Show that Felsensteins [9] map function = 1 2 e2(2)d 1 e2(2)d + 1 (12.22) arises from a stationary renewal model

Show that Felsenstein’s [9] map function

θ = 1 2

e2(2−γ)d − 1 e2(2−γ)d − γ + 1 (12.22)

arises from a stationary renewal model when 0 ≤ γ ≤ 2.

Kosambi’s map function is the special case γ = 0.

Why does (12.22) fail to give a legal map function when γ > 2? Note that at γ = 2 we define

θ = d 2d+1 by l’Hˆopital’s rule.