Demographic genetics / edited by Kenneth M. Weiss and Paul A. Ballonoff.

Date:
[1975]
Catalogue details

Licence: Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)

Credit: Demographic genetics / edited by Kenneth M. Weiss and Paul A. Ballonoff. Source: Wellcome Collection.

    135/440 page 115
    1926.] Natural selection and Mendelian variation. 2!» types are constant, and writing lK,{y, y-t)v{t) = lIU{y, y-t)v{t) = K{t), lK¿y, y-t)v{t) = K'it), (12.11) we have the equations «(?/) +(y) = [ K'{t)u{y — t)dt-\-[ K'\t)v{y — t)dt, (12.2) J Л] Jai (y > v{y)-\-w{y) = \ K'{t)w{y — t)dt-\-[ K{t)v{y — t)df, (12.3) Jxi Jxi u(y)-\-v{y) =Ш+\' K'it) u{y-t)dt+\' K{t) v{y-t)df, (12.21) JXl J Al i\>y> 0) v{y)-\-w{y) =My)+[ K{t)v{y-t)dt-\-[ K'{t)w{y-t)dt, (12.31) J Al J Al where, in the last two équations, the integrals vanish if Ài ^ y, and iire given functions of their arguments. 13. Witli regard to the functions K'it), etc., it will be supposed that they are contniuous and never negative if Л2 ^ t ^ Ai, and that numbers a, ß, e exist such tliat e > Ü, Л2 ^ ^ il t ß, К'ф, Кit), K'{t) > e. Tt will he suj)j)0sed, as in the last part, that a, ß, « have been chosen once for all such that these conditions are satisfied; any function Kiy, a ), which is continuous and never negative in the stri[) 0 < '^0 < ?y < + and is such that, if a ^ y—x ^ ß, K(y, x) > e, will be said to satisfy con¬ ditions A ; and will be used for a constant, which depends only on a, ß, €, Л2, and M, the upper bound of K'it), K{t), K'{t). It will be convenient to define K{t), K'{t) as zero, if f is negative oi- is greater than Л2. There is no objection to regarding tlie functions so defined as continuous. The first question that arises is as to the existence of a solution (12.1.2); but, since Ai > 0, its existence is obvious. If fi{y), f^iy) are arbitrary continuous functions and are never negative, the right- hand sides of (12 . 21 . 31) are given functions for 0 ^ у ^ Ai, and hence, by (12 . 1), niy), v(y), and w(y) are given in that interval. If these are substituted in the right-hand sides of (12 . 21 . 31), the values of u(y), v(y), and ic(y) are found, with the help of (12.1), for 115
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