## L

been just two attempts (Ludwig 1981, Lande 1985) to answer this question. (Gavrilets (2003) has a nice, general review of the topic.) Here, I will walk you through Ludwig's analysis; the problem is highly stylized and the analysis is difficult, but at the end we will have a deepened and sharpened intuition about the general issue. Our starting point is the Ornstein-Uhlenbeck process dX = -Xdt + V"dW

for which we know that the stationary density is Gaussian, with mean 0 and variance e/2, so that the confidence intervals for the stationary density are O(v/e) ; for example the 95% confidence interval is approximately [—>/"> V" ]. Thus the mechanism that we consider consists of deterministic return to the origin with fluctuations superimposed upon that deterministic return.

We shall also consider a larger interval, [— L, L] (Figure 8.7) and metaphorically consider that within this larger interval we have one "fitness peak'' and that outside of it we have another "fitness peak,'' so that escape from the interval [— L, L] corresponds to transition between peaks.

Our first calculation is an easy one. If we replace Eq. (8.47) by the deterministic equation dx/dt = —x, we know that the only behavior is attraction towards the origin.

Figure 8.7. Our understanding of transitions from one fitness peak to another on the adaptive landscape will rely on the metaphor of an Ornstein-Uhlenbeck process dX = -Xdt + pSdW,for which the stationary density is Gaussian with mean 0 and variance e/2. We consider an interval [—L, L] that is much larger than the confidence interval for the stationary density, which is O(^/e), as domain of one adaptive peak and values of X outside of this interval another adaptive peak, so that when X escapes from the interval, a transition has occurred. As described in the text, we are interested in three kinds of times: the deterministic time to return from initial value L to 2 e, the mean time to escape from [— L, L], and the mean time to escape from an initial value X(0) > 0 without returning to 0.

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