A Nonlinear Integral Operator Arising from a Model in Population Genetics I. Monotone Initial Data

We study the asymptotic behavior of the solutions to the recursion un + 1 (x) = Q[uₙ ](x) for n ≥ 0. Here Q[u](x) = (K * g ∘ u)(x) acts on functions bounded between 0 and 1, $K(x)$ is a probability density function with compact support, and g(u) ∈ C¹ [0,1] satisfies certain additional assumptions. It is known that there exists a c_ + ^ * such that for c ≥ c_ + ^ *, there are nonincreasing travelling waves wc (x) facing right. We prove here that if $K(x)$ is the exponential of a concave function and u₀ (x) is monotone nonincreasing and decays to zero rapidly enough, then uₙ (x) converges in a certain sense to wc_ + ^ * (x) as n approaches infinity uniformly in R.