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A Historical Review on Discrete Models of Population Changes and Illustrative Analysis Methods Using Computer Softwares  

Shim, Seong-A (Dept. of Math., Sungshin Women's Univ.)
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Journal for History of Mathematics / v.27, no.3, 2014 , pp. 197-210 More about this Journal
Species like insects and fishes have, in many cases, non-overlapping time intervals of one generation and their descendant one. So the population dynamics of such species can be formulated as discrete models. In this paper various discrete population models are introduced in chronological order. The author's investigation starts with the Malthusian model suggested in 1798, and continues through Verhulst model(the discrete logistic model), Ricker model, the Beverton-Holt stock-recruitment model, Shep-herd model, Hassell model and Sigmoid type Beverton-Holt model. We discuss the mathematical and practical significance of each model and analyze its properties. Also the stability properties of stationary solutions of the models are studied analytically and illustratively using GSP, a computer software. The visual outputs generated by GSP are compared with the analytical stability results.
population dynamics; discrete models; cobweb diagram; discrete logistic model; the Beverton-Holt stock-recruitment model; Ricker model; Shepherd model; Hassell model; Sigmoid type Beverton-Holt model;
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  • Reference
1 N. Kaldor, A Classificatory Note on the Determination of Equilibrium, Review of Economic Studies Vol. I (1934), 122-136.
2 W. C. Allee, Animal Aggregations, University of Chicago Press, Chicago, 1931.
3 R. J. H. Beverton, S. J. Holt, On the Dynamics of Exploited Fish Populations, Fishery Investigations Series II Volume XIX (1957), Ministry of Agriculture, Fisheries and Food.
4 M. P. Hassel, Density-dependence in single-species populations, Journal of Animal Ecology 44 (1975), 283-295.   DOI   ScienceOn
5 T. R. Malthus, An Essay on the Principle of Population, Penguin Books, 1970, Originally published in 1798.
6 J. Maynard Smith, M. Slatkin, The stability of predator-prey sytems, Ecology 54 (1973), 384-391.   DOI   ScienceOn
7 W. E. Ricker, Stock and recruitment, J. Fisheries Res. Board Can. 11 (1954), 559-623.   DOI
8 J. G. Shepherd, A versatile new stock-recruitment relationship for fisheries and the construction of sustainable yield curves, Journal du Conseil International pour l'Exploration de la Mer 40 (1982) 67-75.   DOI
9 G. G. Thomson, A proposal for a threshold stock size and maximum fishing mortality rate, Risk Evaluation and Biological Reference Points for Fisheries Management, Canad. Spec. Publ. Fish. Aquat. Sci. 120 (1993), 303-320.
10 P.-F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance mathematique et physique 10 (1838) 113-121.