# The basic idea escape from a domain of attraction

Central to the computation of extinction times and extinction probabilities or the movement from one peak in a fitness landscape to another is the notion of "escape from a domain of attraction.'' This impressive sounding phrase can be understood through a variety of simple metaphors (Figure 8.1). In the most interesting case, the basic idea is that deterministic and stochastic factors are in conflict - with the deterministic ones causing attraction towards steady state (the bottom of the bowl or the stable steady states in Figure 8.1) and the stochastic factors causing perturbations away from this steady state. The cases of the ball Trait

on the top of the hill or the steady state being unstable or a saddle point are also of some interest, but I defer them until Connections.

We have actually encountered this situation in our discussion of the Ornstein-Uhlenbeck process, and that discussion is worth repeating, in simplified version here. Suppose that we had the stochastic differential equation dX =—Xdt + dW and defined u(x, t) = Pr{X(s) stays within [—A, A] for all s, 0 < s < t|X(0) = xg (8.1) We know that u(x, t) satisfies the differential equation ut = 1 uxx — xux (8-2)

so now look at Exercise 8.1.

Derive Eq. (8.2). What is the subtlety about time in this derivation?

Equation (8.2) requires an initial condition and two boundary conditions. For the initial condition, we set u(x, 0) = 1 if —A < x < A and to 0 otherwise. For the boundary conditions, we set u(— A, t) = u(A, t) = 0 since whenever the process reaches A it is no longer in the interval of interest. Now suppose we consider the limit of large time, for which ut ! 0. We then have the equation 0 = (1/2)uxx — xux with the boundary conditions u(—A) = u(A) = 0.

Show that the general solution of the time independent version of Eq. (8.2) is u(x) = kj J*A exp(s2)ds + k2, where kj and k2 are constants. Then apply the boundary conditions to show that these constants must be 0 so that u(x) is identically 0. Conclude from this that with probability equal to 1 the process will escape the interval [—A, A].

We will thus conclude that escape from the domain of attraction is certain, but the question remains: how long does this take. And that is what most of the rest of this chapter is about, in different guises.