fonstad 2006 cellular automata analysis and synthesis

2006) 217–234www.elsevier.com/locate/geomorph

Geomorphology 77 (

Cellular automata as analysis and synthesis engines at thegeomorphology–ecology interface

Mark A. Fonstad

Department of Geography, Texas State University – San Marcos, San Marcos, TX 78666-4616, United States

Received 8 July 2004; received in revised form 9 June 2005; accepted 5 January 2006Available online 7 February 2006

Abstract

The linkages between ecology and geomorphology can be difficult to identify because of physical complexity and thelimitations of the current theoretical representations in these two fields of study. Deep divisions between these disciplines aremanifest in the methods used to simulate process, such as rigidly physical-deterministic methods for many aspects ofgeomorphology compared with purely stochastic simulations in many models of change in landcover. Practical and theoreticalresearch into ecology–geomorphology linkages cannot wait for a single simulation schema which may never come; as a result,studies of these linkages often appear disjointed and inconsistent.

The grid-based simulation framework for cellular automata (CA) allows simultaneous use of competing schemas. CA use ingeneral geographic studies has been primarily limited to urban simulations models of change for land cover, both highly stochasticand/or expert rule-based. In the last decade, however, methods for describing physically deterministic systems in the CAframework have become much more accurate. The possibility now exists to merge separate CA simulations of differentenvironmental systems into unified “multiautomata” models. Because CAs allow transition rules that are deterministic,probabilistic, or expert rule-based, they can immediately incorporate the existing knowledge rules in ecology and geomorphology.The explicitly spatial nature of CA provides a map-like framework that should allow a simple and deeply rooted connection withthe mapping traditions of the geosciences and ecological sciences.© 2006 Elsevier B.V. All rights reserved.

Keywords: Geomorphology; Ecology; Modeling; Simulation; Cellular automata

1. Introduction

Cellular automata (CAs) are a class of numericalmodels based directly on a discrete space–time grid, andthey are a versatile representation of distributeddynamics that lend themselves well to the integrationof geomorphology and ecology. This versatility resultsfrom a highly general form of simulation that uses localneighborhood operations to produce dynamic response

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from lookup tables, algebraic equations, stochasticprobabilities, or expert rules centered on each discretepart of the grid. This article describes the historicaldevelopment of the cellular automata approach, definesthe major components of cellular automata models,focuses attention on the use of CA in geomorphologyand ecology, and comments on some of the moreadvanced or subtle issues that underlie the use of CA indynamic simulation.

Deep down, at the heart of process description ingeomorphology and ecology, lie the basic mental

218 M.A. Fonstad / Geomorphology 77 (2006) 217–234

representations (schema) that allow conceptual under-standing and quantification of environmental dynamics.Some of the most difficult and stubborn barriers toeffectively integrating geomorphology and ecology arethe vast differences in these conceptual schema. Forexample, the flow of water in a channel is usuallydescribed through continuum mechanics, a branch ofphysics that assumes the response of a fluid atmacroscopic scales to be a continuum (non-particle-like), and uses this assumption to build severalahistorical laws (such as continuity and locality) offluid motion that can then be used to solve fluid flowproblems via the mathematics of partial differentialequations. In contrast, models of the shifts of large-areaplant species often take the form of regressionequations, expert rules, discriminant equations, orother multivariate techniques. They are based on ideassuch as the existence of discrete classes (species),probabilistic descriptions of transition (stochastic be-havior), and the importance of previous occurrences(history) in affecting the outcome of transition. Thesetwo examples show the vast theoretical divides thatmust be crossed if effective interdisciplinary research isto take place, even if the systems under study are highlysimplified.

In ecology and geomorphology cries of anguish areheard over the lack of suitable data to analyze orsynthesize. Such cries are often well-founded, but it isimmediately apparent that both of these disciplines arefar more theory-poor than data-poor. Indeed, in manyways, both of these disciplines are data-rich; think of theenormous amount of information contained in an airphoto, a satellite image, or a digital elevation model. Wehave thousands of such images, but no theories ingeomorphology or ecology can fully explain thepatterns in any of them. In contrast, in some areas ofphysics (such as parts of cosmology), for example,almost no observations exist but vast understanding hasbeen gained through a small number of theories. Thelack of theory is especially apparent at the interfacebetween geomorphology and ecology. Experimentalreductionism can aid partially in this gap, as cananalytical modeling, but a huge amount of theorydevelopment at the interface undoubtedly will comefrom numerical simulation.

Sometimes resistance to simulation occurs in thedevelopment of understanding, because of the lack ofexactitude that can come from numerical description,from the multiplicity of parameters that often accompa-ny simulations, or from the reliance on computers.Nevertheless, the importance of simulation cannot bedenied. Well-defined simulations allow users to ask

“what if?” questions, they allow direct observation andprediction of system dynamics, they can be used toobserve otherwise unobservable space–time scales, andthey often generate unexpected or surprising results.Application of geomorphology and ecology to policyand decision-making often relies on simulation, so itsimportance stretches far beyond the analysis ofhypothetical natural systems.

Although the approach of cellular automata isdecades old, its widespread development and accep-tance across the natural sciences has been relativelyrecent and is rapidly accelerating. The time is approach-ing where CA models will become commonplace, andmay become the archetypal description for certain kindsof systems. To efficiently and effectively include CA inthe study of geomorphology and ecology, researchersshould carefully consider and understand the strengths,weaknesses, current limitations, and future possibilitiesof this approach at a general level.

2. The origins and structure of the approach tocellular automata

2.1. Origins

The origins of the approach of cellular automatastretch back to the 1950s when mathematician John VonNeumann devised the idea of discrete states in discretespace–time in his search for ways to create self-reproducing structures. Fellow mathematician StanislawUlam, who had worked with Von Neumann on theManhattan Project (and who also invented the MonteCarlo simulation technique), made the idea much morepractical by devising the idea of a two-dimensionallattice or grid where the distributed dynamics could takeplace. These grid-based dynamics were to only operatelocally (locality); no global (top-down) dynamics wereset by the modelers. The local dynamics were controlledby transition rules that were the same for every grid cell(universality or homogeneity). Time is viewed asdiscrete frames, as in a movie, where the states of allcells would update simultaneously according to thetransition rules (simultaneous update or parallelism).

The first popular application of cellular automata,and certainly the most well known CA application, wasCambridge mathematician John Conway's “Game ofLife” (Gardner, 1970). Conway's original “game” askedplayers to place checkers on a checkerboard in anydesired configuration. The game consisted of simplerules for these grid squares, whereby the checkers(which were “alive”) could live, die, or multiplydepending on the states of neighbors. Some initial

219M.A. Fonstad / Geomorphology 77 (2006) 217–234

configurations were found to multiply and reformcontinuously, while others would die off quickly orproduce only repetitive patterns. Life was convertedquickly to digital computers so that highly complexcollections of objects could interact with each other on avery large game board. The enormous richness ofdynamics that can be produced in Life was a harbingerof things to come; indeed, the range of spatio-temporaldynamics allowed in even the simplest CAs is verysimilar to the range of allowed dynamics in thecontinuum-mechanical description of the universe(Toffoli, 1984).

In the early 1980s, a veritable explosion occurred inresearch into the possibilities of the cellular automataapproach. Mathematicians and physicists, in particular,began to take seriously the notion that a discreterepresentation of system dynamics may be preferablein many cases to a continuum description. Models offluid response, based only on the single-velocitymovement of “particles” (discrete on or off states) ona lattice with rules for what to do if a collision occurswith other “particles”, known as lattice gases, becamemuch more sophisticated (d'Humieres et al., 1985;Frisch et al., 1986; Margolus et al., 1986; Salem andWolfram, 1986) since invention in the mid-1970s(Hardy et al., 1976). Specialized computers (calledcellular automata machines) built of enormous numbersof extremely simple processors, allowed much moreefficient processing of models of cellular automata(Toffoli and Margolus, 1987). Wolfram (1994) began toexplore systematically the overall dynamics of largeclasses of one-dimensional, single-state (on or off)cellular automata, finding some rules that generatedcompletely random space–time patterns despite beingbased completely on deterministic rules.

In the geosciences as well, cellular automata began tobe used as a simulation world-view. Tobler (1979)introduced the idea of “cellular geography” by showinghow several distributed geographical concepts could besimulated through locally effective rules. His articlebecame the foundation for a small industry ofgeographers who have designed simulations of urbangrowth and land cover change based on cellularautomata, with probabilistic rules for state changes. In1987, physicist Per Bak and his colleagues fromBrookhaven National Laboratory developed the “Sand-pile CA” (Bak et al., 1987); a simple 2-dimensional CAwith integer states (“blocks”) that represented sandgrains on a pile. By using local threshold rules (i.e. ifmore than three blocks are added to a cell, the blocks aredistributed to the cardinal direction neighbor cells), themovements of material around the CA produced power

law magnitude/frequency relationships, similar to thosefound in many natural systems such as landslides. Thisinitial Sandpile CA was enormously important in thedevelopments of simulation capabilities in the pastdecade, as wave after wave of CA-based simulationshave been proposed in many areas of geomorphology.

The current research into simulations of CA ingeomorphology and ecology is accelerating in manydirections. The most obvious research direction hasbeen the simple diffusion of CA models into manysubfields of geomorphology and ecology. Other lines ofresearch include questions such as (1) which formalrepresentations of form and process can best be used insimulation, (2) how to develop transition rules, (3) whattypes of programming structures, software, and com-puter hardware best handle massive models of CA, (4)how to calibrate a model of CA to handle real time andreal space scales, (5) how do simulations of CA relate to(or compete against) alternative techniques such asmean-field statistical approaches, expert systems, dif-ferential equations, and multi-agent systems, and (6)how do we validate the output of such complexsimulations?

2.2. Structure

The range of modeling approaches based on CA iscertainly large and nebulous, but all CAs share certaincommonalities. The first is the definition of the universein CA; the model universe is discrete in space and time.In one-dimensional (1D) CA, this means that theuniverse is represented as a linear series of numbers.Two-dimensional (2D) CA, the most common ingeomorphic and ecological simulations, are describedas a grid (or lattice), much like a checkerboard, althoughthe importance of the boundaries of these grids variesfrom simulation to simulation. Three-dimensional CA(3D) can be represented as a stack of two-dimensionalgrids (a space–time cube). Discrete time slices in the CAuniverse can be added to the spatial descriptions byadding a dimension. Thus, 1D models through time canbe described on a 2D grid, 2D simulations through timecan be described in a 3D cube, and 3D automata modelscan be described by a “stack of cubes”, where each cuberepresents a time slice. Image processing provides ananalogy for the computational power required for CAprocessing; 2D simulations are similar to processing amultispectral image per time step, and 3D simulationsare similar to processing a hyperspectral image per timestep.

The tessellation of these space–time grids iscommonly depicted as a lattice of squares (cells), each


with four cardinal neighbors and four diagonal neigh-bors. Although nearly all CA models are built upon suchsquare-lattice tessellations, alternatives exist; triangularand hexagonal tessellations are also possible (Fig. 1). Inparticular, so-called “lattice–gas” and “lattice–Boltz-mann” CA techniques often employ hexagonal tessella-tions (Frisch et al., 1986). Nevertheless, the simplicity ofthe square-lattice approach and its similarity to the rasterframework for spatial data make it the basis for furtherdiscussions.

Although many authors do not mention explicitlywhat happens at the edges, or boundaries, of the CAgrids, the specification of boundary conditions isimportant in the actual integration of a CA model witha computer (Parsons, 2004). All boundary conditionsinfluence the response of the gridded dynamics to somedegree. Many theoretical studies of CA simply use theno-boundary, or periodic, condition, where cells on oneedge are neighbors with cells on the opposite edge. In a2D lattice, this technically produces a three-dimensionaltorus, or donut, universe shape. Such a no-boundarycondition can produce significant error in models ofspecific real-world examples, however, and is oftenreplaced by some other boundary condition. Forexample, a periodic boundary CA grid containing amountain with water flowing of one slope and off thegrid edge would have that water strangely appear at thebase of the opposite slope, certainly an unrealisticsituation. An alternative includes reflecting (“bounce-back”) boundaries, where no dynamics can go off of orcome onto the grid through the boundary. Anotherpossibility is the absorbing boundary, where materialcan flow off the grid, but the “off-grid” areas have noeffect on the transition rules. One boundary condition,the “no-slip” condition stipulates that the cell statesalong a boundary do not change throughout thesimulation of CA (Massalot and Chopard, 1998).Unfortunately, boundaries generally require an extraIF–THEN rule in the development of the CA transition

Fig. 1. Tesselations of two-dimensional space: (A

rules (IF the cell is a border cell, the cell state neverchanges, as an example). One way to avoid the effects ofboundaries is to make the CA lattice much larger thanthe area where dynamics will be monitored, but thiscomes at the expense of computer time and resources.

In classical cellular automata, the states of the cellsare simple integers, such as 0, 1, 2, etc. Such a highlydescriptive ‘discrete state’ of the CA universe makescomputation enormously simple and effectively restrictstransition rules to expert (IF–THEN) rules and lookuptables (Wolfram, 1994). Many or even most CAs in usetoday, beginning with Kaneko (1985), however, allowfloating point cell state values; the cell states are stilltechnically discrete, because computers have somemaximum floating point number length, but theyallow algebraic and stochastic transition rules. Such“floating point cell state” CAs are commonly referred toas coupled-map lattices (CML), although the namescoupled-lattice maps and continuous cellular automataare also often used to describe this class of CA(Wolfram, 1994).

In the CA framework, dynamics are represented as achange in the state of grid cells from one time step to thefollowing time step. The cell need not, however,necessarily change its state. What happens to each gridcell is defined by a transition rule or transition rules(Wolfram, 1994); these rules are the same for all cells inthe lattice (“rule universality”, also known as homoge-neity). If the transition rule requires that the state of agrid cell is only dependent on its state at a previous timestep, such a model is called a Markov model, and is notconsidered a CA model. Cellular automata models haveone additional feature: the transition rules operate oncells based on the local neighborhood of those cells, aproperty known as locality. For example, in a 2D grid,the state of a cell at time t+1 could be a function of thestates of the cells to the north, south, east, and west ofthe cell of interest at time t. This simple neighborhood,composed of the four cardinal neighbors, is called the

) square grid; (B) triangular; (C) hexagonal.


Von Neumann neighborhood. Other square-latticeneighborhoods include the Moore Neighborhood (alleight surrounding cells), and the extended form of thesetwo neighborhoods (Fig. 2). In 1D CAs, the neighborsare simply the cells to the right and left (X direction) ofthe cell in question, in 2D CAs the neighbors in the Ydirection (above and below) are added, and in the 3Dcase the neighbors on the Z axis “above the X–Y plane”and “below the X–Y plane” must also be added.

The dependence of the transition of a cell upon thestate of its neighbors directly encodes a form of spatialassociation into the system dynamics generated by a CA(Tobler, 1979). While this association does not neces-sarily lead to spatial autocorrelation, it often does, andprovides a new measure (autocorrelation) of validatingthe results of a simulation. The form of this autocorre-lation, or its nonexistence, is controlled by the form ofthe transition rules used in describing the dynamics ofthe CA system.

The power of CA is its ability to handle differentclasses of transition rules. For example, a set of nestedIF–THEN statements could be used to drive thedynamics (for example, “IF all of my neighbors are ofclass ‘tree’, THEN the center cell either remains a ‘tree’or changes to a ‘tree’, ELSE the center cell does notchange”). This is an “expert rule” description of change,a common form of explanation in geomorphology andecology. A set of expert rules can be encoded into a CAas a lookup table with predefined transition rules; a cellwith only two ‘tree neighbors’might change to state ‘X’,one with three ‘tree’ neighbors to state ‘Y’, and so on. Insome situations, algebraic equations are used as thetransition rule: the center cell at time t+1 might be equalto the average of the neighbor cells at time t. Mostsimulations will incur a complex combination of thesedifferent types of transition rules, as well as includingsome level of nonspatial Markov influence, such as inCA models of differential (transformed into a Markov-ian difference equation) growth.

Fig. 2. Some common local neighborhoods for a square-tesselated lattice; (A)Von Neumann neighborhood; (D) extended Moore neighborhood.

The set of transition rules used in CA can be entirelydeterministic, relying only on the initial conditions ofthe grid and the neighboring cells states to produce thedynamics, or it can include a stochastic process.Stochastic CAs require the generation, for each timestep, of a random number grid (for example, Bolliger etal., 2003). For example, a transition rule could bewritten as, “if the random number is greater than 0.3,then change from state X to state Y”, requiring a randomnumber between 0 and 1 to work. Because of the effectof the random number grids, these stochastic CAs oftenrequire Monte Carlo-like simulations; the model mustbe run many times to come up with a probability-distribution for the overall spatio-temporal response ofthe system under study (Kronholm and Birkeland,2004). Coupled map lattices and stochastic CAs areirreversible distributed systems (because of randomnessor round off error), whereas true CAs (with only a fewallowed discrete cell states) are, in principle, reversible.

The transition rules define what happens to a cellfrom one time step to the next, but another importantconcept is that of parallelism, or simultaneous update;all cells are conceived to change at the exact same time.If some cells were to change first, the overall dynamicswould be different than if other cells were to change firstbecause of the effect of the local neighborhoods.Because CAs almost always run on standard seriescomputers (which process one cell at a time), CAprogrammers generally include a “buffer layer” thatholds the new cell states until all the cells have beenupdated, and then use this “buffer layer” in futurecalculations of the time step (Parsons, 2004).

The student of cellular automata modeling finallymay ask “where do the transition rules come from?” Nosimple answers exist to this question. The old modelingmaxim holds true: a model should be made as simple aspossible, but no simpler. Rule-making can proceeddeductively from general rules (such as the conservationlaws), or from induction (we went and measured

Von Neumann neighborhood; (B) Moore neighborhood; (C) extended


empirical changes between these plant species), or somecombination. Most elaborate CAs have induced anddeduced rules. This ability to use either form ofknowledge-building will allow future CA models to“raid” the past 100 years of geomorphic and ecologicalobservations and theories to build and validate newsimulations.

3. Cellular automata in ecology and geomorphology

In ecology, some models of change are based upondeterministic rules, often upon continuous spatio-temporal rates of change. The Lotka–Volterra modelof predator–prey relationships is a commonly usedexample of differential equation based ecologicalmodeling. Another classic example of this model basisis growth modeling applied to ecological organisms orpopulations. Based on controlled experiments andfundamental studies or organisms, models of growthare usually based on differential form equations. Simpleexponential growth is rare, because of spatial limita-tions, food and energetics limits, and competition, socontinuum models of growth incorporate a term or termsfor carrying capacity. These models can vary incomplexity from simple one-parameter exponentialgrowth models (Verhuslt, 1838) to five-parametergeneralized logistic models (Tsoularis and Wallace,2002). These types of growth may be applied to thegrowth of an individual, or the growth of populationsthrough space or time. Much more complex anddynamical systems can be formed by interconnectingthese continuum models of individuals or groups intosystems models (Odum, 1994). Simulations of thesehighly interconnected systems require recasting thedifferential equations into difference form to be solvedat a finite time step.

Nevertheless, growth in organisms or populations isvery rarely a straightforward, continuous process;spatially varying rates of change, growth disconti-nuities caused by competition or death, fragmentationby external, non-biotic factors, or a wealth of otherreasons may contribute to complexity. Alternatively,stochastic models of change, which include some formof probabilistic explanation of change, also provideexplanations of ecological change in many situations.Simple stochastic models would include Markovmodels through time or space, or regression (linear,multiple linear, or logistic) to predict the presence,absence, or density of organisms in an area. Manyother types of probabilistic models of ecologicalchange exist; these are some of the common andsimple tools.

The locality inherent in CAs and the flexibility of themodel-builder's choice in including or not includingcertain rules provide an excellent rationale for using CAin ecology. Ecologists have applied CA to predator–prey relationships (Tobin and Bjørnstad, 2003) andorganism diffusion or spread (Zhou and Liebhold, 1995;Cannas et al., 1999; Seabloom et al., 2001, Alftine andMalanson, 2004). Also, researchers have developedcellular automata models of competition (Silvertown etal., 1992; Galam et al., 1998; Matsinos and Troumbis,2002; van Wijk and Rodriguez-Iturbe, 2002), fragmen-tation and patch formation (Hogeweg, 1988; Bascompteand Sole, 1996; Keymer et al., 1998; Caswell and Etter,1999; Favier et al., 2004), and large models built tocontain many of these processes acting simultaneously(Green, 1982; Green et al., 1985; Satoh, 1989; Sato andIwasa, 1993; Alonso and Sole, 2000; Messina andWalsh, 2001; Bolliger et al., 2003; Kupfer and Runkle,2003). Hogeweg (1988) and Ermentrout and Edelstein-Keshet (1993) provide general overviews to the uses andconsiderations of cellular automata in ecologicalsimulation.

CA also provides a similar level of utility inproducing simulations of the responses of forest fires.One of the very first CAs in the natural sciences was Baket al.'s (1990) “forest-fire CA”, which produced similarmagnitude-frequency distributions to those observed innatural forest fires. This simple CA has been muchsurpassed in recent years; forest fire CA simulations,developed by Clarke et al. (1994), Clar et al. (1996), andKarafyllidis and Thanailakis (1997), have improved onthe basic Bak forest-fire CA by including more realisticvariables and processes in the transition rules.

Some of the early researchers in geomorphology totake advantage of the CA approach were researchers inItaly who built landslide and debris-flow simulations(Barca et al., 1986; Di Gregorio et al., 1994; Segre andDeangeli, 1995). Building on the foundations of Bak etal.'s (1987) original sandpile CA, these groups havemade significant advances in representing mass move-ments in a cellular form (Avolio et al., 1999; Clerici andPerego, 2000; D'Ambrosio et al., 2003, in press; Datilloand Spezzano, 2003; Iovine et al., 2005), and havereached a level of predictive power that these simula-tions now are viable and important for public policy anddecision-makers (Fig. 3). These simulations have sincebeen extended to include lava flows (Crisci et al., 2003;Crosweller, 2003) and pyroclastic flows (Avolio et al.,2002). Smith (1991) provided a fascinating and unusualCA of surface erosion by using a vertical grid, with onlyone dimension in the horizontal plane but explicitlyincluding vertical structure. Using simple transition

Fig. 3. (A) Observed and (B) CA-simulated landslide paths, from D'Ambrosio et al. (2003).


rules, Smith was able to build quite realistic-lookingpatterns of differential erosion.

Research in geomorphic CA simulation has expand-ed to eolian ripples (Anderson, 1990; Forrest and Haff,1992; Anderson and Bunas, 1993; Zhang and Miao,2003), and dune formation (Werner, 1995; Momiji et al.,2000). Werner and Fink (1993) and Ashton et al. (2001)have modeled coastal hydrodynamics and morphody-namics using the CA approach, and have connectedthese simulations to concepts of self-organization.

The direct modeling of fluid and particle transportoriginally began in the 1980s by using lattice gases.These CAs have discrete particles moving from cell tocell at uniform speed. The transition rules in thisapproach are simple enough that large improvements,including three-dimensional space and increased phys-icality of the fluid movement, increased to the pointwhere the flows are very similar to those deduced fromthe Navier–Stokes law (normally approached usingpartial differential equations). Nevertheless, lattice–gasand lattice–Boltzmann approaches, with discrete parti-cles, have not yet made a large impact on geomorphol-ogy. Bahr and Rundle (1995) have developed a model ofglacial ice flow using the lattice–Boltzmann approach,and Massalot and Chopard (1998) have built highlyrealistic simulations of water and particulate flow(1998), as well as the transport and deposition ofsnow. Jimenez-Hornero et al.'s (2003) work hasextended the Lattice–Boltzmann approach to the flowof water and sediment around complex obstacles, andDupuis (2002) has completed a “virtual river”; this

Lattice–Boltzmann reusable simulation produced high-ly realistic three-dimensional fluvial behavior.

Despite the advancements made in the lattice–gasand lattice–Boltzmann approach towards the simulationof geomorphic processes, nearly all geomorphic CAmodels are built using the coupled map latticeframework. Some of the most dramatic examples ofCA modeling in geomorphology and ecology have beencoupled map lattice models of river and watershedhydrology, fluvial sediment transport, and large-scalesimulations of landscape evolution.

In river and watershed CA hydrology, Thomas andNicholas (2002), Thomas et al. (2002) and De Roo et al.(2000) have designed effective 2D simulations ofsteady, gradually varied flow for channels and flood-plains, respectively. Parsons (2004) used time chainingapproaches to simplify these runoff transition rules andto produce accurate 2D unsteady CA simulations ofrainfall–runoff (Fig. 4). Brusik et al.'s (2003) “hydro-dynamic automaton” is an unusual combination of CAand differential equation approaches to solving fluidflow that does produce realistic patterns of landformdenudation. Other work in hydrology includes themovement of pollution in the soil water system (DiGregorio et al., 1998; Bandini et al., 1999) andKronholm and Birkeland's (2004) CA model ofsnowpack stability.

The combination of CA approaches for fluidmovement and sediment erosion, transport, and depo-sition has yielded important insights about the responsesof rivers, watersheds, and landscape evolution. In

Fig. 4. Three-dimensional views of Parsons (2004) CA rainfall–runoff model: (A) landcover patterns (later converted to roughness values); (B) digitalelevation model; (C–F) water depth during an ongoing rainstorm, with lighter colors representing deeper water (images made at time 0 min, 5 min, 20min, and 30 min); (G) a three-dimensional exaggerated view of accumulated water depth at one time during the runoff period; (H) a 3D exaggeratedview showing the distribution of infiltrated runoff water within the basin; (I) a “pull-back” showing that the CA simulation can be used over large areas.


particular, Murray and Paola's work on simulations ofriver planforms (Murray and Paola, 1994, 1996, 2003;Sapozhnikov et al., 1998) have been highly effective inshowing how qualitatively different river responses aretied to specific transition rules that can be simple yetrealistic. Further work in fluvial form simulationsinclude Fonstad and Marcus's (2003) work on demon-strating self-organized criticality in riverbank dynamicsand Favis-Mortlock et al.'s (2000) work simulating thedevelopment of rills. Brown et al.'s (2003) sedimenttransport CA focuses much more closely on localpatterns of sediment sorting and movement in the fluvial

system in the search for larger patterns of material self-organization.

At larger sizes, Coulthard et al.'s (1998) and Craveand Davy's (2001) CA simulations have highlighted thepossibility of CA models to explore the large range ofwatershed geomorphic responses to long-term climateand land use changes. By coupling surface water CAmodels to subsurface flow and surface erosion transitionrules, Luo's (Luo, 2001) LANDSAP simulationsproduce correctly scaled patterns of fluvial andgroundwater sapping forms. Studies of long-termlandscape evolution have benefited enormously from


the CA approach, which allows fast and accuratesimulations of repeated local erosion rules to digitalelevation models. Chase's (1992), Rinaldo et al.'s(1992), and Rodriguez-Iturbe and Rinaldo's (1997)CA simulations of landscape evolution showed thetemporal processes by which overall energy patternswere minimized and fractal networks emerged fromlocal erosion rules. These CA models have beenextended by Caldarelli (2001) and De Boer (2001), aswell as Rebeiro-Hargrave (1999), who have exploredthe evolution of asymmetric drainage basins usingcellular automata simulations.

Although many geomorphic CAs allow vegetationand other layers of information to be used as boundary-

Fig. 5. (A) A no vegetation, CA model of barchan dune formation; and (B) ththrough time (5, 10, 30, and 50 steps), modified from Baas (2002).

type information, very few CAs try to integrateecological and geomorphic processes into viable cellular“multiautomata”. Two notable examples of interactionare the formation of shapes of meandering rivers bycombining Murray and Paola's braided river CA withvegetation growth to strengthen the “riverbanks”(Murray and Paola, 2003). Moreover, efforts to combineearlier barchan dune (Baas, 2002) and ripple (Zhang andMiao, 2003) formation CAs with emergent vegetationgrowth produced highly realistic modified ridge-likeforms (Fig. 5). Even these simulations which haveexplored the intricate relationships between these twogroups of dynamics processes have barely scratched thesurface in what is possible. Large-scale fully formed

e dune formation CA modified by a rule to grow emergent vegetation


CAs exist in both disciplines, and even simple, toymodels with chained multiautomata will likely providevast improvements in our understanding of systembehavior. The model of bog patterning, put forward byRietkerk et al. (2004), is a fairly complex combinationof dense vascular plant growth coupled with spatialvariable subsurface water flow. Both of these sub-systems are based on a set of discretized differentialequations, which allows for the emergence of bandingpatterns in bogs. The framework of numerical solutionin two dimensions is considerably more complex thatwould likely be required if the same physicalformulations were recast as a CA model in the moldof Murray and Paola (2003). Similarly, the model ofdunefield activation and stabilization by Hugenholtzand Wolfe (2005) includes a differential equationframework for the changes in vegetation cover. Thisdetailed model subsystem might be enhanced if thecomponents of sand dynamics model were allowed tohave spatial dynamics based on a CA model of sanddune growth and decay, a framework that would allowthe existing difference-equation basis for the vegetationcomponents. The topic of a geomorphology/ecologyinterface at a radically different scale is the influence ofemergent vegetation in river channels. In this case, theriver flows shape the spatial patterns of submergedplant community and partly alter its temporal dynam-ics, whereas the emergent vegetation contributes flowresistance and causes flow partitioning laterally andvertically. In this complex plant–water interaction, alattice–gas or lattice–Boltzmann approach might wellallow a modeler to determine some of these relation-ships explicitly in three dimensions, from physical firstprinciples rather than relying solely on measured valuesfrom flumes and streams.

Because widespread CA modeling in the geosciencesand ecological sciences is relatively new, its practi-tioners are often criticized for using the approach ratherthan some more traditional approach. Continuummechanics, for example, is so engrained in thegeosciences, that any alternative to strict differentialequation formulations is often branded as “arbitrary” ornon-advantageous. These views, while historicallyunderstandable, are in conflict with the very physicaland mathematical sciences that produced continuummechanics over the past 300 years. A huge number ofrecent physics articles in the past twenty years contain awide range of modeling approaches, including differ-ential equations, cellular automata, CMLs, multi-agentsystems, and many others. For example, at thebeginning of the period of major physics research intocellular automata, Toffoli (1984) carefully compared the

physicality of differential equation approaches inphysics to automata approaches. He found that theautomata approach, even with limited discrete states percell, offered just as large a range (in phase space) ofmodeling possibilities, making it a viable alternative todifferential equation methods. The nature of hiscomparison should be of utility to geoscience modelersas well.

The great advantage of differential equations… isthat we have three centuries' experience withmethods for their symbolic integration. As longas all computation had to be done by hand, it paidto stylize the physics in a certain direction so as tobe able to handle the resulting mathematics. Butfew differential equations have closed-form solu-tion anyway, and the past fifty years have seennumerical computation make bolder and bolderclaims at being recognized as an essential part ofmathematics.

The moment one gives up symbolic manipulation asa major motive for using differential equations, onestarts wondering whether one should still keep themas the starting point for numerical modeling. In fact,they lead to concrete numerical computation (e.g., asrun on a general-purpose computer) that is at leastthree levels removed from the physical world thatthey try to represent. That is, first (a) we stylizephysics into differential equations, then (b) we forcethese equations into the mold of discrete space andtime and truncate the resulting power series, so as toarrive at finite-difference equations, and finally, inorder to commit the later to algorithms, (c) weproject real-valued variables onto finite computerwords (“round-off”). At the end of the chain we findthe computer – again a physical system; isn't there aless roundabout way to make nature model itself?(Toffoli, 1984, p. 121)

Certainly differential equations have enormous valuein the simulation of geomorphology and ecology; mostspatially explicit models are based on them. The critiquehere is not that differential equation methods arenecessarily inappropriate or incorrect, but rather thatthey may not be necessary. Similarly, other schemassuch as blind empiricism, statistical mechanics (asopposed to classical continuum mechanics), multi-agentsystems, and other approaches do not necessarilyprovide the only (or even best) choice for processrepresentation; model choice is fundamentally arbitrary,and the CA approach is simply another tool in thetoolbox of choices.


4. Advanced issues in CA implementation

Despite the strengths and successes of the CAapproach, important issues in implementation andtheory need to be addressed if this schema will beuseful in integrating geomorphic and ecologic model-ing. Some of these issues can be identified, considered,and addressed early in the model-building process,others have rarely been considered, much lessovercome.

The structure of the CA universe as a lattice of locallyconnected sites sometimes can produce strange, unreal-istic effects. For example, in some nondiffusive CAswhere the number of possible cell states is low (forexample the states could be 1 or 0), some dynamics willhave preferred directions of motion. Such anisotropicresponse is the macroscopic realization of the inabilityof a simple lattice and simple neighborhood structure toallow dynamics in any direction (Frisch et al., 1986;Margolus et al., 1986). In strong anisotropic situations,the directional effects may be enough to invalidate themodel completely. Much of the history of lattice–gasmodels is centered on finding a lattice where suchanisotropies are minimized or removed altogether, sothat the emergent fluid dynamics have the so-calledGalilean invariance, where the direction of the latticedoes not affect the dynamics of the CA. Many or evenmost complex CAs used in geomorphology and ecologyavoid this problem because they contain some diffusiveor probabilistic properties. Hence, even with a simpleVon Neumann neighborhood, a CML fluid (Thomas andNicholas, 2002; Thomas et al., 2002; Parsons, 2004)“poured” onto a cell (with no nearby obstructions) willappear to be shaped into a nearly circular fluid frontwithin just a few time steps. Ideally, CAs would beconstructed on a hexagonal lattice or one with an evenhigher degree of symmetry (requiring additional dimen-sions), but most simulations will continue to beconstructed on square-grid lattices because of theinherent raster structure of computer graphics andimage-processing software. Simple observation andtesting after the construction of the model shouldidentify potential anisotropy problems in a newlycreated CA.

Another important issue in the development andapplication of CA simulations is the careful consider-ation of time. As mentioned in earlier sections, somemodels carefully consider what amount of “real time” isbeing spent during each “time step” or “time slice” in aCA simulation. Many current models do not. Whenqualitative response of single-CA systems is all that isrequired, non-real time steps function adequately and

are certainly the most straightforward method ofmodeling (Bak et al., 1987). This approach begins tohave problems, however, when more than one CAmodel will be interlinked (multiautomata), or whenresearchers are interested in truly dynamic temporalprocesses (Favis-Mortlock et al., 2000). An excellentexample of the problem of timing is the observation ofdiffusion in a coupled-map lattice fluid. According tothe principle of locality, the amount of fluid (volume, forexample) in a cell is a function of its neighbor cells in theprevious time step. This would mean, therefore, that therate of overall spread of the fluid will be one pixel pertime step, regardless of the actual rules of volumetransfer. Such a single-velocity response of diffusionalflow is certainly not realistic at macroscopic scales.

A certain amount of temporal control and under-standing can be gained by scaling the observed rates ofchange in a CA to known rates of change in observedreality. Such a scaling procedure (for example, Murrayand Paola, 2003) allows a certain level of coupling ofdifferent CA layers into a multiautomata. In systemswhere overall equilibrium response is the focus of study,scaling is an appropriate method for chaining processestogether with the correct rates of change. If the systemdynamics of interest are fast relative to the time it wouldtake the process to affect the entire CA grid, however,equilibrium scaling may not be appropriate. As analternative, the concept of time chaining may be applied.

The effect of the response of one CA cell cannot, bydefinition, spread past its neighbors faster than one CAtime step. Standard “time-less” CAs also show that theresponse effect of a cell cannot propagate slower thanone cell per time step. The locality principle shows thatthe CA cannot propagate faster than one cell per timestep, but no restrictions for developing rules would keepthe dynamics from moving slower than one cell per timestep. This is essentially the same computational issue aswas early described by Courant et al. (1928) in thesolution of partial differential equations. I term a methodfor slowing the propagation of a CA process to a desiredrate time chaining, and the encoded rule for doing theslowing a timing chain (it has also been called phaselocking, but I dislike the term because it suggests thatprocesses are physically operating at the same rate, orphase).

The simplest example of a timing chain is to considera stochastic CA with one black cell surrounded by ahuge lattice of white cells. An initial transition rulecould be that a white cell to the right of a black cell willbecome black at the next time step, and any cell that isblack will become white at the next step. The overalleffect of this rule is to make the black square “move”, at


one cell per time step, to the right. Now what if wewished to explicitly define time on this grid? Supposewe were modeling the movement of a particle (the blackcell) through a matrix. We wished to specify that eachtime step was equal to one second. Also, the cells are all1 cm wide. This means that, as previously described, theparticle will always move 1 cm/s. Observation of a realsystem of particles moving through a matrix reveals,however, that this sized particle only moves at 0.4 cm/s.How do we include this observation? If we were onlydealing with one size of particle, we could simply scalethe grid so that the cells were 0.4 cm on a side, and thecorrect timing would be produced. Such a method isunwieldy, however, if we were to have more than oneparticle size in the model. Instead, we alter the originaltransition rule: “the black cell moves to the right if (andonly if) a randomly generated number for the black cellis greater than or equal to 0.4.” Now we require a layerof random numbers (between 0 and 1) to be generated ateach time step, but when they are (and the transition ruleis applied) the effect is that the black cell, albeit in arather “jerky” motion, is held back from moving the fullone-cell-for-each-time-step by this stochastic rule thatwe have implemented. Averaged through time, the cellwill appear to propagate at 0.4 cm/s. In this way,stochastic CAs can be time chained to any desiredexplicit length of time by altering the probability oftransition accordingly.

Time chaining of deterministic CAs, however, isslightly more challenging. If we have a grid with 1 mpixels, and a coupled-map lattice fluid on the grid, wemight calculate, for example, that the fluid in a particularcell should move across that cell at the rate of 0.25 m/s.It does not make much physical sense to have astochastic process determine the time chaining (i.e. “theprobability of motion across the pixel is 0.25”). Weknow that the fluid should move over the pixel in fourtime steps. One approach is to have two additionalstates, or layers, for the cell. The first is a “clock”, orcounter, that is reset to 0 when the fluid is removed fromthe cell, and increases its value by one at each time stepthat fluid occupies the cell. The second is an “alarmclock”which holds the value of the amount of time stepsneeded to hold the fluid before it can be moved to thenext cell. In this example, the fluid speed is 0.25 m/s,and the cell size is 1 m, so we would calculate that thealarm clock needs to be set to “4”, so that after four timesteps (four “clicks” of the “clock” layer), we have an IF–THEN rule that allows the fluid to move. This “alarmclock”method is essentially that used by D'Ambrosio etal. (2003) in their SCIDDICA CML models of debrisflows, Favis-Mortlock et al.'s (2000) model of rill

initiation, and Parsons's (2004) CML model of thetiming of rainfall–runoff.

An important consideration in time chaining is theprinciple of locality, which will not allow processes topropagate faster than one cell per time step. If a CMLfluid would be calculated to move at 10 m/s across agrid with 1 m pixels and time step of 1 s, seriousinaccuracies will occur because the fluid will not beable to flow at the speed that is needed. Instead, aresearcher needs to make a best-guess estimate of themaximum speed of propagation possible, and thenadjust the explicit length of time devoted to each timestep, or alternatively adjust the spatial size of the latticecells. This problem is essentially the same as the well-known Courant–Friedrichs–Lewy Condition (or Crite-rion) in the solution of differential equation models ofsystems (Courant et al., 1928), where computationalinstabilities arise when processes are calculated tomove faster over a space than is allowed by the amountof time per time step of computational iteration. Timechaining is an important consideration in using CAs tomodel real-world systems, and more research is neededto provide general chaining techniques for large classesof CAs.

At the interface between geomorphology andecology, widely varying rates of change occur in spaceand time between the interconnected or nested systems.In some cases, models can and should be designed tohold some variables constant and exogenous. This ideacertainly is not new; Schumm and Lichty's (1965)discussion of time, space, and causality posits suchregions of time–space where variables are independentor dependent. In many cases the mutual interaction ofmany variables, geomorphic and ecologic, however,must be considered. Perhaps a reasonable starting pointfor defining which variables to include as endogenousvariables and how to correctly chain them to the rest ofthe model's timing is Phillips's (1999) informationcriterion in geomorphology. Much like the CFLcriterion, the information criterion shows the mathe-matical relationship between the spatial scale, temporalscale, and rate of change of a system component ormultiple system components. Known rates of change inmodel subsystems can be used when specifying the gridresolution of the model and time step interval, andconversely specified model resolution in time and spacewill place limits on the allowed rate of change of asubsystem. The power of the information criterion, inaddition to its utility in time chaining the model, isexposing when two systems change at such vastlydifferent space–time rates that one or the other mightwell be modeled as an exogenous boundary condition.


Verification of a CA model (does the model do whatit is supposed to do) is an inherent part of the modelconstruction, and it is not easy to separate verificationfrom model validation (how well does the modelsimulate a real-world system). Nicholas (2005) pointsout that the two concepts are entangled in CA modelingbecause, for example, the grid structure of the CA is animplicit part of process parameterization. Indeed, it wasa combination of normal verification procedures andlater validation studies that found the original lattice gasmodels of fluid flow to have unexpected and unphysicalanisotropic response because of the grid structure andnonsymmetrical neighborhoods used. Currently, nostandard techniques exist for CA model verificationbecause of the large range of conceptual schemasallowed by the CA framework and because of theentangled verification and validation concepts.

An equally challenging problem in CA use is thecorrect use of a validation technique or techniques(Murray, 2003). In simple, 1D predictions usingempirical approaches or differential equationapproaches, many goodness-of-fit techniques existsuch as the calculating the coefficient of determinationor the index of agreement. For validating prediction ofobject classes without consideration of spatial accuracy,error matrices, or truth tables might be appropriate.Validating a two-dimensional or three-dimensional CAposes a much larger challenge. Nonspatial techniquessuch as the coefficient of determination are far toosimplistic, because many patterns of prediction wouldgive the same R2 value, and many of these potentialpatterns will be far more “wrong” than others. Also,either object-like classes or real-valued cells can exist ina CA, so choosing just truth tables or R2 measures is alsooversimplistic. Geostatistical methods are one avenuelikely to yield promising results in searching for usefulvalidation tools, and they are widely available in over-the-counter GIS software packages. Also, if a modelcorrectly produces an “emergent property” that isexpected, but was not directly programmed into themodel, this is also a reasonable, if qualitative, method ofvalidation. An example would be the production ofbraids in rivers with the correct spatial scaling fromsimple rules of sediment and water flux (Murray andPaola, 1996, 1997; Sapozhnikov et al., 1998). Observ-ing suitable spatial and temporal scaling emerging froma model is also an indication of validity, and researchersshould, in the future, consider incorporating this type ofvalidation. No simple approach seems to be “perfect” forvalidating all CAs, and the search for useful validationtechniques will continue to be an important goal inmodeling.

One of the most fascinating issues in modeling is theissue of material representation. The human brain seemsinnately wired to classify forms into named objects (hill/valley/glacier/cloud/tree/forest), yet a great many men-tal and linguistic descriptions of degree also exist as well(higher/lower/steeper/rougher/smoother/more round).The archetypal representations in science are objectsand fields, where the object is materially distinct fromother objects (Mark et al., 1999), and processes happento and between objects. In the field representation, onlyone object (the “field”) exists, but it varies in degreefrom one place to another and from one time to another.A contour map, for example, describes changes in thedegree (elevation) of the topographic field. To makematters more confusing, models in the same disciplineregularly use completely different representations tomodel the same thing. In geomorphology, for example,rivers can be represented as one-dimensional objects(lines, as they often are for meander-bend studies), two-dimensional raster fields (as they are in CML modelsand remote sensing images), or a set of three-dimensional objects (lattice–gas particles in lattice–Boltzmann simulations). The issue of representation isbound up with issues of the theory-ladenness ofobservation and measurement at different scales(which can augment the field-like or object-likecharacteristics of a system). Efficient communicationof CA models across interdisciplinary lines will requiredirect communication of representational choice, andresearchers will need to accept that some degree ofobject–field duality may influence model-building.

The raster-like gridding and tessellation of CAs makethem ideal for modeling the field representation ofnature. Each cell is a position in space–time that can beobserved for dynamic changes. This type of view, theEulerian frame of reference, is particularly suited to thefield representation because objects move and grid cellsmay not. Objects can move about in CA models (forexample, the aforementioned “black cell” in a sea of“white cells”), but the choice of CAs for a strictassemblage of objects seems unwieldy and inefficient.The multi-agent systems representation of nature (alsoknown as particle systems or agent-based models) mayprovide a more reasonable approach to model a systemof objects than can CAs, as it provides a view of adistinct object as it moves through space–time (theLagrangian frame of reference). Kessler and Werner's(2003) work on modeling structures of sorted patternedground using large particle systems is an ideal exampleof a situation where CAwould be an inelegant approachbecause the necessary representations are almostcompletely object-like.


Approaches using cellular automata can be criti-cized, on occasion, because of the dependence of thelocality principle. What about nonlocal or top-downinfluences? Ecological hierarchy theory postulates thatinfluences can and must flow from top and bottomdirections of influence, and both directions need to beaddressed in any process description. Most CAs dealwith top-down influences as external boundary condi-tions or as aspatial variables in the transition rules.External control programs can also change external ortop-down conditions in lock-step with the CA model,but such control programs are often inelegant andunwieldy. Another approach might be to haveinteracting grids with different cell resolutions tosimulate nonlocality yet maintain spatial elements.Regardless of the approach, CAs are an inherently“bottom-up” approach to modeling, and spatiallyvariable top-down influences are difficult to representefficiently in this schema.

Finally, we come to the issue of computationaldesign. Cellular automata have been previouslydesigned in the raster-grid, image processing mold, orusing an object-oriented approach. Although theparallelism inherent in CAs make them especiallysuitable for multiprocessor computers, nearly all CAs(except some lattice–gas designs) are run on normalseries-type, single processor computers. To analyzewhich approaches are more suitable, more efficient, oreasier to design is a task far beyond this article, and timewill tell whether or not a standardized, generalcomputing method for CAs should be implemented.

5. CA in teaching and general research

In addition to the many pragmatic and usefulbenefits of simulating geomorphology and ecologyusing the cellular automata approach, CA modelsprovide a conveniently simple approach to model-building as an educational exercise. Linkages betweengeomorphology and ecology involve competing con-ceptual schema and different process rates of changethat require explicit description in any model, and theCA framework provides an excellent educationalforum for both issues. Also, the many examples inthis article show that using CA as a modelingapproach does not limit a model to one particularlevel of complexity. Toy models, like the SandpileCA, are simple enough to be played by hand, andcertainly in simple spreadsheet programs. Slightlymore complicated CAs, like the Murray and Paolabraided river CA, require an intermediate level ofprogramming knowledge, or at least a fair competency

of using raster GIS or image processing software formodeling purposes. The most advanced CA modelsoften involve many people, a high degree ofprogramming competency, and a large amount oftime to work through difficulties and to build infunctionality. This progression, from toy models tofully designed realistic simulations, is a continuumthat is well-suited to step-wise environmental scienceeducation. From an applied perspective, Malamud andTurcotte (2000) argue that CA provide an excellentway to understand and predict the response of naturalhazards, making them a reasonable field of study forstudents.

As students learn the fundamentals of model design,they may experience much of the excitement ofdiscovery that is common in science; the importanceof using toy models as a starting point in learningmodeling is to show that complex response can arisefrom simple constructs, and also that fundamentalinsights can arise from only a modicum of modelingexperience. Starting with simple models alsoencourages the “frugality” (Carpenter, 2003) of goodmodeling; a good model should be just complex enoughto be realistic. Rule-based modeling is an important firststep in simple ecosystem simulation (Starfeld, 1990).Also, the “language” of CA modeling is inclusive anduseful in communicating ideas across traditionallyisolating disciplines.

Along with the general knowledge of CA theory,only the lack of simple, standardized tools for buildingCA models limits its immediate use in education. Oneof the most overriding themes of the past ten years ofCA research in geomorphology and ecology has beenthe lack of standardization in modeling approaches ortechnology. To be sure, standardization can potentiallyremove some of the versatility and model understand-ing that serves the model-building community well.Pragmatically, however, the lack of standardizedsoftware for importing spatial data sets, setting CAboundary and neighborhood conditions, constructingand iterating the transition rules, and producing outputthat can be displayed, analyzed, and validated hasprobably hampered overall CA development in thesesciences. Whoever develops a general CA model-building package will have a significant influence inshaping the next period of theory-building in geomor-phology and ecology. An analogy can be made withthe period before STELLA was developed for simpledevelopment of differential equation models, or beforeGISs became widely available for individual researchin the late 1980s and the early 1990s. Park andWagner (1997) called for CAs to be coupled with GIS


for maximum simulation viability. Some software arebeginning to incorporate rudimentary CA interfaces,such as Idrisi, Erdas Imagine (and its SpatialModeler), IDL, PCRaster, and ArcGIS, but CAmodeling is not the primary goal of any of thesepackages, and full-fledged CA capabilities are stilllacking in these programs.

The use of cellular automata in model and theorydevelopment in geomorphology and ecology is justbeginning to flourish as a subfield. Its developers arebeginning to look at CA research as an independent fieldof study worthy of discussion and critical analysis(Malanson, 1999). The number of practitioners fromacross both of these immense fields of study has reacheda critical point where general methods of model designhave begun to leak from one subgroup to another. Thegreatest power of CA is likely its potential to linkmodels that are currently isolated because of use onradically different schema (e.g. differential equations vs.expert rules vs. stochastic mechanisms). This survey ofcellular automata in geomorphology and ecologysuggests that the time has arrived to introduce CAtheory as a general tool to advanced undergraduates andgraduate students studying modeling, just as differentialequations and empirical regression are taught in suchforums today.

Acknowledgements

The author would like to acknowledge Mike Urban,Melinda Daniels, and Martin Doyle for their sponsor-ship of the “Linking Geomorphology and Ecology”session at the 2004 Annual Meeting of the AAG. JayParsons provided time-chained CA watershed simula-tions and valuable discussions on computer implemen-tation of CA models. Andrew Marcus and Carl Legleiterproved to be ideal sounding boards during therecognition of many of the representational andimplementational issues in this paper. Finally, MartinDoyle gave important advice on the utility of ‘fast andfrugal’ modeling in the training of students. BradMurray and two anonymous reviewers helped signifi-cantly tighten the manuscript's structure and clarifiedmany details.

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