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For this kind of random walk model, most trajectories tend to find their way back

to a position not far from the initial starting point (Fig. 7.4). In other words,

walking randomly is not a very effective means of getting anywhere new. This is

a useful null model, however, that sets an extreme standard against which we might

evaluate the movements of real organisms. The random walk model is perhaps most

plausible at large spatial scales, such as for dispersing juveniles, in which animals

have no past experience with local conditions.

We can readily expand this kind of model to a group of individuals (Case 2000).

To keep it simple, we will concentrate on only one spatial dimension, such as for

sea otters dispersing up and down the coast of California. Let’s say that there are 100

individuals released at a central position “0” and that each individual has a 20% prob-

ability of moving left and a 20% probability of moving right, with position along this

axis indicated by the variable x. This probability we will term “d” for dispersal. Local

changes in density of individuals can be modeled in the following manner:

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

The local population loses 2 ×d×Nindividuals due to movement in either direc-

tion, but gains d×Nindividuals from each adjacent site. We need to repeat this

exercise over the full range of distance intervals.

The output of this model demonstrates two important features (Fig. 7.5). First,

the spatial distribution of individuals in the population begins to take on a bell-shaped

or normal distribution over time. Second, the rate of spread is initially fast, but slows

over time. This is because movement away from the release point is balanced to a

considerable degree by movement backwards. This slower movement away becomes

more pronounced over time because the distribution is getting flatter. When dynam-

ics are driven purely by random motion the population range spreads at a rate pro-

portional to √time. If we repeat this simulation with a larger fraction of dispersers

(say d=0.3), the rate of spread will increase accordingly. The rate of spread is pro-

portional to √d.

We can use our random walk model to derive the differential equation that defines

diffusive changes in local density, over a continuous gradient of space and time:

100 Chapter 7

20

0

–20

• 20 – 10 0 10 20
  X

Y

Fig. 7.4Hypothetical
trajectory over 100 time
steps for a single
individual following a
random walk, starting
from the origin (0,0).