Spreading speed and travelling wave solutions of a partially sedentary population

摘要

In this paper, we extend the population genetics model of Weinberger (1978, Asymptotic behavior of a model in population genetics. Nonlinear Partial Differential Equations and Applications (J. Chadam ed.). Lecture Notes in Mathematics, vol. 648. New York: Springer, pp. 47–98.) 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], whereIn 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 ξ ∈ Rd, there exists a c*_g(ξ) which has an explicit formula and is the spreading speed of Q_g in the direction ξ. We also show that for each c ≥ c*_g(ξ), there exists a travelling wave solution in the direction ξ which is continuous if gf ′(0) ≤ 1.