2.2. Models of individual movements

Simple random walks have formed the basis of several animal movement models. Except in unusual circumstances, however, simple random walks cannot adequately describe the movements of individuals; the random walk represents too much of a simplification. On the other hand, if step size is adequately small and the number of individuals is sufficiently large, the spatial dynamics of a group of random walkers shares many similarities with observed population patterns.

The simple random walk model can be presented as follows. First, assume that an individual moves a distance x (in one dimension) during each time interval t. Assume that the individual moves to the right with probability and to the left with probability , with . When the random walk is termed isotropic, and when the random walk is anisotropic or biased. After n moves, let n_r be the number of moves to the right and n_l be the number to the left. The position of the individual in units of movement after n steps is

. (2.1)

The probability of individual occurring at position m after n steps is

; (2.2)

that is, p(m, n) follows a binomial distribution. p(m, n) can also be expressed as forward Chapman-Kolmogorov equation (Okubo, 1980):

. (2.3)

The random walk model is easily expanded to two or three spatial dimensions.

Jones (1977) successfully applied a simple random walk on a grid to describe population patterns of cabbage butterflies (Pieris rapae L.). An added advantage of random walk models is that behaviors such as taxis, kinesis and density dependence can be easily added to a random walk model, as demonstrated by Rohlf and Davenport (1969).

Several workers have extended the simple random walk model on a regular grid to include movements of various lengths in any direction and correlation in the direction of movements (e.g., Siniff and Jensen, 1969; Skellam, 1973; Kitching and Zalucki, 1982; Kareiva and Shigesada, 1983). In these models, for each movement increment, a length and an angle are drawn from distributions, with the new angle of movement based on the previous angle. Othmer, et al. (1988) provide many modifications to random walk and dispersal models.

Individual movement in continuous time and space

In continuous time and space, the position of an individual can be denoted by X(t), with , n = 1, 2 or 3, and t > 0. For ease of notation, I will assume . The change in position of an individual with respect to time can be described by a stochastic differential equation (SDE) (Gardiner, 1983):

. (2.4)

W(t) is white noise and has the following properties:

<W(t)> = 0,

<W(t), W(t + )> = (),

where (t) is the Dirac distribution. Ito calculus is assumed. If the parameters r and are constants and W(t) is Gaussian white noise, then X(t) is the Weiner drift process. The Wiener drift process has the following properties (Ross, 1985):

2. X(0) = 0;
3. for t > 0, X(t) is normally distributed with mean rt and variance ²t;
4. each disjoint segment of an individual path is independent.