Spreading speeds in reducible multitype branching random walk

Abstract

This paper gives conditions for the rightmost particle in the $n$th generation of a multitype branching random walk to have a speed, in the sense that its location divided by $n$ converges to a constant as $n$ goes to infinity. Furthermore, a formula for the speed is obtained in terms of the reproduction laws. The case where the collection of types is irreducible was treated long ago. In addition, the asymptotic behavior of the number in the $n$th generation to the right of $na$ is obtained. The initial motive for considering the reducible case was results for a deterministic spatial population model with several types of individual discussed by Weinberger, Lewis and Li [J. Math. Biol. 55 (2007) 207–222]: the speed identified here for the branching random walk corresponds to an upper bound for the speed identified there for the deterministic model.