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2002). ). New York: Springer-Verlag. Lamb, H. H. (1995). ). New York: Routledge. Pearl, R. (1927). The growth of populations. Quarterly Review of Biology, 2, 532–548. Pratt, D. M. (1943). Analysis of population development in Daphnia at different temperatures. Biological Bulletin, 85, 116–140. Richards, O. W. (1928). Potentially unlimited multiplication of yeast with constant environment, and the limiting of growth by changing environment. Journal of General Physiology, 11, 525–538. Smith, T. , Link, J.

The model given by Eq. (1-25) may also be obtained by modifying the unlimited growth model from Eq. (1-1) to allow net per capita growth rate to vary with population size. In Eq. (1-25), as in Eq. (1-24), K is the carrying capacity for the population; a > 0 is the inherent per capita growth rate; and the arguments for choosing   Pn rðPn Þ ¼ a 1 À will be the same as for the continuous logistic model K developed in Section IV. Although the Verhulst model has the same equilibrium states as the continuous logistic model, it can exhibit radically different long-term behavior.

For most models developed thus far, we have presented analytical solutions. Knowing the analytical form of a solution allows for direct calculation of the predicted value. For example, knowing that the solution of Eq. 1 million for the United States for the year 1825. S. 8 million. It is not always easy to solve a model analytically, and, as the sophistication of the models increases, the mathematics for solving the equations become increasingly more challenging. ) to obtain the actual analytic solution, numerical solutions are used instead.

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