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

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Cellular automata models and Ecology Ecological systems are inherently complex. Ecologists have used this complexity to argue that models must also be correspondingly complex

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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

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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

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Ecological Scenarios Two types of competition –Scramble –Contest Two scales of dispersal –Local –Long

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Possible neighborhood configurations 010001000110101011 111 100 0132Sum

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Transition Rules 0110 Local Dispersal Long distance Dispersal 1 0132Sum 110

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Long distance dispersal

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Local Dispersal

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Totalistic Rule Set 2 states, nearest neighbors 2 8 or 256 possible rules Scramble Contest

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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?

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Totalistic Rule Set 2 states, nearest neighbors 2 8 or 256 possible rules Scramble Contest

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Do plant populations follow simple rules? Cardamine pensylvanica Fast generation time No seed dormancy Self-fertile Explosively dispersed seeds 1-dimensional experimental design Grown at 2 different spacings (densities) Molofsky 1999. Oikos

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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

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Experimental Plant Populations Fast generation time No seed dormancy Self-fertile Explosively dispersed seeds Replicated 1-dimensional plant populations Followed for 8 generations

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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

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Rule system Species 0 Species 1 1414 2323 3232 4141 Positive0011 Negative1100 Allee effect0101 Modified Allee1010

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Biological Scenarios Positive Frequency Dependence Negative Frequency Dependence Allee effect

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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 Ergodic periodic Phase separation Molofsky et al 1999. Theoretical Pop. Biology 00 10 01

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0.98,0.98 0.35, 0 0.31,0 0.27,0 D. Griffeath, Lagniappe U. Wisc. Pea Soup web site

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Probability that a migrant of species 1 arrives on the site Probability that the migrant establishes: H 1 =0.5 + a (F 1 -0.5) Probability that a site is colonized by species 1 Neighborhood shape Frequency dependence Dispersal Moore neighborhoood

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Positive frequency dependence Molofsky et al 2001. Proceedings of the Royal Society

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Spatial Model Determine the probability that a species colonizes a cell One individual per grid cell Update all cells synchronously Stochastic Cellular Automata

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Probability that a migrant of species 1 arrives on the site Probability that the migrant establishes h 1 =0.5 + a (f 1 -0.5) Probability that a site is colonized by species 1 Transition Rule P 1 = h 1 f 1 /(h 1 f 1 + h 2 f 2 + h 3 f 3 + h 4 f 4 + h 5 f 5 + h 6 f 6 + h 7 f 7 + h 8 f 8 + h 9 f 9 + h 10 f 10 ) DispersalFrequency dependence

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1.The neutral case (a=0) Ecological Drift sensu Hubbell 2001

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2. Positive Frequency (a=1) Generation 0 Generation 100,000 Molofsky et al 2001. Proc. Roy. Soc. B.268:273-277.

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3. Positive Frequency 20 % unsuitable habitat Generation 0 Generation 100,000

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4. Positive Frequency Dependence 40% unsuitable habitat Generation 0 Generation 100,000

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The Burren

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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

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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

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Consideration of spatial processes requires that we explicitly consider spatial scale Each process may occur at its own unique scale

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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

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Negative frequency dependence Two species, two processes dispersal frequency dependence Probability that a migrant of species 1 arrives on the site Probability that the migrant establishes: H 1 =0.5 + a (F 1 -0.5) Probability that a site is colonized by species 1

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D1D1 F1F1 Interaction neighborhood Dispersal neighborhood Each process can occur at a unique scale Focal site Molofsky et al 2002 Ecology

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Local Frequency Dependence Local Dispersal Strong Frequency Dependence a = -1 Intermediate Frequency Dependence a = -0. 1 Weak Frequency Dependence a = - 0.01

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For local interactions when frequency dependence is strong (a = -1) random patterns develop because H 1 = 1 - F 1, D 1 = F 1 P 1 = (1- F 1 ), F 1 / (1- F 1 ), F 1 + F 1 (1- F 1 ) = (1- F 1 ), F 1 / 2( (1- F 1 ), F 1 ) = 0.5

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Weak Frequency Dependence a = -0.01 Local Dispersal Local Frequency Dependence Long Dispersal Long Frequency Dependence Dispersal and Frequency Dependence at same scale

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Local Frequency Dependence Long Distance Dispersal Strong Frequency Dependence a = - 1 Weak Frequency Dependence a = - 0.01

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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

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Local Dispersal Long Distance Frequency Dependence Strong Frequency Dependence a = - 1 Weak Frequency Dependence a = - 0.01

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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

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How robust are these results? Boundary Conditions Torus, Reflective or Absorbing Interaction Neighborhoods Square or Circular Updating Synchronous or Asynchronous Disturbance Habitat Suitability

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Effect of Disturbance Local Frequency Dependence Long Distance Dispersal Local Dispersal Long Distance Frequency Dependence Strong Frequency Dependence a = -1 Disturbance = 25 % of cells

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Habitat Suitability Local Frequency Dependence Long Distance Dispersal Local Dispersal Long Distance Frequency Dependence Strong Frequency Dependence a = -1 25 % of cells are unsuitable

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Processes that give rise to patterns… Strong Negative Frequency Dependence only if equal scales Weak 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

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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

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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?

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