Mimicry in Butterflies/Chapter 12

by Reginald Crundall Punnett

APPENDIX I

For the table on p. 155 I am indebted to the kindness of Mr H. T. J. Norton of Trinity College, Cambridge. It affords an easy means of estimating the change brought about through selection with regard to a given hereditary factor in a population of mixed nature mating at random. It must be supposed that the character depending upon the given factor shews complete dominance, so that there is no visible distinction between the homozygous and the heterozygous forms. The three sets of figures in the left-hand column indicate different positions of equilibrium in a population consisting of homozygous dominants, heterozygous dominants, and recessives. The remaining columns indicate the number of generations in which a population will pass from one position of equilibrium to another, under a given intensity of selection. The intensity of selection is indicated by the fractions ¹⁰⁰⁄₅₀, 100/75, etc. Thus 100/75 means that where the chances of the favoured new variety of surviving to produce offspring are 100, those of the older variety against which selection is operating are as 75; there is a 25% selection rate in favour of the new form.

The working of the table may perhaps be best explained by a couple of simple examples.

In a population in equilibrium consisting of homozygous dominants, heterozygous dominants and recessives the last named class comprises 2.8% of the total: assuming that a 10% selection rate now operates in its favour as opposed to the two classes of dominants—in how many generations will the recessive come to constitute one-quarter of the population? The answer is to be looked for in column B (since the favoured variety is recessive) under the fraction ¹⁰⁰⁄₉₀. The recessive passes from 2.8% to 11.1% of the population in 36 generations, and from 11.1% to 25% in a further 16 generations—i.e. under a 10% selection rate in its favour the proportion of the recessive rises from 2.8% to 25% in 52 generations.

Percentage of total population formed by old variety	Percentage of total population formed by the hybrids	Percentage of total population formed by the new variety	Number of generations taken to pass from one position to another as indicated in the percentages of different individuals in left-hand column.
			A: Where the new variety is dominant				B: Where the new variety is recessive
			100⁄50	100⁄75	100⁄90	100⁄99	100⁄50	100⁄75	100⁄90	100⁄99
99.900	.09	.000
98.000	1.96	.008	4	10	28	300	1920	5740	17,200	189,092
90.700	9.00	.030	2	5	15	165	85	250	744	8,160
69.000	27.70	2.800	2	4	14	153	18	51	149	1,615
44.400	44.40	11.100	2	4	12	121	5	13	36	389
25.000	50.00	25.000	2	4	12	119	2	6	16	169
11.100	44.40	44.400	4	8	18	171	2	4	11	118
2.800	27.70	69.000	10	17	40	393	2	4	11	120
.030	9.00	90.700	36	68	166	1632	2	6	14	152
.008	1.96	98.000	170	333	827	8243	2	6	16	165
.000	.09	99.900	3840	7653	19,111	191,002	4	10	28	299

If the favoured variety is dominant it must be borne in mind that it can be either homozygous or heterozygous—that for these purposes it is represented in the left-hand column by the hybrids as well as by the homozygous dominant. In a population in equilibrium which contains about 2% of a dominant form, the great bulk of these dominants will be heterozygous, and the relative proportion of recessives, heterozygous, and homozygous dominants is given in the second line of the left-hand column.

Let us suppose now that we want to know what will be the percentage of dominants after 1000 generations if they form 2% of the population to start with, and if, during this period, they have been favoured with a 1% selection advantage. After 165 generations the proportion of recessives is 90.7, so that the proportion of dominants has risen to over 9%; after 153 further generations the percentage of dominants becomes 27.7 + 2.8 = 30.5; after 739 generations it is 88.8%, and after 1122 generations it is 69.0 + 27.7 = 96.7. Hence the answer to our question will be between 89% and 97%, but nearer to the latter figure than the former.

Mr Norton has informed me that the figures in the table are accurate to within about 5%.

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