Monotone dependence of the spectral bound on the transition rates in linear compartment models

For linear compartment models or Leslie-type staged population models with quasi-positive matrix the spectral bound of the matrix (the eigenvalue determining stability) is studied in the situation where particles or individuals leave a compartment or stage with some rate and enter another with the same rate. Then the matrix carries the rate with a positive sign in some off-diagonal entry and with a negative sign in the corresponding diagonal entry. Hence the matrix does not depend on the rate in a monotone way. It is shown, however, that the spectral bound is a monotone function of the rate. It is all the time strictly increasing or strictly decreasing or it is constant. A simple algebraic criterion distinguishes between the three cases. The results can be applied to linear systems and to the stability of stationary states in non-linear systems, in particular to models for the transmission of infectious diseases, and in population dynamics.     

Existence and stability of local excitations in homogeneous neural fields

Summary Dynamics of excitation patterns is studied in one-dimensional homogeneous lateral-inhibition type neural fields. The existence of a local excitation pattern solution as well as its waveform stability is proved by the use of the Schauder fixed-point theorem and a generalized version of the Perron-Frobenius theorem of positive matrices to the function space. The dynamics of the field is in general multi-stable so that the field can keep short-term memory.     

Instability of non-constant equilibrium solutions of a system of competition-diffusion equations

The system of interaction-diffusion equations describing competition between two species is investigated. By using a version of the Perron-Frobenius theorem of positive matrices generalized to function spaces, it is proved that any non-constant equilibrium solution of the system is unstable both under Neumann boundary conditions (for the rectangular parallelepiped domain) and under periodic conditions. It is conjectured that this result extends to convex domains, and that the simple interaction-diffusion model cannot explain spatially segregated distributions of two competing species in such domains.