giving (KD/P)l/2q0 « 5.9 foxes/km. For p = 0.5 fox/km, (13.59) gives an estimate of tc ~ 33 for these values of the parameters, and so the ratio of the two exponentials exp[-dtc] and exp[-xtc] is approximately 3 x 10-6, which justifies neglecting the smaller exponential in (13.58) in the above analysis.
We now derive an estimate for the break width xc. This involves solving the problem posed by (13.52) with (13.53). The first of (13.52) gives q(x, t) = 2q08(x)e-xt. (13.60) Substituting this into the second equation gives dr d 2r
— = -dr + — + 2q0x8(x )e-xt (13.61) the solution of which, with initial conditions (13.53), is of the form x2
r (x, t) = —p= exp Vn t where r *(x, t) is the solution of dr* ± d2r*
with homogeneous initial data. This equation can be solved using Laplace transforms. Denote the Laplace transform of r * by p, that is, n TO
Then p satisfies the inhomogeneous ordinary differential equation d2P , / ^ , 2q0x8(x)
—- + (x - d - s)p =--, -to < x < to, Re s > 0. (13.63)
We are only interested in the solution for x > 0; it is given by exp[-(s + d — ix)l/2 x ] ^' S) = s(s + d — g)1/2 •
So, inverting the transform, we get r*(xti exp[— (S + d — ^)1/2X^ ds (13 64)
" (X't) = 2^i Jc s(s + d — vl)1/2 dS' (13'64)
where C is the Bromwich contour. The singularities of the integrand are a pole at s = 0 and a branch point at s = —(d — /¿). The branch cut can be taken along the negative real axis to the left of the branch point, and so the contour of integration can be deformed to lie above and below the negative real axis. Since it is only necessary to evaluate r * (x, t) for t = tc, it can be assumed that t ^ 1 in the integral (13.64). If we now use the method of steepest descents (see, for example, Chapter 6 in the book by Murray 1984) the main contribution to the integral is given by the residue at the pole s = 0; the contribution from the branch cut is exponentially small in comparison, provided that
This inequality is shown to hold below. We thus arrive at the asymptotic solution for r (x, t) given by r (x, t) ~ exp
To estimate the break width, note that the formula (13.55) cannot be directly used since, with (13.60), q(x, t) always involves a 5-function. Instead, we replace (13.55) by r (xc, tc) = mr (0, tc)• (13.67)
The assumptions (13.65) and t ^ 1 can again be used to justify neglecting the first term in (13.66) as compared with the second. S0, from (13.67) and (13.66), an estimate for the break width is given by
If we take m = 10 4 together with the parameters used previously to estimate tc, then assumption (13.65) is easily verified to be valid for t = tc and x = xc since (xc/2tc)2 « 0-05 and d — n « 0-38.
Note that, at least to leading order, the formula for xc is independent of the critical time tc. The calculation of tc was only necessary for the purpose of verifying the 't large' assumption that was made throughout the analysis.
In dimensional terms, (13.68) gives, using (13.32),
with typical values for these parameters given in Table 13.1.
In the expression (13.68), the dependence of xc on 8 and m roughly agrees with Figure 13.14. It also suggests that the break width should not be very sensitive to p, which, as shown by Murray et al. (1986) is the case when the carrying capacity in the break is not too close to the critical value.
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