12.2. Diffusion ModelsAdvantage of diffusion models is that they can be applied to any initial distribution of organisms. The most simple diffusion model in 1-dimensional space is:
where N is population density, and D is diffusion coefficient. This equation indicates that the rate of population change is proportional to the curvature of population density. Examples below show that population increases where curvature is positive and decreases where it is negative.
Skellam (1951) combined diffusion equation in a 2-dimensional space with exponential local population growth:
Skellam's model predicts that if a population was released at a single
point, then its spatial distribution will be a 2-dimensional normal
One of the most interesting features of this model is that it predicts the asymptotic rate of expansion of population front. The rate of population expansion, V, is defined as the distance between sites with equal population densities in two successive years:
Skellam's model gives the following equation for the rate of population
Both parameters, r and D, can be estimated in independent
experiments. Intrinsic rate of population increase can be determined
from the life-table. Diffusion coefficient D can be estimated using
mark-recapture experiments. For example, if marked animals are
released within a uniform grid of traps, then diffusion coefficient
is estimated as:
where M(t) is mean displacement of organisms recaptured t units time after their release (Skellam 1973).