[PDF][PDF] On the bounded solutions of a nonlinear convolution equation
O Diekmann, HG Kaper - Nonlinear Anal, 1978 - ir.cwi.nl
Nonlinear Anal, 1978•ir.cwi.nl
CERTAIN deterministic models for the spatial spread of an epidemic or an advantageous
gene among a population along a line can be analyzed in terms of the nonlinear convolution
equation u (x)=(go u)* k (x) XER,(1.1) where go u is the composite of g and u, go u (x)= g (u
(x)), and* denotes convolution,(</>* if/(x)= JR< f>(xy) lf;(y) dy. In the epidemic model, g
typically has the form g (x)= a (l-ex), a some constant (o:> 1), cf. Diekmann [1], while in the
genetic model g is given by g (x)=[o: x2+/h (lx)]/[o: x2+ 2/3x (lx)+ y (lx) 2], o:,{3, and y positive …
gene among a population along a line can be analyzed in terms of the nonlinear convolution
equation u (x)=(go u)* k (x) XER,(1.1) where go u is the composite of g and u, go u (x)= g (u
(x)), and* denotes convolution,(</>* if/(x)= JR< f>(xy) lf;(y) dy. In the epidemic model, g
typically has the form g (x)= a (l-ex), a some constant (o:> 1), cf. Diekmann [1], while in the
genetic model g is given by g (x)=[o: x2+/h (lx)]/[o: x2+ 2/3x (lx)+ y (lx) 2], o:,{3, and y positive …
CERTAIN deterministic models for the spatial spread of an epidemic or an advantageous gene among a population along a line can be analyzed in terms of the nonlinear convolution equation u (x)=(go u)* k (x) XER,(1.1) where go u is the composite of g and u, go u (x)= g (u (x)), and* denotes convolution,(* if/(x)= JR< f>(x-y) lf;(y) dy. In the epidemic model, g typically has the form g (x)= a (l-ex), a some constant (o:> 1), cf. Diekmann [1], while in the genetic model g is given by g (x)=[o: x2+/h (l-x)]/[o: x2+ 2/3x (l-x)+ y (l-x) 2], o:,{3, and y positive constants, cf. Weinberger [2]. In both cases, k is a nonnegative function, normalized such that Jak (x) dx= 1. We consider (1.1) under the following hypotheses:
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