Spreading speeds in reducible multitype branching random walk

TL;DR

This paper establishes conditions and formulas for the spreading speed of the rightmost particle in reducible multitype branching random walks, extending previous results from irreducible cases and linking to deterministic population models.

Contribution

It provides the first explicit conditions and formulas for the spreading speed in reducible multitype branching random walks, generalizing prior irreducible case results.

Findings

Derived conditions for the existence of a limiting speed.

Obtained a formula for the asymptotic speed in terms of reproduction laws.

Analyzed the number of particles to the right of a threshold as n grows.

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 nth 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.