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∆Nx=Nx,t+ 1 −Nx,t=− 2 dNx,t+dNx−1,t+dNx+1,t

We then rearrange terms on the right-hand side of the equation:

∆Nx=d[(Nx−1,t−Nx,t) −(Nx,t−Nx+1,t)]

The rate that individuals accumulate at site xdepends on the degree of difference

between the density gradient below the site and the density gradient above the site.

In other words, it is not the gradient itself, but the rate of change of the density

gradient over space that dictates the rate of diffusive movement. Mathematicians refer

to the rate of change of the density gradient as the second derivative. If this occurs

over short enough intervals of time and space, then the result is the following

differential equation (called the diffusion equation in one dimension):

=DN(x, t)

The solution to this equation is the normal distribution:

where tis the time since the animals were released, μis the initial position (usually 0),

and Dis the diffusion coefficient. It reflects how fast individuals tend to diffuse away

from an initial point of release. We discuss how to calculate it below. This equation

may look familiar – it is closely related to the normal (sometimes called Gaussian)

probability distribution. The variance in spatial locations is given by σ^2 = 2 Dt.

The easiest way to estimate the diffusion coefficient Dis to estimate the mean-

squared displacement of the individuals in the population over time. One simply

measures the distance of a given individual from its original release point, squares

that displacement to get rid of positive versus negative values, sums the squared

Nx t

N

Dt

x

Dt

( , ) exp

( )

=

⎡−−

⎣

⎢

⎤

⎦

⎥

0 2

4 π^4

μ

∂^2

∂x^2

dN(x, t)

dt

DISPERSAL, DISPERSION, AND DISTRIBUTION 101

50

40

30

20

10

0

024 6 810

Distance from release

Density

t = 5 t = 25

t = 45

t = 65

Fig. 7.5Variation in the
population density of
individuals over time,
when those individuals
redistribute themselves
every time step (t)
according to an
unbiased random walk.