Chapter 3 Selection gradients as weighted average slopes

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Figure 3.1a. The directional selection gradient, beta, as the weighted average of first derivatives of the individual selection surface (ISS). The ISS is the orange curve; the trait distribution before selection, p(z), is shown in blue. First derivatives of the ISS are shown at regular intervals as superimposed black, straight line segments. The slopes of those segments correspond to the value of the corresponding first derivatives. The average of all those slopes, weighted by the trait distribution, is the directional selection gradient, β, shown in red at the end of the animation run. The running weighted average is shown by the changing value of the red line segment with the vertical dotted line showing the upper limit of the
integration (the lower bound is minus infinity). In this animation, the ISS is a Gaussian function (with an optimum, theta, at 0 and a width parameter, omega, of 1). The trait distribution before selection is a normal distribution (with a mean of -0.5 and a variance of 1).

Figure 3.1b. The nonlinear selection gradient, gamma, as the weighted average of second derivatives of the individual selection surface (ISS). The ISS is the orange curve; the trait distribution before selection, p(z), is shown in blue. The first derivative of the ISS is the yellow curve. First derivatives of this yellow curve (which are the second derivatives of the ISS) are shown at regular intervals as superimposed black, straight line segments. The slopes of those segments correspond to the value of the corresponding second derivatives of the ISS. The average of all those slopes, weighted by the trait distribution, is the nonlinear selection gradient, gamma, shown in red at the end of the animation run. The running weighted average is shown by the changing value of the red line segment.