Sensitivity of evolutionary entropy in Lefkovitch matrices

Evolutionary entropy, introduced by Demetrius, is a demographic invariant that quantifies the temporal organization of structured populations. Explicit sensitivity expressions for this quantity were derived by Demetrius, Gundlach and Ziehe for age-structured Leslie matrices, establishing the foundations of entropy-based perturbation theory.
In this paper we develop a complete sensitivity theory for evolutionary entropy in irreducible Lefkovitch matrices. Using the Perron--Frobenius representation of the associated Markov chain, we derive explicit closed-form expressions for the stationary distribution, generation time, evolutionary entropy and its partial derivatives with respect to fertility, transition and retention parameters. The resulting identities are expressed directly in terms of demographic coefficients, Perron eigenvectors, the dominant eigenvalue and the reproductive potential.
The entropy representation obtained here gives a natural decomposition into transition and retention components and clarifies the distinct mechanisms through which demographic uncertainty is generated in stage-structured populations. We further show that the theory specializes immediately to open-group Leslie matrices, a class that has been shown to comprise a large fraction of empirical demographic models.
The results extend the entropy sensitivity theory of Demetrius--Gundlach--Ziehe from age-structured to general stage-structured populations and provide practical tools for comparative demographic analysis, perturbation studies, demographic robustness, and the investigation of life-history strategies. Several biological examples are presented, illustrating how entropy decomposition and sensitivity analysis reveal complementary aspects of population organization.