Inequalities on the spectral abscissa for matrices arising in a stage-structured population model

The spectral abscissa of a convex combination of two matrices of special form is compared with the convex combination of their individual spectral bounds. The matrices are essentially nonnegative and both the special form and the form of the comparison arise from a linear stage-structured population model. By using M-matrix theory, the problem is converted to a comparison of the spectral radii of two related matrices. Theoretical results are derived for special cases of an arbitrary number of stage classes, and in particular confirm all previous numerical observations for the model with two stage classes.