Transitivity 1

Transitivity – a FORTRAN program for the analysis of multivariate competitive

interactions Version 1.1

Werner Ulrich

Nicolaus Copernicus University in Toruń

Chair of Ecology and Biogeography

Lwowska 1, 87-100 Toruń; Poland

e-mail: ulrichw @ umk.pl

Latest update: 16.12.2013

1. Introduction

Species differ in their competitive ability, and

these differences may translate to observed inequali-

ties in species’ relative abundances within multi-

species 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 inter-

actions assume that species can be ranked unequivo-

cally (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 intransi-

tive competitive hierarchies, the empirical estimation

of the strength of competition and the frequency of

competitive intransitivity in nature has proven diffi-

cult. The Fortran 95 software Transitivity is based up-

on the construction of patch transition matrices (P),

such as those used in Markov chain models as pro-

posed by Ulrich et al. (2013). The basis is a randomi-

zation test to evaluate the degree of intransitivity from

these P matrices in combination with empirical or sim-

ulated C matrices. Ulrich et al. (2013) related empiri-

cally-derived competition matrices C to an explicit

colonization-interaction model to obtain patch transi-

tion matrices P and used a ‘reverse engineering’ ap-

proach 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 competi-

tion matrices (C) can generate the same patch transi-

tion matrix (P) that will reproduce the A matrix. How-

ever, 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

Ulrich et al. (2013) used a simple Markov chain

model that predicts relative abundances. In this model,

a m m patch transition matrix P describes the proba-

2 Transitivity

bility pij of a transition from a patch occupied by spe-

cies i to a patch occupied by species j in a single time

step.

If the probabilistic outcome of species interac-

tions are fully described by the entries of C, Ulrich et

al.

(2013) calculated the patch transition matrix P in

terms of the competi-

tion matrix C.

The model as-

sumes:

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 coloni-

zation 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 differ-

ent species;

5. During a single time-step, each resident species in a

patch may potentially engage in a pairwise competi-

tive 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;

6. Within a time step, the order in which the potential-

ly invading species encounter the resident is random;

7. During a single time step, one of the invading spe-

cies replaces the resident species, or the resident spe-

cies persists in the patch;

8. Local dispersal limitation is not important and the

set of patches is spatially unstructured.

To generate the formula for pij, i, j = 1, ...,m, for

an arbitrary m, we need the following notation: given a

set A = {a1, ..., an} of species with the corresponding

competition matrix C, let P(A)[aj → ai] denote the

probability that species aj is replaced by ai, i, j = 1, ...,

n.

Within this notation: for 1 ≤ i ≠ j ≤ m

and for 1 ≤ i ≤ m

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 pij

of the transition matrix P from the entries of the com-

petition matrix C needs the evaluation of all combina-

tions of cik (k ≠ i,j). Because this becomes computa-

tionally challenging at higher species richness Transi-

tivity uses the approximation introduced by Ulrich et

al. (2013)

where is the geometric average of the respective cik

values.

To estimate the degree of intransitivity in a giv-

en community, we need first to estimate the transition

matrix P from an observed distribution of species

abundances or occurrences (Ulrich et al. 2013). De-

pending on the data, there are three different scenarios.

The first and most obvious approach relies on appro-

priate 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 N1,t,

which runs from generation 1 to t, and N2,t+1, which

runs from generation 2 to t+1. Combining these two

matrices yields:

P = N2,t+1 N1,tT (N1,tN1,t

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 assem-

blage. To estimate P Transitivity uses a ‘reverse engi-

neering’ approach and generates n = 100,000 random-

ly 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 ran-

domly assembled C matrices into P transition matrices

Transitivity 3

to predict the N2,t+1 matrices from our Markov chain

model. The software uses average rank order correla-

tions between respective columns in the predicted and

observed N2,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 the variance of species abundances

into a part explained by competitive effects and a sec-

ond part explained by the environmental variables:

PU=U+E. Using multiple regression they received in

P = I + XTHTUT(UUT)-1

with U being an mn matrix of species relative abun-

dances among n sites, H being the nh matrix of h

environmental variables, X being the vector of regres-

sion parameters that solves UT=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 E = XTHT to find

those C and P matrices that best mimic the observed

abundance distributions. For the environmental calcu-

lations the abundance data contained in the abundance

vector A are initially ln(x) transformed (leaving zero

counts unchanged) with x being the adjusted absolute

abundances where the least abundant species has at

least abundance x=1. To avoid undesired effects of

differential measurements the environmental variables

are Z-transformed. After multiple regression the re-

sulting vector U is back transformed.

The third approach uses spatial data only and

tries to identify the best fitting C and P matrices di-

rectly 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

cij < cji after the matrix has been sorted to maximize

the number of matrix elements with p > 0.5 in the up-

per right triangle (Ulrich et al. 2013). Accordingly

transitivity in the P matrix is estimated by

(i<k and i,k≠j)

4. Program run

Transitivity first asks for the method to esti-

mate the degree of transitivity. The options are envi-

ronmental date (e), time series data (t), or as the de-

fault 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.

As a standard, Transitivity eliminates empty

rows and columns from the abundance matrix. You

can choose whether to retain these. Of course, the in-

put matrix should in general not contain empty col-

umns, because they have no sound ecological interpre-

tation in the present context. However, in some cases

you may wish to retain empty rows, for instance in

automatic data processing of large data sets or for

comparison of patterns with and without focal species.

The last option regards the input files. Give

them with extension (example: file.txt). In the case of

multiple runs a carriage return results in the question

for the name of the file that contains the matrix file

4 Transitivity

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 delimita-

tion 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 re-

spective variables.

5. Output

Transitivity returns two output files. The first,

Matrix.txt, contains the species abundance matrix, the

abundance distribution, best fit transition and competi-

tion matrices as well as the observed and predicted

(dominant eigenvector of the transition matrix) abun-

Transitivity 5

dance distributions.

The second file, Output.txt, gives the file

name, method used (predicting competition and transi-

tion matrices (b) or transition matrices only (p)), the

benchmark limit and matrix match (see below), the

numbers of best fitting competition or transition matri-

ces used for the calculation of confidence limits, num-

bers of species and sites, and four metrics. The first

(Metric1) is either C (in the case of the method = ’b’

option) or P (in the case of the method = ’p’ option)

of the best performing competition or transition ma-

trix. Next, the mean metric value and the lower

(DownCL) and upper (UpCL) 95% confidence limit of

the NBestFit performing matrices are given (in the

example below only the best fitting: NBestFit =1).

Then, the probability is given that the predicted degree

of transitivity is less than 1.0. Finally, in the case of

the environmental approach the software provides the

average coefficient of determination (r2) of the multi-

ple regressions of each species involved.

An important point regards the BenchM and

Match output. The reverse engineering procedure uses

rank order correlation to compare predicted abundanc-

es and observed abundances and retains as a default

those competition and transition matrices that predict

abundances that are correlated with the observed ones

by r > 0.95. If less than NBestFit (default = 100) out

6 Transitivity

of the 100,000 random test matrices fulfil this criterion

the threshold is reduced by a step of 0.1, thus the sec-

ond threshold is r = 0.85. The software reduces the

threshold as long as sufficient (k = NBestFit) test ma-

trices fulfil the required correlation. In the case shown

in the software window the threshold was r = 0.0 and

the abundances predicted by the single best fit compe-

tition matrix correlated weakly (r = 0.443) with the

observed abundances across the sites. That means even

100,000 random competition matrices were not able to

mimic observed abundances. In such a case the fit is

very low and the predicted degree of intransitivity is

not very reliable. This is a strong indication that other

factors overrule competition and that competitive ef-

fects are not the major driver to determine species

abundances. 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 Tran-

sitivity as follows:

Ulrich W. 2012. Transitivity – a FORTRAN program

for the analysis of bivariate competitive interactions.

Version 1.1. www.keib.umk.pl.

7. System requirements

Transitivity is written in FORTRAN 95, has

been compiled under a 64 bit architecture, and runs

under Windows 8, 7, XP, and Vista. The present ver-

sion is only limited by the computer’s memory.

8. Acknowledgements

Development of this program was supported

by grants from the Polish Science Committee (KBN, 3

P04F 034 22, KBN 2 P04F 039 29 ).

9. References

Chesson, P. 2000. Mechanisms of maintenance

of species diversity. Annual Review of Ecology and

Systematics 31: 343–366.

Gilpin, M. E. 1975. Limit cycles in competi-

tion communities. American Naturalist 109: 51–60.

Laird, L. A. and B. S. Schamp. 2006. Competi-

tive intransitivity promotes species co-existence.

American Naturalist 168: 182-193.

Tilman, D. 1988. Plant Strategies and the Dy-

namics and Structure of Plant Communities. Mono-

graphs in Population Biology 26. Princeton University

Press, Princeton.

Ulrich, W., Soliveres, S., Kryszewski W.,

Maestre F. T., Gotelli, N. J. 2014. Matrix models for

quantifying competitive intransitivity. Oikos, in press.