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).