An extinction/persistence threshold for sexually rep roducing populations: The cone spectral radius

Persistence and local stability of the extinction state are studied for discrete-time population models x n = F(x_n-1), n ∈ N, with a map F on the cone X₊ of an ordered normed vector space X. Since sexual reproduction is accounted for, the first order approximation of F at 0 is an order-preserving homogeneous map B on X₊ that is not additive. The cone spectral radius of B acts as the threshold parameter that separates persistence from local stability of 0, the extinction state. An important ingredient of the persistence theory for the induced semiflow is a homogeneous order-preserving eigenfunctional θ: X+ → R+ of B that is associated with the cone spectral radius and interacts with an appropriate persistence function σ. Applications are presented for spatially distributed or rank-structured populations that reproduce sexually.