The Net Reproductive Value and Stability in Matrix Population Models

The net reproductive value n is defined for a general discrete linear population model with a non‐negative projection matrix. This number is shown to have the biological interpretation of the expected number of offspring per individual over its life time. The main result relates n to the population's growth rate (i.e. the dominant eigenvalue λ of the projection matrix) and shows that the stability of the extinction state (the trivial equilibrium) can be determined by whether n is less than or greater than 1. Examples are given to show that explicit algebraic formulas for n are often derivable, and hence available for both numerical and parameter studies of stability, when no such formulas for λ are available.