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

K. P. Hadeler, Horst Thieme

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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 languageEnglish (US)
Pages (from-to)697-712
Number of pages16
JournalJournal Of Mathematical Biology
Issue number5
StatePublished - Nov 1 2008


  • 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


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