[stratlog.gif] 12.3. Stratified Dispersal

   One 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 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.

   The area near the advancing population front of the pest species can be
   subdivided into 3 zones:
     * Uninfested zone where pest species is generally absent
     * Transition zone where isolated colonies become established and grow
     * Infested zone where colonies coalesced

   [strat1.gif]

   [gypmoth.gif] A good example of a population with stratified dispersal is
   [1]gypsy moth. Gypsy moth egg masses can be transported hundred miles away on
   human vehicles (campers, etc.). New-hatched larvae become air-borne and can be
   transferred to near-by 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 [2]Slow-the-Spread 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 dispersal

   Sharov and Liebhold (1998, Ecol. Appl. 8: 1170-1179. [3][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

                                    [stratmy1.gif]

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

                           [stratmy2.gif] n(a) = n[o]exp(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:

                                        N = 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:

                                    [eqstrat1.gif]

   where n(a) is the number of individuals in a colony of age a. The population
   front is defined by the condition N(0) = K. Thus, the traveling wave equation is

                                    [eqstrat2.gif]

   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:

                                    [eqstrat3.gif]

   The population of each colony increases exponentially

                                  n(a) = n[o]exp(ra)

   After substituting these functions b(x) and n(a) into the traveling wave
   equation, we get the following equation

                                    [eqstrat4.gif]

   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

                                    [barrier1.gif]

   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).

                                    [barrier2.gif]

   This model predicted that barrier zones used in the [4]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

   [5]Cellular automata is a grid of cells (usually in a 2-dimensional 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.

   [ca1.gif]

   [caoutput.gif] The figure above shows 3 basic rules for the dynamics of cellular
   automata that simulates stratified dispersal:
    1. Stochastic long-distance jumps
    2. Continous local dispersal
    3. Population growth (population numbers are multiplied by R)

   Results of cellular automata simulation for several sequential time steps are
   shown at the right figure. It is seen how isolated colonies become established,
   grow, and then coalesce. This model was used for prediction of barrier-zone
   effect on the rate of population spread and the results were similar to those
   obtained with the metapopulation model..

References

   Sharov, A. A., and A. M. Liebhold. 1998. Model of slowing the spread of gypsy
   moth (Lepidoptera: Lymantriidae) with a barrier zone. Ecol. Appl. 8: 1170-1179.
   [6]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: 833-845.
   [7]Get a reprint! (PDF)
   Sharov, A. A., and A. M. Liebhold. 1998. Quantitative analysis of gypsy moth
   spread in the Central Appalachians. Pp: 99-110. In: J. Braumgartner, P.
   Brandmayer and B.F.J. Manly [eds.], Population and Community Ecology for Insect
   Management and Conservation. Balkema, Rotterdam.
   [8]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: 165-174.
   [9]Get a reprint! (PDF)

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     ____________________________________________________________________________

   [13]Alexei Sharov 7/6/99

References

   1. http://www.gypsymoth.ento.vt.edu/Welcome.html
   2. http://alexei.nfshost.com/~sharov/stsdec/stsdec.html
   3. http://home.comcast.net/~sharov/pdf/metpop.pdf
   4. http://alexei.nfshost.com/~sharov/stsdec
   5. http://alife.santafe.edu/alife/topics/cas/ca-faq/ca-faq.html
   6. http://home.comcast.net/~sharov/pdf/metpop.pdf
   7. http://home.comcast.net/~sharov/pdf/bioecon.pdf
   8. http://home.comcast.net/~sharov/pdf/quantit.pdf
   9. http://home.comcast.net/~sharov/pdf/optim.pdf
  10. http://alexei.nfshost.com/PopEcol/lec12/diffus.html
  11. http://alexei.nfshost.com/PopEcol/lec12/dispers.html
  12. http://alexei.nfshost.com/PopEcol/lec12/metpop.html
  13. http://alexei.nfshost.com/~sharov/alexei.html