All the rules stay the same but (iii), which becomes:
(iii') An offspring born at x is sent to a neighbor y chosen at random.
For convenience. we use periodic boundary conditions. In words, sites on the left edge of the square are neighbors of those on the right, and those on the top are neighbors of those on the bottom. More formally, y is a neighbor of x if (y1-x1,y2-x2) is in a set V, where the differences of the components are computed modulo L. Here, we will only be concerned with two kinds of neighborhoods:
four nearest neighbors: V = { (1,0), (0,1), (-1,0), (0,-1) }
5 x 5 neighborhood: V = { all points (x,y) with each coordinate smaller than 2 in absolute value.}
The next two figures give the results of the simulations for the spatial model with the 5 by 5 neighborhood and the four nearest neighbors. In each case the number of mutants per generation on the grid is set equal to 1.
5 x 5 neighborhood
Nearest neighbors<P>
Comparing the first panels of the these figures with that of the metapopulation model, shows that the number of species is about 45 in the first case, 100 in the second, and 120 in the third. This pattern is not surprising. In the metapopulation model all sites are adjacent to each other, so competition is the most fierce. In the spatial models, species can become isolated from each other, lessening competition. Comparing the results for the 5 by 5 neighborhood and the nearest neighbors shows that reducing the dispersal distance increases isolation and decreases competition.
The second panels of the three figures show snapshots of the systems at time 160,000. The first noticeable difference is that although species are scattered over (0,1) in the metapopulation case, they are restricted to roughly (0,2/3) in the nearest neighbor case, and to roughly (0,0.6) for the five by five neighborhood. This pattern occurs since in each system species with a death rate larger than the critical death rate dc cannot survive even in the absence of competition.
The third panels of these three figures which average the distribution of species between 10,000 and 160,000 shows that in each case the distribution of species is approximately uniform over the range of possible values.