Figure 7.16. Simulated trajectories of Eq. (7.97) for the case in which the product vc = 1: (a) v = 1, c = 1; (b) v = 0.2, c = 5; (c) v = 0.1, c = 10; and (d) v = 0.02, c = 50. Other parameters aredt = 0.01, r = 0.05.

What interpretation do we give to Eq. (7.97)? If X(t) were the size of a population, then Eq. (7.97) corresponds to deterministic exponential growth with stochastic jumps of ±v individual, determined by the increment of a Poisson process.

What would we expect trajectories to look like? In Figure 7.16, I show four sample paths, corresponding to different values of v and c, but with their product held constant at 1. When v is large (and thus c is small), transitions occur rarely but when one takes place, it involves a big jump. As c increases (and thus v decreases), the transition is more and more likely, but the size of the transition is smaller and smaller. Indeed, Figure 7.16d almost looks like a deterministic trajectory perturbed by Brownian motion. To make a little bit more sense of this, and to get us ready for the next chapter, let us assume that X(t) satisfies

Eq. (7.97), that there is a population ceiling xmax and a critical level xc corresponding to extinction and define w(x) = Pr{X(t) reaches xmax before xc |X(0) = xg (7.98)

We derive the equation that w(x) satisfies in the usual manner tt(x) = E^ {m(x + dX)} 1 ' 2

= tt(x + rxdt — v) - cdt + tt(x + rxdt)(1 — cdt) (7 99)

Clearly, Eq. (7.99) calls for a Taylor expansion. Note that the first and third terms on the right hand side are already O(dt), so that when we do the Taylor expansion only the first term of the expansion will be appropriate. We thus Taylor expand around x — v, x, and x + v, respectively, to obtain w(x) = w(x — v)1 cdt + [w(x) + wxrxdt](1 — cdt) + w(x + v) ^ cdt + o(dt)

and we now subtract u(x) from both sides, divide by dt and allow dt to approach 0, to be left with a differential-difference equation

which is a formidable equation. But now let us think about the limiting process used in Figure 7.16, in which v becomes smaller and smaller. Then we might consider Taylor expanding w(x — v) and w(x + v) around x according to w(x — v)= w(x) — wxv +1 w^v2 + o(v2) and w(x + v) = w(x)+wxv +1 w^v2 + o(v2). If we do this, notice that cw(x) will cancel, as will the first derivatives from the Taylor expansion. If we assume that cv2 = a and that the terms that are o(v2) approach 0, we are left with

which is exactly the equation we would have derived had we started not with Eq. (7.97) but with the stochastic differential equation dX = rXdt + ^fa&W. Thus, the limiting result when the rate of transitions increases but their size decreases with cv2 constant does indeed look very much like a diffusion. Indeed, this is called the diffusion approximation and it too finds common use in population genetics and conservation biology (see Connections).

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