Sensitivity of Evolutionary Entropy and Reproductive Potential in Lefkovitch and Open-Group Leslie Matrices

We develop a sensitivity theory for the Malthusian parameter $r$ , evolutionary entropy $H$ , and reproductive potential $\Phi$ in Lefkovitch matrices, with open-group Leslie matrices as a special case. Starting from the dominant eigenvalue and the Markov chain associated with the projection matrix, we derive explicit formulas for the long-run stage distribution, generation time, evolutionary entropy, and reproductive potential. Evolutionary entropy and reproductive potential both admit exact transition-retention decompositions, with distinct interpretations: for $H$ the separation concerns sources of demographic uncertainty, whereas for $\Phi$ it partitions the demographic links themselves. For small changes to positive coefficients, we obtain closed-form sensitivities for fertility, progression, and retention. The sensitivity of $r$ follows from the standard formula for a simple dominant eigenvalue, while that of $\Phi$ follows from the exact identity $r=H+\Phi .$ The formulas specialise to open-group Leslie matrices, in which retention occurs only in the terminal stage, and reduce to the classical Leslie theory when retention vanishes. We also distinguish a change to an existing demographic link from the creation of a fertility or retention link whose coefficient was zero. When a new link is created, $r$ still has a finite one-sided derivative, whereas the leading changes in $H$ and $\Phi$ are opposite terms of order $\epsilon log\epsilon$ . Empirical examples show that growth, evolutionary entropy, and reproductive potential can respond differently to the same demographic change. They also show that stage retention may account for most of $H$ without determining whether $H$ rises or falls when retention is increased.