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2D Pattern forming Cellular Automata

Table of Contents








Suitability of Using CA for Modeling Ecological Systems



History of Ecological Systems Modeling



A Simple Example of an Ecological Model based on Cellular Automata



Types of Ecological Systems Which are Frequently Modeled using CA



Modeling Spatial Competition in Ecological Systems using CA



Modeling Invasive or Spreading Behavior using Cellular Automata



Modeling Gap Dynamics Using Cellular Automata



Techniques for Improving the Effectiveness of Cellular Automata in Ecosystem Simulations







Modeling Ecological Systems using Cellular Automata


This paper is an attempt to explore the possible research avenues within the field of Ecology Modeling using Cellular Automata (CA) as the basis for the model, and to study past and present work carried out within this framework.


Ecological systems are generally considered among the most complex because they are characterized by a large number of diverse components, nonlinear interactions, scale multiplicity, and spatial heterogeneity [Wolfram-2002], [Elsevier-2003].

Due to this reason a simple rule-based model, such as a cellular automata model, can be utilized to approach large-scale problems. In CA models, each particular cell is affected by its neighbors in a simple, rule-based manner. CA models form a holistic class of models where space, time and states are discrete. Because CA models are rule, rather than equation based, they allow for the direct consideration of knowledge that is not necessarily restricted to hard data and are particularly useful in consideration of complex systems [Wolfram-1986]. Therefore, CA theory allows for the modeling of simple interactions between organisms through time.

One interesting point to note here is that according to Wolfram [Wolfram-2002] this complexity evident in every ecological system is not a result of adaptation or natural selection, but the result of simple underlying rules in such systems.  This suggestion is discussed by Wolfram [Wolfram-2002] using 3 common occurrences; namely growth of plants, biological pigmentation patterns and growth of mollusk shells. 

Suitability of Using CA for Modeling Ecological Systems

A significant amount of research has already been done in modeling ecological systems using cellular automata as the primary tool, as seen by [Hrab-1995], [Gron-1998], [Sten-1995].  Empirical studies by Wolfram and others show that even the simple linear automata behave in ways reminiscent of complex ecological systems.  This can be seen by the fact that the fate of any cellular automata, irrespective of its initial configuration, will be one of the following:

                                I.      To die out

                             II.      Become stable or cycle with a fixed period

                           III.      To grow indefinitely at a fixed speed

                          IV.      To grow and contract irregularly

And thus it can be argued that CA are ideally suited for the purpose of modeling ecological systems.


One other factor which makes CA a promising candidate for this kind of research is the fact that there is considerable controversy in the field of ecological modeling regarding the suitability of mathematical models for such a complex task.  The hypothesis that there is sufficient low-dimensional order to allow prediction of ecological dynamics has been covered in controversy for nearly a century.  But during the last few decades hypothesis such as ‘population fluctuations are shaped largely by low-dimensional deterministic forces’ has been rigorously and successfully tested.  Resulting in the identification of the low-dimensional deterministic phenomena such as equilibria, bifurcations, multiple attractors, resonance, basins of attraction, saddle influences, stable and unstable manifolds, transient phenomena and even chaos [Brauer-2001].  Cellular automata have been introduced into the arena of ecological modeling in such an environment.


CA models have been used to model a vast array of ecological systems.  They have been used to study such diverse subjects as interactions between sea stars and coral reef (crown-of-thorns outbreaks) and disease spread in human populations.  Although the rules used within these CA models may be quite simple, studies carried out by Hogeweg and Phipps show that the results are often highly dynamic. Recent research using CA models to model large scale phenomena has been successfully achieved.  Examples for such systems would be sub-alpine forest wave regeneration and forest gap-phase dynamics.


History of Ecological Systems Modeling


The classical approach to modeling ecological systems is characterized by mathematical tractability and determinism.  Labeled as the Newtonian Mechanics approach, it was first adopted by population ecologists, and remains very much alive as a legacy of the ‘Golden Age’ of theoretical ecology from 1920s and 1940s.  One of the shortcomings of this approach is that it can handle only cases where a small number of components are involved and hence chosen to deal with ‘organized simplicity’. 


The statistical mechanics approach, on the other hand, is effective for tackling the ‘disorganized complexity’, characterizing systems with a large number of components that each behave more or less randomly. Ecologists are often confronted with the so called ‘medium-number’ systems that exhibit the ‘organized complexity’, which is the subject of systems science. No wonder that, as systems science emerged in the 1950s and 1960s, ecologists were among the most active, applying and contributing to the three major theories in systems science: general systems theory, cybernetics, and information theory.


However, the enthusiasm for systems ecology faded away quietly (notably in North America) during the 1980s in the wake of the failure of several large, monolithic computer models produced by the International Biological Program and with the increasing recognition of the importance of ubiquitous spatial heterogeneity and scale. While the systems modeling approach continued to be dominant in modeling energy flow and matter cycling of various ecosystems, spatial modeling approaches, including diffusion-reaction, patch (or gap) dynamics, cellular automata, and fractal models, seemed to take over the central place in ecological modeling during much of the 1980s and the 1990s.


A Simple Example of an Ecological Model based on Cellular Automata

Consider a CA where the landscape of a forest is modeled by a fixed array where each cell in the array represents a unit of land surface.  The states associated with each cell correspond to ecological characteristics.  Using transition rules we can model processes that involve movement through space (e.g. fire, dispersal) in a "natural" fashion.  


To see how easily the cellular automaton model can be brought to bear on real systems, consider a system whose states are of the form (s,t), where t denotes "time since fire burned out the cell" and s takes one of the values: "bare earth" (E), "grass" (G), "woodland" (W), and "closed forest" (F). Define state transitions rules as follows:


                    (E, 0)           ->                 (G,  1), 
                    (G, 4)           ->                 (W,  5), 
                    (W,49)        ->                 (F, 50),
                    (s, t)            ->                 (E,  0),  if a fire ignites nearby
                                         ->                 (s,t+1),  otherwise

Starting with an arbitrary "landscape", consisting of a 2-D grid of cells in random states, the rules quickly produce patterns. For instance, "forest zones" quickly develop, even though the model contains no assumptions about the environment or site preferences of the vegetation. Admittedly, this model is a mere caricature of true forest succession, but it is only a short jump to management models taking (say) satellite data as inputs [Green-1993].

Since cellular automata are based on homogeneous cells this approach can be incorporated with both pixel-based satellite imagery and with quadrat-based field observations without too much difficulty.


Types of Ecological Systems Which are Frequently Modeled using CA

Space and time have always been recognized as crucial components of ecological change.  Important spatial patterns such as aggregated distributions of plant community species (at one or more scales) and juxtaposed neighborhood competition, has led to continued study of vegetation utilizing spatial methodologies [Silvertown-1992]. Population processes such as reproduction and interspecies competition are affected by the initial spatial arrangement of a species. The manner in which a species evolve in a landscape is influenced by the spatial and temporal correlations that exist in its agents.


According to [Baltzer-1998] the incorporation of spatially explicit models in the arena of ecological modeling are “expected to increase our ability to accurately model populations subject to complex processes”. The inclusion of space in vegetation models introduces more complex rules into a simulation than those created from simple parameters. This can lead to unpredictable chaotic simulation and non-linear dynamics [Silvertown-1992].


The theories regarding the spatial arrangements that many population models are based on were first introduced by Skellam.  Skellam introduced a mathematical model that can be used to simulate population dispersal. This model is based on the assumption of randomly distributing a species in space, and with an associated constant growth rate.


Due to these factors we can identify three types of behavior in ecological systems usually modeled using CA, and they are:

  1. Spatial Competition
  2. Invasive or Spreading Behavior
  3. Gap Dynamics


Within this paper an example of each type of ecological model will be discussed.  And how these techniques can be improved and emerging techniques of improving these mechanisms will also be discussed.


Modeling Spatial Competition in Ecological Systems using CA

Plants and other organisms must secure space in order to obtain access to scarce resources.  When competing organisms are of similar stature, competition primarily occurs between immediate neighbors and is local to the boundaries between the two species.  According to [Silvertown-1992] the recognition of importance of neighborhood competition in plant populations has led to the development of many interspecific neighborhood models for competition.  And models have been developed to analyze individual performance in population dynamics as well as effects of spatial patterns affect the process of competition.

One important point to note here is that modeling spatial competition using CA draws increased scrutiny to initial conditions and their effect and influence on the outcome of interspecific competition.  The term “Interspecific competition” is used to refer to the competition between two or more species for some  specific limiting resource(s), such as sun light.

Silvertown [Silvertown-1992] developed an experimental study using five grass species where the following relationship was found. Based on this a CA model was developed to analyze the invasion process if two grass species were grown adjacent to each other.  Within this model each cell in the CA can be one of many states, and such a state would typically represent vacant space or occupancy by a particular species.  A probabilistic approach was used to determine the local rules for cells which govern state transitions.

When random initial starting conditions were used it was noted that the extinction of the inferior competitors (Lolium, Poa and Cynosurus) was very rapid.  In aggregated models the inferior competitors survived for much longer periods of time.  These observations are compatible with other ecological models developed using traditional approaches.  The fact that the inferior species became rapidly extinct suggests that aggregation (clumping) has a dramatic effect on the rate at which stronger species can eliminate weaker species.  This introduces new factors into the arena of spatial competition, namely configuration of patches and juxtaposition of species.  These factors seem to demonstrate strong effects on community composition in the medium term [Silvertown-1992].

Modeling Invasive or Spreading Behavior using Cellular Automata

In this section we present an ecological model which has a Geographic Information System (GIS) implementation of cellular automata demonstrating the spread for the invasive plant species Rhamnus alaternus [Vivienne-1999].


A Geographic Information System (GIS) is a computer program for storing, retrieving, analyzing, and displaying cartographic data.  A GIS typically is a sophisticated implementation specifically targeted towards deriving interpretations from geographic data.  For example, in town planning a GIS can be used to analyze the town’s traffic patterns and other factors to determine which is the ideal location for a new supermarket.


According to [Vivienne-1999] GIS have been increasingly utilized within modeling projects this decade. The primary advantage of incorporating GIS into ecological modeling has been integration of environmental and biological features across a diverse set of spatial scales within a complex terrain and heterogeneous landscape.


In this case CA are very suitable as a modeling tool because they very effectively deal with the issue of spatial auto correlated parameters and have - by GIS standards - better temporal capabilities.  In cases such as this where CA are coupled with simulation systems they are very suited to the manipulation of the ecological parameters due to their bottom-up approach.  This is analogues to the scaling from individual to community level in this type of research. 


The coupling of the CA to a GIS is done via the rule base of the GIS.  The rules for dispersal and other temporal dynamics for the particular species are obtained from the GIS at runtime based on the spatial and temporal data for a particular cell.  The GIS also governs some aspects of the simulation which require long distance interactions, such as seed dispersal by birds over coastal (geographic) barriers, since the underlying CA is not very suited for such a task. 


One of the major difficulties in this kind of simulation is the run time complexities, to avoid extremely high run times in this approach two layers of grids are used, one coarse grid which is primarily updated and maintained by the GIS in which long range interactions take place and a fine grid in which the active spread and dispersal takes place.  For example if a long range interaction such as a seed dispersal by a bird occurs this will be simulated by the GIS on the coarse grid and once the location of the destination is identified a fine grid is created for this particular locale so that the active spreading mechanism can take place, which is based on the CA.  When the border of a fine grid is reached the adjacent coarse grid cell is opened and subdivided into a suitable number to form the a new fine grid and merged with the initial fine grid.  This allows the system to handle a higher degree of complexity by focusing on the areas of the simulation (governed by the GIS) within which the CA must operate, without exhaustively running a CA throughout the complete heterogeneous landscape.


The spread of the species Rhamnus alateirnus is defined by its neighborhood or the maximum distance between male/female plants for successful pollination.  The flexible manner in which the CA allows the manipulation of parameters allowed the researchers to study a multitude of initial conditions and their effects on the invasive spread of this species.  Species invasion is a good example of a complex and not well understood problem in the area of dynamic spatial modeling. It is also an example of complexity that can be difficult to recreate within the limits of current GIS technologies [Vivienne-1999].  By coupling CA with a GIS the researchers in this case managed to overcome two key weaknesses in both techniques.


Modeling Gap Dynamics Using Cellular Automata

In this section we will study the usage of CA to model gap dynamics in ecological systems.  Gap dynamics is extremely important in certain ecosystems.  For example in the case of a rain forest gap dynamics play a crucial role in maintaining tree diversity [Hubbel-1986]. 


When a tree dies in a closed-canopy forest, an area in which a light filters to lower levels become accessible. As a consequence of the sharp change in local environmental conditions (humidity, temperature, light), trees at the gap boundary (the edge of the newly opened gap) have a higher mortality rate. Such ‘canopy’ disturbances, of different intensity, are responsible for allowing other species to grow. The whole set of gap sites at different successive stages is believed to contain more species per tree than in closed-canopy gap sites which have the same area.  Also openings in the canopy are believed to initiate strong dynamic patterns in such systems. Some researchers have shown that data obtained from the Barro-Colorado Island (BCI) rainforest in Panama displays a fractal distribution of gaps over space.  It was shown that the BCI plot is in fact a fractal set with a box counting dimension of 1.85. Typically fractal structures appear in time and space as a result of strong self-organized large-scale correlations [Alonso-1999].


Cellular automata have been used in many instances to demonstrate gap dynamics in ecosystems.  In this section we present a Forest Simulator which uses a Cellular Automata as the modeling tool which tries to analyze the effects of gap dynamics on a rain forest.  The DivGame simulator [Alonso-1999] is a oversimplified simulator for rain forest dynamics where species diversity is taken into account (i.e. it is not a single species model such as ForestGame simulator).  The CA used for the DivGame simulator is a stochastic CA (there are some complications regarding the design of the CA but these will be ignored for brevity). 


In the CA model a gap is defined by a free cell.  When a tree dies, it creates gap in the canopy. Every tree death is the starting point for a disturbance in the system. Trees at the edge of a gap in the canopy have an increased probability of dying.  This field observation is built into in the DivGame model using some external dynamics.  When an opening is formed as a consequence of a single tree death, the relative environmental change of every neighboring point at the edge of the gap is computed by the CA.  One of the primary observations the authors make based on the output of the CA is that random disturbances can travel vast distances within the system depending on the local neighborhood of the area in which the disturbance took place..  The other key observation is that two similar species are able to perform very different walks (dispersion patterns), indicating that for a species its assemblage (grouping) is a very path dependent, unpredictable process. 


Techniques for Improving the Effectiveness of Cellular Automata in Ecosystem Simulations

If Cellular Automata are used for the modeling of a species dispersion in a given landscape, then the given landscape is divided into a lattice of homogeneous cells.  One of the primary characteristics (as mentioned earlier) of CAs are that they are discrete in time and space.  Furthermore these cells are updated synchronously, i. e., they change their states at the same time and in discrete time steps called time phases. This is done with respect to their own state as well as to those of the neighboring cells (dependent upon a given rule). While the local evolving demonstrated by a system can easily be described by a cellular automaton, problems arise when individual organisms try to disperse in space within one time phase of the system such that they have to diffuse through several cells.


Due to the restrictions of this diffusion model researchers have been looking for a more general solution for this problem. Thus, it should be possible to model several changes of cell states within one time phase of the system without synchronization of the cells. This can be done by describing the cell's behavior with respect to reactions to events. A well-known technique for the description of a technical system with event-based, concurrent, asynchronous state changes is the modeling with Petri nets. They build formal models that allow simulating systems where parts can evolve without synchronization with the rest of the system [Veronika-1998].


Another technique researchers use in this research culture is to couple Cellular Automata with Geographic Information Systems.  This allows them to circumvent issues with dispersion and synchronization, but maintain the CAs superior characteristics. 



It is clear that Cellular Automata offers many features which make them attractive models for ecosystem simulations.  In addition to this one key factor that favors the choice of cellular automata as the tool for modeling is the natural complexity of ecosystems.  Due to that fact that cellular automata are descrete temporally and spatially, there can be certain drawbacks as well, especially when it comes to the areas of dispersion and asynchronous updating.  But these drawbacks can be effectively handled by incorporating techniques such as Petri nets and geographic information systems.  The popularity of cellular automata as a possible modeling tool for ecosystem simulations has been steadily rising, and will most likely continue to grow due to the fact that it is a simple technique which can generate highly complex behavior.




[Wolfram-2002] : Wolram, Stephan – A New Kind of Science


[Elsevier-2003] : www.elsevier.com/locate/ecolmodel


[Brauer-2001] : Fred Brauer and Carlos Castillo-Chavez - Mathematical Models in Population Biology and Epidemiology


[Green-1993]: David G. Green – An Introduction to Cellular Automata


[Silvertown-1992]: Jonathan Silverton, Senino Holtier, Pam Dale – Cellular Automata Models for Interspecific Competition for Space – The effect of Pattern on Process


[Vivienne-1999]: Vivienne Cole, Jochen Albrecht - Exploring Geographic Parameter Space With A GIS Implementation Of Cellular Automata


[Hubbell-1986]: Hubbell, S. P. and Foster, R. B. - Biology, chance and history and the structure of tropical tree communities.

[Alonso-1999]: David Alonso, Ricard V. Slo’e - The DivGame Simulator: A Stochastic Cellular Automata Model of Rainforest Dynamics

[Veronika-1998]: Eva Veronika R´acz, J´anos Karsai - Computer Simulations On Cellular Automata Models Of Metapopulations In Conservation Biology





















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