Projection matrices revisited: a potential-growth indicator and the merit of indication

The mathematics of matrix models for age- or/and stage-structured population dynamics substantiates the use of the dominant eigenvalue $\lambda_1$ of the projection matrix $\boldsymbol L$ as a measure of the growth potential, or of adaptation, for a given species population in modern plant or animal demography. The calibration of $\boldsymbol L=\boldsymbol T+\boldsymbol F$ on the “identified-individuals-of-unknown-parents” kind of empirical data determines precisely the transition matrix $\boldsymbol T$, but admits arbitrariness in the estimation of the fertility matrix $\boldsymbol F$. We propose an adaptation principle that reduces calibration to the maximization of $\lambda_1(\boldsymbol L)$ under the fixed $\boldsymbol T$ and constraints on $\boldsymbol F$ ensuing from the data and expert knowledge. A theorem has been proved on the existence and uniqueness of the maximizing solution for projection matrices of a general pattern. A conjugated maximization problem for a “potential-growth indicator” under the same constraints has appeared to be a linear-programming problem with a ready solution, the solution testing whether the data and knowledge are compatible with the population growth observed.