Inhibition of a linear growth model The model of limited exponential growth can be regarded as an adjustment of the linear growth model to model inhibition of growth. Suppose that a quantity $y$ satisfies the differential equation of linear growth, i.e., $y'(t)=c$ for some positive constant $c$. The solution is a linear function in time $t$, of which the graph is a straight line with slope $c$. This function has no maximum. To get a gradual flattening of the graph you could introduce an inhibition factor into the differential equation:

$$\frac{\dd y}{\dd t}=c\cdot \textrm{inhibition factor}$$ where the inhibition factor is a number between 0 and 1. You can of course use an inhibition factor that is a function of time, for example $\textrm{inhibition factor}=e^{-\lambda\cdot t}$ with positive constant $\lambda$, but for limited exponential growth the following inhibition factor is used $$\mathit{\textrm{inhibition factor}}=1-\frac{y}{a}$$ where $a$ is some positive constant. This inhibition factor decreases linearly from 1 to 0, when the magnitude of quantity $y$ increases linearly from 0 to $a$. This is why the inhibition factor is also called 'braking factor': it slows down and eventually stops growth. The differential equation for limited growth of $y$ is then $$\frac{\dd y}{\dd t}=c\cdot\left(1-\frac{y}{a}\right)$$ alternatively written as $$\frac{\dd y}{\dd t}=r\cdot(a-y)$$ with growth rate constant $r=\frac{c}{a}$. This is the differential equation of limited exponential growth as we have introduced before.

Inhibition of an exponential growth model In a similar manner, the exponential growth model can be adapted to model inhibition of growth. Suppose that a quantity $y$ satisfies the differential equation of exponential growth, i.e., $y'(t)=r\cdot y$ for some positive constant $r$. The solution is an exponential function. To get a gradual flattening of the graph you could introduce an inhibition factor in the differential equation:

$$\frac{\dd y}{\dd t}=r\cdot y\cdot \mathrm{inhibition\;factor}$$ where the inhibition factor is always a number between 0 and 1. When you choose $$\mathrm{inhibition\;factor}=e^{-\alpha\cdot t}$$ for some positive constant $\alpha$, you get the so-called Gompertz model.

We also use the following expression for the inhibition factor:

$$\mathrm{inhibition\;factor}=1-\frac{y}{a}$$ for some positive constant $a$. The differential equation for the inhibited growth of quantity $y$ is then $$\frac{\dd y}{\dd t}=r\cdot y\cdot \left(1-\frac{y}{a}\right)$$ alternatively written as $$\frac{\dd y}{\dd t}=\frac{r}{a}\cdot y\cdot (a-y)$$ This is called the differential equation of logistic growth. If the start value $y(0)$ between 0 and $a$ is, you get an increasing S-shaped graph $y$, i.e., an increasing sigmoid, with positive $r$; you get a decreasing sigmoid for a negative value of $r$. Note that the relative growth rate $y'(t)/y(t)$ is constant at exponential growth, but is a linear function in the logistic model for growth $y(t)$ is.

It is surprising how well the logistic growth model describes many inhibited growth processes. Whether you study the growth of the surface area of a leaf, the weight of a pig, the size of a tumour, the spread of an infectious disease, enzyme kinetics, the growth spurt of boys and girls in puberty, or the growth of fungi, often the logistic growth model gives a very reasonable mathematical description of measured data (or at least first step hereto).