Each square is a star system, and its color shows how full it is, from empty (white) to as many people as it can support (dark red). Every generation, each settled world grows toward its limit and sends some of its people to the systems north, south, east and west of it. How many leave depends on crowding: with the crowding setting above zero, few people leave a world until it is close to full. The aliens start on the homeworld (black square) and the green ring is the Solar System. Click the grid to move either one, then press Start and see whether the colonists get here before the civilization's lifetime runs out.
By Claude Opus
In 1975 Michael Hart argued that a spacefaring civilization could fill the galaxy in a few million years even with slow ships, so the fact that nobody is here means nobody exists. Frank Tipler sharpened the argument in 1980 with self-copying probes. William Newman and Carl Sagan answered in 1981 in the journal Icarus, with a paper titled "Galactic Civilizations: Population Dynamics and Interstellar Diffusion." Their point was that Hart's calculation treats colonization as a matter of ship speed, when it is really a matter of population.
They borrowed their mathematics from population biology, where the spread of an advantageous gene or an invading species is modeled as a wave of diffusion. Every world has a carrying capacity, its population grows toward that limit, and emigration depends on population density. People leave when home gets crowded. A newly founded colony therefore sends out almost nobody until it has grown close to full, and the colonization front advances only as fast as the worlds behind it fill up. The speed of ships hardly enters into it. Run the simulator and compare the front speed with the one system per generation the ships could manage: the wave typically moves at a small fraction of that.
Newman and Sagan worked out two cases. With local population growth, the front becomes a traveling wave that moves at a steady speed set by the growth and emigration rates. With zero population growth, there is no wave of that kind. The population spreads the way heat spreads through a solid, and it slows down the farther it gets. Their estimates came out as required lifetimes. With growth, the wave from the nearest civilization could have reached Earth only if that civilization had lasted more than 2.6 million years, or 20 million years if the worlds are treated as a continuous medium. Under zero population growth it would have to last 13 billion years, about the age of the galaxy. Their conclusion was that Earth is uncolonized because there are too many worlds to fill within any plausible lifetime, not because spacefaring civilizations are rare. They added that the galaxy contains no very old civilizations with a consistent policy of conquering inhabited worlds.
To see the difference between the two cases, run with the defaults and watch the front speed, which settles at a steady value within a few dozen generations and stays there. Then check Zero population growth and run again. The front speed starts low and drifts lower the longer it runs, and the Solar System is out of reach. Setting Crowding to 0 turns the model into ordinary diffusion, where people leave at the same rate whether home is crowded or not, and the front speeds up. Raising the growth rate also speeds it up, because the worlds behind the front fill faster. Trials show how often a civilization with a given lifetime gets here. If you turn off random positions, the trials come out nearly the same every time, because the wave averages over thousands of worlds and the luck of any single world hardly matters. That is a real difference from the Landis model, where a few colonies that stay home can stop everything. With random positions, the trials answer a more useful question: how far a civilization with a given lifetime can reach.
Some features here are not in the paper. Newman and Sagan solved equations for a continuous, three-dimensional galaxy. This is a flat grid of identical cells with hard edges, where colonists move only to the four neighboring systems. The lifetime cutoff, the world-to-world variation, and the trials are additions, and in the zero growth case the homeworld is held at full population so the civilization doesn't simply thin out to nothing. A system counts as settled once it holds 1% of its capacity.
The model has its critics. The slow wave depends on the assumption that people emigrate only to escape crowding. A civilization that sends out colonists for its own reasons, or builds self-copying probes, spreads at close to ship speed, and that puts Hart's question back on the table. Later simulations have also pointed out that stars move. Over a few hundred million years the galaxy's own motion stirs settled and unsettled systems together, which would carry even a slow civilization a long way.
The other tools in the tools/fermi/ directory look at the same question in different ways: whether colonization can keep going at all, and whether a given star system ever gets reached.