12.3. Stratified Dispersal

Passive transportation mechanisms are most important for discontinuous dispersal. They include wind-borne transfer of small organisms (especially, spores of fungi, small insects, mites); transportation of organisms on human vehicles and boats. Discontinuous long-distance dispersal usually occurs in combination with short-distance continuous dispersal. This combination of long- and short-distance dispersal mechanisms is known as stratified dispersal (Hengeveld 1989).

Stratified dispersal includes:

• establishment of new colonies far from the moving population front;
• growth of individual colonies;
• colony coalescence that contributes to the advance of population front.

• Uninfested zone where pest species is generally absent
• Transition zone where isolated colonies become established and grow
• Infested zone where colonies coalesced

Metapopulation model of stratified dispersal

Population numbers in a colony, N(a), increases exponentially with colony age, a

n(a) = n_oexp(ra)

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 Slow-the-Spread 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 dispersal

1. Stochastic long-distance jumps
2. Continous local dispersal
3. Population growth (population numbers are multiplied by R)