Figure 28.7. A differential equation is a good approximation to a discrete recursion when change is gradual. (A) The series of triangles shows the solution to the recursion pt + 1 = pt + sptqt, for generations 0, 1, 2, ... . The continuous red curve is the solution to dp/dt = sp(t)q(t). This has the same slope at time t = 0, when p(0) = p0. Compare the slope of the red curve with the diagonal line, which has slope sp0q0. However, because the differential equation allows for a continuous acceleration (i.e., an upward curve) as allele frequency increases, it increases slightly faster than the discrete recursion (black steps). (B) The discrete recursion (black dots) with WP/WQ = 1.1 compared with the continuous solution (red curve) to dp/dt = spq with s = 0.1; allele frequency is initially 0.01. Agreement is close at first, but errors build up over time; nevertheless, the error is never more than 4%.
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