Stationary solutions and their stability for Kimura’s diffusion model with intergroup selection II
Yukio Ogura, Norio Shimakura · Kyoto journal of mathematics · 1987
Introduction.T h e p re se n t p a p e r is the second part of o u r p re v io u s o n e [ 7 ] , w here w e studied the diffusion model o f gene frequency proposed by M .K im ura [3 ], [4 ] a n d [ 5 ] .W e d iv id e d th e m odel into seven cases (a), (1:1), ••• , (g) according to the regions o f p a ra m e te rs .F or each o f th e s e c ases, w e g a v e th e num ber of stationary solutions and in sp e c te d th e ir sta b ility .B u t, a s fo r th e stability, w e h a d t o s e t a re s tric tio n o n in itia l m easures in the cases (d) and (e), and we could obtain no result in the case (g).T h e first purpose o f th is article is to establish th e sta b ility th e o re m in a d e fin itiv e fo rm (se e T h e o re m C in §1 and T heorem 1 in § 2).The second is to obtain tw o com parison theorem s for the moment sequences of the solutions.One of them (Theorem 2 in § 2) ensures th e o rd e r p re se rv in g p ro p e rty o f th e moment sequences under th e tim e evolution, and the other (T heorem 3 in § 3) shows the monotone dependence of the m om ent sequences o n th e parameters.T h e la s t p u rp o s e is to im p ro v e s lig h tly o u r p re v io u s T h e o re m 5 in [7 ] on K im ura's property (Theorem 4 in § 2).Our methods in th is article are basically the sam e a s in [ 7 ] .B ut for the sta b ility th e o re m in the cases (d) and (e), w e in addition m ake use of a comparison theorem on the solutions of stochastic differential equations.F urther, in the case (g), w e introduce a kind of energy of the solutions, and show that it d e c a y s a s t im e e la p s e s .In th e la tte r c a se (g ), th e se t o f th e stationary solutions is not discrete but in one-to-one correspondence to th e interval [0, 1 ] .S o it w ould be rem arkable to show th a t th e y a ll a re sta b le in o u r se n se (se e § 3 below for details).I n § 1 , w e r e v ie w K im u r a 's diffusion model and our main results in [7] (Theorems A , B and C ) .W e sta te o u r main r e s u lts o f t h e p re se n t article in § 2. T h e rem aining § § 3 and 4 are devoted to the proof o f Theorems.T h e au thors w ould like to express their sincere gratitude to P rofessors M. K im ura a n d T .Shiga for their valuable discussions and comments.