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Community and gradient analysis: Matrix approaches in macroecology The world comes in fragments

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Statistical inference means to compare your hypothesis H 1 with an appropriate null hypothesis H 0. Type I error Type II error Simple examples in ecology are The correlation between species richness and area (H 0 : no correlation, t-test) Differences in productivity between plots of different soil properties. (H 0 : no difference between means, ANOVA) But what about more complex patterns: Relative abundance distributions Productivity – diversity relationship Succession Community assembly

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Galapagos Islands But: Your variance estimator comes from the underlying distribution of species and individuals. Does the variance stem from Species interactions? Random processes? Evolutionary history? Ecological history? In fact we do not have an appropriate null hypothesis. Bootstrapped or jackknifed variance estimators only catch the variability in the underlying distribution. We compare diversities on islands A t-test points to significant differences in diversity.

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Statistical inference Is species co-occurrence random or do species have similar habitat requirements? A simple regression analysis points to joint occurrences. P F (r=0) < 0.00001 Abundance scale exponentially. Extreme values bias the results Spearman’s r = 0.67, P F (r=0) < 0.001 Classical Fisherian testing relies on an equiprobable null assumption. All values are equiprobable. In ecology this assumption is often not realistic.

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Species do not have the same abundances in the meta- community and sites differ in capacity. Statistical testing should incorporate such differences in occurrence pobabilities. Ecologists often have a good H 1 hypothesis. Much discussion is about the appropriate null assumption H 0. What do we expect if colonization of these three islands is random? Ecology is interested in the differences between observed pattern and random expectation. Our statistical tests should deal with these differences and not with raw pattern! If we use classical Fisherian testing nearly all empirical ecological matrices are significantly non-random. Thus we can’t separate ecological interactions from mass effects.

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Theory of Island biogeography Galapagos Islands tries to understand diversity from a stochastic species based approach. We treat the theory as H 1 The theory gives us expectations that have to be confirmed by observation. 95% confidence limits We treat the theory as H 0 The theory gives us random expectations. Residuals need ecological interpretation.

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Multispecies metapopulation and patch occupancy models Islands in a fragmented landscape Random dispersal of individuals between islands results in a stable pattern of colonization The change of occupancy p in time depends on patch size and distance according to a logistc growth equation. Metapopulation models are single species equivalents of the island biogeography model. Multispecies metapopulation models give null expectations on community structure.

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The neutral theory of biodiversity Neutral models try to explain ecological patterns by five basic stochastic processes: - Simple birth processes- Simple death processes - Immigration of individuals- Dispersal of individuals - Lineage branching Neutral models are the individual based equivalents to the species based theory of island biogeography! Although they make predictions about diversities they do not explicitly refer to species! Diversities refer to evolutionary lineages Ecological drift The main trigger of neutrality is dispersal. A high dispersal rates species specific traits are of minor importance for the shape of basic ecological distributions.

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Used as H 1 Neutral models make explicit predictions about Shape and parameters of species rank order distributions Species – area relationships Abundance - range size relations Local diversity patterns Patterns of succession Local and regional species numbers Branching patterns of taxonomic lineages Used as H 0 residuals from model predictions are measure of ecological interactions The model contains a number of hidden variables (dispersion limitation, branching mode, dispersal probability, isolation, matrix shape… CPU times are a limiting resource Variable carrying capacities are needed to obtain realistic evolutionary time scales

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The neutral, metapopulation and island biogeography models contain too many hidden variables to be of use as null hypothesis. Ecological realism without too many parameters We need null models that are ecologically realistic and rely on few assumptions that apply to all species. Gradient of null model assumptions including more and more constraints. Null models only use information given in the matrix. Theses are matrix fill, marginal totals, and degree distributons.

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Gradient of null model assumptions including more and more constraints. Retain fill Retain fill and row totals Retain fill and column totals Retain fill and row degree distribution Retain fill and column degree distribution Retain fill and row and column degree distribution Retain row and column totals Possible constraints Rows Columnsequiprobable proportional to marginal totals Marginal totals fixed equiprobablexxx proportional to marginal totalsxxx marginal totals fixedxxx Degree distribution Marginal totals Start from an empty matric and fill it randomly without or according to some constraints

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Gradient of null model assumptions including more and more constraints. Equiprobable - equiprobable Proportional - proportional Equiprobable - fixed Fixed - Equiprobable Fixed - proportional Fixed - fixed Includes mass effects Most liberal Identifies nearly all empirical matrices as being not random Low discrimination power Partly includes mass effects Appropriate if species abundances or site capacities are equal Identifies most empirical matrices as being not random Partly excludes mass effects Appropriate if species abundances or site capacities are proportional to metapopulation abundances or sites capacities Identifies many empirical matrices as being not random Excludes most mass effects Appropriate if column totals are proportional to sites capacities Identifies many empirical matrices as being random Excludes mass effects Appropriate if nothing is known about abundances and capacities Identifies most empirical matrices as being random

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An initial empty matrix is filled step by step at random. If after a placement violates the above constraints it steps back and places elsewhere. The process continues until all occurrences are placed. Major drawbacks: Long computation times Potential dead ends Fill algorithm Swap algorithm The algorithm screens the original matrix for checkerboards and swaps them to leave row and columns sums constant. Use at least 10*species*sites swaps. Major drawbacks: Generates biased matrices in dependence on the original distribution The algorithm starts with a random matrix according to the row and column constraints and sequentially swaps all 2x2 submatrices until only 1 and 0 remain. Major drawbacks: Randomized matrices have a low variance that are prone to type II errors. Trial algorithm (Sum of squares reduction) Algorithms for the fixed fixed null model

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The Swap algorithm is most often used 1.Sequential swap: First make a burn in and swap 30000 times and then use each further 5000 swaps as a new random matrix 2.Independent swap: Generate each random matrix from the original matrix using at least 10*species*sites swaps. Compare the observed metric scores with the simulated ones (100 or more randomized matrices) Z-score lower CL = -0.37 Z-score upper CL = -3.00 0 500 1000 1500 2000 3.573.673.783.893.994.1 Scores Frequency Observed score upper CL Lower CL

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Using abundances Abundance Species Populations equiprobable proportional to observed totals marginal totals fixed proportional to marginal totals equiprobable marginal totals fixed proportional to marginal totals equiprobable populations fixed Including abundances into null models increases the number of possible null models These 27 combinations regard rows, columns, and row and columns.

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CAMorisitaVarianceMantel LCLUCLLCLUCLLCLUCLLCLUCL Prop-prob, total abundance fixed039583200 Prop-prob, row/column abundances fixed20102122000 Prop– prop, row/column richenss fixed432503909802 Prop-prob, total richnes fixed656910734490 Row/column richenss and abundance fixed811251234500 Occurences fixed5825482170166 Occurrences and row/cloumn abundances fixed07282420560164 Populations fixed1823019401960158 Populations per column fixed17111145118115142 Populations per row fixed180101930 0158 Testing of null models and metrics using proportional random matrices. The metrics shouldn’t detect these matrices as being non-random. 200 random matrices

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Abundance matrices are more often detected as being non-random Fraction of 185 matrices detected as being significantly (two-sided 95% CL) segregated (dark bars) or aggregated (white bars).

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