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Spreading speed and travelling wave solutions of a partially sedentary population. (English) Zbl 1128.92026
Summary: We extend the population genetics model of H. F. Weinberger [Asymptotic behavior of a model in population genetics. Lect. Notes Math. 648, 47–96 (1978; Zbl 0383.35034)] to the case where a fraction of the population does not migrate after the selection process. Mathematically, we study the asymptotic behaviour of solutions to the recursion \(u_{n+1} = Q_g[u_n]\), where
\[ Q_g[u](x)=(1-g)\int_{\mathbb R^d}K(x-y)f(u(y))\,dy+gf(u(x)),\quad 0\leq g\leq 1. \]
In the above definition of \(Q_g\), \(K\) is a probability density function and \(f\) behaves qualitatively like the Beverton-Holt function. Under some appropriate conditions on \(K\) and \(f\), we show that for each unit vector \(\xi\in \mathbb R^d\), there exists a \(c^*_g(\xi)\) which has an explicit formula and is the spreading speed of \(Q_g\) in the direction \(\xi\). We also show that for each \(c\geq c^*_g(\xi)\), there exists a travelling wave solution in the direction \(\xi\) which is continuous if \(gf '(0)\leq 1\).

MSC:
92D15 Problems related to evolution
92D40 Ecology
35B40 Asymptotic behavior of solutions to PDEs
60E99 Distribution theory
35K99 Parabolic equations and parabolic systems
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D10 Genetics and epigenetics
47N60 Applications of operator theory in chemistry and life sciences
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