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Structured-Population Models in Marine, Terrestrial, and Freshwater Systems pp 19–58Cite as

Matrix Methods for Population Analysis

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Part of the Population and Community Biology Series book series (PCBS,volume 18)

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).

Keywords

  • Matrix Model
  • Hopf Bifurcation
  • Vital Rate
  • Killer Whale
  • Stable Fixed Point

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature Cited

  • Beddington, J. 1974. Age distribution and the stability of simple discrete time population models. Journal of Theoretical Biology 47: 65–74.

    PubMed  CrossRef  CAS  Google Scholar 

  • Brault, S., and H. Caswell. 1993. Pod-specific demography of killer whales (Orcinus orca). Ecology 74: 1444–1454.

    CrossRef  Google Scholar 

  • Caswell, H. 1986. Life cycle models for plants. Lectures on Mathematics in the Life Sciences 18: 171–233.

    Google Scholar 

  • Caswell, H. 1989a. Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer, Sunderland, Mass.

    Google Scholar 

  • Caswell, H. 1989b. The analysis of life table response experiments. I. Decomposition of treatment effects on population growth rate. Ecological Modelling 46: 221–237.

    CrossRef  CAS  Google Scholar 

  • Caswell, H. 1989c. Life history strategies. Pp. 285–308 in J. M. Cherrett, ed., Ecological Concepts. Blackwell, Oxford, Engl.

    Google Scholar 

  • 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.

    Google Scholar 

  • Caswell, H.. 1996b. Second derivatives of population growth rate: calculation and applications. Ecology 77: 870–879.

    CrossRef  Google Scholar 

  • Caswell, H.. In press. Analysis of life table response experiments. II. Alternative parameterizations for size-and stage-structured models. Ecological Modelling.

    Google Scholar 

  • Caswell, H. and M. C. Trevisan. 1994. The sensitivity analysis of periodic matrix models. Ecology 75: 1299–1303.

    CrossRef  Google Scholar 

  • Chapman, F.M. 1932. Handbook of Birds of Eastern North America. 2nd, rev. ed. D. Applet on, New York.

    Google Scholar 

  • Cushing, J. M. 1988. Nonlinear matrix models and population dynamics. Natural Resource Modeling 2: 539–580.

    Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • Getz, W. M., and R. G. Haight. 1989. Population Harvesting: Demographic Models of Fish, Forest, and Animal Resources. Princeton University Press, Princeton, N.J.

    Google Scholar 

  • Goodman, L. A. 1967. On the reconciliation of mathematical theories of population growth. Journal of the Royal Statistical Society A 130: 541–553.

    CrossRef  Google Scholar 

  • Guckenheimer, J., G. Oster, and A. Ipaktchi. 1977. The dynamics of density dependent population models. Journal of Mathematical Biology 4: 101–147.

    CrossRef  Google Scholar 

  • Hale, J., and H. Koçak. 1991. Dynamics and Bifurcations. Springer-Verlag, New York.

    CrossRef  Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • Horn, R. A., and C. A. Johnson. 1985. Matrix Analysis. Cambridge University Press.

    Google Scholar 

  • Keyfitz, N. 1967. Reconciliation of population models: Matrix, integral equation and partial fraction. Journal of the Royal Statistical Society A 130: 61–83.

    CrossRef  Google Scholar 

  • Keyfitz, N. 1968. Introduction to the mathematics of population. Addison-Wesley, Reading, Mass.

    Google Scholar 

  • Lefkovitch, L. P. 1965. The study of population growth in organisms grouped by stages. Biometrics 21: 1–18.

    CrossRef  Google Scholar 

  • Leslie, P. H. 1945. On the use of matrices in certain population mathematics. Biometrika 33: 183–212.

    PubMed  CrossRef  CAS  Google Scholar 

  • Leslie, P. H. 1948. Some further notes on the use of matrices in population mathematics. Biometrika 35: 213–245.

    Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • Logofet, D.O. 1993. Matrices and Graphs: Stability Problems in Mathematical Ecology. CRC Press, Boca Raton, Fla.

    Google Scholar 

  • May, R. M. 1979. Bifurcations and dynamic complexity in ecological systems. Annals of the New York Academy of Sciences 316: 517–529.

    CrossRef  Google Scholar 

  • McDonald, D. B., and H. Caswell. 1993. Matrix methods for avian demography. Current Ornithology 10: 139–185.

    CrossRef  Google Scholar 

  • Mesterton-Gibbons, M. 1993. Why demographic elasticities sum to one: A postscript to de Kroon et al. Ecology 74: 2467–2468.

    CrossRef  Google Scholar 

  • Neubert, M. G., and M. Kot. 1992. The subcritical collapse of predator populations in discrete-time predator-prey models. Mathematical Biosciences 110: 45–66.

    PubMed  CrossRef  CAS  Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • Seneta, E. 1981. Non-Negative Matrices and Markov Chains. 2nd ed. Springer-Verlag, New York.

    Google Scholar 

  • Silva, J. A. L., and T. G. Hallam. 1992. Compensation and stability in nonlinear matrix models. Mathematical Biosciences 110: 67–101.

    PubMed  CrossRef  CAS  Google Scholar 

  • 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.

    CrossRef  Google Scholar 

  • Whit ley, D. 1983. Discrete dynamical systems in dimensions one and two. Bulletin of the London Mathematical Society 15: 177–217.

    CrossRef  Google Scholar 

  • Wiggins, S. 1990. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York.

    Google Scholar 

  • Wikan, A., and E. Mjolhus. 1995. Periodicity of 4 in age-structured population models with density-dependence. Journal of Theoretical Biology 173: 109–119.

    CrossRef  Google Scholar 

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Authors
  1. Hal Caswell
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Editors and Affiliations

  1. President, Mountain View Research, Los Altos, California, USA

    Shripad Tuljapurkar

  2. Biology Department, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, USA

    Hal Caswell

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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

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  • DOI: https://doi.org/10.1007/978-1-4615-5973-3_2

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