# 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:

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