Matrix models for quantifying competitive intransitivity from species abundance data
Werner Ulrich, Santiago Soliveres, Wojciech Kryszewski, Fernando T. Maestre, Nicholas J. Gotelli
Worked example of the benchmark testing procedure, which consists of four steps. (1) translation of a competition matrix Ctest into a patch transition matrix
Ptest. (2) Calculation of the column vectors of the Utest matrix that simulate the abundance distribution of species (5 rows) among sites (7 columns) by
randomly filling the Utest matrix proportion to the values of the dominant eigenvector V of the Ptest matrix. (3) Calculation of the dot products between each
Utest column vector and the matrix Ptemp matrices (obtained from Ctemp) generates a predicted species × sites matrix; the average Spearman rank correlation
between the respective sites serves as a measure of goodness of fit. We repeat this procedure at least 10,000 times and retain the best fitting matrix Cpred.
(4) Finally, we generate the associated matrices of the numbers of intransitivities, and calculate for both Ctest and Cpred the degrees of transitivity (C and P),
the proportions of species engaged in intransitive species pairs (fC and fP). Except for first step translation from the C to the P matrices, the same
benchmark procedure applies when starting directly from a P matrix.
A
B
C
D
E
A
1.0
0.4
0.2
0.3
0.9
Competition
Ctest matrix
B
C D
0.6 0.8 0.7
1.0 0.3 0.8
0.7 1.0 0.8
0.2 0.2 1.0
0.3 0.1 0.5
E
0.1
0.7
0.9
0.5
1.0
A
B
C
D
E
A
0.1
0.1
0.0
0.0
0.8
Transition
Ptest matrix
B
C D
0.3 0.7 0.2
0.2 0.1 0.3
0.4 0.2 0.3
0.0 0.0 0.1
0.1 0.0 0.1
V
E
0.0
0.2
0.7
0.1
0.1
0.3
0.3
0.3
0.1
0.1
1
2
Utest matrix
3
4
5
6
7
0.3
0.3
0.2
0.1
0.0
0.3
0.3
2.0
0.1
0.0
0.3
0.3
0.2
0.1
0.0
0.3
0.2
0.3
0.1
0.0
0.4
0.2
0.2
0.1
0.0
0.3
0.3
0.3
0.1
0.1
0.3
0.3
0.2
0.1
0.0
r 0.90 0.90 0.90 0.70 0.70 0.90 1.00
mean r 0.86
A
B
C
D
E
A
B
C
D
E
A
0
0
0
0
1
B
0
0
1
0
0
C
0
0
0
0
0
D
0
0
0
0
0
E
0
0
0
0
0
A
B
C
D
E
A
0
0
0
1
3
B
0
0
1
0
1
C
0
0
0
0
0
D
0
1
1
0
0
E
0
0
2
1
0
1
0.37
0.27
0.21
0.10
0.05
2
0.37
0.26
0.22
0.09
0.05
3
0.30
0.29
0.24
0.12
0.05
4
0.35
0.26
0.23
0.12
0.04
5
0.39
0.21
0.24
0.11
0.06
6
0.32
0.20
0.21
0.11
0.05
7
0.35
0.21
0.30
0.10
0.04
Sum
2.2
1.9
3.4
0.7
0.1
Dot
product
Sum
2.43
1.81
1.65
0.76
0.35
A
B
C
D
E
A
0.35
0.25
0.21
0.13
0.06
A
B
C
D
E
A
0
0
0
0
0
Ptemp matrix
B
C D
0.34 0.40 0.30
0.24 0.20 0.30
0.29 0.24 0.23
0.09 0.12 0.11
0.04 0.04 0.06
B
0
0
0
0
0
C
0
0
0
0
0
D
0
0
0
0
0
A
C temp matrix
B
C D
10000 times
E
0.30
0.28
0.28
0.09
0.05
A
B
C
D
E
1.00
0.42
0.37
0.27
0.14
0.58
1.00
0.54
0.27
0.15
0.63
0.46
1.00
0.33
0.15
0.73
0.73
0.67
1.00
0.36
0.86
0.85
0.85
0.64
1.00
E
0
0
0
0
0
A
B
C
D
E
A
0
0
0
0
0
B
0
0
1
0
0
C
0
0
0
0
0
D
0
0
0
0
0
E
0
0
0
0
0
Ctest
0.80
Ptest
0.27
Ppred
1.00
C pred
0.90
fCtest
0.40
fPtest
0.80
fPpred
0.00
fC pred
0.20
E