200 Brother and Sister Mathig 



Let Vn, Sn, tn represent the proportions of the three types of zygotes 

 in the nth. generation, and let 



Tfi + S,i + fc,j = 1 . 



The analysis of the formation of the zygotes of the nth generation gives 

 the following recurrence relations : 



Vn = a- (r„_i + s„_i/4) + (1 - o") (2r,,„j + .s-„_i)V4, (1) 



S,, = a Sn-J2 + (1 - a) (2r„_i + S„_,) (2tr,-^ + Sn-i)/2, (2) 



tn = a- {tn-i + Sn-J4<) + (1 - a) {^n-r + Sn-,fl^ (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,, + s,i = 2r„_i + s„_i = =2ro + So- 



Thus we have that 2r„ + s„ is constant. Let 



p = 2ro-l-5o. 

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



r„ = o-r„_i/2 + [o-p + (l-(7)/D2]/4, (4) 



6-„ = crs„_i/2 + /o(l-o-)(2-p)/2, (5) 



tn = (Ttn-^l'2. + [a{2-p) + {l-a-){2-pf]l4> (6) 



The solutions of these equations are, 



■cr^"^ , /30"(1 — p)-\- p- 

 2(2- a) 



^""UJ ^""^ 2(2"^^"^ ^^^ 



-(1)"".+"^^^^^ («) 



.^(i)%.. <^--'>^-:-^-^- > («) 



The constants Co, d^, e^ are determined by the initial conditions and are 



c„ = ro + p(/jo--p-o-)/2(2-o-), (10) 



rf„ = So + p(2-p)(o--l)/(2-o-), (11) 



eo = 4 + (2-p)(p + cr-po--2)/2(2-o-) (12) 



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

 selfing and random mating. Incidentally we may specialize these 



