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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)

- 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)

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)

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

- 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)

We consider discrete time random perturbations of monotone cellular automata (CA) in two dimensions. Under general conditions, we prove the existence of half–space velocities, and then establish the validity of the Wulff construction for asymptotic shapes arising from finite initial seeds. Such a shape converges to the polygonal invariant shape of the… (More)

We explore a variety of two-dimensional continuous-valued cellular automata (CAs). We discuss how to derive CA schemes from differential equations and look at CAs based on several kinds of non-linear wave equations. In addition we cast some of Hans Meinhardt's activator-inhibitor reaction-diffusion rules into two dimensions. Some illustrative runs of CAPOW,… (More)

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

In the discrete threshold model for crystal growth in the plane we begin with some set A 0 ⊂ Z 2 of seed crystals and observe crystal growth over time by generating a sequence of subsets A 0 ⊂ A 1 ⊂ A 2 ⊂ · · · of Z 2 by a deterministic rule. This rule is as follows: a site crystallizes when a threshold number of crystallized points appear in the site's… (More)