rock average growth rate (Dlog Asi ), intrinsic growth rate (rsi ), or strength of density dependence (asi ).
APPENDIX D Table D1. Result of model selection for mean-variance relationship: models for Taylor’s power law with the top five AIC. Predictor variable
AIC
DAIC
AIC weight
log10 m Region, log10 m Season, log10 m Season, Region, log10 m Season, Region, log10 m, Season 3 log10 m
75.72 76.95 79.63 81.28 86.93
0.00 1.23 3.91 5.56 11.21
0.57 0.31 0.08 0.04 0.00
Notes: log10 m: log of average population size; Season: an indicator variable that represents spring or autumn; Region: a categorical variable that represents each studied region. Random intercepts and slopes, which varied for each shore, rock, and combination of season and shore, were included in each model.
SUPPLEMENT BUGS code of the fitted model (Ecological Archives C004-007-S1).
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April 2013 v Volume 4(4) v Article 48
