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On Species Persistence-Time Distributions

We present new theoretical and empirical results on the probability distributions of species persistence times in natural ecosystems. Persistence times, defined as the timespans occurring between species' colonization and local extinction in a given geographic region, are empirically estimated from local observations of species' presence/absence. A connected sampling problem is presented, generalized and solved analytically. Species persistence is shown to provide a direct connection with key spatial macroecological patterns like species-area and endemics-area relationships. Our empirical analysis pertains to two different ecosystems and taxa: a herbaceous plant community and a estuarine fish database. Despite the substantial differences in ecological interactions and spatial scales, we confirm earlier evidence on the general properties of the scaling of persistence times, including the predicted effects of the structure of the spatial interaction network. The framework tested here allows to investigate directly nature and extent of spatial effects in the context of ecosystem dynamics. The notable coherence between spatial and temporal macroecological patterns, theoretically derived and empirically verified, is suggested to underlie general features of the dynamic evolution of ecosystems.


Introduction
Extinction rates, pointedly from habitat loss, have been defined as a crucial conservation problem of this century (Raup and Sepkoski, 1984;Wilcox and Murphy, 1985;Pimm et al., 1988;Fahrig, 1997;Ferraz et al., 2003). Direct and reliable estimation of true extinction rates remains problematic, however, and a lively debate exists over the possible overestimation of extinctions incurred through the standard use of methods based on the reversing of the species-area relationship (SAR), i.e. the number of species observed at increasing sampled areas (Green and Ostling, 2003;He and Hubbell, 2011).
Recently Bertuzzo et al. (2011) have proposed empirical and theoretical evidence for a new macroecological pattern, namely the distribution of persistence times for trophically equivalent, co-occurring species. The species persistence time (SPT) is defined as the time-span incurred between a species' emergence and its local extinction in a given geographic region. Using two different long-term ecological databases, related namely to the inventory of North American breeding birds and to Kansas grasslands, Bertuzzo et al. (2011) suggested that the empirical SPT distribution is characterized by a power-law function limited by an exponential cut-off determining the maximum observed persistence time, which in turn is related to the spatial extent of the ecosystem. The advantage to concentrate on SPTs, is that SPT distributions have proved to be a robust measure of species turnover at different spatial scale. Although SPT distributions cannot provide predictions about extinction rates of specific species, they can describe the global evolution of the diversity of an ecosystem and give robust estimates for mean SPTs (Bertuzzo et al., 2011).
The problem of the scale of observation becomes central in this framework. In fact, at the local scale, say an observational site for the presence/absence of breeding birds, the persistence time of a species is controlled by ecological processes operating at short timescales, like birth/death, dispersal, and contraction/expansion of geographic ranges. At this scale local extinctions are dynamically balanced by colonizations (MacArthur and Wilson, 1967;Tilman, 1994;Ricklefs and Scheuerlein, 2003;Muneepeerakul et al., 2008). At the global scale, species emergence and extinction are controlled by mechanisms acting on evolutionary timescales (Brown and Kodric-Brown, 1977;Diamond, 1989). Interestingly, the scaling behavior proposed by Bertuzzo et al. (2011) to govern the transition from local to global scales is capable of effectively describing the overall dynamical evolution of the ecosystem di-versity. Scaling features also allow to predict SPT distributions for wide geographic areas from measures of persistence on smaller areas, which are much easier to monitor.
To provide a theoretical explanation for the SPT pattern, Bertuzzo et al. (2011) resorted to a spatially explicit model based on neutral theory. Neutral theory has been proposed as a unifying theoretical context for understanding ecological patterns (Hubbell, 2001). Since its formulation, several studies have focused their attention on it (e.g. Volkov et al., 2003;Houchmandzadeh and Vallade, 2003;Chave, 2004;Alonso and McKane, 2004;Azaele et al., 2006;Muneepeerakul et al., 2008;Chisholm and Lichstein, 2009). Neutral models are based on the assumption that, within a single trophic level, individual birth and death rates are species-independent. The main advantages of neutral models are that they are falsifiable, and that they are able to generate predictions for many different macroecological patterns. This is the case, for instance, of the relative species abundance (RSA) distribution (McGill, 2003;Volkov et al., 2005), the species-independent beta diversity patterns (Condit et al., 2002;Zillio et al., 2005), species geographic range (Bertuzzo et al., 2009) and the species-area relationship (Brown, 1995;Zillio et al., 2008;O'Dwyer and Green, 2010). Remarkably, this is accomplished by invoking only basic ecological processes such as birth, death, migration and dispersal limitation. Although certain emergent ecological patterns are independent of the fine ecological details and often well predicted by neutral theory (Bell, 2001), this does not imply that the underlying ecological processes are neutral (Harte, 2003;Purves and Pacala, 2005). Although the SPT distribution framework does not necessarily require neutral processes as well, we shall keep them as a reference modelling frame for our theoretical speculations.
This paper explores the subject of the SPT distribution further by extending the empirical and theoretical analysis presented in Bertuzzo et al. (2011). In particular, we report here on the study of two new long-term datasets to test the robustness of the SPT macroecological pattern. Moreover, we study in details the connections of the SPT distribution with the structure of the SAR and the endemics-area relationship (EAR, which gives the number of species that are confined within the sampled area; see Kinzig and Harte, 2000). We also investigate the role of spatial interactions in the new SPT data and show that the effect of the environmental matrix (Ricketts, 2001) on SPT distributions depends on the overall geographic area in which the ecosystem is embedded: if the typical dispersal range is compa-rable to the total area, possibly because of dispersal limitation, then spatial interactions may be neglected. Finally we present and solve analytically a new particular case of a length-biased sampling problem (Cox, 1969) that arises in SPT data analysis; namely we find how to relate the SPTs derived from finite observational windows to the inherent SPT distribution of the ecosystem.
The paper is organized as follows: in the next section we briefly summarize the spatially explicit model for SPTs. In subsection 2.1 we discuss the effects of data sampling on SPT distributions. We present a general mathematical framework to relate SPT theoretical distributions to those obtained from finite samples. In the third Section, we present novel empirical SPT distributions regarding two different ecosystems (a forest in New Jersey and an assembly of estuarine fish in the Bristol Channel) and compare them with the distributions predicted by the neutral theory. The role of spatial scales and their interactions with SPT distributions, and the intimate relation between the SPT distribution and the SAR/EAR are also shown. A set of conclusions closes the article.

The Model
Neutral theory (Hubbell, 2001), which assumes that all the individuals within a given trophic level are competitively equivalent, offers a benchmark dynamics suggesting that many aspects of real biotic patterns may not require a more complex model (Muneepeerakul et al., 2008;Bertuzzo et al., 2009). According to the assumption of neutrality (Hubbell, 2001), the dynamics of a species in the ecosystem is fully specified by its effective birth and death rates b(n) and d(n), which depend exclusively on the population size n. Species abundance dynamics is described by the so called birth-death master equation (ME) (Volkov et al., 2003;Houchmandzadeh and Vallade, 2003): where P (n, t) is the probability, for a given species, of having a population of n individuals at time t. We note that b(n) and d(n) take into account several ecological processes that may increase or decrease the number of individuals in a species over time as, for instance, immigration or emigration. Assuming absorbing boundary conditions in n = 0, the probability density function (pdf) of the theoretical persistence time p τ can be expressed as (Pigolotti et al., 2005): Considering b(n) = n(1 − ν) and d(n) ∝ n, the above model describes analytically the mean field dynamics (in the limit of infinite number of nodes) of the following lattice model, known as the voter model (Holley, 1975;Durrett and Levin, 1996;Zillio et al., 2005). ν is termed diversification rate and it measures the frequency of appearance of new species in the system. Let us consider, for simplicity, a system with N nodes (N → ∞). Each node may be occupied by a single individual belonging to a species which is represented by a given color. The dynamics at each time step is as follows. A randomly selected individual dies. With probability ν this site is occupied by a new species; with probability 1 − ν, the site is colonized by an offspring of a node randomly chosen in the lattice (mean field case). We note that in this framework we require that the probability that a new species enters the system be uniformly distributed in time. This seems like a reasonable assumption in the absence of strong environmental or historical gradients within the considered timescale.
The spatial version of this model allows colonizations to occur only from one of the neighbors of the empty patch (i.e. dispersal limitation). The same model can be applied to 1D/3D lattices, or to a networked environment where dispersal is constrained directionally. By tracking every individual up to a statistically stationary state, it is possible to build the time series of species abundances and, from it, to compute numerically the SPT distribution. SPTs prove crucially dependent on the type of spatial connectivity interactions in the voter model. Exact and numerical results for the spatial voter model (Bertuzzo et al., 2011), in fact, show that the SPT distribution is consistent with power-law shapes of the type: where the scaling exponent α strictly depends on the connectivity structure and C = ν 1−α /Γ(1 − α) is the normalization constant (Γ(x) is the complete Gamma function of argument x ). The neutral model thus predicts the same functional shape for the STP distribution, a power law with the exponent α that is solely determined on the connectivity structure of the environment (Figure 1). Therefore α may be seen as a critical exponent (Stanley, 2000) that is, it describes the behavior of physical systems near phase transitions. In general it is argued that scaling exponents do not depend on the details of the physical system, but only on the dimension D of the system (e.g. D = 1 on a line, D = 2 on an isotropic lattice and so on) or the range of the interactions (Stanley, 2000). This is not the first time that similarities between ecosystem dynamics and critical systems in physics have been pointed out (Banavar et al., 1999;Zillio et al., 2008). From Monte Carlo simulations of the spatial voter model, α exponents for different geometric cases can be evaluated. It has been shown (Bertuzzo et al., 2011) that α ≈ 1.5 for the 1D lattice, α ≈ 1.62 for the networked landscape, α ≈ 1.82 and α ≈ 1.92 respectively for the 2D and 3D lattice ( Figure 1). In the mean field case, the scaling limit (ν 1) of p τ (t) can be calculated exactly (Pigolotti et al., 2005) The robustness of the scaling behavior as a function only of the structure of the environmental matrix of interactions is deemed remarkable.

An Ecological Length-Biased Sampling Problem
Empirical data on persistence times for different species are available, especially from succession studies and long-term ecological databases (Pigolotti et al., 2005;Bertuzzo et al., 2011). However, for further empirical analysis, sampling effects on SPTs need to be taken into account. This type of sampling problem belongs to the class of so called length-biased problems, that arise in a wide range of different disciplines (Patil and Rao, 1978;Vardi, 1982).
We start with a thought experiment which we refer to as the stick-length sample problem. Consider an infinite number of sticks of different random length L distributed according to some unbiased pdf, say p L . Imagine to place randomly these sticks on a plane x-y, aligned in the same direction (e.g. along the x direction). The process is thus defined in a 1D space, and the y axis is just a way to visualize different realizations of the same process. We denote by X 0 the position of the left side of a given stick. X 0 is a random variable uniformly distributed in (0, ∞). Consider only those sticks that cross a vertical line x = x * and measure the lengths of those sticks ( Figure 2).
We will denote by p Lc the length distribution of this sticks sample. The stick-length sample problem can be formulated with the following questions: a) how is p Lc related to p L ? b) How are the left and right ends of the sticks that cross x * (indicated by L p , Figure 2) distributed? We now show how length measurements on particular samples of the stick population give length distributions which are different from that of the whole stick population.
Using tools of probability theory, it can be shown (Appendix A) that the pdf of the length of the sticks that cross x * , in the limit x * → ∞ (to avoid boundary effects), is where L = +∞ 0 Lp L (L)dL is the average stick length. Thus the length pdf of the sample constituted by those sticks that cross the observation point x * , p Lc , is not the same as the overall population stick length pdf, i.e. p L .
Let us now introduce a second vertical line x =x < x * . We now want to measure the length of the portions of sticks L p crossed by the two vertical lines ( x =x and x = x * ) that are comprised inside the space window ∆x = x * −x (see Figure 2). If we denote by p Lp the pdf of L p , then it can be shown (Appendix A) that: Finally, we can calculate the pdf p L of the random variable L describing the length of a stick that is totally or partially comprised within the sampling window ∆x. We note that in this case the maximum measured length of a stick is max{L } = ∆x. L can be expressed as a function of the other independent random variables L and X 0 , that have already been probabilistically characterized. in fact, we can distinguish four different cases (see Figure 3): • the stick is completely inside the window, then L = L; • X 0 >x and the stick crosses • the stick is longer that the space window, then L = ∆x.
As explained in detail in Appendix A, we find that: where N is the normalization constant for p L , and δ(x) and Θ(x) are the Dirac delta distribution and the Heaviside step function, respectively. The last term of equation (6) gives an atom probability in L = ∆x corresponding to the fraction of sticks longer than the total length of the observational window (max{L } = ∆x). The probability distribution p L I of the sticks' length inside the window ∆x follows directly from the first term of equation (6): where N is a proper normalization constant. The analytical results provided by Eqs. (6) and (7) are tested via numerical simulation, as shown in the bottom panel of Figure 2. First we generate 10 5 segments along the x − axis of random length drawn from an exponential pdf, i.e. p L (l) = λe −λl . Afterwards, we consider only those segments that are within the spatial window ∆x = 1/λ and we reconstruct the stick length pdf p L I . Analogously, we build p L . Finally we compare the numerical distributions with the the solution of Eqs. (6) and (7). As expected the agreement is perfect.

Application to SPTs.
Several real ecological sampling problems, both in time and space, can be mapped into the thought experiment described in the previous Section. In particular we now apply these results to SPTs.
Assume that species abundance (or presence-absence) time series for a given ecosystem characterized by a single trophic level are available from field campaigns carried on in a time period of span ∆t w = t f − t 0 years. From the collected data, SPTs can be measured and the empirical SPT distribution likewise obtained. This distribution does not correspond to the theoretical SPT distribution, p τ , in the same way as p L = p L in the stick sample problem. In particular, p τ (t) is affected by the fact that species persistence can be recorded only within a finite temporal window ∆t w . Thus, given the theoretical persistence-time pdf described by Eq. (3), we are interested in the pdfs of the variables that we can actually measure, i.e. the persistence times τ of those species emerging before or during the observations and that are still present or go locally extinct within the time window ∆t w (and the persistence time τ I of only those species that emerge and go locally extinct within the observed time window).
In our theoretical framework, the probability of a diversification event in a time step is constant. Thus the emergences of new species in the system obey a Poisson process, say U (t), with rate λ = ν N , where N is the total number of individuals in the system and λ has the dimensions of the inverse of (generation) time. Consequently the emergence times T 0 of species in the system are uniformly distributed. Therefore, we can map exactly this situation into the previous stick-length sample problem, with SPTs replacing stick lengths, i.e. L → τ , L → τ , L I → τ I and X 0 → T 0 . Eqs. (6) and (7) allow us to test the theoretical SPT pdf p τ (t) obtained by the spatial model against the empirical SPT distribution obtained from the measurements.
It is interesting to note how our thought experiment uses similar approaches to the model known as the mid-domain effect, which describes how species ranges that are randomly shuffled within a bounded geographical domain overlap increasingly toward the center of the domain, creating a middomain peak of species richness (Colwell et al., 2004;Arita, 2004). In fact, SPTs are essentially distributions of species temporal ranges, and therefore, the sampling problem just presented may be interpreted as a sort of middomain problem in time. However there is a crucial difference between the two effects: while for spatial ranges there are physical geometric constraints imposed by hard boundaries in space (i.e. geographic boundaries), and thus species spatial ranges must be contained within the given geographic region (Colwell et al., 2004;Arita, 2004), temporal ranges do not have constraints in time. It is in fact to be reminded that species temporal ranges are not limited by the observational time window, rather is our capacity to measure real species temporal ranges that is limited.

Empirical SPTs and Comparison with Model Results.
Bertuzzo et al. (2011) presented a comparison between the theoretical model and the empirical SPT distributions for two different ecosystems (North American breeding birds and Kansas grassland) which is shown in Figure 4a,b for purpose of completeness. We now provide new evidence of the existence and robustness of the SPT distribution pattern by presenting empirical data on persistence times taken from two different ecological databases and comparing them with the analytical results on SPT distributions. Specifically, we empirically characterize SPT distributions of two long-term datasets concerning ecosystems with very different characteristics: 1) a 44-year long study from the Buell-Small Succession (BSS) Study of plants in New Jersey ; and 2) a 28-year long database of British estuarine fish collected at Hinkley Point (headland on the Bristol Channel coast of Somerset, England).
The BSS study (Institute of Ecosystem Studies, 2008) includes ten fields that were released from agriculture and used to study succession dynamics. Permanent plots, measuring 48 m 2 each, were established in every field. The permanent plots were sampled every year from 1958 to 1979, after which they were sampled alternatively every other year to the present day. The fields differ in the year of release, thus we consider only data collected after the latest field abandonment year (1968). In order to avoid missing data in alternate years, the minimum sampled area (afterwards named cell) considered in the calculation of the empirical SPT distributions comprises two adjacent fields (96 m 2 ). In this way each cell is populated with presence-absence records for each year. We repeat the same analysis for 3 other different scales: 4,6,8 adjacent fields (A=192, 288, 384 m 2 , respectively). We also analyze the whole system, which corresponds to an area of A = A * = 480 m 2 . For every scale A of analysis, the presented results refer to the average over all possible combinations of adjacent plots within the system (moving average procedure). A three-dimensional presence-absence matrix P is in this way built. Each element P stc of the matrix is equal to 1 if species s is observed during year t in a cell c, otherwise P stc = 0. The empirical SPT is defined as the number of consecutive years in which the measurements reveal the presence of the species in that geographic region (see Figure 3). For every cell c and every species s we measure persistence times from presence-absence time series derived from the second dimension/index of matrix P . The presence-absence time series thus form a vector of length T , where T is the total number of years of observation, that has as j-th component a one if species s is observed in cell c during the j-th year. Persistence time is defined as the length of a contiguous sequence of ones in the time series (Figure 3). For every scale of analysis we consider all the measured persistence times regardless of the species they belong to, and we build the SPT empirical pdf at every cell scale ( Figure 4c).
As for the second database, fish samples were collected from the coolingwater filter screens at Hinkley Point B Power Station, situated on the southern bank of the Bristol Channel in Somerset (England) from 1980 to 2008. A full description of the intake configuration and sampling methodology is given in Magurran and Henderson (2003). We only consider estuarine fish species (not crustaceous organisms). Empirical SPT pdf's can be computed as described for the BSS dataset ( Figure 4d). Note, however, that in this dataset there is no spatial scale variability, because the samples were collected in a single point in space. Therefore we limit our analysis to a single spatial scale.
Using Eqs. (6) and (7) and the observational data we performed the best fit of α and ν, the parameters of the theoretical SPT pdf. At any spatial scale, the scaling exponent and the diversification rate for the empirical SPT distributions are determined with a simultaneous nonlinear fit of observational and analytical pdfs of τ and τ I . Table 1 summarizes the results of the fit. Remarkably, in both cases the scaling exponents derived empirically are consistent with those predicted by the neutral voter model. In particular, for the fishes we find α = 1.97 ± 0.06, which is compatible with the exponent α = 2 predicted by the mean field voter model (see Figure 4d). In fact, no spatial information is embedded in our empirical data and thus we do not expect to see the effect of dispersal limitation on the relevant ecosystem dynamics. Figure 4c shows the case of the New Jersey plant communities, for which the best fit of p τ isᾱ = 1.97 ± 0.12 (whereᾱ is the average of the exponents over all the cell scales), which is also compatible with the mean field prediction α = 2. In this case the negligible effects of dispersal limitation on the ecosystem dynamic suggests that the average plant dispersal radius R is comparable with the linear dimension of the total sampled area, i.e. R ≈ √ A * , and thus the net result is a mean field dynamics not affected by spatial interactions (Bertuzzo et al., 2011). The fact that the total sampled area of the ecosystem is very small (A * = 480 m 2 ) strongly supports this interpretation of the data.
While studying SPTs in a given ecosystem on the basis of presence/absence (or count) data, imperfect detection of species is a source of concern (Nichols et al., 1998). Species, in particular animal species, are, in fact, routinely sampled with a detection probability much smaller than one. Although in general the problem of imperfect detection is somewhat less relevant for herbaceous plant datasets, in the case of the BSS forest sampling occurs either annually or biannually, e.g. implying that seasonal variability in plant composition may be not picked up (and this varies from year to year). However, plants that go undetected because of yearly sampling affect the estimates of persistence times shorter than one year, thus not an issue for the scaling of SPTs. Moreover, plants in vegetative state are not identifiable, nor are small recruits, typically. For the fish database the power station intakes at Hinkley Point are an effective sampler because of their location at the edge of a large inter-tidal mudflat in an estuary. The filter screens have a solid square mesh of 10 mm and start to retain fish of standard length of about 25 mm (Henderson and Holmes, 2001). A 99% retention for many species occurs at the standard length of 40 mm. Nevertheless, incomplete sampling is present because only a small fraction of the water in the Bristol channel is filtered, and filtering thus only catches fish that cannot avoid being sucked by the water turbines. Hence, even in intensive sampling programs is difficult to tell whether absences are real or caused by incomplete detection/sampling. Note that the impact of imperfect sampling on SPTs of North America breeding birds has been analyzed in details in (Bertuzzo et al., 2011). However, we acknowledge that to show that SPTs are generally a more robust alternative to estimating extinction rates, a systematic and widespread analysis of the effect of incomplete sampling and detection on SPTs needs to be undertaken. This is still lacking and will be tackled in details in future works.

Relations among SPT, SAR and EAR
The SPT distribution is also particularly interesting because it generates spatial patterns that can be related, within our theoretical framework, to other important macroecological patterns, as the SAR and the EAR relationships. We can use the BSS data at different spatial scales to investigate how SPTs can give information also on the biogeography of the BSS plant ecosystem. In fact, the scale-invariant character of p τ indicates that α depends only on the spatial connectivity of the environment (i.e. networked, 2D, ..), and not on the sampling area. On the contrary ν, which accounts for immigration processes from species outside the local community, is argued to decrease with increasing sampling area, thus representing a clear signature of the geographic scale of the analysis. The SAR characterizes how the average number of observed species increases with increasing sample area and it is usually characterized by a power-law behavior, i.e. S ∼ A z (Brown, 1995).
Our theoretical framework allows for the linkage of the cut-off timescale 1/ν with the spatial scale of analysis, and the SPT distribution to the SAR. In fact, in the neutral model, species emerge as a point Poisson process with rate νN and last for a lifetime τ . Therefore, the mean number of species in the system at stationarity is S = νN τ (Bertuzzo et al., 2011). By deriving the scaling behavior of ν, N and τ with respect to A, we can thus make a connection between SPTs and the SAR.
The last term in Eq. (6) represents the analytical expression of the fraction of species that are present during the whole observational time period (S(∆t w )). It provides a tool to quantitatively estimate the diversification rate from the empirical SPT pdf. In fact, in order to infer with better precision ν from the observational data, we numerically find the value of ν that minimizes S(∆t w ) at every different spatial scales A (see Appendix C for details) using the scaling exponentᾱ. The best fit of the scaling law between ν and A obtained in this way gives ν ∼ A −β , with β = 1 ± 0.1 (Figure 5a), thus confirming that the diversification time ν −1 scales almost linearly with the area (Zillio et al., 2005).
Once the scaling law between ν and A is established, in order to derive the SAR, we need to estimate the scaling between the total number of individuals N and the sampled area A. In Bertuzzo et al. (2011) an isometric relation N ∝ A (MacArthur and Wilson, 1967) was assumed. However the abundance data on the New Jersey plant communities show that this is not the case for that ecosystem. In fact the average total species coverage percentage over all the plots is 165%. Samplers start with the topmost layer, usually the trees, and work their way down to the species closest to the ground (Institute of Ecosystem Studies, 2008). Thus the exceedance in the coverage percentage is due to the overhanging among species. In order to evaluate the scaling relation between N and A, we then build a minimalist model, in which species are classified by their typical average size. Assuming biotic saturation, i.e. that each species covers completely the sampling region, in a plot of area A we can find at most [A/a] individuals of characteristic size area a A ([x] denotes the smallest integer not greater that x), [A/2a] individuals of characteristic area 2a and so forth. In general, the total number of individuals N in a region of area A scales as where H q = q n=1 1/n is the harmonic number and H A/a ∼ ln(A/a). The constant of proportionality in Eq. (8) depends on the arbitrary choice of a. However since H A/a ∼ ln(A/a)+ sub-leading terms, a change of a in λa leads to H λa = 1 λ H A/a + sub-leading terms and so its precise value is irrelevant since it can be absorbed in the proportionality constant (see below) to be determined from the best fit of the data.
Finally, we calculate the asymptotic behavior of the mean SPT τ = tp τ (t)dt (see Appendix C for mathematical details) and we get: Putting all the above scaling results together, we find that for the BSS plant ecosystem (which is characterized by α ≈ 2) the mean-field neutral framework predicts a power-law SAR with logarithmic correction: where K is the constant of proportionality. We determine the constant K in three different ways: 1) through the best fit of Eq. (10) on the empirical SAR data; 2) by imposing that in the total area A * we have the total number of species S * ; 3) by finding the value of K that gives the exact number of species in the smallest area (A = 96 m 2 ) and then predicting the complete SAR curve (upscaling). Remarkably, all three methods yield K ≈ 3.46. We also performed the best fit of the power-law SAR (S ∼ A z ) on the empirical data, obtaining z = 0.34. We note that in our range of areas, this result it is not distinguishable from the SAR given by Eq. (10) as shown by Figure 5c. In the special case of mean field interactions, we can also calculate the EAR. EAR is defined as the relation between the number of species endemic to a region and the area of that region (Kinzig and Harte, 2000;Green and Ostling, 2003). The EAR has been recently proposed by He and Hubbell (2011) as the correct approach in order to estimate extinction rates from habitat loss. The fact that the mean-field approach well describes the empirical SPT distribution suggests that in the BSS ecosystem the phenomenon of spatial aggregation or clustering is negligible. In fact mean-field dynamics indicates that species composition is well mixed in the ecosystem and thus it does not depend on spatial features. Therefore, in this case, we can evaluate the EAR curve just by reversing the SAR (He and Hubbell, 2011): where E is the number of endemic species. Figure 5 summarizes all the above results and shows the comparison between the empirical and theoretical results for the SAR and the EAR.

Conclusions
We have presented new theoretical and empirical evidence supporting the broad validity of a recently proposed macroecological pattern, the SPTs distribution. Specifically, we have completed (and described in detail) the theoretical treatment of the relevant ecosystem dynamics and of the derived sampling problem, and analyzed empirically two further rather diverse ecosystems (hosting respectively herbaceous plants and estuarine fishes). In both cases the observed SPT distributions display a power-law shape with a cut-off due to the finiteness of the observational time window, which is reproduced by the theoretical model. These results are expected to be robust with respect to relaxations of specific assumptions on ecological neutrality (Bertuzzo et al., 2011), thus confirming our expectation that power laws, observed for SPT pdfs, are the result of emergent behaviors independent of fine details of the system dynamics.
The specific values of the scaling exponents of the SPT distributions depend only on a few key factors, namely the spatial dimension of the embedding ecosystem matrix and the nature of dispersal limitation. In particular, our results show that dispersal limitation does not occur in the ecological dynamics of both ecosystems under study. While this result could be expected for the estuarine fishr inventory, where data were collected at a single point in space, quite surprisingly the BSS plant community does not show signs of dispersal-limited spatial interactions. We interpret this result as a consequence of the fact that the total sampled area in the BSS forest is very small. This suggests that the average plant dispersal radius is comparable to the characteristic size of the ecosystem, thus complying to mean-field-like dynamics.
Because of the assessed, robust scale invariance character of the SPT distribution (limited only by biogeographical finite-size effects), and because of its relation with other macro-ecological patterns, we have proposed that the SPTs distribution is a powerful synthetic descriptor of ecosystem biodiversity and of its associated dynamics.
New data would allow for the investigation of other implications of the theoretical predictions. For instance, the neutral voter model predicts that the scaling exponents of the SPT distributions of riparian ecosystems (i.e. networked environments where directional, anisotropic dispersal is forced by the structure of the fluvial environmental matrix) should be lower than those of 2D, 3D or mean-field ones. This result, if proven, would have remarkable consequences for conservation ecology, because it suggests that species that disperse isotropically have shorter average persistence times than species that are constrained to disperse along spatially constrained and anisotropic ecological corridors, like those provided by river networks. This, in turn, calls for long-term analysis of riparian ecosystems to test empirically the effects of river morphology on ecosystem dynamics.
For these reasons we suggest that field biologists and ecologists should perhaps invest renewed efforts in collecting improved long-term datasets of ecosystem dynamics at different spatial and temporal scales and under different dispersal conditions (for instance, archives beyond presence/absence of species within networked environments like riverine ecosystems for freshwater fish) as they prove crucial for our deeper understanding of universal patterns in macroecology. The present study shows that long-term datasets can be profitably used to highlight crucial properties and spatial effects of ecosystem dynamics, including the role played by the underlying connectivity structure in shaping SPT distributions.

Appendix A.
In this appendix we derive the analytical results of the stick-length sample problem presented in the main text. The set up of the thought experiment is fully described in the main text and in Figure 2. Specifically, the stick-length sample problem can be characterized by five random variables, namely: i) X 0 is the position of the left side of a given stick; ii) L describes the length of a stick; iii) L c is the length of those sticks that cross x * ; iv) L p is the length of that part of the stick that crosses x * or x and fall within ∆x; v) L is the length of a stick that is totally or partially comprised inside ∆x. X 0 and L are the two independent random variables of the problem, while L c , L p and L can all be expressed in terms of X 0 and L, as we will show in details. Assuming that p L is the unbiased length stick distribution, we now present a methodology that enable us to derive the length pdf p Lc , and the pdf p Lp as a function of p L . The same methodology may be applied to obtain p L The random variable L p can be expressed as a function of the random variables L and X 0 . In fact, we have that the sticks that cross x * are those for which X 0 < x * and X 0 + L > x * (see Figure 2). Therefore, if Θ is the Heaviside step function, the length of a stick that crosses x * is given by the random variable We know that L is distributed (∼) as p L and that X 0 is uniformly distributed along the x-axis (in particular X 0 ∼ U (0,x * ) ). Therefore we have that the probability density function (pdf) p Lc is where δ is the Dirac delta function. The numerator of Eq. (A.2) simply gives the ensemble average, taken over the random variables X 0 and L, of different realizations of the event that occurs when a stick of length L intersects x * , while the denominator in (A.2) assures the normalization condition.
In order to solve the ensemble average of the latter equation, we split the calculation into two steps. First we calculate the conditional probability function p Lc (l|L) of a stick crossing the vertical line x * , given the fact that it has a length L, i.e. we perform only the average over X 0 As a second step, we marginalize Eq. (A.3) over L, i.e. we perform the average also over L: Assuming that we do not have boundary effects (i.e. x * → +∞) we obtain Eq. (4) in the main text. The case of L p can again be obtained using the same technique. Writing the random variable L p as a function of L and X 0 (see figure 2) we get Following the same steps presented before, we obtain and marginalizing over L we obtain that, in the limit x * → ∞, results in Eq. (5). The case of a stick crossing x =x is perfectly symmetric to the one discussed above and would lead to the same result.
The results for L and L I follow similarly.

Appendix B.
In this section we present the calculations used to estimate the diversification rate ν from the empirical SPT distributions of the BSS plant community. We denote by E(∆t w ) the fraction of species, evaluated from the data, that are present during the whole observational time period ∆t w .
The last term in Eq. (6), denoted here as A ∆tw , gives the theoretical value of the latter quantity, i.e. the atom probability of the SPT distribution for t = ∆t w . By using the theoretical SPT distribution described by Eq.
(3) we can estimate A(∆t w ): where Γ[a, b] is the incomplete Gamma function, and t min is the minimum SPT observed in the ecosystem (in general t min → 0). If α < 2, then the dependence of Eq. (B.1) on νt min is weak, and thus, as νt min 1, we can set νt min ≈ 0. Therefore we calculate the values of the diversification rate ν for a given ∆t w (that in turn depends on the specific sample area A) by numerically minimizing the quantity where the average scaling exponentᾱ obtained by the best fit of the SPT pdf p τ and p τ I has been used for α . Eventually, repeating the same procedure using different time windows (corresponding to the different sampled areas) we obtain the scaling relation between ν and A presented in the main text.
Appendix C.
Therefore we find that τ scales as the logarithm of ν and, in turn, as the logarithm of the sampled area A (in fact we found that ν ∼ A −1 ). The logarithmic correction in the scaling behavior of τ is supported by the data (see Figure 5 in the main text) confirming the hypothesis that the evolution of the BSS ecosystem is well mimicked by mean field dynamics. The variables that can be measured from the sample of sticks in ∆x are L and L I , both with a different distribution with respect to p(L). Bottom panel: comparison between analytical SPT pdfs p L and p L I given by Eqs (6)- (7), and those obtained through numerical simulation of the stick length problem starting from an unbiased stick length distribution p L (l) = λe −λl , with ∆x = 1/λ (in this particular example λ = 0.025). p L , p L I and p L have been shifted in the semi-log y plot for clarity. The species persistence-time (SPT) within a geographic region is defined as the time incurred between a species' emergence and its local extinction. Recurrent colonizations of a species define different SPTs times. The SPTs τ p , τ I and τ , that can be measured within the observational window, are in general different from the unbiased SPT τ . The four different cases are discussed in the main text. The spatial scale of analysis is A = 10, 000 km 2 and ∆t w = 41 years for (a), A=1 m 2 and ∆t w = 38 years for (b), A=480 m 2 and ∆t w = 44 years for (c) and ∆t w = 28 years for (d) (for details on the analysis of the databases relevant to (a) and (b) see supplementary material in Bertuzzo et al., 2011). The finiteness of the time window imposes a cut-off to p τ (t) and an atom of probability in t = ∆t w to p τ (t), which corresponds to the fraction of species that are always present during the observational time. p τ I (t) and p τ (t) have been shifted in the log-log plot for clarity. (3), and logarithmic relation τ ∼ ln(A) (dashed red line) predicted by the neutral model for the mean field case (α = 2); c) empirical SAR and EAR relationships (black points), power law fitting of the SAR and the corresponding EAR (gray continuous line) and the mean field SAR and EAR given by Eqs. (10) and (11) (7) on the empirical SPT distributions. They suggest that both ecosystems are driven by a mean field dynamics. The value ofᾱ given in the text corresponds toᾱ = ( 5 i=1 α i )/5 = 1.97 ± σᾱ, where σ 2 α = 5 i=1 σ 2 αi /5 = 0.015 (σ 2 αi is the variance for the i−th α exponent from the fit). Values of ν for the BSS plants are obtained by the minimization procedure explained in the text and in Appendix B.