2.3. Group movements

The diffusion equation has formed the cornerstone of many models of animal dispersal (Okubo, 1980). While simple passive diffusion is appropriate in some cases, diffusion is often combined with other terms such as population drift or attraction to particular environmental conditions. Also, the simple diffusion equation may be modified to account for factors such as density dependence or variable diffusivity based on environmental conditions.

basic diffusion equation

Ordinary (Fickian) diffusion is a process where the flux J of particles is from high to low concentrations and is proportional to the gradient of concentration. If p(x,t) is concentration (or density), one-dimensional flux is expressed as

, (2.5)

with D determining the diffusivity of the particles. Based on equation (2.5), the change in population density through time is

. (2.6)

In order to solve equation (2.6) for p(x,t), boundary conditions and initial conditions must be specified. The simplest case is to have natural boundaries where X can take on values from to , and a point release at t = 0 and X = x₀. Formally, this is stated as:

,

.

In this case, the unique p(x,t) that satisfies equation (2.6), assuming x₀ = 0 and assuming the parameter D is a constant, is:

(2.7)

(Goel and Richter-Dyn 1974, Gardiner 1983). This solution is a normal distribution with respect to x for fixed t, with mean 0 and variance 2Dt; note that the variance increases linearly with time.

Both equations (2.6) and (2.7) can be derived from a simple random walk. In the first case, the random walk is expressed as a forward Chapman-Kolmogorov equation with step length x and time step t. This is then expanded in a Taylor series, higher order terms are ignored, and the diffusion limit is taken, resulting in equation (2.6) (Okubo, 1980). In the second case, the probability of a particle occupying the mth position after n steps, p(m,n), is expressed as a binomial distribution. Using Stirling's formula, p(m,n) is approximated with a normal distribution. The step length and time step are then allowed to become arbitrarily small, and equation (2.7) is obtained (Murray, 1989).

In a classic experiment, Dobzhansky and Wright (1943) released mutant flies (Drosophila pseudoobscura) with orange eyes (to distinguish them from the wild flies) from a point, and then recaptured the individuals along linear transects emanating from the release point. They then compared the observed distribution of flies to equation (2.7). Kareiva (1983) gathered data from mark recapture experiments of 12 species of herbivorous insects, and compared the data to a passive diffusion model. He concluded that in 8 out of 12 cases, a passive diffusion model is consistent with the data.

advection-diffusion

The advection-diffusion equation is appropriate when a population is not only spreading but also "drifting" in a particular direction. This equation can be formulated in one dimension as

, (2.8)

where r determines the rate of drifting. With a point release at x = 0 and natural boundaries, the solution of equation (2.8) is:

. (2.9)

For fixed t this is a normal distribution with mean rt and variance 2Dt. These two equations are derived in a similar manner to the corresponding ordinary diffusion equations but starting with a biased random walk - a random walk where the probability of moving to the right is not equal to the probability of moving to the left.

The advection-diffusion equation has been used most commonly as a model of migration. Wilkinson (1952) used this as a basis of a model of bird migration. Saila and Shappy (1963) present a model migration based on a random walk with a directed movement component and apply the model to migrating adult salmon. They concluded that -very little oriented movement is necessary to achieve the observed migratory patterns. Recently Hiramatsu and Ishida (1989) modified Saila and Shappy's model in terms of the advection-diffusion equation

, (2.10)

where the x axis is aligned in the direction of orientation, and r_x is the drift in that direction. The advection-diffusion equation forms the basis of the travel time models used in chapters 4-6.

spatial heterogeneity

As mentioned in the previous section, environmental heterogeneity can affect the dispersal behavior of animals. There are several ways to incorporate this into a model. One way is to assume that the diffusion coefficient, D, is related to some environmental factor and thus varies spatially.

When there is spatial heterogeneity in the diffusion coefficient, it is important to categorize the response to the heterogeneity as "attractive", "neutral", or "repulsive" (Skellam, 1973; Aronson, 1985; Okubo, 1986). In other words, is the diffusiveness of the individual determined by conditions of the current location (repulsive), conditions at the location of the next move (attractive), or an average of both these (neutral)? As shown by Skellam (1973) and Okubo (1986), these distinctions have a drastic effect on the resulting distribution. If we let D = D(x), the following equations describe a repulsive system, a neutral system and an attractive system, respectively:

(2.11)

(2.12)

. (2.13)

In a closed system (i.e., a system with reflecting boundaries), equation (2.11) results in , equation (2.12) results in , and equation (2.13) results in , where C is a constant determined by the system (Skellam, 1973).

Returning to the site of Dobzhansky and Wright's (1943) original experiment, Dobzhansky et al. (1979) examine the effect of habitat heterogeneity on the dispersal of Drosophila spp. They found that a dispersal model with diffusion coefficients related to habitat type was better able to describe observed patterns than one with constant coefficients.

models of chemotaxis

Originally developed to describe the response of cells to a chemical gradient (Keller and Segel, 1971), the chemotaxis model is an alternative way to describe an organism's response to environmental heterogeneity. In chemotaxis, variability in the concentration of a critical chemical produces an advection velocity in the direction of the gradient of concentration. The equation for chemotaxis is of the form:

, (2.14)

where U is the concentration of the chemical and is the chemotactic coefficient. In ecological applications, U(x) can be viewed as an environmental potential function (Teramato and Seno, 1988). Equation (2.14) can be rewritten as:

. (2.15)

Thus the effect as the environmental potential is to induce an advective velocity of magnitude . Few studies have actually attempted to apply these models to field data, perhaps because of the difficulty in defining the environmental potential function. In chapter 8, I apply this type of model to the vertical distribution of fish along an environmental gradient.

density dependence

The standard form of the diffusion equation with density dependence is:

(2.16)

(Gurney and Nisbet, 1975; Gurtin and MacCamy, 1977). If D`(u) > 0 for u > 0, then equation (2.16) models interference among individuals (Alt, 1985). If, on the other hand, D`(u) < 0 for u < 0, then equation (2.16) models attraction among individuals. Random movement can be included in addition to density dependent movement by formulating the diffusion term as:

(2.17)

similar to Shigesada, et al. (1979). Here is a constant and represents density independent diffusion, and (u) represents density dependent diffusion. When D(u) is of the form D(u) = (u/u₀)^m, equation (2.16) is called the porous medium equation (Gurtin and MacCamy, 1977; Murray, 1989), and an analytical solution is available. Note that when m = 0, the porous medium equation reduces to simple diffusion. A feature of the porous medium equation is that the population disperses as a front - there is no infinite tail as there is in the simple diffusion equation. This is because D(u) = 0 when u = 0, as is the case just beyond the dispersing front. Shigesada (1980) applied the porous medium equation with m = 1 to dispersing ant lions. Included in her model is a settling phase of the organism.

Density dependent diffusion involving attraction among individuals is more difficult mathematically. Since the diffusion term is negative, the problem is not well posed (Alt, 1985; Aronson, 1985). A problem is well posed if there is a unique solution that varies continuously with the initial conditions (Haberman, 1987). Slight perturbations result in only slight changes in the unique solution. Because of the compound effect of higher densities attracting more density, spikes of density form, and the position of these spikes is highly sensitive to the initial conditions. This a case, however, where the underlying discrete model is well posed and can be used to simulate density dependence with attraction among individuals.