Entropy and convergence in dynamics and demography.

Demographic dynamics is formally equivalent to the dynamics of a Markov chain, as is true of some nonlinear dynamical systems. Convergence to demographic equilibrium can be studied in terms of convergence in the Markov chain. Tuljapurkar (1982) showed that population entropy (Kolmogorov-Sinai entropy) provides information on the rate of this convergence. This paper begins by considering finite state Markov chains, providing elementary proofs of the relationship between convergence rate and entropy, and discusses in detail the uses and limitations of entropy as a convergence measure; these results also apply to Markovian dynamical systems. Next, new qualitative and quantitative arguments are used to discuss the demographic meaning of entropy. An exact relationship is established giving population entropy in terms of the eigenvalues of the Leslie matrix characteristic equation. Finally, the significance of imprimitive and periodic limits is discussed in relation to population entropy.