Volume 18 - Article 3 | Pages 59–116  

Perturbation analysis of nonlinear matrix population models

By Hal Caswell

References

Almany, G.R., Berumen, M.L., Thorrold, S.R., Planes, S., and Jones, G.P. (2007). Local replenishment of coral reef fish populations in a marine reserve. Science 316(5825): 742-744.

Weblink:
Download reference:

Bartholomew, D.J. (1982). Stochastic models for social processes. New York, USA: Wiley.

Download reference:

Baudisch, A. (2005). Hamilton’s indicators of the force of selection. Proceedings of the National Academy of Sciences USA 102(23): 8263-8268.

Weblink:
Download reference:

Bonneuil, N. (1994). Special issue: nonlinear models in demography. Mathematical Population Studies 5(1).

Download reference:

Bullock, J.M., Silvertown, J., and Hill, B.C. (1996). Plant demographic responses to environmental variation: distinguishing between effects on age structure and effects on age-specific vital rates. Journal of Ecology 84(5): 733-743.

Weblink:
Download reference:

Carey, J.R. (2003). Longevity: The Biology and Demography of Life Span. Princeton, New Jersey, USA: Princeton University Press.

Download reference:

Carey, J.R., Liedo, P., and Vaupel, J.W. (1995). Mortality dynamics of density in the Mediterranean fruit fly. Experimental gerontology 30(6): 605-629.

Weblink:
Download reference:

Carey, J.R. and Vaupel, J.W. (2005). Biodemography. In: Poston, D.L. and Micklin, M. (eds.). Handbook of population. New York, USA: Springer: 625-658.

Download reference:

Caswell, H. (1978). A general formula for the sensitivity of population growth rate to changes in life history parameters. Theoretical Population Biology 14(2): 215-230.

Weblink:
Download reference:

Caswell, H. (2006). Applications of Markov chains in demography. In: Langville, A.N. and Stewart, W.J. (eds.). MAM2006: Markov Anniversary Meeting. North Carolina, USA: Boson Books, Raleigh: 319-334.

Download reference:

Caswell, H. (2007b). Evolutionary demography: the invasion exponent and the effective population density in nonlinear matrix models. In: Rooney, N., McCann, K.S., and Noakes, D.L.G. (eds.). From energetics to ecosystems: the dynamics and structure of ecological systems. Dortrecht, Netherlands: Springer: 237-256 .

Download reference:

Caswell, H. (2001). Matrix population models: construction, analysis, and interpretation. Sunderland, Massachusetts, USA: Sinauer Associates.

Download reference:

Caswell, H. (1982). Optimal life histories and the maximization of reproductive value: a general theorem for complex life cycles. Ecology 63(5): 1218-1222.

Weblink:
Download reference:

Caswell, H. (2007a). Sensitivity analysis of transient population dynamics. Ecology Letters 10(1): 1-15.

Weblink:
Download reference:

Caswell, H., Takada, T., and Hunter, C.M. (2004). Sensitivity analysis of equilibrium in density-dependent matrix population models. Ecology Letters 7(5): 380-387.

Weblink:
Download reference:

Caswell, H. and Weeks, D.E. (1986). Two-sex models: chaos, extinction and other dynamic consequences of sex. American Naturalist 128(5): 707-735.

Weblink:
Download reference:

Chu, C.Y.C. (1998). Population dynamics: a new economic approach. New York, USA: Oxford University Press.

Download reference:

Chung, R. (1994). Cycles in the two-sex problem: an investigation of a nonlinear demographic model. Mathematical Population Studies 5(1): 45-73.

Weblink:
Download reference:

Clutton-Brock, J.H., Illius, A.W., Wilson, K., Grenfell, B.T., MacColl, A.D.C., and Albon, S.D. (1997). Stability and instability in ungulate populations: an empirical analysis. American Naturalist 149(2): 195-219.

Weblink:
Download reference:

Coale, A.J. (1957). How the age distribution of a human population is determined. Cold Spring Harbor Symposia on Quantitative Biology 22: 83-90.

Download reference:

Coale, A.J. (1972). The growth and structure of human populations. New Jersey, USA: Princeton University Press.

Download reference:

Cohen, J.E. (1995). How many people can the earth support? New York, USA: W.W. Norton.

Download reference:

Costantino, R.F. and Desharnais, R.A. (1991). Population dynamics and the Tribolium model: genetics and demography. New York, USA: Springer-Verlag.

Download reference:

Cushing, J.M. (1998). An introduction to structured population dynamics. Philadelphia, Pennsylvania, USA: SIAM.

Download reference:

Cushing, J.M., Costantino, R.F., Dennis, B., Desharnais, R.A., and Henson, S.M. (2003). Chaos in ecology: experimental nonlinear dynamics. San Diego, California, USA: Academic Press.

Download reference:

Dennis, B., Desharnais, R.A., Cushing, J.M., and Costantino, R.F. (1995). Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments. Ecological Monographs 65(3): 261-281.

Weblink:
Download reference:

Easterlin, R. (1961). The American baby boom in historical perspective. American Economic Review 51(5): 869-911.

Download reference:

Ekamper, P. and Keilman, N. (1993). Sensitivity analysis in a multidimensional demographic ppojection model with a two-sex algorithm. Mathematical Population Studies 4(1): 21-36.

Weblink:
Download reference:

Emecki, M., Navarro, S., Donahaye, J.E., Rindner, M., and Azrieli, A. (2001). Respiration of stored product pests in hermetic conditions. In: Donahaye, E.J., Navarro, S., and Leesch, J.G. (eds.). Proceedings of the International Conference on Controlled Atmosphere and Fumigation in Stored Products. Clovis, California, USA: Executive Printing Services: 26-35.

Download reference:

Feichtinger, G. (1971). Stochastische Modelle demographischer Prozesse. Lecture notes in operations research and mathematical systems. Berlin: Springer-Verlag.

Download reference:

Fennel, W. and Neumann, T. (2004). Introduction to the modelling of marine ecosystems. Amsterdam, Netherlands: Elsevier.

Download reference:

Frank, P.W., Boll, C.D., and Kelly, R.W. (1957). Vital statistics of laboratory cultures of Daphnia pulex DeGeer as related to density. Physiological Zoology 30: 287-305.

Download reference:

Frauenthal, J. and Swick, K. (1983). Limit cycle oscillations of the human population. Demography 20(3): 285-298.

Weblink:
Download reference:

Gaines, S. and Roughgarden, J. (1985). Larval settlement rate: a leading determinant of structure in an ecological community of the marine intertidal zone. Proceedings of the National Academy of Sciences USA 82: 3707-3711.

Weblink:
Download reference:

Gani, J. (1963). Formulae for projecting enrolments and degrees awarded in universities. Journal of the Royal Statistical Society A 126(3): 400-409.

Weblink:
Download reference:

Gillman, M., Bullock, J.M., Silvertown, J., and Hill, B.C. (1993). A density-dependent model of Cirsium vulgare population dynamics using field-estimated parameter values. Oecologia 96(2): 282-289.

Weblink:
Download reference:

Grant, A. and Benton, T.G. (2003). Density-dependent populations require density-dependent elasticity analysis: an illustration using the LPA model of Tribolium. Journal of Animal Ecology 72(1): 94-105.

Weblink:
Download reference:

Grant, A. and Benton, T.G. (2000). Elasticity analysis for density dependent populations in stochastic environments. Ecology 81(3): 680-693.

Hamilton, W.D. (1966). The moulding of senescence by natural selection. Journal of Theoretical Biology 12(1): 12-45.

Weblink:
Download reference:

Hauser, C.E., Cooch, E.G., and Lebreton, J.D. (2006). Control of structured populations by harvest. Ecological Modelling 196(3-4): 462-470.

Weblink:
Download reference:

Hinde, A. (1998). Demographic methods. London, United Kingdom: Arnold.

Download reference:

Hsu, S.B., Hubbell, S.P., and Waltman, P. (1977). A mathematical theory for single-nutrient competition in continuous cultures of microorganisms. SIAM Journal of Applied Mathematics 32(2): 366-383.

Weblink:
Download reference:

Iannelli, M., Martcheva, M., and Milner, F.A. (2005). Gender-structured population modeling: mathematical methods, numerics, and simulations. Philadelphia, Pennsylvania, USA: SIAM.

Download reference:

Iosifescu, M. (1980). Finite Markov processes and their applications. New York, USA: Wiley.

Download reference:

Keyfitz, N. (1977). Applied mathematical demography. New York, USA: Wiley.

Download reference:

Keyfitz, N. (1971). Linkages of intrinsic to age-specific rates. Journal of the American Statistical Association 66(334): 275-281.

Weblink:
Download reference:

Keyfitz, N. (1972a). Population waves. In: Greville, T.N.E. (ed.). Population dynamics. New York, USA: Academic Press: 1-38.

Download reference:

Keyfitz, N. (1972b). The mathematics of sex and marriage. Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability 4: 89-108.

Download reference:

Keyfitz, N. and Caswell, H. (2005). Applied mathematical demography. New York, USA: Springer-Verlag.

Download reference:

Keyfitz, N. and Flieger, W. (1990). World population growth and aging: demographic trends in the late twentieth century. Chicago, Illinois, USA: University of Chicago Press.

Download reference:

Keyfitz, N. and Flieger, W. (1968). World population: an analysis of vital data. Chicago, Illinois, USA: The University of Chicago Press.

Download reference:

Kirkland, S.J. and Neumann, M. (1994). Convexity and concavity of the Perron root and vector of Leslie matrices with applications to a population model. SIAM Journal of Matrix Analysis and Applications 15(4): 1092-1107.

Weblink:
Download reference:

Klepac, P., Caswell, H., and Neubert, M.G. (2007). Matrix models for stage-classified epidemics. Paper presented at the Contributed paper, Ecological Society of America Annual Meeting, San Jose, CA, August, 2007.

Download reference:

Kokko, H. and Rankin, D.J. (2006). Lonely hearts or sex in the city? Density-dependent effects in mating systems. Philosophical Transactions of the Royal Society B 361(1466): 319-334.

Weblink:
Download reference:

Lee, C.T. and Tuljapurkar, S. (2007). Population and prehistory I: Food-limited population growth in constant environments. (Submitted).

Download reference:

Lee, R.D. (1976). Demographic forecasting and the Easterlin hypothesis. Population and Development Review 2(3/4): 459-468.

Weblink:
Download reference:

Lee, R.D. (1986). Malthus and Boserup: a dynamic synthesis. In: Coleman, D. and Schofield, R. (eds.). The state of population theory: forward from Malthus. Oxford, United Kingdom: Blackwell: 96-130.

Download reference:

Lee, R.D. (1987). Population dynamics of humans and other animals. Demography 24(4): 443-465.

Weblink:
Download reference:

Lee, R.D. (1974). The formal dynamics of controlled populations and the echo, the boom, and the bust. Demography 11(4): 563-585.

Weblink:
Download reference:

Legendre, S., Clobert, J., Møller, A.P., and Sorci, G. (1999). Demographic stochasticity and social mating system in the process of extinction of small populations: the case of passerines introduced to New Zealand. American Naturalist 153(5): 449-463.

Weblink:
Download reference:

Lenz, T.L., Jacob, A., and Wedekind, C. (2007). Manipulating sex ratio to increase population growth: the example of the Lesser Kestrel. Animal Conservation 10(2): 236-244.

Weblink:
Download reference:

Li, N. and Tuljapurkar, S. Dependency ratios: past, future, and optimal. (Unpublished manuscript).

Download reference:

Lindström, J. and Kokko, H. (1998). Sexual reproduction and population dynamics: the role of polygyny and demographic sex differences. Proceedings of the Royal Society London B 265(1395): 483-488.

Weblink:
Download reference:

MacArthur, R.H. (1972). Geographical ecology: patterns in the distribution of species. New York, USA: Harper and Row.

Download reference:

Magnus, J.R. and Neudecker, H. (1988). Matrix differential calculus with applications in statistics and econometrics. New York, USA: John Wiley & Sons.

Download reference:

Magnus, J.R. and Neudecker, H. (1985). Matrix differential calculus with applications to simple, Hadamard, and Kronecker products. Journal of Mathematical Psychology 29(4): 474-492.

Weblink:
Download reference:

McFarland, D.D. (1972). Comparison of alternative marriage models. In: Greville, T.N.E. (ed.). Population dynamics. New York, USA: Academic Press.

Download reference:

Mueller, L.D. and Joshi, A. (2000). Stability in model populations. Princeton, New Jersey, USA: Princeton University Press.

Download reference:

Murdoch, W.W., Briggs, C.J., and Nisbet, R.M. (2003). Consumer-resource dynamics. Princeton, New Jersey, USA: Princeton University Press.

Download reference:

Nel, D.G. (1980). On matrix differentiation in statistics. South African Statistical Journal 14: 137-193.

Download reference:

Neubert, M.G. and Caswell, H. (2000). Density-dependent vital rates and their population dynamic consequences. Journal of Mathematical Biology 41(2): 103-121.

Weblink:
Download reference:

Neudecker, H. (1969). Some theorems on matrix differentiation with special reference to Kronecker matrix products. Journal of the American Statistical Association 64(327): 953-963.

Weblink:
Download reference:

Nussbaum, R.D. (1988). Iterated nonlinear maps and Hilbert’s projective metric. Memoirs of the American Mathematical Society 75(391) (Providence, Rhode Island, USA).

Download reference:

Nussbaum, R.D. (1989). Iterated nonlinear maps and Hilbert’s projective metric, II. Memoirs of the American Mathematical Society 79(401) (Providence, Rhode Island, USA).

Download reference:

Pascual, M. and Caswell, H. (1991). The dynamics of a size-classified benthic population with reproductive subsidy. Theoretical Population Biology 39(2): 129-147.

Weblink:
Download reference:

Pearl, R., Miner, J.R., and Parker, S.L. (1927). Experimental studies on the duration of life. XI. Density of population and life duration in Drosophila. American Naturalist 61(675): 289-318.

Weblink:
Download reference:

Pennycuick, L. (1969). A computer model of the Oxford Great Tit population. Journal of Theoretical Biology 22(3): 381-400.

Weblink:
Download reference:

Pollak, R.A. (1987). The two-sex problem with persistent unions: a generalization of the birth matrix-mating rule model. Theoretical Population Biology 32(2): 176-187.

Weblink:
Download reference:

Pollak, R.A. (1990). Two-sex demographic models. Journal of Political Economy 98(2): 399-420.

Weblink:
Download reference:

Pollard, J.H. (1968). A note on the age structures of learned societies. Journal of the Royal Statistical Society A 131(4): 569-578.

Weblink:
Download reference:

Pollard, J.H. (1977). The continuing attempt to incorporate both sexes into marriage analysis. Paper presented at the International Population Conference, International Union for the Scientific Study of Population: 291-308.

Download reference:

Pollard, J.H. (1982). The expectation of life and its relationship to mortality. Journal of the Institute of Actuaries 109: 225-240.

Download reference:

Preston, S.H., Heuveline, P., and Guillot, M. (2001). Demography: measuring and modeling population processes. Oxford, United Kingdom: Blackwell.

Download reference:

Promislow, D.E.L., Montgomerie, R., and Martin, T.E. (1992). Mortality costs of sexual dimorphism in birds. Proceedings of the Royal Society of London B 250(1328): 143-150.

Weblink:
Download reference:

Roth, W.E. (1934). On direct product matrices. Bulletin of the American Mathematical Society 40(1934): 461-468.

Weblink:
Download reference:

Roughgarden, J., Iwasa, Y., and Baxter, C. (1985). Demographic theory for an open marine population with space-limited recruitment. Ecology 66(1): 54-67.

Weblink:
Download reference:

Sarrazin, F. and Legendre, S. (2000). Demographic approach to releasing adults versus young in reintroductions. Conservation Biology 14(2): 488-500.

Weblink:
Download reference:

Scheltema, R.S. (1971). Larval dispersal as a means of genetic exchange between geographically separated populations of shallow-water benthic marine gastropods. Biological Bulletin 140(2): 284-322.

Weblink:
Download reference:

Scott, S. and Duncan, C.J. (1998). Human demography and disease. Cambridge, United Kingdom: Cambridge University Press.

Weblink:
Download reference:

Silva Matos, D.M., Freckleton, R.P., and Watkinson, A.R. (1999). The role of density dependence in the population dynamics of a tropical palm. Ecology 80(8): 2635-2650.

Skalski, J.R., Ryding, K.E., and Millspaugh, J. (2005). Wildlife demography: analysis of sex, age, and count data. San Diego, California, USA: Academic Press.

Download reference:

Solbrig, O.T., Sarandon, R., and Bossert, W. (1988). A density-dependent growth model of a perennial herb, Viola fimbriatula. American Naturalist 131(3): 385-400.

Weblink:
Download reference:

Takada, T. and Nakajima, H. (1992). An analysis of life history evolution in terms of the density-dependent Lefkovitch Matrix model. Mathematical Biosciences 112(1): 155-176.

Weblink:
Download reference:

Takada, T. and Nakajima, H. (1998). Theorems on the invasion process in stage-structured populations with density-dependent dynamics. Journal of Mathematical Biology 36(5): 497-514.

Weblink:
Download reference:

Tanner, J.E. (1999). Density-dependent population dynamics in clonal organisms: a modelling approach. Journal of Animal Ecology 68(2): 390-399.

Weblink:
Download reference:

Tilman, D. (1982). Resource competition and community structure. Princeton, New Jersey, USA: Princeton University Press.

Download reference:

Tuljapurkar, S. (1997). Stochastic matrix models. In: Tuljapurkar, S. and Caswell, H. (eds.). Structured-population models in marine, terrestrial, and freshwater systems. New York, USA: Chapman and Hall: 59-87.

Download reference:

Tuljapurkar, S. and Horvitz, C.C. (2006). From stage to age in variable environments: life expectancy and survivorship. Ecology 87(6): 1497-1509.

Tuljapurkar, S., Puleston, C.O., and Gurven, M.D. (2007). Why men matter: mating patterns drive evolution of human lifespan. PLoS ONE 2(8): e785.

Weblink:
Download reference:

Vaupel, J.W. (1986). How change in age-specific mortality affects life expectancy. Population Studies 40(1): 147-157.

Weblink:
Download reference:

Vaupel, J.W. and Romo, V.C. (2003). Decomposing change in life expectancy: a bouquet of formulas in honor of Nathan Keyfitz’s 90th birthday. Demography 40(2): 201-216.

Weblink:
Download reference:

Vaupel, J.W. and Romo, V.C. (2002). Decomposing demographic change into direct vs. compositional components. Demographic Research 7(1): 1-14.

Weblink:
Download reference:

Verdy, A. and Caswell, H. (2008). Sensitivity analysis of reactive ecological dynamics. Bulletin of Mathematical Biology (in press).

Weblink:
Download reference:

Wachter, K.W. and Bulatao, R.A. (2003). Offspring: human fertility behavior in biodemographic perspective. Washington, DC, USA: National Academies Press.

Download reference:

Wachter, K.W. and Finch, C.E. (1997). Between Zeus and the salmon: the biodemography of longevity. Washington DC, USA: National Academy Press.

Download reference:

Wachter, K.W. and Lee, R.D. (1989). U.S. births and limit cycle models. Demography 26(1): 99-115.

Weblink:
Download reference:

Weiss, K.M. and Smouse, P.E. (1976). The demographic stability of small human populations. Journal of Human Evolution 5(1): 59-73.

Weblink:
Download reference:

Willekens, F.J. (1977). Sensitivity analysis in multiregional demographic models. Environment and Planning A 9(6): 653-674.

Weblink:
Download reference:

Wilson, C. (2004). Fertility below replacement level. Science 304(5668): 207-209.

Weblink:
Download reference:

Wood, J.W. and Smouse, P.E. (1982). A method of analyzing density-dependent vital rates with an applicaion to the Gainj of Papua New Guinea. American Journal of Physical Anthropology 58(4): 403-411.

Weblink:
Download reference:

Yearsley, J.M., Fletcher, D., and Hunter, C.M. (2003). Sensitivity analysis of equilibrium population size in a density-dependent model for short-tailed shearwaters. Ecological Modelling 163(1-2): 119-129.

Weblink:
Download reference:

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