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- W2079661452 abstract "We study the asymptotic behavior of the solutions to the recursion $u_{n + 1} (x) = Q[u_n ](x)$ for $n geq 0$. Here $Q[u](x) = (K * g circ u)(x)$ acts on functions bounded between 0 and 1, $K(x)$ is a probability density function with compact support and $g(u) in C^1 [0,1]$ satisfies certain additional assumptions. It is known that there exist $ - c_ - ^ * < c_ + ^ * $ such that for $c geq c_ + ^ * $, there are nonincreasing travelling waves $w_c (x)$ facing right, while for $c leq - c_ - ^ * $, there are nondecreasing travelling waves $bar w_c (x)$ facing left. We prove here that if $K(x)$ has certain variation diminishing property, ${{g(u)} / u}$ is nonincreasing and $u_0 (x)$ has compact support, then $u_n (x)$ develops uniformly in x, as n approaches infinity, into a pair of diverging waves, $w_{c_ + ^ * } (x)$ on the right and $bar w_{c_ + ^ * } (x)$ on the left. This paper is a sequel to [R. Lui, SIAM J. Math. Anal., 13 (1982), pp. 913–937], which proved convergence to a single travelling wave when $u_0 ( - infty ) > 0$." @default.
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- W2079661452 date "1982-11-01" @default.
- W2079661452 modified "2023-10-18" @default.
- W2079661452 title "A Nonlinear Integral Operator Arising from a Model in Population Genetics II. Initial Data with Compact Support" @default.
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- W2079661452 doi "https://doi.org/10.1137/0513065" @default.
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