Plotting the Value of a Variable at Time t 1 versus at Time t

Consider the haploid model of natural selection with the recursion equation (3.9),

Figure 4.9 plots p(t + 1) as a function oip{t) when WA is greater than Wa (solid curve in Figure 4.9a) and when WA is less than Wa (solid curve in Figure 4.9b). In both cases, we have also drawn a dashed diagonal 1:1 line. The diagonal line represents those special cases where p(t + 1) = p(t). Wherever the recursion curve crosses the diagonal line, the allele frequency in the next time step p(t + 1) will equal the allele frequency in the previous time step p(t). Such a point is called an "equilibrium" because it remains unchanged over time (see Chapter 5). Thus, if the system starts at an equilibrium value for the allele frequency, it will remain there forevermore.

Figure 4.9 plots p(t + 1) as a function oip{t) when WA is greater than Wa (solid curve in Figure 4.9a) and when WA is less than Wa (solid curve in Figure 4.9b). In both cases, we have also drawn a dashed diagonal 1:1 line. The diagonal line represents those special cases where p(t + 1) = p(t). Wherever the recursion curve crosses the diagonal line, the allele frequency in the next time step p(t + 1) will equal the allele frequency in the previous time step p(t). Such a point is called an "equilibrium" because it remains unchanged over time (see Chapter 5). Thus, if the system starts at an equilibrium value for the allele frequency, it will remain there forevermore.

Allele frequency at time t, p(t)

Allele frequency at time t, p(t)

Allele frequency at time /, p(t)

Figure 4.9: Allele frequency recursion in the haploid model of selection. The frequency of allele A at t + 1 is plotted against the frequency of allele A at t, using the recursion equation (3.9). The

Allele frequency at time /, p(t)

diagonal line (dashed) represents the case where p(t + 1) = p(t). At any point where the recursion curve falls above the diagonal line, the allele frequency increases over time, as in (a) where WA = 1 and Wa = 0.5. At any point where the recursion curve falls below the diagonal line, the allele frequency decreases, as in (b) where = 0.5 and Wa = 1. The vertical and horizontal lines starting at p{0) = 0.5 illustrate "cobwebbing," a procedure that can be used to determine changes to the variable over time (Box 4.2).

In the haploid model, there are two places where the recursion function crosses the diagonal: when the frequency of A is zero and when it is one (Figure 4.9). Again, we know that this is true only for the particular fitnesses used in the figure, but we will see in Chapter 5 that these two points represent the only two equilibria of the haploid model with constant fitnesses.