, (8.10)where z is depth, I0 is the light intensity at the surface, and 
is the decay coefficient. This function is plotted in Figure 8.2.
Now assume that there is a desirable light level, 
. The environmental potential function can then be expressed as:
As stated above, the difficulty lies in finding the appropriate function f. A plot of 
versus z (Figure 8.2) reveals an abrupt change in the slope of the curve at I. This carries through to modeled distribution of fish, n(z), and this abrupt change in distribution is not observed in the data. A simple modification that produces a smooth curve is
This is plotted in Figure 8.2. In this equation I introduce a constant 
that determines the intensity of the chemotactic response and is often termed the chemotactic coefficient. The steady state distribution of organisms along a light gradient is then
which is also plotted in Figure 8.2.
The squared term in equation (8.12) might be justified because the light gradient experienced by the fish is not simple. As a fish looks upwards or downwards, it is not experiencing the local gradient but an integration of light levels above or below based on its "line of sight" (Pitcher, 1986); this has the effect of intensifying the gradient. Obviously direct studies would be necessary to justify this term (or some other form), but in the mean time, it produces a tractable model that is consistent with the data.
. (8.11)
. (8.12)
, (8.13)