Transitivity a FORTRAN program for the analysis of bivariate competitive interactions Version 1.1

Transcription

1 Transitivity 1 Transitivity a FORTRAN program for the analysis of bivariate competitive interactions Version 1.1 Werner Ulrich Nicolaus Copernicus University in Toruń Chair of Ecology and Biogeography Gagarina 9, Toruń; Poland umk.pl Latest update: Introduction Species differ in their competitive ability, and these differences may translate to observed inequalities in species relative abundances within multispecies assemblages. Ecologists have devoted much effort inferring competitive processes from observed patterns of species abundances and morphology, and from changes in the temporal and spatial distribution of species (Chesson 2000). Many theoretical models of competitive interactions assume that species can be ranked unequivocally (A>B>C >Z) according to their competitive strength or resource utilization efficiency (Tilman 1988). However, intransitive competitive networks (Gilpin 1975) can generate loops in the hierarchy of competitive strength (e.g. the rock-scissors-paper game, in which A>B>C>A). Theoretical studies have shown that competitive intransitivity can moderate the effects of competition, allowing weak competitors to coexist with strong ones (Laird and Schamp 2006). Despite the conceptual simplicity of intransitive competitive hierarchies, the empirical estimation of the strength of competition and the frequency of competitive intransitivity in nature has proven difficult. The Fortran 95 software Transitivity is based upon the construction of patch transition matrices (P), such as those used in Markov chain models as proposed by Ulrich et al. (2013). The basis is a randomization test to evaluate the degree of intransitivity from these P matrices in combination with empirical or simulated C matrices. Ulrich et al. (2013) related empirically-derived competition matrices C to an explicit colonization-interaction model to obtain patch transition matrices P and used a reverse engineering approach to infer the structure of the competition (C) and the transition (P) matrices from an empirical (temporal or spatial) A matrix. There is no unique solution to this problem because a large number of different competition matrices (C) can generate the same patch transition matrix (P) that will reproduce the A matrix. However, by simulating a large set of stochastically created C matrices, the set of matrices that provide the best fit to an empirical A matrix can be analyzed with respect to their transitivity patterns. 2. Metrics and Reversed engineering

2 2 Transitivity Ulrich et al. (2013) used a simple Markov chain 6. The competition in a given patch stops when an model that predicts relative abundances. In this model, invading species wins over the resident and over the a m m patch transition matrix P describes the probability p ij of a transition from a patch occupied by spe- all of the potential invaders. rest of invaders, or when the original resident defeats cies i to a patch occupied by species j in a single time To generate the formula for p ij, i, j = 1,...,m, for step. If the probabilistic outcome of species interactions are fully described by the entries of C, Ulrich et set A = {a 1,..., a n } of species with the corresponding an arbitrary m, we need the following notation: given a al. (2013) calculated the patch transition matrix P in competition matrix C, let P(A)[a j a i ] denote the terms of the competition matrix C. probability that species a j is replaced by a i, i, j = 1,..., m. Within this notation: for 1 i j m) and for 1 i m The model assumes: 1. There are many homogeneous patches, each of which can be colonized and occupied by individuals of a set of m species; 2. All species produce a large number of potential propagules, so there is a propagule rain, and colonization is never limited by dispersal limitation; 3. Only a single species can occupy one patch at a time; 4. In a single time-step, a species occupying a patch either retains its occupancy or is replaced by a different species; 5. During a single time-step, each resident species in a patch will engage in a pairwise competitive encounter with all remaining (m 1) species that do not occupy that patch. The (m 1) invading species may all interact with the resident species and each one can potentially replace it, depending on the probabilistic outcome of competition between the resident and the invader. The order in which these encounters with the resident species occur is not important, and the probability that the resident species will engage in a pairwise competition with other (m 1) remaining species is uniformly distributed and equals 1/( m 1). These equations generate the required transition matrix P for an arbitrary number m of species in terms of competitive strength matrices for sets consisting of (m 1) species. The calculation of the total probability for all p ij of the transition matrix P from the entries of the competition matrix C needs the evaluation of all combinations of c ik (k i,j). Because this becomes computationally challenging at higher species richness Transitivity uses the approximation introduced by Ulrich et al. (2013) where is the geometric average of the respective c ik values. To estimate the degree of intransitivity in a given community, we need first to estimate the transition matrix P from an observed distribution of species abundances or occurrences (Ulrich et al. 2013). Depending on the data, there are three different scenarios. The first and most obvious approach relies on appropriate time series data. If data are available from at least t+1 time steps, the single abundance vectors of each step can be converted into two matrices N 1,t,

3 Transitivity 3 which runs from generation 1 to t, and N 2,t+1, which runs from generation 2 to t+1. Combining these two matrices yields: P = N 2,t+1 N T 1,t (N 1,t N T 1,t ) -1 where T denotes the transpose. This approach allows for the estimation of the P matrix from an A matrix of consecutive temporal censuses of an assemblage. To estimate P Transitivity uses a reverse engineering approach and generates n = 100,000 randomly assembled C matrices, in which each entry above the diagonal in the C matrix is chosen from a random uniform [0,1] distribution. It then transformed the randomly assembled C matrices into P transition matrices to predict the N 2,t+1 matrices from our Markov chain model. The software uses average rank order correlations between respective columns in the predicted and observed N 2,t+1 matrices to assess goodness of fit, and selects those P and C matrices that generated the best fit to the observed vector of relative abundances. The second approach is based on spatial abundance data for m species collected at i = 1 to n sites for which environmental variables are available. Ulrich et al. (2013) partitioned te variance of species abundances into a part explained by competitive effects and a second part explained by the environmental variables. Using multiple regression they received in P = I + X T H T U T (UU T ) -1 with U being an m n matrix of species relative abundances among n sites, H being the n h matrix of h environmental variables, X being the vector of regression parameters that solves U T =HX, and I being the m m identity matrix. As with time series data, the software uses the reverse engineering approach to compare predicted and observed environmental terms X T H T to find those C and P matrices that best mimic the observed abundance distributions. The third approach uses spatial data only and tries to identify the best fitting C and P matrices directly from the matrix of observed relative abundances at n sites using reversed engineering (Ulrich et al. 2013). 3. Metrics of transitivity Transitiv estimates transitivity of C matrices by (i<j) Where N is a count of species pairs for which c ij < c ji after the matrix has been sorted to maximize the number of matrix elements with p > 0.5 in the upper right triangle (Ulrich et al. 2013). Accordingly transitivity in the P matrix is estimated by (i<k and i,k j) As and auxiliary metric Transitivity calculates the fraction of species that is engaged in intransitivitive loops. 4. Program run Transitivity first asks for the method to estimate the degree of transitivity. The options are environmental date (e), time series data (t), or as the default abundance data only (n). Next the program asks whether to calculate bivariate competitive strength matrices C (option b) or only transition matrices P (option p). In the first case the output contains also the respective P matrices. Then give the names of the output and matrix file names. Both are shown in the two Figures below. Carriage returns assign the default names Output.txt and Matrix.txt. The default number of random matrices C or P is 100,000. You can change this number with the next option. The last option regards the input files. Give them with extension (example: file.txt). In the case of

4 4 Transitivity multiple runs a carriage return results in the question for the name of the file that contains the matrix file names for multiple analysis (cf. the example above). The first line of the batch file has to be a comment line. All of the files have to be in the same directory. Input files have to be space delimited. Tab delimitation is not allowed. In the case of the time series and abundance only approaches you need a single input files as shown above. For the environmental variable approach the software asks for a second file that contains the respective variables. 5. Output Transitivity returns two output files. The first, Matrix.txt, contains the best fit transition and competi-

5 Transitivity 5 tion matrices as well as the observed and predicted (right eigenvector of the transition matrix). The second file, Output.txt, gives the method used (predicting competition and transition matrices (b) or transition matrices only (p)), numbers of species and sites, and four metrics. The first is either TrC (in the case of the method = b option) or TRP (in the case of the method = p option) of the best performing competition or transition matrix. Next, the mean metric value and the lower (DownCL) and upper (UpCL) 95% confidence limit of the 100 best performing matrices are given. Lastly, the probability is given that the predicted degree of transitivity is less than 1.0. An important point regards the BenchM output. The reverse engineering procedure uses rank order correlation to compare predicted abundances and observed abundances and retain as a default those competition and transition matrices that predict abundances that are correlated with the observed ones by r > If less than 100 out of the 100,000 random test matrices fulfil this criterion the threshold is reduced by a step of 0.05, thus the second threshold is r = The software reduces the threshold as long as sufficient (n = 100) test matrices fulfil the required correlation. In the case below the threshold was r = In such a case the fit is very low and the predicted degree of intransitivity is not very reliable. In other words it is probable that there is no competition matrix that would be able to generate the observed abundance distribution and thus other factors than competition shape the observed abundance distribution. The correlation for the right eigenvector can be inferred from the Matrix.txt file where observed and predicted abundances of the best fit are given. 6. Citing Transitivity Transitivity is freeware but nevertheless if you use Turnover in scientific work you should cite Transitivity as follows: Ulrich W Transitivity a FORTRAN program for the analysis of bivariate competitive interactions. Version

6 7. System requirements 6 Transitivity Gilpin, M. E Limit cycles in competi- Transitivity is written in FORTRAN 95, has tion communities. American Naturalist 109: been compiled under a 64 bit architecture, and runs Laird, L. A. and B. S. Schamp Competitive intransitivity promotes species co-existence. under Windows 7, XP, and Vista. The present version is only limited by the computer s memory. American Naturalist 168: Tilman, D Plant Strategies and the Dy- 8. Acknowledgements Development of this program was supported by grants from the Polish Science Committee (KBN, 3 P04F , KBN 2 P04F ). 9. References ralist, submitted. Chesson, P Mechanisms of maintenance of species diversity. Annual Review of Ecology and Systematics 31: namics and Structure of Plant Communities. Monographs in Population Biology 26. Princeton University Press, Princeton. Ulrich, W., Soliveres, S., Kryszewski W., Maestre F. T., Gotelli, N. J Matrix models for quantifying competitive intransitivity. American Natu-