NAT ORE 
[JUNE 1, 1899 
Variation of Species. 
On p. 181 of Wallace’s ‘‘ Darwinism,” ed. 1889, this passage 
occurs :—‘‘ Let us suppose that a given species consists of 
100,000 individuals of each sex, with only the usual amount of 
fluctuating external variability. Let a physiological variation 
arise, so that 10 percent. of the whole number—10,000 in- 
dividuals of each sex—while remaining fertile z7fer se become 
quite sterile with the remaining 90,000. This peculiarity is not 
correlated with any external differences of form or colour, or 
with inherent peculiarities of likes or dislikes leading to any 
choice as to the pairing of the two sets of individuals. We have 
now to inquire, What would be the result ?” 
I have here attempted to investigate this question alge- 
braically. 
A. We shall suppose, as Dr. Wallace does, that the number 
of males in the species is the same as the number of females. 
Each of these numbers we shall denote by unity. For con- 
venience, we shall speak of the number of either sex as the 
number of the species. Let then 
x = the number of the normal species, 
and 
y = the number of the variant variety, 
at any given stage of the change above described. 
If in any given generation (x, 7), the ratio, which the number 
of the variant individuals, born in a family of the normal 
species, bears to the total number of young born in that family, 
be denoted by %; and if («’, y’) denote the generation which 
succeeds (1, y); then must 
Ky! 22 xT —2) 3 gy? phx? 5 
with the relation 
By — at y— 1s 
because the total species remains constant in number of in- 
dividuals. 
Now, if the variants succeed in establishing themselves as a 
new species, and the above two generations belong to the per- 
manently settled state of the whole species, we must have x’ =x, 
and 7/=y. 
Consequently, to determine 
have the equations 
(1=£)a? _ y+ ha? 
x 
.. (I-—A)x=2x2—2x4+13 
vw. £=4(3-24+ N22 62 +1). 
To take Dr. Wallace’s example, put =*1. Then 
B—nOOl Osis. iey.— eA 2) eras 
x, y under this condition we 
> X+yVY=15 
or 
H='56492)..., j= "43508)...- 
Thus, taking Dr. Wallace’s number 100,000, we find that, 
ultimately, the normal species will number 88,508 and the 
variant 11,492 These numbers differ but little from Dr. 
Wallace’s, but they represent the final distribution of the 
original species into twe species. Another possible distribu- 
tionis given by the numbers 56,492 and 43,508. If by any 
chance the first permanent distribution be disturbed materially, 
then the total species might reach the second permanent state. 
If, however, at any time the parent species were to cease to 
produce the variants, then the latter would quickly disappear. 
They could be saved from extinction only by the ceasing of 
intermarriage between the two species. For, if (x,, y,,) denote 
the wth generation from the one (x, y) in which the variants 
ceased to be produced by the normal species, then 
25 SURE 
If 2=4, and x="9, y="I, as in Dr. Wallace’s case, then 
X42 Ys 2: (81)* : ‘oo000001 ; 
so that the original species of 100,000 would have no variants 
left at all. The disappearance of the variants is due to the two 
facts, (1) that the total number of the two species together is 
constant, (2) that the number of unfruitful unions is very large 
in proportion to the number of unions possible to the smaller 
species. For example, if the variants be ‘1 of the whole species, 
the probability will be that ‘9 of their unions will be unfruitful ; 
but that “9 of the unions of the normal species will not be 
unfruitful. 
B. We shall now consider the case when the unions between 
the two varieties are not sterile, and the hybrids are also fertile 
znter se and with the parent varieties. 
NO. 1544, VOL. 60] 
xm: yn 
Let the relative, effective, fertility of the hybrids and 
mongrels, z7z/er se and with the parent varieties, be denoted by 
the factor £, which we shall assume to be always less than unity. 
Also, let the effective fertility of the normal species in the pro- 
duction of variants be denoted by the factor 4 ; and let 2 denote 
the number of the hybrid variety in the generation (., y, =). 
Then the equations which determine the stable and permanent 
condition, if there be one, are 
(Lam)? 9 + x? {ot y #2)? — 0" yf Xk 
= 2 
x y z 
otk Nahas) PECL <A. alls) 
Put 
a=1-—2, #=1-£, pw =1—-p; 
then 
(heli 2 N= eee oso 6 oe) 
and 
ieee (Qa), 0) oll tease oN) 
Put 
B=(1-fa)wa; . - (4) 
then 
B? —2(3 — w)B + 2h" =o. - (5) 
Whence ls 
I A'B 
Cy Oe i= i\ BE ee > . (6) 
and 
Af a \ 
B=5\3 ~HENH— Ou+15¢ - ein! alt) 
The roots of u7—64+1=O are "17158... and 5°82842.. . 
As we suppose u less than 1, it follows that in no case must 
exceed the lower root, "17158 .. . 55 (0) 
From (7) it follows that 8 must >. 
From (6) it follows that 1-4 must >42'B, or =42'B ; 
*, I-—p must >4hk'; 
, 
Also must not >“, . 
4h 
-". & must not >°5 
- (9) 
. . (10) 
To take Dr. Wallace’s example, put «= ‘I. 
We find, then, that £2” must not > ‘225. If we put £4’= "225, 
and solve for %, we find that 
h—2AQe , or 658 
But £2’ must not >*225; hence *# must not lie between 
242%, . . and 658)... siemsibya(Q)i£ must not —sa2 ieee 
Take, for example, = 2. 
Then by (7), B='225969. . . or "354031 . 
By (10) we must reject the second value of B. 
Adopting the first value, we find from equations (2) and (1) 
the following two solutions, 
= +7OSiciis yn LOA REE 2 OOGTe 
and 
ea eetots) 
OL e052 
Here the effective fertility of the hybrids, zzte se and with 
the parent varieties, must not exceed 34 per cent. of that of 
the parent varieties ; and in no case must it exceed 50 per cent. 
of the latter. Also, in no case must the parent species supply 
more than, or even as much as, 18 per cent. of its total progeny 
to the variant species. 
C. If no hybrid unions occur, and the two varieties supply 
individuals to each other in such a way that, taking the progeny 
of the generation («, v), a fraction A of the x progeny belongs to 
the y variety, while a fraction p of the y progeny belongs to the 
x variety (where A and w are proper fractions), it is easy to 
prove that in the ultimate, established, state of the total species 
aHrysi era. 
Therefore, if A=, the species will be, in its final state, 
equally divided between the two varieties. The equations for 
the established state are, since now there is no intermarriage, 
XA AKT MY _I = MY TEAK 
x 
ia — las 
whence 
AX= My 3 
Z.é. 
Xiyprimia. 
This may also be proved by direct calculation, 
Woodroffe, Bournemouth. J. W. SHARPE. 
