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A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case. (English) Zbl 0585.45001
[For parts I and II see ibid. 13, 913–937, 938–953 (1982; Zbl 0508.45006, Zbl 0508.45007)]. The author continues his series of papers concerning the asymptotic behavior of the solutions to the recursion \(u_{n+1}=Q[u_ n]\) for \(n\geq 0\). Here \(Q[u](x)=\int k(x-y)g(u(y))\,dy\), \(K(x)\geq 0\), \(\int K(x)\,dx=1\). This model proposed by H. F. Weinberger [Lect. Notes Math. 648, 47–96 (1978; Zbl 0383.35034)] to describe the spread of advantageous genes, is similar to the model of R. A. Fisher [The advance of advantageous genes, Ann. Eugenics 7, 355–369 (1937; JFM 63.1111.04)]. The gene fraction is given by \(g(u)=\frac{su^ 2+u}{1+su^ 2+\sigma (1-u)^ 2}\).
Reviewer: L. A. Sakhnovich

MSC:
45G10 Other nonlinear integral equations
92D10 Genetics and epigenetics
45M05 Asymptotics of solutions to integral equations
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