Towards a
maximum entropy
theory of food webs
Francis Banville
Dominique Gravel
Timothée Poisot
2025 • 07 • 08
2
This theory serves two purposes:
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3
Better understand and quantify the uncertainty within food webs
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This theory serves two purposes:
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Predict food-web structure with parsimony
4
This theory serves two purposes:
Better understand and quantify the uncertainty within food webs
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Let’s draw a piece of art...
…to illustrate this theory
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piece of art
artistic process
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piece of art
artistic process
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piece of art
artistic process
material
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artistic process
few material
(a lot of constraints)
simple piece of art
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artistic process
complex piece of art
a lot of material
(few constraints)
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food webs
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principle of maximum entropy
food webs
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versions of
the theory
principle of maximum entropy
food webs
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Maximum entropy
theory of food webs
principle of maximum entropy
2
3
1
versions of
the theory
food webs
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Maximum entropy
theory of food webs
principle of maximum entropy
2
3
versions of
the theory
food webs
1
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Where does the uncertainty of
food webs come from?
1
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?
Where does the uncertainty of
food webs come from?
1
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?
Where does the uncertainty of food webs come from?
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?
Where does the uncertainty of food webs come from?
The local probability depends on
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Where does the uncertainty of food webs come from?
The local probability depends on
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Where does the uncertainty of food webs come from?
The local probability depends on
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Where does the uncertainty of food webs come from?
The local probability depends on
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?
Where does the uncertainty of food webs come from?
The local probability depends on
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?
Where does the uncertainty of food webs come from?
The local probability depends on
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?
Where does the uncertainty of food webs come from?
The local probability depends on
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?
Where does the uncertainty of food webs come from?
The local probability depends on
27
?
Where does the uncertainty of food webs come from?
The local probability depends on
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?
Where does the uncertainty of food webs come from?
The local probability depends on
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?
Where does the uncertainty of food webs come from?
The local probability depends on
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?
Where does the uncertainty of food webs come from?
The local probability depends on
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?
Where does the uncertainty of food webs come from?
The local probability depends on
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1
?
We want to build a theory of
food webs that…
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1
?
We want to build a theory of
food webs that…
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1
?
We want to build a theory of
food webs that…
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1
?
We want to build a theory of
food webs that…
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3
1
versions of
the theory
food webs
principle of maximum entropy
2
Maximum entropy
theory of food webs
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How does the maximum entropy
theory of food webs work?
2
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How does the maximum entropy theory of food webs work?
Entropy is a measure of uncertainty of a random process
?
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How does the maximum entropy theory of food webs work?
Entropy is a measure of uncertainty of a random process
and a measure of
network complexity
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How does the maximum entropy theory of food webs work?
Random network that satisfies some constraints
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How does the maximum entropy theory of food webs work?
Random network that satisfies some constraints
Entropy
N1
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How does the maximum entropy theory of food webs work?
Find another network that satisfies the constraints
Entropy
N1
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How does the maximum entropy theory of food webs work?
Find another network that satisfies the constraints
Entropy
N2
N1
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How does the maximum entropy theory of food webs work?
Repeat until we find the network of maximum entropy
Entropy
N2
N1
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How does the maximum entropy theory of food webs work?
Entropy
N2
N1
N3
Repeat until we find the network of maximum entropy
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How does the maximum entropy theory of food webs work?
Entropy
N2
N1
N3
This is our
least-biased prediction
Repeat until we find the network of maximum entropy
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2
Our predictions of network
structure depend on the constraints
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2
These constraints determine different versions of the theory
Our predictions of network
structure depend on the constraints
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principle of maximum entropy
2
1
food webs
Maximum entropy
theory of food webs
3
versions of
the theory
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What determines
food-web structure?
3
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What determines food-web structure?
number of prey and predators of each species
Constraints of version 2
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What determines food-web structure?
number of prey and predators of each species
Constraints of version 2
These constraints predict several measures of food-web structure
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Mangal
the ecological interactions database
257 local
food webs
What determines food-web structure?
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diets
specialist species
generalist species
What determines food-web structure?
measure 1: nestedness
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data
version 2 MaxEnt
nestedness
What determines food-web structure?
0.4
0.5
0.6
0.7
0.8
0.9
0.4
0.5
0.6
0.7
0.8
0.9
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data
version 2 MaxEnt
nestedness
What determines food-web structure?
0.4
0.5
0.6
0.7
0.8
0.9
0.4
0.5
0.6
0.7
0.8
0.9
Nestedness is determined by the number of prey and predators of each species
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What determines food-web structure?
measure 2: motifs
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What determines food-web structure?
S1 : Trophic chain
S2 : Omnivorie
S3 : Boucle trophique
S4 : Compétition apparente
S5 : Compétition d’exploitation
measure 2: motifs
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What determines food-web structure?
measure 2: motifs
S1 : Trophic chain
S2 : Omnivorie
S3 : Trophic loop
S4 : Apparent competition
S5 : Exploitative competition
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What determines food-web structure?
measure 2: motifs
S1 : Trophic chain
S2 : Omnivorie
S3 : Trophic loop
S4 : Apparent competition
S5 : Exploitative competition
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What determines food-web structure?
measure 2: motifs
S1 : Trophic chain
S2 : Omnivorie
S3 : Trophic loop
S4 : Apparent competition
S5 : Exploitative competition
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What determines food-web structure?
S1 : Trophic chain
S2 : Omnivorie
S3 : Trophic loop
S4 : Apparent competition
S5 : Exploitative competition
measure 2: motifs
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proportion
What determines food-web structure?
0%
20%
40%
60%
80%
100%
data
null model
version 2 MaxEnt
motifs
the null model is based on the same constraints as the version 2 of the theory, but without the optimization
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proportion
What determines food-web structure?
0%
20%
40%
60%
80%
100%
data
null model
version 2 MaxEnt
trophic chain
trophic chain
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proportion
What determines food-web structure?
0%
20%
40%
60%
80%
100%
data
null model
version 2 MaxEnt
omnivorie
omnivorie
trophic chain
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proportion
What determines food-web structure?
0%
20%
40%
60%
80%
100%
data
null model
version 2 MaxEnt
trophic
loop
trophic
loop
omnivorie
trophic chain
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proportion
What determines food-web structure?
0%
20%
40%
60%
80%
100%
data
null model
version 2 MaxEnt
apparent competition
apparent competition
omnivorie
trophic chain
trophic
loop
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proportion
What determines food-web structure?
0%
20%
40%
60%
80%
100%
data
null model
version 2 MaxEnt
exploitative competition
exploitative competition
omnivorie
trophic chain
trophic
loop
apparent competition
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proportion
What determines food-web structure?
0%
20%
40%
60%
80%
100%
data
null model
version 2 MaxEnt
Alone, these constraints are not enough
We need
maximum entropy
omnivorie
trophic chain
trophic
loop
apparent competition
exploitative competition
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3
What we learned about
food-web structure
Food-web structure is highly determined by the number of prey and predators of each species and maximum entropy
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Predict food-web structure with parsimony
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The maximum entropy theory of
food webs serves two purposes:
Better understand and quantify the uncertainty within food webs
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What’s to come in Computational Biodiversity Science and Services?
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principle of maximum entropy
food webs
Towards a maximum entropy
theory of food webs
versions of
the theory
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Maximum entropy
theory of ecology (METE)
number of individuals
number of species
total energy
number of interactions
Maximum entropy
theory of food webs
area
A next version could become
a general theory of food webs
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Monitoring biodiversity for
human, animal and environmental health
Kunming-Montreal
Global Biodiversity
Framework
ne Health
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Thank you! Questions?
francis.banville@umontreal.ca
Don’t hesitate to
reach out!
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Additional Slides
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Intro
Figure 1.1
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Article 2
Figure 3.1
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Article 2
Figure 3.4
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Article 2
Figure 3.5
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Article 2
Figure 3.6
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Entropie de Shannon
Article 1
Table 2.1
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Entropie de Shannon
Article 2
Table 3.1 and table 3.2