### Logistic growth: More examples of logistic growth

### Autocatalysis in chemistry

We look into the kinetics of the following autocatalytic chemical reaction \[\text{A}+\text{B}\longrightarrow 2\,\text{B}\] in which the reaction rate is proportional to the product of the concentrations of the substances A and B, say \[\frac{d[\text{A}]}{dt}=-k\, [\text{A}]\, [\text{B}]\] for some constant \(k\). The rate of formation of substance B satisfies the differential equation \[\frac{\dd[\text{B}]}{\dd t}=k\, [\text{A}]\, [\text{B}]\] Because of the stoichiometry of the reaction, the sum of the concentrations of the substances A and B is constant, say \[[\text{A}]+[\text{B}]=K\] Then we can rewrite the equation for the formation of substance B \[\frac{\dd[\text{B}]}{\dd t}=k\, [\text{B}]\bigl(K-[\text{B}]\bigr)\] So, the concentration of substance B satisfies a logistic differential equation and is explicitly described by a logistic function. Below is a simulation of this kinetic model. Play with parameter choices and verify that the concentration of B is limited form above and that this upper bound is reached even when there is only a little bit of substance B present at the start of the reaction.

Such autocatalytic model is not a theoretical model, suitable only for chemical reactions: three examples of other types of autocatalytic processes are

- coagulation of milk during the cheese-making;
- coagulation of blood;
- production of a protein that activates a transcription factor for its own expression (positive feedback)