Growth rates and the ``bell'' view

Just as the differential equation (1) reveals the mechanism propelling its integral equation (2), the rates of change of the component logistics provide clues to the mechanisms propelling the composite logistic. Analyzing the rates of change is often useful when yearly or percent per year tabulations are applicable, as in the case of economic data. Recall that the analytic form of the three-parameter logistic is:

$$N(t)= \frac{\kappa}{1 + \text{exp} \left[ {-\frac{\ln(81)}{\Delta t}}(t -t_{m}) \right] }$$

The instantaneous rate of growth of the logistic function is given by its derivative with respect to time:

$$\frac{d N(t)}{d t}= \frac{\frac{\ln(81)}{\Delta t} \kappa \... ...1 + \text{exp} \left( {-\frac{\ln(81)}{\Delta t}}(t -t_{m}) \right) \right]^2 }$$ (8)

Figure 6: The rates of change of the component logistics (``The Bell View'')

Plotting (8) produces a bell-shaped curve similar, but not identical, to the normal distribution function. Naturally, the bell-shaped curve peaks at the midpoint $t_{m}$; analytically, this is the point of inflection, and thus it is an extremum of $N(t)$. Panel B of Figure 6 shows the derivative of the component logistics of our test function (Panel A is shown again for comparison purposes).

The rate of change of the data subsets $D_j$ is computed discretely, creating the sets $\overline{D_j}$, in which each point $E_i$ is

$$E_i = \left(\frac{t_i + t_{i+1}}{2} + t_i, \frac{\bold{c}_i(t_{i+1}) - \bold{c}_i(t_{i})}{t_{i+1} - t_i} \right)$$

where $1 \leq j \leq m-1$. In other words, $E_i$ contains the discrete derivative from $(t_i,d^\ast_i)$ to $(t_{i+1},d^\ast_{i+1})$.

Loglet Lab can decompose a logistic curve into its discrete components; each component can be transformed using the Fisher-Pry transform, or their rate of change can be plotted.