Because natural systems contain many components whose states are dependent on numerous interactions, it is difficult to construct realistic models that match the natural level of complexity (Clark 1993). It is for this reason that simple rule-based models, such as cellular automata models, can be utilized to approach large-scale problems. In cellular automata (CA) models, each particular cell is affected by its neighbors in a simple, rule-based manner (Wolfram 1986). Cellular automata 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 (Jeltsch et al. 1996). Therefore, CA theory allows for the study of simple interactions between organisms through time.
The applications of CA models are quite diverse.
They have been used to study such diverse subjects as interactions between
sea stars and coral reef (Green et al. 1990) and disease
spread in human populations (Sieburg and Clay 1991).
Although the rules may be quite simple, the results are often highly dynamic
(Phipps 1993, Hogeweg 1988). CA results
have been shown to resemble large-scale natural phenomenon such as subalpine
forest wave regeneration (Sato and Iwasa, 1993), and
forest gap-phase dynamics in Barro Colarado Island, Panama
(Manrubia and Sole 1996).
Cellular automata date back to 1948, however, they
became popular in the 80s.
They are characterized by a few salient features:
1) the system substrate is comprised of a 1-, 2-, or 2-dimensional cell network.
2) each cell interacts with its neighborhood (i.e., a limited number of cells chosen according to rules)
3) each cell can adopt any one of m possible discrete states
4) the system follows a discrete time dynamic
5) at each time unit, each cell updates its state according to a transition rule that takes into account the state of the neighboring cells
6) an initial configuration is given at the beginning of a simulation run
7) the system is allowed to evolve under the same conditions over a finite number of time units
Deterministic Cellular Automata
In deterministic CA, the new state of a cell is determined rigidly on the basis of its actual state and states present in neighboring cells.
Example of a one dimensional CA. There are two states and a neighborhood size of 3
Local State transitions of elementary cellular automata
States=2; Neighborhood size k=3, So 23 = 8 possible combinations
In 2D CA, cells are found commonly in hexagonal or more commonly square lattices
A CA neighborhood can be defined in a number of ways, 2 common ways are:
1)von Neumann neighborhood, which considers the four adjacent cells sharing a common edge with the central cell
2) Moore neighborhood, which considers diagonals, i.e., eight surrounding cells
In addition, each cell considers its own state. Thus,
the von Neumann consists of 5 cells and the Moore consists of nine.
EXAMPLE
Consider a model that uses a square lattice and a
Moore neighborhood.
This example is called The Game of Life (GOL)
There are two states in the GOL, 1-alive, 2-dead.
The rules are as follows:
-A cell may have two possible states, i.e., alive (L) or dead (D).
-Any L cell dies unless it has 2 or 3 L cells in its neighborhood of 8.
-Any D cell can become L only if it has 3 L cells
in its neighborhood.
TRY IT OUT:
Online JAVA Game of Life (GOL): http://www.xs4all.nl/~ranx/gameoflife/
Or a much cooler version at: http://www.mindspring.com/~alanh/life/
Also GOL patterns can be browsed and JAVA applet activated at:
http://www.cs.jhu.edu/~callahan/patterns/contents.html
*Try random life patterns, then change the rules. You will see the range of behavior that is exhibited from these simplistic rule-based beginnings.
Stochastic Cellular Automata
Another kind of CA models are stochastic CA, where a transition rule can also incorporate some stochastic or probabilistic element. Such as if the neighborhood is occupied by 5 cells there is a 60% chance that the cell will change from occupied to unoccupied.
Stochastic CA Example 1.
Inghe (1989) developed a 3D stochastic automaton to study survivorship in a plant population composed of individuals produced by either sexual reproduction (genets) or by clonal growth (ramets)
Each neighborhood consists of 12 sites
Phalanx Mode
Guerilla Mode
-For each site in the neighborhood, there is a certain probability, P, that a ramet grows there. On average each ramet will produce 3 daughters
-Also, if it is the only ramet in the neighborhood to colonize the central site from time t to t+1 (one generation), if it is unoccupied.
-If more than one ramet compete for colonizing an empty site, the probability for a ramet in position I to colonize the empty place PI' is given by:
where Pi (i=1, 2, 3,..., 12) is a value from one
of the 12 positions.
-Phalanx represents a clonal herb with most of its
daughter ramets placed very near the original
-Guerilla mode tends to place the daughter ramets
further away from the original
-Undisturbed growth yields a roughly circular clone
with a nearly closed advancing front
-Disturbances are randomly imposed on the system
to produce mortality.
Stochastic CA Example 2: Use of CA for interspecific
competition for space
Silvertown et al. (1992) developed a CA model based on an experimental study.
In the experimental study using five grass species, the following relationship was found
From this study
a CA model was generated to show how invasion would proceed if two grass
species were grown adjacent to each another.
Four behaviors that can be generated by CAs
Class 1: Evolution leads to a homogenous state. After a certain number of time units, all cells have a similar state irrespective of the initial configuration
Class 2: Evolution leads to a set of separated simple stable or periodic structures. In this case a spatial organization appears with distinct spatial domains in which two types of temporal patterns may exist: homogenous and stable state or periodical changes of states.
Class 3: Evolution leads to chaotic pattern. Irrespective of initial conditions, CA yield aperiodic patterns indistinguishable from the initial pattern
Class 4: Evolution leads to complex localized structures, sometimes long-lived. In this class, some very complex spatial patterns may arise and reproduce over long periods of time; these patterns may also exhibit intriguing spatial propagation despite a perfect conservation of their shape.
Thus, surprisingly complex behaviors can arise from
the action of local processes that are not globally directed.
CA models can be used to depict:
-spatial competition
-invasive or spreading behaviors,
-gap dynamics
ALIFE at the Sante Fe Institute: Online information on cellular automata.
Clark, J.S. 1993. Scaling the population level: Effects of species composition in gaps and structure of a tropical rain forest. Pages 255-285. In: J.R. Ehleringer and C.B. Field [eds]. Scaling Physiological Processes Leaf to Globe. Academic Press, New York.
Green, D.G. 1990. Cellular automata models of crown-of-thorns outbreaks. Pages 157-169. In: R.H. Bradbury [ed]. Acanthaster and the Coral Reef: A Theoretical Perspective. Springer-Verlag, New York.
Hogeweg, P. 1988. Cellular automata as a paradigm for ecological modeling. Applied Mathematics and Computation 27: 81-100.
Inghe, O. 1989. Genet and ramet survivorship under different mortality regimes- a cellular automata model. Journal of Theoretical Biology, 138: 257-270.
Jeltsch, F., S.J. Milton, W.R.J. Dean, and N. VanRooyen. 1996. Tree spacing and coexistence in semiarid savannas. J. of Ecol. 84: 583-595.
Manrubia, S.C. and R.V. Sole. 1996. Self-organized criticality in rainforest dynamics. Chaos, Solutions & Fractals 7 (4): 523-541.
Phipps, M.J. 1993. From local to global: The lesson of cellular automata. Pages 165-187. In: J.R. Ehleringer and C.B. Field [eds]. Scaling Physiological Processes Leaf to Globe [eds]. Academic Press, New York.
Sato, K. and Y. Iwasa. 1993. Modeling wave regeneration in subalpine Abies forests: Population dynamics with spatial structure. Ecology 74: 1538-1550.
Sieburg, H.B. and O.K. Clay. 1991. The cellular device machine development system for modeling biology on the computer. Complex Systems 5: 575-601.
Silvertown, J., S. Holtier, J. Johnson, and P. Dale. 1992. Cellular automaton models of interspecific competition for space- the effect of pattern on process. Journal of Ecology 80: 527-534.
Wolfram, S. 1986. Theory and applications of cellular automata. World Scientific, Singapore.