The spectral radius of homogeneous order-preserving maps and the persistence of dispersing two-sex populations (partially joint work with Wen Jin)
- Horst Thieme (Arizona State University, USA)
In discrete-time population models, the year to year development of the population is described by a map. Usually, the spectral radius of its derivative at the origin (the extinction state) acts as a threshold parameter that separates population extinction from population persistence. If it is taken into account that many populations have males and females that need to mate in order to reproduce, the first order approximation of the map at the origin is not longer additive but just homogeneous. A spectral radius can also be defined for homogeneous order-preserving maps, and it acts as a threshold parameter in a completely analogous way. Establishing population persistence relies on finding conditions for the spectral radius to be a lower eigenvalue with a positive lower eigenvector. As in the linear case, the spectral radius can be characterized and approximated by Collatz-Wielandt numbers and bounds, but different techniques are required. As motivation and illustration, a two-sex population is considered with very short reproductive seasons and diffusion of both males and females (impulsive PDE).