The goal of this thesis was to apply a two-state Markov chain to grasshopper population dynamics in Wyoming using a Geographic Information System (GIS).

This Markov Chain analysis produced several maps that can be used to facilitate pest management decisions, especially those pertaining to the allocation of limited survey and control resources.

Data Grids (provided via USDA-APHIS-PPQ rangeland grasshopper surveys) Metadata for Data Grids

Since there are two conditions (states), uninfested and infested, a two-state Markov chain can be applied to the data grids from 1944 to 1996. There are 4 possible transitions: uninfested to uninfested, uninfested to infested, infested to uninfested, and infested to infested. For purposes of rangeland grasshopper management, the most important transitions are from infested or uninfested conditions to infested conditions, and the most important time-scale is the 1-year transition, see Markov.htm.

2) An ArcView extension was designed using Avenue and Spatial Analyst, that performs the steps of the two-state Markov chain.

The Markov extension adds a new menu choice to the menu bar within ArcView called "Markov Chain." Below this new menu are all the options to perform the steps of the Markov chain.

The menu options perform the following:

*Outbreak Frequency*

Download Outbreak Frequency Shapefile Metadata

Outbreak Frequency computes the frequency with which each cell was infested
with ³ 9.6 grasshoppers per m^{2}
by summing up the values in each cell over the time period.

From this map the user can get a sense of whether a particular region has had a serious problem with rangeland grasshoppers, but this does not address the concern of future conditions.

*One Year Transition Probabilities*

Download Transition Infested to Infested Shapefile Metadata

Download Transition Infested to Uninfested Shapefile Metadata

Download Transition Uninfested to Infested Shapefile Metadata

Download Transition Uninfested to Uninfested Shapefile Metadata

The menu choice "One Year Transition Probabilities" creates four grids,
in a new view, that represent the one-step transition matrix,
where 0 = uninfested or <9.6 grasshoppers per m^{2} and 1 =
infested or ³ 9.6 grasshoppers per m^{2}.
These four grids are used to answer the question "If a cell is uninfested/infested
this year, what are the chances it will be uninfested/infested next year?".

This menu choice sorts the grids into alphanumeric order; it was assumed that the user would name the grids similar to g1, g2, and so on. The grid g1 would represent the cell values for year 1. The extension then creates temporary grids that track the number of transitions from 0 to 1 and 1 to 0. Temporary grids are also used to count the number of transitions that started in 0 and 1. For example, to compute the probability of the transition from 0 to 1 the temporary grid tracking the number of transitions from 0 to 1 was divided by the temporary grid tracking the number of transitions that started at 0. The resulting grids are displayed in a view and labeled.

The probabilities of each transition were determined according to :

It was only necessary to compute the transition probabilities for P_{01}
and P_{10 }and use those values to derive the other two probabilities,
by subtracting from 1.

*Expected Number of Years*

Download Expected Time Infested Shapefile Metadata

Download Expected Time Uninfested Shapefile Metadata

Download 95% Confidence Interval (Infested) Shapefile Metadata

Download 95% Confidence Interval (Uninfested) Shapefile Metadata

Download Variance (Infested) Shapefile Metadata

Download Variance (Uninfested) Shapefile Metadata

Once the four, one-step transition grids have been computed, the user can then use the menu option "Expected Number of Years" to compute the time that each cell will be in each state/condition before switching to the other. The grids computed by "One Year Transition Probabilities" are used to compute the expected number of years, variances, and 95% confidence intervals. The grids of and () are used in the following equations:

and variances

.

and confidence interval :

.

A new view is created that displays the computed grids. These grids
can be used to answer the question, "How long will the uninfested or infested
conditions last?".

*Probabilities in 2, 3, 4, and
5 Future Years*

Download nth-Step Transitions n=2 Shapefile Metadata

Download nth-Step Transitions n=3 Shapefile Metadata

Download nth-Step Transitions n=4 Shapefile Metadata

Download nth-Step Transitions n=5 Shapefile Metadata

To compute the probability of each cell being in state 0 or 1 at times beyond the next year, the user would use the menu option "Probabilities in 2, 3, 4, and 5 Future Years." This option computes the n-step transition matrix in the form of 16 grids and displays the information in a new view. The sixteen grids that represent the four transition matrices are computed by taking the grids computed by the "One Year Transition Probabilities" and multiplying them in a matter to satisfy:

.

*Percent of Years in 5 Years in
Each State/Condition*

Download Number of Years During Next 5 Years Metadata

Download Percentage of Next 5 Years Metadata

The last menu option, "Percent of Years in 5 Years in each Transition" creates a view that displays the number of years and percent of time each cell is expected to be in each state in the next five years, given that it started in 0 or 1. The grids computed by the menu choice "Probabilities in 2, 3, 4, and 5 Future Years" are used to compute:

and .

Infested = ³ 9.6 grasshoppers/m

Uninfested = < 9.6 grasshoppers/m

0 = Uninfested

1 = Infested

Transitions:

Uninfested => Uninfested (0=>0)

Uninfested => Infested (0=>1)

Infested => Uninfested (1=>0)

Infested => Infested (1=>1)

Generic Markov Extension. (Requires the Grid files, legends, and grid Metadata for the maps produced for Wyoming.

Shapefiles of the maps produced
for Wyoming.

Copyright © 2000 Kiana Zimmerman.

Entomology Section

Department of Renewable Resources

College of Agriculture

University of Wyoming

Most recent update: 11/01/10