Alfred Russel Wallace : Alfred
Wallace : A. R. Wallace : Russel Wallace : Alfred Russell
Wallace (sic)
Difficult
Cases of Mimicry (S359: 1883)
Editor Charles H. Smith's Note: A continuation of the
discussion on the new ideas on protective mimicry advanced by Fritz Müller,
printed as a letter to the Editor in the 22 March 1883 number of the journal
Nature. Original pagination indicated within double brackets. To
link directly to this page, connect with: http://people.wku.edu/charles.smith/wallace/S359.htm
[[p. 481]] I have
received from Mr. Thos. Blakiston, of Tokio, Japan, a communication to
the Japan Mail by himself and Prof. Alexander, [[p.
482]] commenting on my article in Nature, vol. xxvi. p.
86, and pointing out some errors as to the estimated advantage derived
by the mimicking butterflies. On referring to my article, I find that
I have, by an oversight, misstated the mathematical solution of the problem
as given by Dr. Fritz Müller and confirmed by Mr. Meldola, and have
thus given rise to some confusion to persons who have not the original
article in the Proceedings of the Entomological Society to refer
to. Your readers will remember that the question at issue was the advantage
gained by a distasteful, and therefore protected, species of butterfly,
which resembled another distasteful species, owing to a certain number
being annually destroyed by young insectivorous birds in gaining experience
of their distastefulness. Dr. Müller says: "If both species are equally
common, then both will derive the same benefit from their resemblanceeach
will save half the number of victims which it has to furnish to the inexperience
of its foes. But if one species is commoner than the other, then the benefit
is unequally divided, and the proportional advantage for each of
the two species which arises from their resemblance is as the square
of their relative numbers." This is undoubtedly correct, but in my article
I stated it in other words, and incorrectly, thus: "If two species, both
equally distasteful, resemble each other, then the number of individuals
sacrificed is divided between them in the proportion of the square of
their respective numbers; so that if one species (a) is twice as
numerous as another (b), then (b) will lose only onefourth
as many individuals as it would do if it were quite unlike (a);
and if it is only onetenth as numerous, then it will benefit in the proportion
of 100 to 1."
This statement is shown by Messrs. Blakiston
and Alexander to be untrue; but as some of your readers may not quite
see how, if so, Dr. Müller's statement can be correct, it will be
well to give some illustrative cases. Using small and easy figures, let
us first suppose one species to be twice as numerous as the other, a
having 2000 and b 1000 individuals, while the number required
to be sacrificed to the birds is 30. Then, if b were unlike a
it would lose 30 out of 1000, but when they become so like each other
as to be mistaken, they would lose only 30 between them, a losing
20, and b 10. Thus b would be 20 better off than before,
and a only 10 better off; but the 20 gained by b is
a gain on 1000, equal to a gain of 40 on 2000, or four times as much in
proportion as the gain of a. In another case let us suppose
c to consist of 10,000 individuals, d of 1000 only,
and the number required to be sacrificed in order to teach the young birds
to be 110 for each species. Then, when both became alike, they would lose
110 between them, c losing 100, d only 10. Thus c
will gain only 10 on its total of 10,000, while d will gain 100
on its total of 1000, equal to 1000 on 10,000, or 100 times as much proportional
gain as c. Thus, while the gain in actual numbers is inversely
proportional to the numbers of the two species, the proportional
gain of each is inversely as the square of the two numbers.
I am, however, not quite sure that this way
of estimating the proportionate gain has any bearing on the problem.
When the numbers are very unequal, the species having the smaller number
of individuals will presumably be less flourishing, and perhaps on the
road to extinction. By coming to be mistaken for a flourishing species
it will gain an amount of advantage which may long preserve it as a species;
but the advantage will be measured solely by the fraction of its own
numbers saved from destruction, not by the proportion this saving
bears to that of the other species. I am inclined to think, therefore,
that the benefit derived by a species resembling another more numerous
in individuals is really in inverse proportion to their respective numbers,
and that the proportion of the squares adduced by Dr. Müller, although
it undoubtedly exists, has no bearing on the difficulty to be explained.
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