The Red and the Gray (Squirrels)

 

(Adapted from Mooney and Swift, A Course In Mathematical Modeling.)

 

Background

 

The American gray squirrel (Sciurus carolinensis Gmelin) was introduced in Great Britain by a series of releases from various sites starting in the late nineteenth century.  In 1876, the first gray squirrels were imported from North America, and have subsequently spread throughout England and Wales, as well as parts of Scotland and Ireland.

 

Simultaneously, the native red squirrel (Sciurus vulgaris L.), considered the endemic subspecies, has disappeared from most of the areas colonized by gray squirrels.  Originally, the red squirrel was distributed throughout Europe and eastward to northern China, Korea, and parts of the Japanese archipelago.  During the last century, the red squirrel has consistently declined, becoming extinct in many areas of England and Wales, so that it is now confined almost solely to Northern England and Scotland.  A few isolated red squirrel populations exist on offshore islands in southern England and mountainous Wales.

 

The introduction of the American gray squirrel continued until the early 1920's, by which time the gray squirrels had rapidly spread throughout England.  By 1930 it was apparent that the gray squirrel was a pest in deciduous forests, and control measures were attempted.  Once the pest status of the gray squirrel was recognized, national distribution surveys were undertaken.  The resulting distribution maps clearly showed the tendency for the red squirrel to be lost from areas that had been colonized by the gray squirrel during the preceding 15 to 20 years.

 

Since 1973, an annual questionnaire has been circulated to foresters by the British Forestry Commission.  The questionnaire concerns the presence or absence of the two squirrel species.  It also includes questions on the changes of squirrel abundance, details of tree damage, squirrel control measures, and the number of squirrels killed.  Using the data collected by the Forestry Commission, we wish to construct a model to predict the trends in the distribution of both species of squirrels in Great Britain.

 

                        (from A Course In Mathematical Modeling, pp. 127-128)

 

Data

 

We will use data extracted by Usher et. al. [UCB] from the distribution maps compiled by the Forestry Commission.  Each 10 km square region (100 sq. km.) was classified into one of four states:

 

            R:  only red squirrels recorded in that year

            G:  only gray squirrels recorded in that year

            B:  both species of squirrels recorded in that year

            N:  neither species of squirrels recorded in that year

 

Notice that we are not counting the number of squirrels, just the number of regions in which they were sighted.  For each pair of years, from 1973-74 to 1987-88, each region was allocated to one of 16 classes depending on its state in each year, e.g., R®R, G®R, B®N, R®N, etc.  The table below summarizes these transition counts.

 

 

R

G

B

N

R

2529

35

257

5

G

61

733

20

91

B

282

25

4311

335

N

3

123

310

5930

 

Problems

 

1.      Compute the matrix of transition probabilities, T.  I.e., compute the percentage of regions in state R, for example, which were in each of the other states the following year.  Be careful when rounding - you must make sure that the entries in each column add up to 1 (Why?).

2.      Write down the system of linear equations giving the number of regions with only red squirrels, only gray squirrels, both and neither in a given year in terms of the numbers of each type of region in the previous year.  I.e., if r(t), g(t), b(t) and n(t) are the number of regions of each type in year t, find an equation for each of these in terms of r(t-1), g(t-1), b(t-1) and n(t-1).

3.      The transition matrix for n time steps (years) is just Tn.  Compute this matrix for n = 2, 10, 50, 100, 200, 500.  Interpret your results.

4.      Make up three different initial distributions of the regions (i.e. the percentage of regions which fall into each state R, G, B, and N).  In each case, what will the distribution be after 2, 10, 50, 100, 200, and 500 years?  Interpret your results.

5.      What can you conclude about the long-term behavior of the squirrel population in Great Britain?  Are the red squirrels in danger of extinction?

 

References

 

[UCB]   M. Usher, T. Crawford, and J. Bunwell.  "An American Invasion of Great Britain:  The case of native and alien squirrel species."  Conservation Biology, vol. 6, no. 1, pp. 108-115, 1992

 

Graded out of 35 points