A Spatial Model For Markov Chain Analysis Of Grasshopper Population Dynamics In Wyoming

   

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.

Steps to the project:

1) All locations in Wyoming were converted to one of two conditions "infested" (³ 9.6 grasshoppers/yard2) or "uninfested" (< 9.6 grasshoppers/m2). (These are the only population densities reported by surveys for the last 50 years.)

data.jpg (45185 bytes)

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

outbreak.jpg (92850 bytes)

Download Outbreak Frequency Shapefile                              Metadata

Outbreak Frequency computes the frequency with which each cell was infested with ³ 9.6 grasshoppers per m2 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

prob1uu.jpg (96176 bytes)prob1ui.jpg (95617 bytes)prob1iu.jpg (155423 bytes)prob1ii.jpg (154472 bytes)

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 m2 and 1 = infested or ³ 9.6 grasshoppers per m2. 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 P01 and P10 and use those values to derive the other two probabilities, by subtracting from 1.

Expected Number of Years

expectu.jpg (105332 bytes)95u.jpg (89971 bytes)vu.jpg (90961 bytes)expecti.jpg (93488 bytes)95i.jpg (66138 bytes)vi.jpg (61956 bytes)

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

prob2.jpg (115415 bytes)prob3.jpg (114142 bytes)prob4.jpg (113237 bytes)prob5.jpg (113063 bytes)

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

futuu.jpg (100755 bytes)futui.jpg (102558 bytes)futiu.jpg (100735 bytes)futii.jpg (102569 bytes)

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 .


Terms:

    Infested = ³ 9.6 grasshoppers/m2
    Uninfested = < 9.6 grasshoppers/m2
    0 = Uninfested
    1 = Infested 
 
 
Transitions:

    Uninfested => Uninfested (0=>0)
    Uninfested => Infested (0=>1)
    Infested => Uninfested (1=>0)
    Infested => Infested (1=>1) 

To Download:

The Markov Extension for Grasshopper analysis with the legend files.  (Requires the Spatial Analyst extension and the legend files.  Set the working directory in ArcView to the directory that contains the legend files.)

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. 

To read the entire thesis.
Entomology Section
Department of Renewable Resources
College of Agriculture
University of Wyoming
Most recent update: 11/01/10