Bi- and multi-logistic curves

As it turns out, many growth and diffusion processes are actually made up of several subprocesses. First, let us consider the case of a system which experiences growth in two discrete growth phases. Then, we will extend this to an arbitrary number of phases.

Systems with two growth phases follow what we call the ``Bi-logistic'' model [12]. In this model, growth is the sum of two discrete ``wavelets'' , each of which is a three-parameter logistic:

$$N(t) = N_1(t) + N_2(t),$$ (5)

where

$$N_1(t)= \frac{\kappa_1}{1 + \text{exp} \left[ {-\frac{\ln(81)}{\Delta t_1}}(t -t_{m1}) \right] }$$

$$N_2(t)= \frac{\kappa_2}{1 + \text{exp} \left[ {-\frac{\ln(81)}{\Delta t_2}}(t -t_{m2}) \right] }$$

Naturally, we can examine system-level behavior (i.e., $N(t)$), or we can decompose the model and examine the behavior of the discrete components (either $N_1(t)$ or $N_2(t)$). In fact, we can plot $N_1(t)$ and $N_2(t)$ on the same axes, and moreover we can compare two disparate wavelets by normalizing them with the Fisher-Pry transform.