— = v[1 + mq ( p)]-$q, where the fluxes are given by

Jcu =-uCu (x - Xu, q), Jdu = -du (u)Vu, Jau = au (q)uVq (14.33) Jcv = -vCv(x - Xv, p), Jdv = -dv(v)Vv, Jav = av(p)vVp (14.34)

and where the functions cu, cv, du, dv, au, av are all nonnegative functions (or constants) as described above.

The boundary conditions (14.22) and (14.23) are unchanged and an appropriate nondimensionalisation of the initial data is u0 =

which leaves the initial conditions (14.24) also unchanged after omitting the asterisks. Note too that the nondimensionalisation of space has made the dimensionless domain Πequal to unity. Also, with this nondimensionalisation

and so, at any given time, u(x, t) and v(x, t) are probability density functions for the location of wolves.

We now have to specify appropriate forms for the interaction functions in the model equations. Lewis et al. (1997) showed that if the increased marking function m is typically as we described above (specifically a concave down function for the scent-marking density) then the time-independent solutions of (14.29)-(14.32) satisfy a system of ordinary differential equations with space as the independent variable. The integral conditions (14.35) are transformed into initial conditions for the ordinary differential equations. The resulting expected wolf density functions decrease mono-

tonically with distance away from the den site. A sufficient condition for the buffer zone (that is, a minimum in the value of u + v between the den sites) is that the movement function, cu , is also a concave down function of foreign scent-mark density.

For analytical simplicity and demonstration they considered a one-dimensional system with dens at opposite ends of the domain (xu = 0, xv = 1) and the movement response to foreign RLUs omitted. So, steady state solutions of (14.29)-(14.32) satisfy

where ru, rv are distance measured from the respective dens. Boundary conditions (14.22) are now

Jv, Ju = 0 at x = 0,1 (14.40) and conservation conditions (14.35)

0 0

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