The  population sizes of five perennial vascular plant species confined to old unimproved dry grasslands were  assessed, viz. &lt;i&gt;Anthericum ramosum&lt;/i&gt;, &lt;i&gt;Filipendula vulgaris&lt;/i&gt;,  &lt;i&gt;Silene nutans&lt;/i&gt;, &lt;i&gt;Thymus pulegioides&lt;/i&gt;, and &lt;i&gt;Thymus serpyllum&lt;/i&gt;.  All populations within the region were included.  Only for &lt;i&gt;Filipendula vulgaris&lt;/i&gt; and &lt;i&gt;Thymus serpyllum&lt;/i&gt;,  significant relationships between habitat area and  population size were found. Thus, apparently perennial  vascular plants have a limited ability to respond  to large habitat areas by forming large populations.  This puts a question mark on the use of incidence-function  models for the study of plant metapopulations, because  these models are based on an assumed positive relationship between habitat area and population size

Lee, Ji

Schnelli, Kevin

Ji Oon Lee

Kevin Schnelli

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Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population

We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that σ is diagonal and that the entries of X have a sub-exponential decay

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Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population

H. H. Bruun

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1 WEB ECOLOGY 5, 2005Web Ecology 5: 1–5.Accepted 1 January 2005Copyright © EEFISSN 1399-1183Empirical studies of metapopulation dynamics (Hanski1999) have gained considerable momentum over the pastdecade. Parallel to this, there has been much debate aboutthe applicability of the concept of the metapopulation toplants and many empirical studies (e.g. Freckleton andWatkinson 2002, Bullock et al. 2002, Ehrlén and Eriksson2003, Ouborg and Eriksson 2004). Two of the most im-portant reasons for interest in metapopulations, both ingeneral and for plants in particular, have been the apparentsuitability of the metapopulation concept to conservationefforts in landscapes with increasingly fragmented habi-tats, and the development of practically applicablemetapopulation models. While some models requirepopulation censuses in each population (e.g., Akçakaya2001), one of the less data demanding, and therefore morepopular, methods is the incidence function model (Hanskiet al. 1996, ter Braak et al. 1997). This type ofmetapopulation model allows for the simplifying assump-tion that extinction probability increases linearly with de-creasing habitat area. Making this assumption allows oneto replace laborious population censuses with the inci-dence, i.e. presence or absence, of the focal species fromeach patch of suitable habitat.The assumption of a habitat area-extinction probabilityrelationship has the following rationale: 1) extinction riskmust increase with decreasing population size, at least be-low a certain threshold (Barrett and Kohn 1991, Ellstrandand Elam 1993, Fischer and Stöcklin 1997, but see Hus-band and Barrett 1996), and 2) population size is likely todecrease with decreasing habitat area. Assumption 2 is jus-tified if populations respond immediately to changes inresources, and if no dispersal limitation exists (Ouborg andEriksson 2004). Therefore, it applies better to fast-repro-ducing (and short-lived) and highly motile organisms,than to long-lived sessile organisms. Most vascular plantsare found in the latter end of this continuum. Species con-A field test of the relationship between habitat area andpopulation size for five perennial plant speciesHans Henrik BruunBruun, H. H. 2005. A field test of the relationship between habitat area and populationsize for five perennial plant species. – Web. Ecol. 5: 1–5.The population sizes of five perennial vascular plant species confined to old unim-proved dry grasslands were assessed, viz. Anthericum ramosum, Filipendula vulgaris, Si-lene nutans, Thymus pulegioides, and Thymus serpyllum. All populations within the re-gion were included. Only for Filipendula vulgaris and Thymus serpyllum, significant rela-tionships between habitat area and population size were found. Thus, apparently peren-nial vascular plants have a limited ability to respond to large habitat areas by forminglarge populations. This puts a question mark on the use of incidence-function modelsfor the study of plant metapopulations, because these models are based on an assumedpositive relationship between habitat area and population size.H. H. Bruun (hans_henrik.bruun@ekol.lu.se), Lund Univ., Dept of Ecology, Section forPlant Ecology and Systematics, Ecology Building, SE-223 62 Lund, Sweden.2 WEB ECOLOGY 5, 2005fined to natural and semi-natural habitats of conservationinterest often have strong propensities to form remnantpopulations (Eriksson 1996).The present paper aims at testing whether populationsize for five perennial vascular plant species inhabitingScandinavian grassland patches is well predicted by habitatpatch area.Materials and methodsStudy areaThe studied area (483 km2) is situated on the island Zea-land (Sjælland) in eastern Denmark (12°E, 55°45´N). Theclimate is temperate (mean annual temperature 7.7°C;mean annual precipitation 500–600 mm). Semi-naturalgrasslands occur scattered throughout the area. Theymostly occur on areas inaccessible to agriculture due to to-pography, i.e. often on steep slopes, mainly fixed coastalslopes from the Littorina sea transgression. Another im-portant group of patches constitute Bronze-age burialmounds, which are circular vaulted mounds 5–15 m indiameter and 2–5 m high, originally built of sods (Holst etal. 1998). Grazing has ceased on most burial mounds inthe 1930s due to a general protection act, whereas it has con-tinued until around 1970 or later on all other grassland sites.Assessment of population sizeThe species composition in 97 sites of old unimprovedsemi-natural grassland, comprising all available siteswithin the study area, was investigated in the summers1996–1997 (Bruun 2000a). Species also occurring fre-quently in other habitats were identified by use of threeexternal data sets: 1) a data set of species observed duringthe field survey at other sites than old grasslands, 2) a dataset on ground floor species in deciduous forest in the sameregion (Graae and Sunde 2000), and 3) data on speciesfrom road verges in the island of Sjælland (Hansen andJensen 1972). The deletion of these generalist species left76 perennial herbaceous grassland specialist species out ofa total of 210 species. Of these five species were selected soas to differ as much as possible on the number of habitatpatches occupied. The same species were subjected topopulation genetic studies (Rosquist and Prentice 2000,Weidema et al. 2000, unpubl.), so the final selection ofspecies was based on their suitability to allozyme electro-phoresis. For these species, Anthericum ramosum  L.,Filipendula vulgaris  Moench,  Silene nutans  L.,  Thymuspulegioides L., and Thymus serpyllum L., population size(adult plants) on each occupied habitat patch was esti-mated, for small populations by simple counts, for largepopulations by extrapolating counts on unit area to theoccupied area within the grassland patch as estimated us-ing measuring bands. The total area of grassland patcheswas estimated from the digitised polygons using geo-graphical information systems. Based on the focal species’occurrence in classes of soil acidity and water availability ina large external data set (2133 records of vegetation andlocal environment in Danish grasslands and related com-munities) and on the range of measured soil acidity andwater availability at each of the focal grassland site, all fivespecies were expected to occur potentially in all 97 sites(data from Bruun 2000b).Relationships between patch area and population sizeof each of the five species, with all variables subjected tologarithmic transformation, was sought for by two differ-ent methods: conventional linear regression and parti-tioned regression (Thomson et al. 1996). Initial visual in-spection of the bivariate double-log scattergrams (Fig. 2)showed an apparent ceiling, i.e. that y < a*x, where y is logpopulation size, a is a constant, and x is patch area. Thus,population size is constrained to low values at small sites,whereas it can take a broader range of values at large sites.Partitioned regression was used in an attempt to parame-terize this ceiling. It was done by first fitting a regressionline through the entire point cloud by linear least-squaresregression, then dividing the data into two subsets accord-ing to the sign of their residuals, and then to repeat thatprocedure within the subsets.ResultsThe five species investigated exhibit highly varying re-gional abundance; F.  vulgaris  occurs at 61 sites, T.pulegioides at 23, T. serpyllum at 17, S. nutans at 11, and A.ramosum at 8 sites (Fig. 1). Scatterplots of population sizeversus patch area seemed in all five cases to have a ceiling.In  other words small sites supported only smallpopulations, whereas large sites could support populationsof any size. However, linear regression only showed a re-sponse of population size to patch area in the cases of F.vulgaris (y = 0.61x–0.08; r2 = 0.34; F = 30.8; p < 0.001)and T. serpyllum (y = 0.56x–1.20; r2 = 0.35; F = 8.13; p =0.012) (Fig. 2). In addition, partitioned regression analysisdetected an apparent ceiling for T. serpyllum (r2 = 0.63; F =8.61; p = 0.033) (Fig. 2). For the other three species, nosignificant relationships were found. In the case of T.pulegioides an outlying data point above the ceiling was re-moved, but that did not change the regression result. Like-wise, for S. nutans an outlier representing a population of afew individuals in a very large grassland site was removed,but again it did not change the results.DiscussionRelationships between population size and habitat areawere found for F. vulgaris and T. serpyllum. Thus, only two3 WEB ECOLOGY 5, 2005Fig. 2. Scatterplots of population size (number of individuals) against habitat patch area (m2) (both transformed by the natural logarithm) for Filipendula vulgaris, Thymus pulegioides, T.serpyllum, Silene nutans and Anthericum ramosum. Significant relationships are indicated by “envelopes” drawn according to a (partitioned) regression analysis.Fig. 1. Distribution and population size of Filipendula vulgaris, Thymus pulegioides, T. serpyllum, Silene nutans and Anthericum ramosum within the study area. Marine areas are shaded. Thedelimitation of the study area on land is indicated by a thin line. The smallest dots indicate unoccupied grassland sites. Occupied sites are indicated by circles with diametres indicatingpopulation size (:1000, :500, :100). The situation of the study area within Denmark is inset in the leftmost panel.4 WEB ECOLOGY 5, 2005out of five investigated species seem to be able to respondto large habitat areas by forming large populations.Jacquemyn et al. (2002) found a significant relationshipbetween forest area and population size for the forest herbPrimula elatior. However, if the results of the present inves-tigation apply more generally, a fundamental assumptionof incidence function models is not met with. This sheddoubt on their general applicability (contrary to Hanski1999) and on empirical studies of plant metapopulations(e.g. Quintana-Ascencio and Menges 1996). All five spe-cies are long-lived perennials with moderate reproductiveoutput. Yet, they seem to differ in one or more vital at-tributes, which might be potential population growth rate.Alternatively, variation among the species in other at-tributes might explain results, e.g. dispersal ability andniche breadth. These will be discussed in turn.The dispersal among habitat patches should not influ-ence the relationship between population size and habitatarea. However, dispersal within habitat patches may wellbe limited (Ehrlén and Eriksson 2000), and the effect ofsuch dispersal limitation on population size should in-crease with habitat area. Dispersal ability and potentialpopulation growth rate are probably both governed in partby population-level reproductive output. Thus, the region-ally and locally most abundant species should also be ex-pected to have the highest ability to disperse into emptyhabitat. Filipendula vulgaris was the most abundant of allgrassland specialist species within the study area, occurringin 67% of the sites. This is similar to observations in cen-tral Sweden reported by Eriksson and Kiviniemi (1999). Inaddition, F. vulgaris had larger local populations (two pop-ulations consisting of ca 4000 individuals) than did T.pulegioides, T. serpyllum (both up to 500 individuals), andS. nutans (up to > 800 individuals). However, A. ramosumalso had large local populations (maximum 4500 individu-als), but was regionally sparse, thus contradicting the pat-tern of local and regional abundance being related. In con-trast to the other investigated species, A. ramosum is closeto its northern distribution range in the study area. Thismay cause seed production and/or recruitment to be ham-pered by climate in some, if not most, years (Pigott 1992,Jump and Woodward 2003). These population processeswere not investigated in the present study, however.Niche breadth and potential population growth ratecould probably be linked. If the local environment in thegrassland patches would fit suboptimally to the persistenceniche or the regeneration niche of a species, populationgrowth would be halted. The five study species are all grass-land specialist species, and according to a modelling ap-proach based on soil pH and potential insolation they allshould have the potential to inhabit all investigated grass-land patches (Bruun 2000b). However, a more accuratemodel, or sowing experiments, might identify environ-mental gradients than those mentioned along which thegrassland patches would vary. In any case, the total diversi-ty of grassland specialist plant species is higher in thenorthern part of the study area for reasons of both present-day landscape configuration and land-use history back tothe Iron Age (Bruun et al. 2001).Among the investigated species, the relationship be-tween patch area and population size appeared to bestronger for more frequent species. Thus, perhaps too fewdata points were available to detect a relationship for theless frequent species. There is no doubt that a lower detec-tion limit exists, but T. pulegioides with 23 sites was clearlyabove it and still showed no significant relationship, and11 and 8 sites for A. ramosum and S. nutans, respectively,should also be enough if the patch area-population size re-lationship was strong. Moreover, incidence function meta-population models are mainly of interest to practical con-servation of plant species that are scarce. Thus, plant meta-population studies cannot escape the problem of smallsample size for infrequent species, even at investigationscales similar to the present (ca 500 km2).In conclusion, it is not justified to assume a positiverelationship between habitat area and population size forall these grassland species, and probably not for perennialplants in general. This challenges uncritical usage of inci-dence models to investigate plant metapopulations, butnot the idea that plants may form metapopulations.Acknowledgements – I am deeply indebted to Inger Weidema,Line Magnussen and Gabrielle Rosquist for access to populationcensus data.ReferencesAkçakaya, H. R. 2001. Linking population-level risk assessmentwith landscape and habitat models. – Sci. Total Environ.274: 283–291.Barrett, S. C. H. and Kohn, J. R., 1991. Genetic and evolution-ary consequences of small population size in plants: implica-tions for conservation. – In: Falk, D. A. and Holsinger, K. E.(eds), Genetics and conservation of rare plants. Oxford Univ.Press, pp. 3–30.Bruun, H. H. 2000a. Patterns of species richness in patches ofdry grassland in an agricultural landscape. – Ecography 23:641–650.Bruun, H. H. 2000b. Deficit in community species richness asexplained by area and isolation of sites. – Div. Distribut. 6:129–135.Bruun, H. H. et al. 2001. Plant species richness in grasslands:the relative importance of contemporary environment andland-use history since the Iron Age. – Ecography 24: 569–578.Bullock, J. M. et al. 2002. Plant dispersal and colonisation proc-esses at local and landscape scales. – In: Bullock, J. M., Ken-ward, R. E. and Hails, R. S. (eds), Dispersal ecology. Black-well, pp. 279–302.Ehrlén, J. and Eriksson, O. 2000. Dispersal limitation and patchoccupancy in forest herbs. – Ecology 81: 1667–1674.Ehrlén, J. and Eriksson, O. 2003. Large-scale spatial dynamics ofplants: a response to Freckleton & Watkinson. – J. Ecol. 91:316–320.5 WEB ECOLOGY 5, 2005Ellstrand, N. C. and Elam, D. R. 1993. Population genetic con-sequences of small population size. – Annu. Rev. Ecol. Syst.24: 217–242.Eriksson, O. 1996. Regional dynamics of plants: a review ofevidence for remnant, source-sink and metapopulations. –Oikos 77: 248–258.Eriksson, O. and Kiviniemi, K. 1999. Site occupancy, recruit-ment and extinction thresholds in grassland plants: an ex-perimental study. – Biol. Conserv. 87: 319–325.Fischer, M. and Stöcklin, J. 1997. Local extinction of plants inremnants of extensively used calcareous grasslands 1950–1985. – Conserv. Biol. 11: 727–737.Freckleton, R. P. and Watkinson, A. R. 2002. Large-scale spatialdynamics of plants: metapopulations, regional ensemblesand patchy populations. – J. Ecol. 90: 419–434.Graae, B. J. and Sunde, P. B. 2000. The impact of forest conti-nuity and management of forest floor vegetation evaluatedby species traits. – Ecography 23: 720–731.Hansen, K. and Jensen, J. 1972. The vegetation on roadsides inDenmark. Qualitative and quantitative composition. –Dansk Bot. Arkiv 28: 21–61.Hanski, I. 1999. Metapopulation ecology. – Oxford Univ.Press.Hanski, I. et al. 1996. The quantitative incidence functionmodel and persistence of an endangered butterfly metapop-ulation. – Conserv. Biol. 10: 578–590.Holst, M. K., Breuning-Madsen, H. and Olsson, M. 1998. Soilforming processes in and below a bronze age burial moundat Lejrskov, southern Jutland. – Dan. J. Geogr. 98: 46–55.Husband, B. C. and Barrett, S. C. H. 1996. A metapopulationperspective in plant population biology. – J. Ecol. 84: 461–469.Jacquemyn, H., Brys, R. and Hermy, M. 2002. Patch occupan-cy, population size and reproductive success of a forest herb(Primula elatior) in a fragmented landscape. – Oecologia130: 617–625.Jump, A. S. and Woodward, F. I. 2003. Seed production andpopulation density decline approaching the range-edge ofCirsium species. – New Phytol. 160: 349–358.Ouborg, N. J. and Eriksson, O. 2004. Towards a metapopula-tion concept for plants. – In: Hanski, I. and Gaggiotti, O.E. (eds), Ecology, genetics, and evolution of metapopula-tions. Elsevier, pp. 447–469.Pigott, C. D. 1992. Are species distributions determined by fail-ure to set seed? – In: Marshall, C. and Grace, J. (eds), Fruitand seed production: aspects of environmental physiologyand ecology. Cambridge Univ. Press, pp. 203–216.Quintana-Ascencio, R. F. and Menges, E. S. 1996. Inferringmetapopulation dynamics from patch-level incidence ofFlorida scrub plants. – Conserv. Biol. 10: 1210–1219.Rosquist, G. and Prentice, H. C. 2000. Habitat fragmentationand the structure of genetic diversity within disjunct iso-lates of Anthericum ramosum L. (Anthericaceae) in Scandi-navia. – Biol. J. Linn. Soc. 69: 193–212.ter Braak, C. J. F., Hanski, I. and Verboom, J. 1997. The inci-dence function approach to modeling of metapopulationdynamics. – In: Bascompte, J. and Sole, R. V. (eds), Modelingspatiotemporal dynamics in ecology. Springer, pp. 167–188.Thomson, J. D. et al. 1996. Untangling multiple factors in spa-tial distributions: lilies, gophers, and rocks. – Ecology 77:1698–1715.Weidema, I. R., Magnussen, L. S. and Philipp, M. 2000. Geneflow and mode of pollination in a dry-grassland species,Filipendula vulgaris (Rosaceae). – Heredity 84: 311–320.

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A field test of the relationship between habitat area and population size for five perennial plant species

ISTA Research Explorer (Institute of Science and Technology Austria)

Abstract. The population sizes of five perennial vascular plant species confined to old unimproved dry grasslands were assessed, viz. Anthericum ramosum, Filipendula vulgaris, Silene nutans, Thymus pulegioides, and Thymus serpyllum. All populations within the region were included. Only for Filipendula vulgaris and Thymus serpyllum, significant relationships between habitat area and population size were found. Thus, apparently perennial vascular plants have a limited ability to respond to large habitat areas by forming large populations. This puts a question mark on the use of incidence-function models for the study of plant metapopulations, because these models are based on an assumed positive relationship between habitat area and population size.
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We consider sample covariance matrices of the form Q = (Sigma X-1/2)((EX)-X-1/2)*, where the sample X is an M x N random matrix whose entries are real independent random variables with variance 1/N and where Sigma is an M x M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest resealed eigenvalue of Q when both M and N tend to infinity with N/M -&gt; d is an element of (0, infinity). For a large class of populations Sigma in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that Sigma is diagonal and that the entries of X have a sub-exponential decay.

A field test of the relationship between habitat area and population size for five perennial plant species

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