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Hill-function

As we have seen in the previous sections, the logistic equation can behave strangely when used in EBM. Therefore, in many practical modeling cases replacements are used that show similar dynamics, but behave less capricious.

One of these replacements is the Hill-function which is a logarithmic transform of the logistic equation.

Hill-functions are used to model density dependent growth or decline. They are convenient since their results always range between 0 and 1 and thus can be simply multiplied with population size.

They have the form

\[f(x)=\frac{x^\lambda}{h^\lambda+x^\lambda}\]

with \(\lambda\) defining the delay at the limit, and \(h\) defining the \(x\)-value where the development reaches half of its saturation.

Example: Population growth in the United States - II

As seen in the first part of this example, human populations do not just grow exponentially. In 1940, the biologist Raymond Pearl rediscovered Verhulst's equation and used it to model the growth of the US-population since 1790. As seen before, the logistic equation \(\frac{xt}{dt}=a*x*(1 - \frac{x}{K})\) has an exact solution in \(𝑥_𝑡=\frac{K*x_0}{x_0+(K − x_0) e^{−a*t}}\), implying that the development of the US population could be fitted with a curve of the form

\[x_t=\frac{K*C*e^a(t-1790)}{1+C*e^a(t-1790)}\]

For this the average exponential growth rate \(\alpha_i\) for each decade \((t_{i-1},t_i)\) is calculated as \(\alpha_i=\frac{log x_i - log x_{i-1}}{10}\). This yields an average population size of \(\overline{x_i}=\frac{x_i-x_{i-1}}{log x_i - log x_{i-1}}\). Results are given in the last two columns of the following table:

US-Population

Pearl, R. / Reed, L. J. / Kish, J. F. (1940). The logistic curve and the census count of 1940 Science 92, 486 - 488.