Figure 3.—

Cones in space–time. A single 1D example is shown as a space–time diagram. (I) Mutations arise and spread forward in time. (II) Mutations can spread only if a cone backward in time is free from other mutations. Mutational origins are marked with stars; A and B are successful. Mutation A arises at time s; the probability that it is the first at that point depends on the area of the cone stretching backward from it (shaded); this cone has base f(s) and area h(s). Mutation C  does not spread because it occurs where B  has already reached; we can see this because point B lies in the cone stretching backward from C. The point x₀ is first occupied at time τ; two quantities of interest are X, the distance from x₀ to the origin of the mutation that eventually encloses it, and R = f(τ). Note that the “cones” will not generally have straight sides unless f(t) = νt.