Cellular automata models : Null Models for Ecology Jane Molofsky Department of Plant Biology University of Vermont Burlington, Vermont 05405.

Cellular automata models :Null Models for Ecology

Jane MolofskyDepartment of Plant Biology

University of VermontBurlington, 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 10

0 010 00

1 10

1 01

0 11 1 11

1 00

0 1 32Sum

Transition Rules

0 1 1 0Local Dispersal

Long distance Dispersal 1

0 1 32Sum

1 1 0

Long distance dispersal

Local Dispersal

Totalistic Rule Set

2 states, nearest neighbors 28 or 256 possible rules

Scr

am

ble

Conte

st

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

2 states, nearest neighbors 28 or 256 possible rules

Scr

am

ble

Conte

st

Do plant populations follow simple rules?

Cardamine pensylvanica

Fast generation timeNo seed dormancySelf-fertileExplosively dispersed seeds

1-dimensional experimental design

Grown at 2 different spacings (densities)

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

Fast generation timeNo seed dormancySelf-fertileExplosively dispersed seeds

Replicated 1-dimensional plant populationsFollowed for 8 generations

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

1,0

0,1

1,1

0,0

P1

P2

P2= probability that the target cell becomes a 1 given that the neighborhood sum equals 2

P1= probability that the target cell becomes a 1 given that the neighborhood sum equals 1(0.2,0.4)”Voter rule”

clustering

Ergodicperiodic

Phase separation

Molofsky et al 1999. Theoretical Pop. Biology

00

00

10

01

0.98,0.98

0.35, 0

0.31,0

0.27,0

D. Griffeath, Lagniappe U. Wisc.Pea Soup web site

Probability that a migrant of species 1 arrives on the site

Probability that the migrant establishes:H1=0.5 + a (F1-0.5)

Probability that a site is colonized by species 1

2211

111 DHDH

DHP

Neighborhood shape

Frequency dependenceDispersal

Moore neighborhoood

Positive frequency dependence

Molofsky et al 2001. Proceedings of the Royal Society

Spatial Model

Determine the probability that a

species colonizes a cell

One individual per grid cell

Update all cells synchronously

Stochastic Cellular Automata

Probability that a migrant of species 1 arrives on the site

Probability that the migrant establishesh1=0.5 + a (f1-0.5)

Probability that a site is colonized by species 1

Transition Rule

P1 = h1f1/(h1f1 + h2f2+ h3f3+ h4f4+ h5f5+ h6f6+ h7f7+ h8f8+ h9f9+ h10f10 )

DispersalFrequency dependence

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 Frequency20 % unsuitable habitat

Generation 0

Generation 100,000

4. Positive Frequency Dependence40% 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 scaleEach 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 dependenceTwo species, two processes

dispersalfrequency

dependenceProbability that a migrant of species 1 arrives on the site

Probability that the migrant establishes:H1=0.5 + a (F1-0.5)

Probability that a site is colonized by species 1

2211

111 DHDH

DHP

D1

F1

Interaction neighborhoodDispersal neighborhood

Each process can occur at a unique scale

Focal site

Molofsky et al 2002 Ecology

Local Frequency DependenceLocal Dispersal

Strong Frequency Dependence

a = -1

Intermediate Frequency Dependencea = -0. 1

Weak Frequency Dependence

a = - 0.01

For local interactions when frequency dependence is strong (a

= -1)random patterns develop because

H1 = 1 - F1,

D1 = F1

P1 = (1- F1 ) , F1 / (1- F1 ) , F1 + F1 (1- F1 )= (1- F1 ) , F1 / 2( (1- F1 ) , F1 )

= 0.5

Weak Frequency Dependencea = -0.01

Local DispersalLocal Frequency Dependence

Long DispersalLong Frequency Dependence

Dispersal and Frequency Dependence at same scale

Local Frequency Dependence Long Distance Dispersal

Strong Frequency Dependencea = - 1

Weak Frequency Dependencea = - 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 DispersalLong Distance Frequency

Dependence

Strong Frequency Dependencea = - 1

Weak Frequency Dependencea = - 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 ConditionsTorus, Reflective or Absorbing

Interaction NeighborhoodsSquare or Circular

UpdatingSynchronous or Asynchronous

DisturbanceHabitat Suitability

Effect of Disturbance

Local Frequency DependenceLong Distance Dispersal

Local Dispersal Long Distance Frequency Dependence

Strong Frequency Dependence a = -1Disturbance = 25 % of cells

Habitat Suitability

Local Frequency DependenceLong Distance Dispersal

Local Dispersal Long Distance Frequency Dependence

Strong Frequency Dependence a = -125 % of cells are unsuitable

Processes that give rise to patterns…

Strong Negative Frequency Dependence only if equal scalesWeak Negative Frequency Dependence only if long distance dispersal

Strong Negative Frequency Dependence only if unequal scales

Weak Negative Frequency Dependence only if dispersal is local

Weak Positive Frequency Dependence only if dispersal is local

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?