Spreading Speed and Traveling Wave Solutions of a Partially Sedentary Population

In this paper, we extend the population genetics model of [5] to the case where a fraction of the population does not migrate after the selection process. Mathematically, we study the asymptotic behavior of solutions to the recursion un+1 = Qg[un] where Qg[u](x) = (1− g) ∫ R K(x− y)f(u(y))dy + gf(u(x)), 0 ≤ g ≤ 1 . In the above definition of Qg, 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 ξ ∈ R, there exists c∗g(ξ) which has an explicit formula and is the spreading speed of Qg in the direction ξ. We also show that for each c ≥ c∗g(ξ), there exists a traveling wave solution in the direction ξ which is continuous if gf ′(0) ≤ 1.