12.3. Stratified DispersalOne of the major limitations of diffusion models is the assumption of continuous spread. In nature many organisms can move or can be transferred over large distances. If spread was continuous, then islands would never be colonized by any species. Discontinuous dispersal may result in establishment of isolated colonies far away from the source population.Passive transportation mechanisms are most important for discontinuous dispersal. They include windborne transfer of small organisms (especially, spores of fungi, small insects, mites); transportation of organisms on human vehicles and boats. Discontinuous longdistance dispersal usually occurs in combination with shortdistance continuous dispersal. This combination of long and shortdistance dispersal mechanisms is known as stratified dispersal (Hengeveld 1989). Stratified dispersal includes:

The area near the advancing population front of the pest species can be subdivided into 3 zones:

A good example of a population with stratified dispersal is gypsy moth. Gypsy moth egg masses can be transported hundred miles away on human vehicles (campers, etc.). Newhatched larvae become airborne and can be transferred to nearby forest stands. Distribution of gypsy moth counts in pheromone traps in the Appalachian Mts. (Virginia & West Virginia) in 1995 is shown in the right figure. Isolated populations are clearly visible. The US Forest Service SlowtheSpread project has a goal to reduce the rate of gypsy moth expansion by detecting and eradicating isolated colonies located just beyond the advancing population front. 
Metapopulation model of stratified dispersalSharov and Liebhold (1998, Ecol. Appl. 8: 11701179. [Get a PDF reprint!]) have developed a metapopulation model of stratified dispersal. This model is based on two functions: colony establishment rate and colony growth rate. The probability of new colony establishment, b(x), decreases with distance from the moving population front, x
Population numbers in a colony, N(a), increases exponentially with colony age, a
where n_{o} are initial numbers of individuals in a colony that has just established, and r is the marginal rate of population increase. The population front is defined as the farthest point where the average density of individuals per unit area, N, reaches the carrying capacity, K: The rate of spread, v, can be determined using the traveling wave equation. We assume that the velocity of population spread, v, is stationary. Then, the density of colonies per unit area m(a,x) of age a at distance x from the population front is equal to colony establishment rate a time units ago. At that time, the distance from the population front was x + av. Thus, m(a,x) = b(x + av). The average numbers of individuals per unit area at distance x from the population front is equal to:
This equation can be used for estimating the rate of population spread. To estimate this integral we need to define explicitly functions b(x) and n(a). We assume a linear function of the rate of colony establishment:
After substituting these functions b(x) and n(a) into the traveling wave equation, we get the following equation
where V = v/x_{max} is the relative rate of population spread. This equation can be solved numerically for V; and then the rate of spread is estinmated as v = V x_{max}. This model can be used to predict how barrier zones (where isolated colonies are detected and eradicated) reduce the rate of population spread. We will assume that the barrier zone is placed in the transition zone at some particular distance from the population front. Because new colonies are eradicated in the barrier zone, we can set the colony establishment rate, b(x), equal to zero within the barrier zone as it is shown in the figure below
If this new function b(x) is used with the traveling wave equation, we get the rate of spread with the barrier zone. The figure below shows the effect of barrier zone on the rate of population spread. Relative width of the barrier zone is measured as its proportion from the width of the transition zone. Relative reduction of population spread is measured as 1 minus the ratio of population spread rate with the barrier zone to the maximum rate of population spread (without barrier zone).
This model predicted that barrier zones used in the SlowtheSpread project should reduce the rate of gypsy moth spread by 54%. This prediction was close to the 59% reduction in the rate of gypsy moth spread in Central Appalachian Mountains observed since 1990 (when the strategy of eradicating isolated colonied has been started).
Cellular automata models of stratified dispersalCellular automata is a grid of cells (usually in a 2dimensional space), in which each cell is characterized by a particular state. Dynamics of each cell is defined by transition rules which specify the future state of a cell as a function of its previous state and the state of neighboring cells. Traditional cellular automata considered close neighborhood cells only. However, in ecological applications it is convenient to consider more distant neighborhoods within specified distance from the cell.

The figure above shows 3 basic rules for the dynamics of cellular automata that simulates stratified dispersal:

ReferencesSharov, A. A., and A. M. Liebhold. 1998. Model of slowing the spread of gypsy moth (Lepidoptera: Lymantriidae) with a barrier zone. Ecol. Appl. 8: 11701179. Get a reprint! (PDF)Sharov, A. A., and A. M. Liebhold. 1998. Bioeconomics of managing the spread of exotic pest species with barrier zones. Ecol. Appl. 8: 833845. Get a reprint! (PDF) Sharov, A. A., and A. M. Liebhold. 1998. Quantitative analysis of gypsy moth spread in the Central Appalachians. Pp: 99110. In: J. Braumgartner, P. Brandmayer and B.F.J. Manly [eds.], Population and Community Ecology for Insect Management and Conservation. Balkema, Rotterdam. Get a reprint! (PDF) Sharov, A. A., A. M. Liebhold and E. A. Roberts. 1998. Optimizing the use of barrier zones to slow the spread of gypsy moth (Lepidoptera: Lymantriidae) in North America. J. Econ. Entomol. 91: 165174. Get a reprint! (PDF) 