Eigenfunctionals of Homogeneous Order-Preserving Maps with Applications to Sexually Reproducing Populations

Homogeneous bounded maps B on cones \(X_+\) of ordered normed vector spaces X allow the definition of a cone spectral radius which is analogous to the spectral radius of a bounded linear operator. If \(X_+\) is complete and B is also order-preserving, conditions are derived for B to have a homogeneous order-preserving eigenfunctional \(\theta : X_+ \rightarrow { \mathbb {R}}_+\) associated with the cone spectral radius in analogy to one part of the Krein–Rutman theorem. Since homogeneous B arise as first order approximations at 0 of maps that describe the year-to-year development of sexually reproducing populations, these eigenfunctionals are an important ingredient in the persistence theory of structured populations with mating.