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Cellular automata models :
Null Models for Ecology
Jane Molofsky
Department of Plant Biology
University of Vermont
Burlington, Vermont 05405
Cellular automata models and
Ecology
• Ecological systems are inherently
complex.
• Ecologists have used this complexity to
argue that models must also be
correspondingly complex
Cellular automata models in
Ecology
• Search of ISI web of science ~ 64 papers
• Two main types
– Empirically derived rules of specific systems
– Abstract models
– Many more empirical models of specific
systems than abstract models
1-dimensional totalistic rule of
population dynamics
• Individuals interact primarily locally
• Each site is occupied by only one
individual
• Rules to describe transition from either
occupied or empty
• 16 possible totalistic rules to consider
Molofsky 1994 Ecology
Ecological Scenarios
• Two types of competition
– Scramble
– Contest
• Two scales of dispersal
– Local
– Long
Possible neighborhood
configurations
0
0
0
0
1
0
0
1
1
0
0
1
1
1
0
1
0
1
1
Sum
0
0
1
0
2
1
1
1
3
Transition Rules
Sum
Local
Dispersal
Long
distance
Dispersal
0
1
2
3
0
1
1
0
1
1
1
0
Long distance dispersal
Local Dispersal
Totalistic Rule Set
Contest
Scramble
2 states, nearest neighbors
28 or 256 possible rules
Totalistic Rule Set
• How often we expect complex dynamics to
occur?
– Ignore the 2 trivial cases
– 6/14 result in “chaos”, 6/14 periodic, 2/14
fixation
• How robust are dynamics to changes in
rule structure?
Totalistic Rule Set
Contest
Scramble
2 states, nearest neighbors
28 or 256 possible rules
Do plant populations follow
simple rules?
1-dimensional experimental design
Grown at 2 different spacings (densities)
Cardamine pensylvanica
Fast generation time
No seed dormancy
Self-fertile
Explosively dispersed seeds
Molofsky 1999. Oikos
Do plant populations follow simple
rules?
• In general, only first and second neighbors
influenced plant growth.
• However, at high density, long range
interactions influenced final growth
Experimental Plant Populations
Replicated 1-dimensional plant populations
Followed for 8 generations
Fast generation time
No seed dormancy
Self-fertile
Explosively dispersed seeds
Two dimensional totalistic rule
•
•
•
•
•
Two species 0, 1
von Neumann neighborhood
Dynamics develop based on neighborhood sum
64 possible rules
Rules reduced to 16 by assuming that when only
1 species is present ( i.e sum of 0 or 5), it
maintains the site the next generation
• 16 reduced to 4 by assuming symmetry
Rule system
Species 0
Species 1
1
4
2
3
3
2
4
1
Positive
0
0
1
1
Negative
1
1
0
0
Allee effect
0
1
0
1
Modified Allee
1
0
1
0
Biological Scenarios
Positive Frequency Dependence
Negative Frequency Dependence
Allee effect
System Behavior
00
01
1,1
1,0
P1= probability that the
target cell becomes a 1
given that the
neighborhood sum equals 1
periodic
Ergodic
(0.2,0.4)”Voter rule”
P2
clustering
Phase separation
0,0
P2= probability that the
target cell becomes a 1
given that the
neighborhood sum equals 2
0,1
P1
00
10
Molofsky et al 1999. Theoretical Pop. Biology
0.98,0.98
0.27,0
0.31,0
0.35, 0
D. Griffeath, Lagniappe U. Wisc.
Pea Soup web site
Frequency dependence
Dispersal
Probability that the migrant
establishes:
H1=0.5 + a (F1-0.5)
Probability that a migrant
of species 1 arrives on the
site
H1 D1
P1 
H1 D1  H 2 D2
Probability that a
site is colonized by
species 1
Neighborhood
shape
Moore neighborhoood
Positive frequency dependence
Molofsky et al 2001. Proceedings of the Royal Society
Spatial Model
Stochastic Cellular Automata
One individual
per grid cell
Determine the
probability that a species
colonizes a cell
Update all cells
synchronously
Transition Rule
Frequency dependence
Probability that the
migrant establishes
h1=0.5 + a (f1-0.5)
Dispersal
Probability that a
migrant of species 1
arrives on the site
P1 = h1f1/(h1f1 + h2f2+ h3f3+ h4f4+ h5f5+
h6f6+ h7f7+ h8f8+ h9f9+ h10f10 )
1. The neutral case
(a=0)
Ecological Drift sensu Hubbell 2001
2. Positive Frequency
(a=1)
Generation 0
Generation 100,000
Molofsky et al 2001. Proc. Roy. Soc. B.268:273-277.
3. Positive Frequency
20 % unsuitable habitat
Generation 0
Generation 100,000
4. Positive Frequency Dependence
40% unsuitable habitat
Generation 0
Generation 100,000
The Burren
Interaction of the strength of frequency
dependence and the unsuitable habitat
Number of species
after 100 000
generations
Molofsky and Bever 2002. Proceedings of the Royal Society of London
Invasive species
Local interactions: Yes,
reproduces clonally
Exhibits positive
frequency dependence:
Yes
High levels of
diversity: Yes
No obvious
explanation: Yes
Lavergne, S. and J. Molofsky 2004. Critical Reviews in Plant Sciences
Consideration of spatial processes
requires that we explicitly consider
spatial scale
Each process may occur at
its own unique scale
Competition may occur over short
distances but dispersal may occur
over longer distances
Grazing by animals in grasslands may
occur over long distances while seed
dispersal occurs over short distances
Negative frequency dependence
Two species, two processes
dispersal
frequency
dependence
Probability that the
migrant establishes:
H1=0.5 + a (F1-0.5)
Probability that a migrant
of species 1 arrives on the
site
H1 D1
P1 
H1 D1  H 2 D2
Probability that a site is
colonized by species 1
Each process can occur at a unique scale
Interaction neighborhood
D1
F1
Dispersal neighborhood
Focal site
Molofsky et al 2002 Ecology
Local Frequency Dependence
Local Dispersal
Weak Frequency Dependence
a = - 0.01
Intermediate
Frequency
Dependence
a = -0. 1
Strong Frequency Dependence
a = -1
For local interactions when frequency
dependence is strong (a = -1)
random patterns develop because
H1 = 1 - F1,
D 1 = F1
P1 = (1- F1 ) , F1 / (1- F1 ) , F1 + F1 (1- F1 )
= (1- F1 ) , F1 / 2( (1- F1 ) , F1 )
= 0.5
Weak Frequency Dependence
a = -0.01
Dispersal and Frequency Dependence at same scale
Local Dispersal
Local Frequency Dependence
Long Dispersal
Long Frequency Dependence
Local Frequency Dependence
Long Distance Dispersal
Strong Frequency Dependence
a=-1
Weak Frequency Dependence
a = - 0.01
Why bands are stable?
Local, strong, frequency
dependence
(over the large dispersal scale, the
two species have the same
frequency: D1=D2)
Because the focal,
blue, cell is mostly
surrounded by
yellow, is stays blue
Local Dispersal
Long Distance Frequency Dependence
Strong Frequency Dependence
a=-1
Weak Frequency Dependence
a = - 0.01
Why bands are stable?
Local dispersal
(over the large interaction
scale, the two species have
the same frequency: H1=H2)
Because the focal,
blue, cell is mostly
surrounded by yellow,
is becomes yellow
How robust are these results?
Boundary Conditions
Torus, Reflective or Absorbing
Interaction Neighborhoods
Square or Circular
Updating
Synchronous or Asynchronous
Disturbance
Habitat Suitability
Effect of Disturbance
Strong Frequency Dependence a = -1
Disturbance = 25 % of cells
Local Frequency Dependence
Long Distance Dispersal
Local Dispersal
Long Distance Frequency Dependence
Habitat Suitability
Strong Frequency Dependence a = -1
25 % of cells are unsuitable
Local Frequency Dependence
Long Distance Dispersal
Local Dispersal
Long Distance Frequency Dependence
Processes that give rise to patterns…
Strong Negative Frequency
Dependence
only if equal scales
Weak Negative Frequency
Dependence
only if long distance
dispersal
Weak Negative Frequency
Dependence
only if dispersal is local
Weak Positive Frequency
Dependence
only if dispersal is local
Strong Negative Frequency
Dependence
only if unequal scales
Strong Positive Frequency
Dependence
most likely if local scales only
Negative frequency dependence
If dispersal and frequency dependence
operate over different scales, strong
patterning results
Striped patterns may explain sharp boundaries
between vegetation types
Need to measure both the magnitude and
scale of each process
Next step
• Non symmetrical interactions
• For non-symmetrical interactions, what is
necessary for multiple species to coexist?
Most multiple species interactions fail but
we can search the computational universe
and ask, which constellations are
successful and why?