Algorithms for ecological network analysis
Marco Scotti [email protected]
OUTLINE
From a graphical representation to matrices and vectors (linear algebra notation)
Network construction and evaluation of steady- state conditions
Ecological network analysis (ENA)
From a graphical representation to matrices and vectors FOOD WEBS UNWEIGHTED FOOD WEBS
A crudely simple point of view: who eats whom?
Chesapeake Bay food web 39 nodes (species or trophospecies), 117 links (feeding relations)
Baird, D., and Ulanowicz, R.E. (1989). The seasonal dynamics of the Chesapeake Bay ecosystem. Ecological Monographs, 59, 329-364 WEIGHTED FOOD WEBS
A little less crudely simple point of view: (1) Who eats whom? (2) At which rate?
Chesapeake Bay ecological network 39 nodes (species or trophospecies), 117 weighted links (feeding relations) ECOLOGICAL NETWORK ANALYSIS
1986, Ulanowicz - Ecological network analysis (ENA)
A systematic analysis of ecosystem flow networks that is composed of several techniques ECOLOGICAL NETWORK ANALYSIS
In ecological networks we can Ei identify four classes of flows 1) Import from outside - 2 Z T i i ij j 2) Export to outside - 3 3) Respiration (or Dissipation) - 5 4) Intercompartmental exchange - 8
Ri
Cone Spring ecosystem 5 NODES 18 LINKS (FLOWS)
Tilly, L.J. (1968). The structure and dynamics of Cone Spring. Ecological Monographs, 38, 169-197 ECOLOGICAL NETWORK ANALYSIS
This figure can be associated to a matrix of intercompartmental exchanges [T] and to the corresponding Input (Z), Export (E) and Respiration (R) vectors
0 8881 0 0 0 11184 300 2003 635 860 3109 0 0 5205 2309 0 T = 0 1600 0 75 0 Z = 0 E = 255 R = 3275 1814 0 200 0 0 370 0 0 203 0 167 0 0 0 0 0 EXERCISES
(1) Identify the four types of flows (2) Convert the graphical representation into vectors and a matrix (3) Evaluate steady-state conditions
Odum, H.T. (1957). Trophic structure and productivity of Silver Springs, Florida. Ecological Monographs, 27, 55-112 EXERCISES
(1) Construct the network from linear algebra notation (2) Check steady-state from matrix and vectors
Odum, H.T. (1957). Trophic structure and productivity of Silver Springs, Florida. Ecological Monographs, 27, 55-112 Ecological network analysis (ENA) ECOLOGICAL NETWORK ANALYSIS
The systematic analysis of ecological flow networks is composed of several techniques:
(1) Input-output analysis (2) Trophic analysis (3) Cycling analysis (4) Whole system indices derived from information theory INPUT-OUTPUT ANALYSIS: PARTIAL FEEDING MATRIX
It allows to quantify how one compartment depends on any other to obtain its requisite medium
Partial Feeding Matrix [G] Total input to C = 20 (100%) whose elements gij represent Input from A to C = 5 (5/20 = 25%) the fraction of the total input to Input from B to C = 15 (15/20 = 75%) compartment j that comes directly from i 0 1 0.25 0 0 0 0.75 0 G = 0 0 0 1 0 0 0 0
20 15 15 20 20 A B C D
5 INPUT-OUTPUT ANALYSIS: PARTIAL HOST MATRIX
It allows to quantify how the energy leaving each compartment is redistributed in the network
Partial Host Matrix [F] Total output from A = 20 (100%) whose elements fij Output from A to B = 15 (15/20 = 75%) represent the fraction of Output from A to C = 5 (5/20 = 25%) the total output from compartment i that enters 0 0.75 0.25 0 directly compartment j 0 0 1 0 F = 0 0 0 1 0 0 0 0
20 15 15 20 20 A B C D
5 APPLICATION TO CONE SPRING ECOSYSTEM
= (Z) = = =
T matrix of Z vector of Imports intercompartmental E vector of Exports exchanges R vector of Respirations Partial feeding Partial host matrix [G] matrix [F]
8881 f = 12 8881+2003+300
8881 g12 = 8881+1600+200+167+635 PARTIAL HOST MATRIX - INDIRECT FLOWS
Fraction of i outflow that f f 12 23 directly flows to j f13
f 24 Fraction of i outflow that f f 14 43 flows to j with two step paths
0 0 0
4 0 0
0
[F]n = fraction of i outflow that flows to j with paths of n steps
Which is the total fraction that goes out from i and flows to j?
S [F]n = [F] + [F]2 + [F]3 + … n=1 OUTPUT STRUCTURE MATRIX
The Augustinovics inverse A = [I-F]-1
S [F]n = [F] + [F]2 + [F]3 + … n=1
S [F]n = [I-F]-1 n=1
The Augustinovics inverse A = [I-F]−1 represents the average number of times that a quantum of matter exiting the row compartment has passed through the corresponding column compartment
TOTAL CONTRIBUTION LEONTIEF STRUCTURE MATRIX
[G] = fraction of j inflow that directly flows from i
[G]2 = fraction of j inflow that flows from i with two step paths [G]n = fraction of j inflow that flows from i with n step paths
Which is the total fraction of flows that enters j and comes from i?
S [G]n = [G] + [G]2 + [G]3 + … n=1 S [G]n = [I-G]-1 n=1 The Leontief inverse L = [I−G]−1 represents the average number of times that a quantum of matter entering a column compartment will pass the corresponding row compartment
TOTAL DEPENDENCY - =
In [I-G]-1 each coefficient shows the total dependency of the column compartment from the row one
Herbivores (compartment #2) totally depends (100%) on Plants (compartment #1) All the compartments totally depend on Plants (100%) There is a partial dependence of Compartment #4 from the Herbivores (40%)
- =
Omnivores (4) totally depend on primary producers, because Plants supply energy to the whole system. But Omnivores has a direct dependency equal to 60% (30/50) on Plants compartment
Therefore, dependency of Omnivores from Herbivores is equal to 40%: in part direct dependency (15/50 = 30%), and in part indirect dependency mediated by Carnivores (5/50 = 10%) TROPHIC POSITIONS OF OMNIVORE SPECIES
Effective trophic level 1.2 + 0.9 + 0.4 = 2.5
Omnivores (compartment #4) receive 60% of energy from Plants (this compartment behaves as 0.6 x 2 = 1.2 a herbivore for the 60% of its feeding activity - herbivore behavior, with trophic level 2)
It receives directly 30% of its energy from 0.3 x 3 = 0.9 Herbivores, so it has a 30% activity at trophic level 3
And it depends directly from Primary Carnivores 0.1 x 4 = 0.4 for 10% of its energy intake. 10% of the diet sets Omnivores to trophic level 4 TROPHIC ANALYSIS: THE LINDEMAN SPINE
It yields to different output: 1) Reduction of complex webs to linear chains (in a Lindeman sense: Plants -> Herbivores -> Carnivores1 -> Carnivores2) 2) Trophic Position (TP) of each compartment
Theoretical Lindeman Spine
Plant Herbiv. Carniv.1 Carniv.2
100% A 100% B 75% C 75% D 25% C 25% D
20 15 15 20 20 A B C D
5 TROPHIC ANALYSIS: SPECIES TROPHIC POSITION
This procedure apportions each species’ feeding activity to a series of discrete trophic levels sensu Lindeman. Single fractions by which each compartment feeds at a particular trophic level, weighted by the value of that trophic level, are summed, arriving to the effective trophic position of that given species
Plant Herbivore 20 15 15 20 20 A B C D
5
Compartment C receives: 25% (5/20) of energy from A (Plant), operating as herbivore 75% (15/20) of energy from B (Herbivore), operating as carnivore
Trophic Position (TP) of C = 0.25 x 2 + 0.75 x 3 = 2.75 EXAMPLE OF A TROPHIC CHAIN
Linear Chain
Changes in the values of the index of trophic position could be indicators of ecosystem response to stress
INDEX OF TROPHIC POSITIONS TROPHIC AGGREGATION
10 25 25 2 10 20 10 20 15 Z = T = 5 5 20 25 5 40 40 1 4 0 25 40
10 20 3 20
5
(1.0,
To see how aggregate trophic levels are defined, one begins by dividing the magnitude of each external input by the throughput of the receiving compartment to give the fraction of each species that acts as a “primary
producer”, dj = Zj/(Zj + STij). One assigns this fraction to the first trophic level. Call (d1) a vector with elements (dj)
Ulanowicz, R.E., and Kemp, W.M. (1979). Toward canonical trophic aggregations. American Naturalist, 114, 871-883 After that, multiplying the matrix [G] by the vector (d1) yields the fraction of the newly introduced medium that each species receives along all single-length pathways; that is, that fraction of Ti being supplied at the second trophic level. Call the vector of these fraction (d2) so that:
(1.0 = In the same way, the fraction of each species that feeds at the third trophic level is:
=
And the fourth is:
= Finally, we obtain the “Trophic 10
2 Transformation Matrix” [A]: 10 20 15 5 25 5 40 1 4 25 40
10 20 3 20
5
1 1.5 1.875 2.625
Row-weighted column sums
40 I 40 II 22.5 III 2.5 IV
17.5 20 2.5 TROPHIC POSITIONS IN CONE SPRING
=
Compartments 1 2 3 4 5 Trophic Position Compartment #4 0.969 x 3 + 0.031 x 4 = 3.031
INDEX OF TROPHIC POSITION
Problems with cycles CYCLING ANALYSIS
Cycling analysis considers the overall amount of currency involved in cycles (magnitude of cycles) and it also describes how cycles are organized within the ecosystem (structure of cycles)
Finn cycling index (FCI) calculates the fraction of whole system activity that constitutes recycling
A further method of matrix manipulation lists the number, the length (number of nodes involved) and the quality (list of the nodes) of cycles CYCLING ANALYSIS
(1) The [F] matrix describes the fraction of the total activity of (the row- compartment) i that flows directly to (the column) element j (2) The [F]2 matrix describes the fraction of the total activity of i that flows to the column-compartment j, with two step pathways (3) The [F]n matrix summarizes the fraction of the total activity of i that contributes to the intake of the column-compartment j, through n step pathways
For each matrix, the elements on the diagonal stand the for fraction of matter (or energy) that goes out from i and there come back, after 1, 2, 3,… ,n steps We have no cycles if all the powers of [F] present zeros on the diagonal
f12 f23
f13
f24
f14 f43 TOTAL CONTRIBUTION S [F]n = [F] + [F]2 + [F]3 + … n=1 S [F]n = [I-F]-1 = S n=1
The Augustinovics inverse A = [I − F]−1 represents the average number of times that a quantum of matter exiting the row compartment has passed through the corresponding column compartment
We have recyling when at least one element on the diagonal of [I − F]−1 is > 1 Fraction recycled in each compartment:
(Sii – 1)
Recycling efficiency for each compartment:
(Sii – 1)
Sii
Multiplying this element ((Sii-1)/Sii) by the total output of each compartment (Ti) we obtain the amount of matter (or energy) recycled by each comparment:
Ti (Sii – 1)
Sii Tc = Total energy recyled
FCI = Finn Cycling Index
Compartment Ti 1 – Plants 11,184 2 – Detritus 11,483 3 – Bacteria 5,205 4 – Detritivores 2,384 5 – Carnivores 370 1 2 3 4 5 Compartments No Yes Yes Yes Yes Recycling A HIERARCHICAL VIEW
Host coefficients [F] for energy flows in Cone Spring
11,184(1-1)/1 + 11,483(1.207-1)/1.207 + 5,205 (1.169–1)/1.169 + +2,384(1.039-1)/1.039 + 370(1.018-1)/1.018 = 0 + 1969.3 + 752.5 + 89.5 + 6.5 = 2817.835 2,817.8 kcal/m2/yr
(2,817.8/42,445) = 6.64% Finn Cycling Index A HIERARCHICAL VIEW INFORMATION THEORY INDICES
System level indices are used to quantify ecosystem growth and development (global attributes). Their calculation is based on information theoretical analysis:
(1) Total system throughput (TST) (2) Average mutual information (AMI) (3) Ascendency (A) (4) Development capacity (C) (5) Overhead (F) TOTAL SYSTEM THROUGHPUT
Total system troughput (TST) is calculated by summing all flows in the network. It is a measure of the ecosystem size (activity)
(A) 6 (B) (C) 2 2 2
6 6 12 24 24 12 6 12 6 6 6 12 1 3 1 3 1 3 6 6 6 6 12 12 6 6 12 12 24 24 6 4 6 4 4
6 TST = 96 TST = 96 TST = 96 AVERAGE MUTUAL INFORMATION
Average mutual information (AMI) measures the degree of specialization deriving from the amount of constraints on the medium circulation. With maximum uncertainty AMI = 0, while when the fate of currency is completely determined AMI is maximum
(A) 6 (B) (C) 2 2 2
6 6 12 24 24 6 12 6 6 12 6 12 1 3 1 3 3 6 1 6 6 6 12 12 6 6 12 12 24 24 6 4 6 4 4
6 AMI = min = 0 AMI = 1 AMI = max = 2 ASCENDENCY AND DEVELOPMENT CAPACITY
AMI x TST= A (Ascendency)
Ascendency expresses the fraction of matter that is efficiently managed by the system
If all of the matter was managed in the most efficient way, the ascendency would match the development capacity (C)
0 < A < C (C is the upper limit of A) ASCENDENCY AND DEVELOPMENT CAPACITY
Ascendency (A) quantifies what fraction of the total activity has been converted into organized complexity. It is measured as the product of TST by AMI What remains once ascendency is subtracted from development capacity is called system overhead (F = C – A), and it is the fraction of development capacity that has not yet been organized. Overhead is made of four contributions:
(1) Overhead on import (FI)
(2) Overhead on export (F E)
(3) Overhead on dissipation (F D)
(4) Redundancy (FR)
It has conflicting interpretations: (1) system inefficiencies at processing medium; (2) degrees of freedom available for reconfiguration, in response to external disturbance ASCENDENCY AND OVERHEAD
2 3 Development Capacity 100% 1 4
Overhead Encumbered complexity
Ascendency Organized complexity
2 3 0% 1 4 ASCENDENCY AND OVERHEAD
Development Capacity 100% Redundancy
Dissipative overhead Overhead Overhead on export 2 3
Overhead on import
Ascendency 1 4
0% Algorithms for ecological network analysis
Marco Scotti [email protected]