Identifying Equilibria Graphically

A system at "equilibrium" does not change over time.

• In a plot of n(t + 1) versus n(t), any point where the recursion equation crosses the diagonal line represents an equilibrium, because n(t + 1) = n(t).

• In a plot of An versus n{t), any point where the difference equation crosses the horizontal axis represents an equilibrium, because An = 0.

• In a plot of dn/dt versus n(t), any point where the differential equation crosses the horizontal axis represents an equilibrium, because dn/dt = 0.

Now, focus your attention on whether the recursion lies above or below the diagonal. Before reading on, think about what this means. Whenever the recursion lies above the diagonal, p(t + 1) is greater than p(t), and allele A increases in frequency. For the parameter values in Figure 4.9a (where WA > Wa), the recursion is above the diagonal for all values of the allele frequency (0 < p(t) < 1). As a result, allele A rises in frequency every generation until it approaches one ("fixation"). Conversely, whenever the recursion falls below the diagonal (as in Figure 4.9b, where WA < Wa), p(t + 1) is always lower than p(t). As a result, allele A decreases in frequency until it is lost from the population. You can even use these figures to run your own iterations by hand using a technique known as "cobwebbing" (Figure 4.9; see Box 4.2).

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