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IMA Journal of Applied Mathematics Advance Access originally published online on October 5, 2007
IMA Journal of Applied Mathematics 2007 72(6):801-816; doi:10.1093/imamat/hxm025
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Spreading speed and travelling wave solutions of a partially sedentary population

Darko Volkov{dagger} and Roger Lui{ddagger}

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA

{dagger} Email: darko{at}wpi.edu

{ddagger} Corresponding author. Email: rlui{at}wpi.edu

Received on June 7, 2006; Accepted on May 8, 2007

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 un+1 = Qg[un], where

Formula
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 {xi} isin Rd, there exists a c*g({xi}) which has an explicit formula and is the spreading speed of Qg in the direction {xi}. We also show that for each c ≥ c*g({xi}), there exists a travelling wave solution in the direction {xi} which is continuous if gf '(0) ≤ 1.

Keywords: monostable; spreading speed; wave speed; travelling wave solutions; order preserving.


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