Continuity and approximability of competitive spectral radii

TL;DR

This paper introduces the competitive spectral radius, extending joint spectral radius to a two-player setting, and demonstrates its continuity, approximability, and applications in population dynamics.

Contribution

It establishes the continuity and Lipschitz continuity of the competitive spectral radius for cone-preserving operators and provides a method for its approximation.

Findings

Competitive spectral radius depends continuously on matrix sets.

It can be approximated arbitrarily accurately.

Application demonstrated in age-structured population dynamics.

Abstract

The competitive spectral radius extends the notion of joint spectral radius to the two-player case: two players alternatively select matrices in prescribed compact sets, resulting in an infinite matrix product; one player wishes to maximize the growth rate of this product, whereas the other player wishes to minimize it. We show that when the matrices represent linear operators preserving a cone and satisfying a "strict positivity" assumption, the competitive spectral radius depends continuously - and even in a Lipschitz-continuous way - on the matrix sets. Moreover, we show that the competive spectral radius can be approximated up to any accuracy. This relies on the solution of a discretized infinite dimensional non-linear eigenproblem. We illustrate the approach with an example of age-structured population dynamics.