8.2. The model

(8.1)

where n(x,t) is the population density and is the flux. Note that the spatial component, x, can be multi-dimensional. If we consider simple diffusion along an environmental gradient, the population flux can be expressed as

(8.2)

where U(x) is the environmental potential function (Teramato and Seno, 1988). In the one dimensional case, equation (8.2) can be written as

. (8.3)

The first term on the right side is the diffusion term, with determining the magnitude of the diffusion relative to the second term. The second term introduces an advection that is dictated by the gradient of the environmental potential function. Next assume that the fish reach some stationary distribution during the daytime and nighttime periods. To find a steady-state solution, set

, (8.4)

or equivalently,

. (8.5)

At the steady-state there is no longer time dependence, so we can rewrite equation (8.5) in terms of ordinary differential equations:

. (8.6)

Assuming that U(x) is provided, we can solve for n:

(8.7)

where c is a constant of integration.

The problem comes in determining U(x). First assume that there is some measurable environmental stimulus E(x), and that U(x) is a function of this; that is:

. (8.8)

I also assume that there is a desirable level of the stimulus, , and the advective term of the chemotaxis equation is toward this desirable level:

. (8.9)