# 4.5: Generalization

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In summary, the macroscale model for population dynamics of a single species, in its simplest form, is

\(\frac{1}{N}\,\frac{∆N}{∆t}\,=\,r\,+\,sN\)

This is a straight-line form of a more general form presented by Hutchinson,

\(\frac{1}{N}\,{∆N}{∆t}\,=\,r\,+\,sN\,+\,s_2N^2\,+\,s_3N^3\,+\,s_4N^4\,+\,...\)

and of the most general form proposed by Kolomogorov, where f (N ) can be any function of the population density N.

\(\frac{1}{N}\,\frac{∆N}{∆t}\,=\,f(N)\)

The higher-order terms in the second equation could refine population projections if there were enough data to determine them. They are not really needed, however, because straight-line parts can be pieced together to form a general population growth curve, as in Figure 4.4.1. And as human population growth in Figure 6.3.1 will show, a piecewise approach can more closely approximate the real situation.

Moreover, blending separate versions of the first equation can generalize to either the Hutchinson or Kolomorgov forms as you will see in Chapter 18.

Figure \(\PageIndex{1}\) *Trumpeter swans—the largest North America birds, with wingspans reaching ten feet—were nearing extinction until deliberate protection and reintroduction programs brought their r values back to viable levels.*