Applied Probability - Free

14 1. Basic Principles of Population Genetics5. Moran [12] has proposed a model for the approach of allele frequenciesto Hardy-Weinberg equilibrium that permits generations to overlap.Let u(t), v(t), and w(t) be the relative proportions of the genotypesA 1 /A 1 , A 1 /A 2 , and A 2 /A 2 at time t. Assume that in the small timeinterval (t, t+dt) a proportion dt of the population dies and is replacedby the offspring of random matings from the residue of the population.In effect, members of the population have independent, exponentiallydistributed lifetimes of mean 1. The other assumptions for Hardy-Weinberg equilibrium remain in force.(a) Show that for small dtu(t + dt) = u(t)(1 − dt)+[u(t)+ 1 2 v(t) ] 2dt+ o(dt).Hence,[u ′ (t) =−u(t)+ u(t)+ 1 ] 2.2 v(t)(b) Similarly derive the differential equationsv ′ (t) = −v(t)+2[u(t)+ 1 ][ 1]2 v(t) 2 v(t)+w(t)w ′ (t) = 1] 2.−w(t)+[2 v(t)+w(t)(c) Let p(t) =u(t)+ 1 2 v(t) be the allele frequency of A 1. Verify thatp ′ (t) = 0 and that p(t) =p 0 is constant.(d) Show thatand so(e) Similarly prove[u(t) − p 2 0] ′ = −[u(t) − p 2 0],u(t) − p 2 0 = [u(0) − p 2 0]e −t .v(t) − 2p 0 (1 − p 0 ) = [v(0) − 2p 0 (1 − p 0 )]e −tw(t) − (1 − p 0 ) 2 = [w(0) − (1 − p 0 ) 2 ]e −t .(f) If time is measured in generations, then how many generationsdoes it take for the departure from Hardy-Weinberg equilibriumto be halved?6. Consider an X-linked version of the Moran model in the previousproblem. Again let u(t), v(t), and w(t) be the frequencies of the threefemale genotypes A 1 /A 1 , A 1 /A 2 , and A 2 /A 2 , respectively. Let r(t)and s(t) be the frequencies of the male genotypes A 1 and A 2 .