Abstract
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.
Original language | English (US) |
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Pages (from-to) | 697-712 |
Number of pages | 16 |
Journal | Journal Of Mathematical Biology |
Volume | 57 |
Issue number | 5 |
DOIs | |
State | Published - Nov 1 2008 |
Keywords
- Adaptive dynamics
- Basic reproduction number
- Bifurcation of stationary points
- Epidemics
- Invasion thresholds
- Irreducible matrix
- Perron-Frobenius
- Quasipositive matrix
ASJC Scopus subject areas
- Modeling and Simulation
- Agricultural and Biological Sciences (miscellaneous)
- Applied Mathematics