Sensitivity of evolutionary entropy and reproductive potential in Lefkovitch and open-group Leslie matrices

Abstract

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 \(\varepsilon\log\varepsilon\). 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.