482 
NATURE 
[March 22, 1883 
4 
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 solu- 
tion of the problem as given by Dr. Fritz Miller 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 remem- 
ber that the question at issue was the advantage gained by a dis- 
tasteful, 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. Miiller says: ‘‘If both 
species are equally common, then both will derive the same 
benefit from their resemblance—each will save half the number 
of victims which it has to furnish t> 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 (4), then (4) will lose only one-fourth as many individuals 
as it would do if it were quite unlike (a) ; and if it is only one- 
tenth 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. Miiller’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 ¢ 1000 individuals, while the number 
required to be sacrified to the birds is 30. ‘Then, if 4 were 
unlike @ 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, @ losing 20, and 4 10. Thus 4 would be 20 
better off than before, and a only 10 better off; but the 20 
gained by d is a gain on 10co, equal to a gain of 40 on 2000, or 
four times as much zz proportion as the gain of a. In another 
case let us suppose ¢ to consist of 10,000 individuals, @ 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, ¢ losing 100, 
donly 10. Thuse will gain only 10 on its total of 10,000, while 
d will gain roo on its total of 1009, equal to 1000 on 10,000, or 
100 times as much proportional gain asc. Thu-, while the gain 
in actual numbers is inversely proportional to the numbers of the 
two species, the Zrofortional gain of each is inversely as the 
sguare 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 fiourishing, 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 zfs own numbers saved from 
destruction, not by the proportion this saving bears to that of the 
other species. I am inclined to think, theref re, 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. 
Miiller, although it undoubtedly exists, has no bearing on the 
difficulty to be explained. ALFRED R. WALLACE 
Mr. A. R. WALLACE has been so good as to forward me the 
extract from the /afan Mail above referred to, together with 
his reply. The article in question bears the title, “‘ Protection 
by Mimiery—a Problem in Mathematical Zoology.” The 
authors, while admitting the broad principles involved in Dr. 
Fritz Miiller’s theory, fail to see why the advantage derived by 
the mimicking species, in cases where the litter is less numer- 
ou; than the model, should be as the square of the relative 
numbers. They admit that ‘‘the ingenious explanation seems 
perfectly satisfactory,” but the proportional benefit appeared to 
them exaggerated. Mr. Wallace has now, I think, cleared up 
the misunderstanding with reference to this part of the question, 
but it may be of use in assisting towards the further discussion 
of the problem if I here give the simple algebraical treatment 
adopted in the original paper. 
Let a, and a, be the numbers of two distasteful species of 
butterflies in some definite district during one summer, and let 
x be the number of individuals of a distinct species which are 
destroyed in the course of a summer before its distastefulness 
is generally known. If both species are totally dissimilar, then - 
each loses 7 individuals. If, however, they are undistinguish- — 
ably similar, then the first loses = and the second loses 
a + a, | 
Ayn 
aaeR The absolute gain by the resemblance is therefore for 
rete 22) 
j 
ayn 
the first species, 7 — = 
a+ ay 
3 and ina similar manner 
for the second species, —1” —. 
a, + a | 
with the total numbers of the species, gives for the first (A), _ 
sone —" | We thus have 
a4(4y + ay (a, + ay) 
the proportion, A, :A, = a2: @,". 
With reference to Mr. Wallace’s concluding paragraph, I may 
point out that the advantage of the mimic is ‘* measured solely 
by the fraction of zts own members saved from destruction.” 
Thus, taking his last example, the speciesc saves only 1/1000 of 
its whole number, and d saves 1/10 of its whole number by the’ 
resemblance to ¢, The fact that these numbers stand to one| 
another in the ratio of 1: 10°, whilst c:¢d= 10:1, is a me | 
matical necessity from which I do not see how we can escape. 
As the numerical disproportion betwen the species increases, the 
advantage derived by the more abundaat insect is practically a 
vanishing quantity ; whilst, on the other hand, if the two species 
are equal in numbers, it is obvious that they both derive the 
same advantage, each losing only half the number that it would 
if there was no resemblance between them. 
It must not be forgotten in considering the question of 
mimicry between two nauseous species that the foregoing calcu- 
lations apply only to the case where the resemblance is perfect, 
z.e. so exact that the insects are absolutely undistinguishable by 
their foes. The initial steps may be hastened in these cases by 
the near blood-relationship of the species, and it is a remarkabl 
circumstance that large numbers of species belonging to differen 
distasteful genera havea close similarity of wing-pattern, althoug 
the distinctness of the genera has never been called in question. 
But the genera concerned, although distinct, are very closely 
related, and this is quite in accordance with the views here 
advocated. 
, and for the second (Ay), 
The general question as to the persecution of distasteful butter 
flies by young inexperienced birds, &c., is certainly one or 
which much work remains to be done, and-very great servic 
could be rendered if naturalists residing in the tropics would 
undertake some systematic experiments in this direction. M 
friend, Mr. W. L. Distant, the author of the ‘‘Rhopalocera 
Malayana,” has already given reasons in these columns (vol, 
xxvi. p. 105) tor disbelieving in any such want of experience, 
and I have discussed this phase of the question with him else: 
where (Axx. and Mag. Nat. Hist., December, 1882). | 
Rk. MELDOLA 
On the Value of the ‘‘ Neoarctic” as One of the eB 
Zoological Regions 
In the Proceedings of the Academy of Natural Sciences a 
Philadelphia (December, 1882) Prof. Angelo Heilprin has a1 
article under the above title, in which he seeks to show that ih 
Neoarctic and Palearctic should form one region, for which h¢ 
proposes the somewhat awkward name “ Triarctic Region,” of 
the region of the three northern continents. The reasons fojff 
this proposal are, that in the chief vertebrate classes the propoxy 
tion of peculiar forms is less in both the Nearctic and Palzearctijff 
than in any of the other regions; while, if these two regions ary 
combined, they will, together, have an amount of peculiarit) 
greater than some of the tropical regions. 
This may be quite true without leading to the conclusioj 
argued for. The best division of the earth into zoological 
regions is a question not to be settled by looking at it from on 
point of view alone; and Prof. Heilprin entirely omits two cout 
siderations—peculiarity due to the absence of widespre: 
groups, and geographical individuality. The absence of thi! 
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