Abstract
The diffusion equation model and the Lefkovitch matrix model have been employed independently in plant population ecology in order to analyze the dynamics of growth and size structure. The two models describe the dynamics of size structure in biological populations, and thus there must be some relationship between them. In the present paper, we examine the theoretical relationship between these two models. We demonstrate, on a certain assumption, that the one-step Lefkovitch matrix model corresponds to a difference equation of the diffusion equation and that the two- and three-step Lefkovitch matrix model correspond to difference equations of the 4th- and 6th-order Kramers-Moyal expansions, respectively. It is also shown that 2n moments (the first to the 2n-th moments) of growth rate are necessary and sufficient to rewrite uniquely the n-step Lefkovitch matrix model in terms of the linear combination of the moments. We finally discuss the relationship between the species characteristics of census data and the appropriate types of the Lefkovitch matrix.
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References
Bierzychudek, P.: The demography of jack-in-the-pulpit. A forest perennial that changes sex. Ecol. Monoger. 52, 335–351 (1982)
Burns, B. R., Ogden, J.: The demography of the temperate mangrove (Avicennia marina (Forsk.) Vierch.) at its southern limit in New Zealand. Aust. J. Ecol. 10, 125–133 (1985)
Caswell, H.: Matrix population model. Construction, analysis, and interpretation, pp. 27–30. Sunderland, Mass: Sinauer Associates Inc. Publishers 1989
Gardiner, C. W.: Handbook of stochastic methods (2nd ed.) pp. 249–250, Berlin Heidelberg New York: Springer 1990
Goel, N. S., Richter-Dyn, N.: Stochastic models in biology, Chap. 3. New York: Academic Press 1974
Hara, T.: A stochastic model and the moment dynamics of the growth and size distribution in plant populations. J. Theor. Biol. 109, 173–190 (1984a)
Hara, T.: Dynamics of stand structure in plant monocultures. J. Theor. Biol. 110, 223–229 (1984b)
Hara, T.: A model for mortality in a self-thinning plant population. Ann. Bot. 55, 667–674 (1985)
Hara, T.: Effects of density and extinction coefficient on size variability in plant populations. Ann. Bot. 57, 885–892 (1986b)
Hara, T.: Dyamics of size structure in plant populations. Trends Ecol. Evol. 3, 129–133 (1988)
Hara, T., Kimura, M., Kikuzawa, K.: Growth patterns of tree height and stem diameter in populations of Abies veitchii and Betula ermanii. J. Ecol. 79, 1085–1098 (1991)
Harcombe, P. A.: Stand development in a 130-years-old spruce-hemlock forest based on age structure and 50 years of mortality data. Forest Ecol. Manage. 14, 41–58 (1986)
Harcombe, P. A.: Tree life tables. Simple birth, growth and death data encapsulate life histories and ecological roles. Bioscience 37, 557–574 (1987)
Hartshorn, G. S.: A matrix model of tree population dynamics. In: Golley, F. B., Medin, E. (eds.), Tropical Ecological Systems: Trends in Terrestrial and Aquatic Research, pp. 41–51. Berlin Heidelberg New York: Springer 1975
Kawano, S., Takada, T., Nakayama S. and Hiratsuka A.: Demographic differentiation and life-history evolution in temperate woodland plants. pp. 153–181. In: Urbanska, K. M. (ed.) Differentiation patterns in higher plants. New York: Academic Press, 1987
Kinoshita, E.: Sex change and population dynamics in Arisaema (Araceae) I. Arisaema serratum (Thumb.) Schott. P1. Sp. Biol. 2, 15–28 (1987)
Kohyama, T.: Stand dynamics in a primary warm-temperature rain forest analyzed by the diffusion equation. Bot. Mag. Tokyo 100, 305–317 (1987)
Kohyama, T.: Simulation of the structural development of warm-temperate rain forest stands. Ann. Bot. 63, 625–634 (1989)
Kohyama, T.: Simulating stationary size distribution of trees in rain forests. Ann. Bot. 68, 173–180 (1991)
Kohyama, T.: Size-structured multi-species model of rain forest trees. Funct. Ecol. 6, 206–212 (1992)
Kohyama, T., Hara, T.: Frequency distribution of tree growth rate in natural forest stands. Ann. Bot. 64, 47–57 (1989)
Kramers, H. A.: Brown motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940)
Lefkovitch, L. P.: The study of population growth in organisms grouped by stages. Biometrics 21, 1–18 (1965)
Lubbers, A. E., Lechowicz, M. J.: Effects of leaf removal on reproduction vs. below ground storage in Trillium grandiflorum. Ecology 70, 85–96 (1979)
Meagher, T. R.: The population biology of Chamaelirium luteum a dioecious member of the lily family: two-sex population projections and stable population structure. Ecology 63, 1701–1711 (1982)
Metz, J. A. J., de Roos, A. M., van den Bosch, F.: Population models incorporating physiological structure: a quick survey of the basic concepts and an application to size-structured population dynamics in waterfleas. In: Ebenman, B., Persson, L. (eds.) Size-structured populations. Berlin Heidelberg New York: Springer 1988
Moyal, J. E.: Stochastic processes and statistical physics. J. R. Stat. Soc. 11, 151–210 (1949)
Nagano, M.: Dyanmics of stand development. In: Kira, T., Ono, Y., Hosokawa, T. (eds.) Biological Production in a Warm-Termperate evergreen Oak Forest of Japan, pp. 21–31. JIBP Synthesis 18, Tokyo: University of Tokyo Press 1978
Nakashizuka, T.: Population dynamics of coniferous and broad-leaved trees in a Japanese temperate mixed forest. J. Veget. Sci. 2, 413–418 (1991)
Petersen, T. D.: Effects of inteference from Calamagrostis rubescens on size distributions in stands of Pinus ponderosa. J. Appl. Ecol. 25, 265–272 (1998)
Petersen, T. D., Ning Z., Newton, M.: Dynamics of size structure in seedling stands to Fraxinus mandshurica in Northeast China. Ann. Bot. 66, 255–263 (1990)
Sarukhán, J. and Gadgil, M.: Studies on plant demography: Ranunculus repens L., R. bulbosus L. and R. acris L. III. A mathematical model incorporating mulitple modes of reproduction. J. Ecol. 62, 921–936 (1974)
Shinko, J. W., Streifer, W.: A new model for age-size structure of a population. Ecology 48, 910–918 (1967)
Snow, A. A., Whigham, S. F.: Costs of flower and fruit production in Tipularia discolor (Orchidaceae). Ecology 70, 1286–1293 (1989)
Sohn, J. J., Policansky D.: The costs of reproduction in the mayapple Podophyllum peltautum (Berberidaceae). Ecology 58, 1366–1374 (1977)
Takada, T., Iwasa, Y.: Size distribution dynamics of plants with interaction by shading. Ecol. Modell. 33, 173–184 (1986)
West, P. W., Jackett, D. R., Borough, C. J.: Competitive processes in a monoculture of Pinus raidata D. Don Oecologia 81, 57–61 (1989)
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Takada, T., Hara, T. The relationship between the transition matrix model and the diffusion model. J. Math. Biol. 32, 789–807 (1994). https://doi.org/10.1007/BF00168798
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DOI: https://doi.org/10.1007/BF00168798