Abstract
Matrix models for structured populations were introduced by P. H. Leslie in the 1940’s (Leslie 1945, 1948). Although they are in some ways the simplest of the mathematical approaches to structured population modeling (see Chapter 1), their analysis requires computational power. For this reason, and because ecologists of the day viewed matrix algebra as an esoteric branch of advanced mathematics, they were largely neglected until the late 1960’s, when they were rediscovered by ecologists (Lefkovitch 1965) and human demographers (Goodman 1967; Keyfltz 1967). In the 1970’s, matrix models were adopted by plant ecologists, who discovered that they could easily handle the complexity of plant life cycles in which size or developmental stage was more important than chronological age in determining the fate of individuals (Sarukhán & Gadgil 1974; Hartshorn 1975; Werner & Caswell 1977).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Literature Cited
Beddington, J. 1974. Age distribution and the stability of simple discrete time population models. Journal of Theoretical Biology 47: 65–74.
Brault, S., and H. Caswell. 1993. Pod-specific demography of killer whales (Orcinus orca). Ecology 74: 1444–1454.
Caswell, H. 1986. Life cycle models for plants. Lectures on Mathematics in the Life Sciences 18: 171–233.
Caswell, H. 1989a. Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer, Sunderland, Mass.
Caswell, H. 1989b. The analysis of life table response experiments. I. Decomposition of treatment effects on population growth rate. Ecological Modelling 46: 221–237.
Caswell, H. 1989c. Life history strategies. Pp. 285–308 in J. M. Cherrett, ed., Ecological Concepts. Blackwell, Oxford, Engl.
Caswell, H. 1996a. Demography meets ecotoxicology: Untangling the population level effects of toxic substances. Pp. 255–292 in M. C. Newman and C. H. Jagoe, eds., Ecotoxicology: A Hierarchical Treatment. Lewis Publishers, Boca Raton, Fla.
Caswell, H.. 1996b. Second derivatives of population growth rate: calculation and applications. Ecology 77: 870–879.
Caswell, H.. In press. Analysis of life table response experiments. II. Alternative parameterizations for size-and stage-structured models. Ecological Modelling.
Caswell, H. and M. C. Trevisan. 1994. The sensitivity analysis of periodic matrix models. Ecology 75: 1299–1303.
Chapman, F.M. 1932. Handbook of Birds of Eastern North America. 2nd, rev. ed. D. Applet on, New York.
Cushing, J. M. 1988. Nonlinear matrix models and population dynamics. Natural Resource Modeling 2: 539–580.
DeAngelis, D. L., L. J. Svoboda, S. W. Christensen, and D. S. Vaughan. 1980. Stability and return times of Leslie matrices with density-dependent survival: Applications to fish populations. Ecological Modelling 8: 149–163.
Dennis, B., R. A. Desharnais, J. M. Cushing, and R. F. Costantino. 1995. Nonlinear demographic dynamics: Mathematical models, statistical methods, and biological experiments. Ecological Monographs 65: 261–281.
Getz, W. M., and R. G. Haight. 1989. Population Harvesting: Demographic Models of Fish, Forest, and Animal Resources. Princeton University Press, Princeton, N.J.
Goodman, L. A. 1967. On the reconciliation of mathematical theories of population growth. Journal of the Royal Statistical Society A 130: 541–553.
Guckenheimer, J., G. Oster, and A. Ipaktchi. 1977. The dynamics of density dependent population models. Journal of Mathematical Biology 4: 101–147.
Hale, J., and H. Koçak. 1991. Dynamics and Bifurcations. Springer-Verlag, New York.
Hartshorn, G. S. 1975. A matrix model of tree population dynamics. Pp. 41–51 in F. B. Golley and E. Medina, eds., Tropical Ecological Systems. Springer-Verlag, New York.
Horn, R. A., and C. A. Johnson. 1985. Matrix Analysis. Cambridge University Press.
Keyfitz, N. 1967. Reconciliation of population models: Matrix, integral equation and partial fraction. Journal of the Royal Statistical Society A 130: 61–83.
Keyfitz, N. 1968. Introduction to the mathematics of population. Addison-Wesley, Reading, Mass.
Lefkovitch, L. P. 1965. The study of population growth in organisms grouped by stages. Biometrics 21: 1–18.
Leslie, P. H. 1945. On the use of matrices in certain population mathematics. Biometrika 33: 183–212.
Leslie, P. H. 1948. Some further notes on the use of matrices in population mathematics. Biometrika 35: 213–245.
Levin, S. A. 1981. Age-structure and stability in multiple-age spawning populations. Pp. 21–45 in T. L. Vincent and J. M. Skowronski, eds., Renewable Resource Management. Springer-Verlag, Heidelberg.
Logofet, D.O. 1993. Matrices and Graphs: Stability Problems in Mathematical Ecology. CRC Press, Boca Raton, Fla.
May, R. M. 1979. Bifurcations and dynamic complexity in ecological systems. Annals of the New York Academy of Sciences 316: 517–529.
McDonald, D. B., and H. Caswell. 1993. Matrix methods for avian demography. Current Ornithology 10: 139–185.
Mesterton-Gibbons, M. 1993. Why demographic elasticities sum to one: A postscript to de Kroon et al. Ecology 74: 2467–2468.
Neubert, M. G., and M. Kot. 1992. The subcritical collapse of predator populations in discrete-time predator-prey models. Mathematical Biosciences 110: 45–66.
Sarukhan, J., and M. Gadgil. 1974. Studies on plant demography: Ranunculus repens L., R. bulbosus L. and R. acris L. III. A mathematical model incorporating multiple modes of reproduction. Journal of Ecology 62: 921–936.
Seneta, E. 1981. Non-Negative Matrices and Markov Chains. 2nd ed. Springer-Verlag, New York.
Silva, J. A. L., and T. G. Hallam. 1992. Compensation and stability in nonlinear matrix models. Mathematical Biosciences 110: 67–101.
Werner, P. A., and H. Caswell. 1977. Population growth rates and age versus stage-distribution models for teasel (Dipsacus sylvestris Huds.). Ecology 58: 1103–1111.
Whit ley, D. 1983. Discrete dynamical systems in dimensions one and two. Bulletin of the London Mathematical Society 15: 177–217.
Wiggins, S. 1990. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York.
Wikan, A., and E. Mjolhus. 1995. Periodicity of 4 in age-structured population models with density-dependence. Journal of Theoretical Biology 173: 109–119.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Caswell, H. (1997). Matrix Methods for Population Analysis. In: Tuljapurkar, S., Caswell, H. (eds) Structured-Population Models in Marine, Terrestrial, and Freshwater Systems. Population and Community Biology Series, vol 18. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5973-3_2
Download citation
DOI: https://doi.org/10.1007/978-1-4615-5973-3_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-412-07271-0
Online ISBN: 978-1-4615-5973-3
eBook Packages: Springer Book Archive