Counting butterflies

[Michael Healy]

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All transect-walkers - or perhaps only number-obsessionals like me - ask
themselves how repeatable their counts are, what would happen if, at the end
of a walk, they went round again.  To throw some light on this, I recently 
walked a 2 km stretch of disused railway line (the Nicky Line at Harpenden, 
Hertfordshire, between Tl124147 and TL114135) five times in each direction 
on a clear sunny day between 1030 and 1445.  The results were as follows -

Trip no 	1     2     3     4     5     6     7     8     9    10
Start time   1027  1051  1117  1142  1208  1233  1300  1325  1349  1412
Temperature (C)24    26    25    26    25    25    25    24    23    24

T sylvestris
  /lineola      0     0     1     1     2     2     1     2     0     5
O venata        0     2     8     3     6     2     1     4     6     4
P brassicae     3     4     4     3     6     9     3     6     2     5
P rapae        24    24    19    27    27    28    22    31    30    21
P napi          5     3     2     2     6     6     4     5     1     7
A urticae       1     2     4     2     0     2     2     0     1     1
P tithonus      1     7     5     9     4    10     7     8     6     5
M jurtina       6    11    14    10    13    12    17     8    10    12

Total          40    53    58*   57    64    71    58*   65*   57*   60

*Plus singletons -
Trip 3	P icarus
Trip 7	P aegeria
Trip 8	V atalanta
Trip 9	M galathea

The first trip gives a somewhat lower count than the others, in accordance 
with the standard instruction not to undertake transect walks before 1100.  
There is a slight tendency for the even-numbered northbound trips to give 
higher counts than the southbound ones - the line runs approximately
north-south and it is easier to count the butterflies with the sun at your 
back.  Apart from this, the counts, both for the total and for the commoner 
species, are reasonably consistent.

It is in fact, using a rather simplistic statistical argument, possible to 
make a rough estimate of the fraction of the total population which is 
included in the counts, assuming this to be constant over the period.  If 
this fraction is very small, statistical theory suggests that the variance 
of the counts (the square of the standard deviation) is expected to equal 
the mean.  If all the butterflies are seen on every occasion, all the counts 
will be the same and the variance will be zero.  Between these extremes the 
expected variance is given by the mean times (1 - f) where f is the fraction 
observed.  Applying this argument to the overall totals and to the counts 
for P rapae and M jurtina (omitting trip no 1) gives f = 0.51, 0.32 and 0.42
respectively, suggesting that the total population might be between two and 
three time the numbers actually observed.  Even apart from the over-simple 
statistical model, these figures should not be taken too seriously, since an 
estimated variance based on no more than 9 observations is necessarily 
very imprecise.

It is of some interest that a regular transect walk the following day, under 
similar weather conditions, gave a total count of 87, including 37 P rapae 
and 20 M jurtina.  On the evidence, this represents a real increase in the 
population over 24 hours.

Michael Healy
31 July 1977