Model of Slowing Gypsy Moth Spread

Gypsy moth (Lymantria dispar) is a forest pest insect that feeds on a large number of tree species. Oaks are among most preferred host species. Gypsy moth was accidentally introduced to North America in 1868-69 near Boston and since that time it has been spreading mainly to the west and south. Since 1980, the spread of gypsy moth is accurately documented using data on male moth counts in pheromone-baited traps.

Gypsy moth populations spread via stratified dispersal which is a combination of (1) long-distance dispersal which results in establishment of small isolated colonies beyond the population front, and (2) short-distance dispersal which results in growth of isolated colonies until they coalesce. Long-distance dispersal results mainly from inadvertent transportation of egg masses and other life-stages by humans (e.g., on campers, logs, etc.). Short-distance dispersal results from larval dispersal.

It was suggested that detection and eradication of isolated colonies bejond the population front may reduce the rate at which gypsy moth populations are spreading. The idea was partially implemented in the Appalachian IMP project, and later fully implemented in the Slow-the-Spread project. A barrier zone was set just ahead of the population front with a dense grid of pheromone traps. Detected isolated colonies were treated.

The notion of a barrier-zone is usually associated with a constant area which prevents further progression of the population front. Here we are talking about the shifting barrier zone which only slows the spread of gypsy moth. Ok, why not to stop its spread? Stopping of gypsy moth spread is feasible but not rational because it will require spraying of large forest areas with strong pesticides. This is environmentally dangerous and unreasonably expensive.

I have developed a model of stratified dispersal which quantified the effect of a barrier zone on the rate of population spread. However this model is too abstract to provide guidelines for optimization of the management of the barrier zone. Here I present a discrete version of this model which can be directly applied to pest management. The model is written in Microsoft Excel (ver. 5.0) for Windows. It can be run on Macintosh computers as well.

To optimize the allocation of pheromone traps you will need a Solver which is a tool that comes with Microsoft Excel. Solver may be not installed on your computer if you used standard installation of Excel. In this case, use installation disks to add the Solver.

To download the model, click on the icon below.

Excel spreadsheet "stsopt4.xls"

Model Description

The model consists of 2 tables. The upper table describes gypsy moth spread without barrier zones, and the lower table describes population spread with a barrier zone.

Upper table

We know that the rate of spread without barrier zones is ca. 21 km/yr. Population front will be considered at locations where defoliation first occurs because we assume that the major source of long-distance dispersal are defoliation areas.Then the area beyond the population front (beyond defoliation) can be separated into bands of 21 km width. These bands are represented as rows in the table. For example, row 23 is 21 km away from defoliation front (B23). Thus, there is 1 year left until first defoliation will occur. In the row above, 2 years are left until defoliation and so on.

The 3-rd column (C) shows colonization rate, which is a linear function of distance from population front (see fig. at the right). Two parameters are used: xmax is the maximum distance from defoliation front at which new colonies can become estblished, and cmax is maximum colonization rate. Columns to the right from colonization rate correspond to different colony ages: 1, 2, 3 years, etc.

Numbers in the table (D5:T23) show the number of colonies (colony centers) per 1 sq.km. These numbers are shifted diagonally because the number of colonies in age a at time t is the same as the number of colonies at age a+1 at time t+1.

At the bottom of the table (C25:U25) there are population numbers (total egg massess in a colony of specific age) which are assumed to increase exponentially. The last column in the table (V) shows the average population density as a function of distance from defoliation front.

Lower table

The lower table simulates the reduction of gypsy moth spread rate due to barrier zone. This table has a similar structure as the upper table, however there are 2 major differences. First, we set the target rate of population spread (Z13) as 9 km/yr which is smaller than the uncontrolled rate of spread (14 km/yr). As a result, the numbers in column B are different in the upper and lower tables. Second, the number of colonies per 1 sq.km. is not simply shifted diagonally, but rather decreased because some colonies become eradicated.

The probability to detect a colony is proportional to colony area (C81:U81) and to the density of pheromone traps (AA29:AA80). Initial density of pheromone traps is assumed to be 0.02 per sq.km. at all distances from the population front. This allocation of traps is not optimal: it is clear that we don't need traps too close to the population front and too far from it. Later we will find the optimum allocation of traps.

We assume that all detected colonies are eradicated using pesticides. Knowing the area of colonies (C81:U81) it is easy to estimate the area where pesticides are applied for eradication. Eradication costs are $3,500 per sq.km and the cost of 1 trap is $50. Average expences on trapping and eradication within 1 sq.km. are shown in (W29:W80) and (X29:X80), respectively. Total sums (W81 and X81) are multiplied by the rate of spread to get the total annual cost of maintaining 1 km of a barrier zone measured along the population front. Total cost of 1 km of the barrier zone (prior to optimization) is $33,869 to maintain 1 km of the barrier zone. Then to maintain a 100-km barrier zone it will cost annually $3,386,900. This sum of money is very large because we have not optimized trap allocation yet.

The graph below the lower table shows the density of traps and the proportion of area treated with pesticides at different distances from the befoliation front.

The most interesting part is the optimization of trap allocation. We start from the uniform density of traps. Then we use the Solver to find the best trap allocation. Open "Tools/Solver". Let’s examine the Solver dialog box. At the top we put the cell that contains total costs (Y81). The value of this cell should be minimized (thus we select "min"). Now we need to specify model parameters which can be modified automatically by the computer in order to minimize the costs. In this field we put the entire column of trap densities (AA29:AA80). The last thing is to specify 2 constraints: (1) trap density cannot be negative, and (2) the average population density at the defoliation front should be the same in both tables (Z80=V24). The second constraint is better to write in the form Z80 ≤ V24 because this will improve the stability of optimization process. However, this will not affect the final result (you may check it later): after optimization the value of Z80 will be equal to the value of V24.

Now let’s run the Solver. At the bottom we see how costs are reduced with each iteration.The Solver finally converged to a minimum of $9,078 per 1 km length of population front annually (there may be small variations in this number depending on initial conditions and solver options). Now you can see on the graph below the table that the traps should be set from 81 to 225 km from the defoliation front. Both trap density and treatment area are higher at the proximal portion of the barrier zone than in the distant portion.

You can change parameters of the model (Z6:Z13) and run optimization again. I did these simulations for different target spread rates from 5 to 17 km/yr, and the results are presented in the Sheet 2.

Alexei Sharov 06/30/97