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- J Molofsky, R Durrett, J Dushoff, D Griffeath, S Levin
- Theoretical population biology
- 1999

In sessile organisms such as plants, interactions occur locally so that important ecological aspects like frequency dependence are manifest within local neighborhoods. Using probabilistic cellular automata models, we investigated how local frequency-dependent competition influenced whether two species could coexist. Individuals of the two species were… (More)

- David Griffeath, Cristopher Moore
- Complex Systems
- 1996

It is shown that if a cellular automaton (CA) in two or more dimensions supports growing “ladders” which can turn or block each other, then it can express arbitrary boolean circuits. Thus the problem of predicting the CA for a finite amount of time becomes P-complete, the question of whether a finite configuration grows to infinity is P-hard, and the… (More)

- Janko Gravner, David Griffeath
- Physical review. E, Statistical, nonlinear, and…
- 2009

We introduce a three-dimensional, computationally feasible, mesoscopic model for snow-crystal growth, based on diffusion of vapor, anisotropic attachment, and a boundary layer. Several case studies are presented that faithfully replicate most observed snow-crystal morphology, an unusual achievement for a mathematical model. In particular, many of the most… (More)

- Janko Gravner, David Griffeath
- Experimental Mathematics
- 2006

Digital snowflakes are solidifying cellular automata on the triangular lattice with the property that a site having exactly one occupied neighbor always becomes occupied at the next time. We demonstrate that each such rule fills the lattice with an asymptotic density that is independent of the initial finite set. There are some cases in which this density… (More)

Start by randomly populating each site of the two-dimensional integer lattice with any one of N types, labeled 0,1, , 1 The type at site can the type at neighboring site ( . y y eat x x (i.e., replace the type at with ) that mod We describe the dynamics x y provided y x 1 N. of , discrete-time deterministic systems which follow the rule: cyclic cellular… (More)

Assume that a cellular automaton (CA) rule enlarges subsets of Z 2 , and does so in such a way that a larger set results in a larger outcome. Such models are called monotone solidi cation CA. In the critical case, these dynamics cannot cover the lattice starting from any nite set, but are able to do so from any set with nite complement. We assume that the… (More)

We introduce and analyze a simple probabilistic cellular automaton which emulates the flow of cars along a highway. Our Traffic CA captures the essential features of several more complicated algorithms, studied numerically by K. Nagel and others over the past decade as prototypes for the emergence of traffic jams. By simplifying the dynamics, we are able to… (More)

- Richard Durrett, David Griffeath
- Experimental Mathematics
- 1993

This research was partially supported by research grants to each of the authors from the National Science Foundation We study two families of excitable cellular automata known as the Greenberg–Hastings model and the cyclic cellular automaton. Each family consists of local deterministic oscillating lattice dynamics, with parallel discrete-time updating,… (More)

In the Exactly 1 cellular automaton (also known as Rule 22 ), every site of the one-dimensional lattice is either in state 0 or in state 1, and a synchronous update rule dictates that a site is in state 1 next time if and only if it sees a single 1 in its three-site neighborhood at the current time. We analyze this rule started from finite seeds, i.e.,… (More)

- Janko Gravner, David Griffeath
- Theor. Comput. Sci.
- 2012

We study one-dimensional cellular automata (CA) with values 0 and 1. We assume that such CA are started from semi-infinite configurations (those that have 0’s to the left of some site), andwe focus on the identification of robust periodic solutions (RPS), which,when observed from the left edge of the light cone, advance into any environment with positive… (More)