×

Entropy and convergence in dynamics and demography. (English) Zbl 0782.92014

The paper starts with a review of some results by S. Goldstein and O. Penrose [J. Stat. Phys. 24, 325-342 (1981; Zbl 0516.70021)] on the relation between convergence rate, Kullback information, and Kolmogorov-Sinai entropy for irreducible, aperiodic finite Markov chains (including the effects of time-scaling and of coarse-graining of the state space). Elementary proofs of such results are provided.
The transformation of the standard demographic Leslie projection model into a Markov chain is reviewed. An expression for the entropy in terms of the eigenvalues of the Leslie matrix is obtained. When there is a stable age distribution, the demographic meaning of the entropy is discussed in connection with the convergence rate. Finally, imprimitive and periodic limits and the relation to population entropy are discussed.

MSC:

92D25 Population dynamics (general)
91D20 Mathematical geography and demography
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)

Citations:

Zbl 0516.70021
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Artzrouni, M.: The rate of convergence of a generalized stable population. J. Math. Biol. 24, 405-422 (1986) · Zbl 0609.92031 · doi:10.1007/BF01236889
[2] Auger, P.: Stability of interacting populations with class-age distributions. J. Theor. Biol. 112, 595-605 (1985). · doi:10.1016/S0022-5193(85)80025-4
[3] Caswell, H.: Matrix population models. Sunderland, MA: Sinauer Associates 1989
[4] Chirikov, B. V.: A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263-379 (1979) · doi:10.1016/0370-1573(79)90023-1
[5] Coale, A. J.: The growth and structure of human populations: A mathematical investigation. Princeton: Princeton University Press 1972
[6] Crutchfield, J. P., Packard, N. H.: Symbolic dynamics of noisy chaos. Physica 7D, 201-223 (1983)
[7] Curry, J. H.: On computing the entropy of the Henon attractor. J. Stat. Phys. 26, 683-695 (1981) · Zbl 0512.28013 · doi:10.1007/BF01010933
[8] Demetrius, L.: Demographic parameters and natural selection. Proc. Natl. Acad. Sci., USA 71, 4645-4647 (1974) · Zbl 0303.62075 · doi:10.1073/pnas.71.12.4645
[9] Demetrius, L.: Relations between demographic parameters. Demography 16, 329-338 (1979) · doi:10.2307/2061146
[10] Demetrius, L., Schuster, P., Sigmund, K.: Polynucleotide evolution and branching processes. Bull. Math. Biol. 47, 239-262 (1985). · Zbl 0576.92020
[11] Farmer, D., Crutchfield, J., Freehling, H., Packard, N., Shaw, R.: Power spectra and mixing properties of strange attractors. Ann. N.Y. Acad. Sci. 357, 453-472 (1980) · Zbl 0475.58011 · doi:10.1111/j.1749-6632.1980.tb29710.x
[12] Fill, J. A.: Eigenvalue bounds on convergence to stationarity for nonreversible Markov Chains, with an application to the Exclusion problem. Ann. Appl. Probab. 1, 62-87 (1991) · Zbl 0726.60069 · doi:10.1214/aoap/1177005981
[13] Fraser, A. M.: Information and entropy in strange attractors. IEEE Trans. Inf. Theory. IT-35 (1989) · Zbl 0712.58038
[14] Fraser, A. M., Swinney, H. L.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134-1140 (1986) · Zbl 1184.37027 · doi:10.1103/PhysRevA.33.1134
[15] Goldstein, S.: Entropy increase in dynamical systems. Isr. J. Math. 38, 241-256 (1981) · Zbl 0463.28014 · doi:10.1007/BF02760809
[16] Goldstein, S., Penrose, O.: A non-equilibrium entropy for dynamical systems. J. Stat. Phys. 24, 325-343 (1981) · Zbl 0516.70021 · doi:10.1007/BF01013304
[17] Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Berlin Heidelberg New York: Springer 1983 · Zbl 0515.34001
[18] Hamilton, I., Brumer, P.: Relaxation rates for two dimensional deterministic mappings. Phys. Rev. A 25, 3457-3459 (1982) · doi:10.1103/PhysRevA.25.3457
[19] Hamilton, I., Brumer, P.: Intramolecular relaxation in N = 2 Hamiltonian systems: the role of the K entropy. J. Chem. Phys. 78, 2682-2690 (1983) · doi:10.1063/1.445027
[20] Keyfitz, N.: Introduction to the mathematics of populations. Reading, MA: Addison Wesley 1968
[21] Keyfitz, N.: Applied mathematical demography. Berlin Heidelberg New York: Springer 1985 · Zbl 0597.92018
[22] Kim, Y. J.: On the speed of convergence to stability. (Unpublished manuscript, 1991)
[23] Krieger, W.: On entropy and generators of measure preserving transformations. Trans. Am. Math. Soc. 199, 453-464 (1970). · Zbl 0204.07904 · doi:10.1090/S0002-9947-1970-0259068-3
[24] Marcus, M., Mine, H.: A Survey of Matrix Theory and Matrix Inequalities. Rockleigh, NH: Allyn and Bacon 1964
[25] Ornstein, D.: Ergodic theory, Randomness, and Dynamical Systems. New Haven: Yale University Press 1974 · Zbl 0296.28016
[26] Penrose, O.: Entropy and irreversibility. Ann. N.Y. Acad. Sci. 373, 211-219 (1981) · doi:10.1111/j.1749-6632.1981.tb51149.x
[27] Petersen, K.: Ergodic theory. Cambridge: Cambridge University Press 1983 · Zbl 0507.28010
[28] Pollard, J. H.: Mathematical models for the growth of human populations. Cambridge: Cambridge University Press 1973 · Zbl 0295.92013
[29] Schlögl, F.: Mixing distance and stability of steady states in statistical nonlinear thermodynamics. Z. Phys. B 25, 411-421 (1976). · doi:10.1007/BF01315257
[30] Schoen, R., Kim, Y.: Movement toward stability is a fundamental principle of population dynamics. Paper presented at Population Association of America annual meeting. Washington, D.C.: 1991
[31] Seneta, E.: Entropy and martingales in Markov Chain models. J. Appl. Probab. 19A, 367-381 (1982). · Zbl 0492.60080 · doi:10.2307/3213576
[32] Shaw, R.: Strange attractors, chaotic behavior, and information flow. Z. Naturforsch. 36a, 80-112 (1981) · Zbl 0599.58033
[33] Shaw, R. S.: The dripping faucet as a model of chaotic system. Santa Cruz, CA: Aerial Press 1985 · Zbl 0842.58059
[34] Sinai, Ya. G.: A weak isomorphism of transformations having an invariant measure. Sov. Math. Dokl. 3, 1725-1729 (1962). · Zbl 0205.13501
[35] Tuljapurkar, S. D.: Why use population entropy? It determines the rate of convergence. J. Math. Biol. 13, 325-337 (1982). · Zbl 0478.92011 · doi:10.1007/BF00276067
[36] Wachter, K. W.: Lotka’s roots under rescalings. Proc. Natl. Acad. Sci., USA 81, 3600-3604 (1984) · Zbl 0568.92014 · doi:10.1073/pnas.81.11.3600
[37] Wightman, A. S.: Statistical mechanisms and ergodic theory. In: Cohen, E. G. D. (ed.) Statistical mechanics at the turn of the decade, pp. 1-32. New York: Marcel Dekker 1971
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.