:2(>0 Jirotlwr and Sister JJatiiuj 



Let r,i, s,j, t,i represent the proportions uf the three types of zygotes 

 in the ;(th generation, and let 



r,,. +*•-! +t„ = 1. 



The analysis of the formation of the zygotes of the ;(th generation gives 

 the f)llowing recurrence relations: 



r„ = cT (r„_, + .s-„_i/4) + (1 - 0-) Cir.^, + ,v_,)V4, (1) 



s„ = a A'„,_,/2 + ( 1 - a ) (2r„_, + 6-„_, ) ( 2*„_i + *■„_, )/2, (2) 



t„=o- (<„_, + s„_,/4) + (1 - 0-) (2^,_, + 6-„^,)V4 (3) 



On the right hand side of each of these equations, the first term is the 

 contribution of self-fertilization, and the second term is the contribution 

 of random mating. To solve these equations we notice that 



2r„ + *•„, = 2r,,_i + «„_, = = 2/-o + s„. 



Thus we have that 27',, + s„ is constant. Let 



p = 2r, + ti,. 



Then equations (1), (2), (3) simplify giving, 



/•„=<T;-„,_,/2 + [ap + (l-(7)p=]/4, (4) 



.„ = as,,^J-2 + p(l-a){2-p)/2, (.5) 



t,, = at,_,l2+[a(2-p) + (l-a)(2-pyy4> (6) 



The solutions of these equations are, 



_ , a\"- ^ pa- (1 — p) + p 



•'-iij^»+^2^r ''^ 



-®"''»-^^^^^' ^«) 



^"~ \2) "^ 2(2-0") ^ ' 



(2-p)(2 -. 

 \2J ^" ' 2(2-0-) 



The constants c„, fZ„, e„ are determined by the initial conditions and are 



c„ = r„ + p(pa - p -o-)/2(2- a) (10) 



d„ = s„ + p(2 - p)(cT - \)/C2 - a), (11) 



e, = t„ + (2-p){p + a-pa- 2)/2 (2 - a) (12) 



Equations (7), (8), (9) give the results for our problem of combined 

 selfing and random mating. Incidentally we may specialize these 



