Evenness and Abundance Models James A DanoffBurg Dept

Evenness and Abundance Models James A. Danoff-Burg Dept. Ecol. , Evol. , & Envir. Biol. Columbia University

Today: Evenness and Abundance Models Evenness – a review Evenness – new issues Introduction to the Models Geometric Series Log-Normal Series Broken-Stick Model All series / models will have some associated worked examples and computer time Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Diversity of Diversities Difference between the diversities is usually one of relative emphasis of two main envir. aspects Two key features þ þ Richness Abundance – our emphasis today Each index differs in the mathematical method of relating these features þ þ One is often given greater prominence than the other Formulae significantly differ between indices Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Evenness Definition of Evenness þ þ How equally abundant are each of the species? A simple way to combine abundance and richness Rarely are all species equally abundant þ Some are better competitors, more fecund, more abundant in general than others Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Evenness increases diversity Increasing evenness greater diversity þ True for all indices S=4 N=8 Higher Evenness, Diversity Site 1 Lecture 3 – Evenness & Species Abundance Models Site 2 © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Evenness as an Indicator For many ecosystems, high evenness is a sign of ecosystem health þ þ þ Don’t have a single species dominating the ecosystem Often invasives dominate Paradox of enrichment • E. g. , polluted / enriched Lake Okeechobee, Florida þ Disturbed areas are mostly edge species • Simple biodiversity • Dominance of a few species ecologically, numerically Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Evenness Across Locations Between ecosystem comparability is usually not possible þ Some areas have lower biodiversity naturally than others • Tiaga is naturally much less even than the deciduous forest • Tiaga is often dominated by a single species (e. g. , Blue Spruce) þ Seasonality may confound the comparison as well • Earlier in temperate growing season, less even than later þ This is a general principle for most all indices this term When would you want to compare across locations? þ þ Trying to prioritize areas for conservation Based largely on biodiversity (not ecol. uniqueness) Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Today: Evenness and Abundance Models Evenness – a review Evenness – new issues Introduction to the Models Geometric Series Log-Normal Series Broken-Stick Model Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

New Ideas on Evenness Types of evenness Consequences of those types of evenness (a. k. a. species abundance models) Methods of testing and evaluation Introduction to each series Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Types of Evenness Types of evenness patterns are called Species abundance models Have four main types of abundance models 1. 2. 3. 4. Geometric series Log-normal series Broken stick Decreasing dominance of a single species from #1 to #4 þ Possibly both numerical and ecological dominance Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Today: Evenness and Abundance Models Evenness – a review Evenness – new issues Introduction to the Models Geometric Series Log-Normal Series Broken-Stick Model Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Hypothetical Model Curves How would this appear if: Maximal evenness? Minimal evenness? 100 10 Minimal Evenness (one species) Maximal Evenness 1 Per Species Abundance 0. 1 0. 01 Next to Minimal Evenness (two species) 0. 001 10 20 30 40 Species Addition Sequence Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Hypothetical Model Curves 100 10 Broken Stick Model Per 1 Species Abundance 0. 1 Log-Normal Series 0. 01 0. 001 Geometric Series 10 Log Series 20 30 40 Species Addition Sequence Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Abundance Models Use all the available data þ Most thorough representation of the data Observations þ þ þ Evenness increases from Geometric Lognormal Broken Stick models Dominance of any one species decreases from Geometric Log-normal Broken Stick models Broken stick is the closest nature gets to maximal evenness Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Other Methods of Abundance Curves How else could these graphs be constructed? þ How would the data thereby be interpreted? Possible ideas: þ þ Biomass Number of species per trophic level • Trophic level Species • Number of species Abundance þ Number of species per feeding guild • Feeding Guild Species • Number of species Abundance þ Others? Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Testability Simple visual inspection of a single curve is insufficient How to test each abundance model differs between each model þ þ Geometric – simple rank / abundance plots with abundance on log scale on Y axis (as already seen) Log – frequency distribution of # of spp vs. abundance Log-normal – similar to Log, but use a log scale on X axis Broken stick – rank / abundance plot, using ranks, rather than abundance (more in a moment) Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Today: Evenness and Abundance Models Evenness – a review Evenness – new issues Introduction to the Models Geometric Series Log-Normal Series Broken-Stick Model Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Hypothetical Model Curves 100 10 Broken Stick Model Per 1 Species Abundance 0. 1 Log-Normal Series 0. 01 Log Series 0. 001 Geometric Series 10 20 30 40 Species Addition Sequence Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Geometric Series Niche pre-emption is structuring the ecosystem þ Species 1 takes a certain percentage of the resources and prevents others from using them • Assumes competitive exclusion and resource exhaustion þ þ Species 2 takes a bit more Continues with other species until all resources are used and all species are included Minimal cooperation in ecosystem Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Geometric Series Assumes that the species abundance is roughly proportional to total resource use þ þ Linear increase in abundance linear increase in resource use Interspecific per-individual resource use is comparable Mostly commonly found in species poor communities þ þ þ Early succession Degraded ecosystems (Enriched, Invaded) Harsh Ecosystems Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Testing the Geometric Series Model – An Exercise Should have: straight line species plot & statistical test 100 10 Broken Stick Model Per 1 Species Abundance 0. 1 Log-Normal Series 0. 01 Log Series 0. 001 Geometric Series 10 20 30 Species Addition Sequence Lecture 3 – Evenness & Species Abundance Models 40 © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Worked Example 2 Magurran p. 130 -131 Use the plant feeder data from one of the gardens Work individually Create a Rank / Abundance graph as in Fig. 2. 4 of Magurran þ Only a gross approximation of whether it actually fits Estimate k þ Use Excel and do so iteratively Conduct Chi-square goodness of fit test (GOF) þ Use SPSS to do this Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Today: Evenness and Abundance Models Evenness – a review Evenness – new issues Introduction to the Models Geometric Series Log-Normal Series Broken-Stick Model Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Hypothetical Model Curves 100 10 Broken Stick Model Per 1 Species Abundance 0. 1 Log-Normal Series 0. 01 0. 001 Geometric Series 10 Log Series 20 30 40 Species Addition Sequence Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Log Series Model Closely related to geometric series þ þ Some studies have found both fitting the same data Similar to geometric in hypotheses about origin of community • Arrival of species to a novel environment • Both say that a few factors predominantly structure the community • Both say that one (geometric) or a few (log) species dominate a species þ Log differs from Geometric in assumptions about arrival • Arrivals are randomly arranged – Can get some clumped, some long intervals between arrivals • In geometric series, the arrivals are regular and continual Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Log Series Model Mathematics Base equations þ þ Best fit to a GOF test between expected & obs data Base form of log series: , ( x 2 / 2), ( x 3 / 3), … ( xn / n) Observed S = [-ln(1 -x)] x is calculated iteratively • • þ Using the following equation S / N =(1 -x) / x[-ln(1 -x)] Solve S / N, then plug numbers in for x to determine its value 0. 9 > x > 1. 0 If N / S > 20, then x > 0. 99 (a diversity index) = N(1 -x) / x • Plug in x once obtained and get Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Testing the Log Series Model – Worked Example 3 Procedure: þ Construct a rank abundance plot as in the geometric • Estimate rough estimate of how well it fits to theoretical þ þ þ þ Determine log 2 based classes (octaves = doubling abd) Determine numbers of each class in observed data Estimate x Solve for Calculate expected abundance for each abd level Group into classes Conduct GOF test Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Today: Evenness and Abundance Models Evenness – a review Evenness – new issues Introduction to the Models Geometric Series Log-Normal Series Broken-Stick Model Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Hypothetical Model Curves 100 10 Broken Stick Model Per 1 Species Abundance 0. 1 Log-Normal Series 0. 01 0. 001 Geometric Series 10 Log Series 20 30 40 Species Addition Sequence Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Log Normal Series Most communities fit the Log Normal Series þ þ Usually large, mature communities E. g. , temperate forest trees Ubiquity may be b/c of simple mathematics þ þ Normal distribution is often a consequence of large numbers Central Limit Theorem • Large # factors random variation will result in normal distribution • Central assumption behind parametric statisticss • probability with # of factors Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Log Normal Series Species are grouped into classes þ þ Octaves – most common (Log 2) Any log base can be used Number of Species Number of Individuals Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Log Normal Community Assembly Assumption about community formation þ Sequential breaking of empty niche space (Sugihara 1980) • Each species that arrives splits the niche space • Occupies a niche space proportional to its relative abundance • Probability of niche space being subdivided is independent of its sizes • Breakages occur successively þ Mechanism can be through an ecological or evolutionary process Fit to model: not necessarily supports assumption Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Other Explanations Central Limit Theorem þ Not necessarily a biological explanation (May 1981) Ugland & Gray 1982 þ Species can be divided into three abundance classes • Rare (65%), Intermediate (25%), Common (10%) þ þ þ Communities are composed of patches Abundance of species = sum of abd in all patches enough to result in Log Normal distribution Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Log-Normal Miscellany Missed species þ þ Very rare species will not be sampled Those less abundant than the critical number are beind the veil line Need to estimate how many there should be there Smaller the sample increased number behind veil line • Because have a higher veil line, relative to larger samples Simplicity of calculations þ þ Would be there, but for the veil line Pielou (1975) created a fit to truncated log normal Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Testing the Log-Normal Series Model – Worked Example 4 Use Pielou’s Truncated Log Normal process þ Estimating # of spp missed to the left of veil line Process (14 Steps!) þ þ þ þ þ Sort species, from most to least abundant, ln transform Calculate mean and variance of community Determine observed class abundance Calculate Gamma & Sobs Estimate Theta Calculate Mu, Vx, zo Lookup po of zo Estimate total S (including those behind veil line) Lookup po of zo of each class abundance Conduct GOF test Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Today: Evenness and Abundance Models Evenness – a review Evenness – new issues Introduction to the Models Geometric Series Log-Normal Series Broken-Stick Model Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Hypothetical Model Curves 100 10 Broken Stick Model Per 1 Species Abundance 0. 1 Log-Normal Series 0. 01 0. 001 Geometric Series 10 Log Series 20 30 40 Species Addition Sequence Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Broken Stick Model Sometimes called random niche boundary hypothesis þ “broken stick” – Mac. Arthur (1957) A stick randomly and simultaneously broken into S pieces No real relationship between earlier species presence and niche size of subsequent arrivals þ Unlike all earlier models Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Examples of fits to Broken Stick Successfully fit in past þ þ Passerine birds (Mac. Arthur 1960) Minnows and Gastropods (King 1964) In General: þ Best fit in narrowly defined communities of taxonomically related organisms No adequate diversity index needed if data fit Broken Stick þ S is adequate measure of diversity Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Broken Stick Model Most equitable species abundance as ever happens naturally þ Most biologically realistic “uniform” distribution Theoretically, only looking at one resource þ E. g. , space Strongly subject to sample size þ Don’t have crowding limitations between species Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Limitations of Broken Stick Model Not really applicable to a single sample þ þ Usually conceived of as the average spp abd. Distribution Can be misleading to test fit of a single sample to theory of equal resource partitioning Fine to use as we will use it þ Adherence to a species abundance model Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Testing the Broken Stick Model – An Exercise Procedure (5 steps) þ þ þ Calculate N and S Determine Observed species in each Log 2 abundance classes Calculate Expected species for each abundance level (1 -2000 or so) Determine Expected species in each Log 2 abundance class Conduct GOF test Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Our Data This Term I Relationship between plant biodiversity, pest insect biodiversity, and beneficial insect biodiversity þ þ þ Read website at http: //www. columbia. edu/itc/cerc/danoff -burg/web-pages/gardens_main. htm Has a pretty good amount of background on the topic Field sites were in Manhattan and Brooklyn community gardens Data collected during summer 2001 I will also email you the data matrix • Please begin looking it over so that you are comfortable with it Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Our Data This Term II Influence of Hemlock Woolly Adelgid on carrion beetle biodiversity þ þ þ Separated by many (at least 3) trophic levels Adelgid is a phloem-feeding insect Carrion beetles are detritivores or predators on fly larvae on carrion Field sites at Black Rock Brook, Black Rock Forest Data collected during summer 2001 I will also email you these data • Please begin looking over the data set Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu

Next week: Abundance, An Introduction Read þ þ þ Magurran Ch 2 Magurran Worked Examples 1 -6 Southwood & Henderson 2. 1, 2. 2, 13. 1 We will conduct a few evenness and species abundance models next week Decide which of the two projects on which you are interested in working collaboratively þ 3 people per group Lecture 3 – Evenness & Species Abundance Models © 2003 Dr. James A. Danoff-Burg, jd 363@columbia. edu