The name is absent

case with one CBD. These convex combinations we construct via distance deterrence functions.
Such functions were introduced by the authors in Thorsen et al. (1999), see this paper for a
motivation and discussion of the concept. As an example of this sort, we will make use of a logistic
function

(4.1)

Here a is defined as the marginal level of interaction. If x is very small, D(x)'0 (i.e., no deterrence),
and if x is large, then D(x)~1 (i.e., full deterrence). The parameter d₀ signifies the distance at which D
is marginally close to no deterrence, and d8 signifies the distance at which D is marginally close to
full deterrence.

Now consider the situation where there are two CBDs, CBD1 and CBD2. Consider a point in space
at distances d1 to CBD1 and d2 to CBD2. We let R1[d] and R2[d] be the functions found from (3.9)
using the construction in Section 2 and 3. R12[d] is the corresponding expression when we merge
CBD1 and CBD2 together. Now we propose to model E/L using the expression
(4.2)

R[dι, d2]= D(d)D(d2)R₀ + (1 - D(d1))D(d2)R[d1]

+(1 -D(d2))D(d1)R2[d2] + (1 - D(c⅞))(1 - D(d2))R⅞₂[(d + d2)/2]

As the reader may wish to verify, the four numbers D(d1)D(d2), (1-D(d1))D(d2), (1-D(d2))D(d1), and
(1-D(d1))(1-D(d2)) always add to 1. Hence E/L is modeled as a convex combination of the four states
R8,R1,R2, and R12. From (4.2) we obtain the following:

• If d1,d2 are both large, the model suggest the expression R8.

• If d1 is small and d2 is large, the model suggest the expression R1[d1]. (All other terms are small in
this case).

• If d2 is small and d1 is large, the model suggest the expression R2[d2].

• If d1 and d2 are both very small, the model suggests that CBD1 and CBD2 are acting as a single
unit.

At first sight the inclusion of a merged state R12 may seem somewhat superficial. One application of
this kind of model is, however, to provide predictions for a change in the spatial pattern of retailing
facilities when the internal distances within the system are subject to change. Consider the network
shown in Figure 7. If a new road connection is introduced between the nodes B and E, one would
expect a stronger kind of response if B and E are both CBDs than if any other pair of nodes has this
property.

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