untitled

displacements for all individuals, and divides this sum by the total sample size to

estimate mean-squared displacement. Dis then calculated by dividing mean-squared

displacement by 2t.

In the more typical case of diffusion in two dimensions (xand y, centered at the

release point), these equations are slightly altered:

where ris the distance (i.e. radius) from the release point. Despite the slight change

in formula, this equation also predicts that the range occupied by the population is

proportional to √time. This is a very useful prediction that differs from other models

of population spread, as we shall shortly see.

As we discussed in Chapter 6, a newly reintroduced population is likely to have plenty

of resources with which to grow and multiply. This logically leads to geometric or

exponential growth, at least in the initial period following release. Unrestricted popu-

lation growth can be readily incorporated in our random walk model of population

spread. We simply multiply the local population by the finite rate of growth λ(in

this case, let us say that λ=1.05):

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

The rate of spread now seems to be much more consistent over time (Fig. 7.6) than

was the case for diffusive movement alone (Fig. 7.5). In fact, the population range

now spreads at a rate proportional to tand dbecause the population grows fastest

where density is highest. This relationship tends to create a rapid rate of change in

Nr t

N

Dt

r

Dt

( , ) = exp

⎛−

⎝

⎜

⎞

⎠

⎟

0 2

44 π

d

d

Nx y t

t

D

x

Nx y t

y

Nx y t

(, , )

=+(, , ) (, , )

⎡

⎣

⎢

⎤

⎦

⎥

∂

∂

∂

∂

2

2

2

2

102 Chapter 7

7.6.2Spread of
reintroduced species:
diffusion +
exponential growth

0

024 6810

Distance from release

Density

t = 5

t = 25

t = 45

t = 65

150

100

50

Fig. 7.6Variation in the
population density of
individuals over time,
when those individuals
redistribute themselves
every time step (t)
according to an
unbiased random walk.
Unlike Fig. 7.5, the
population is also
growing at an annual
rate of λ=1.05.