### 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