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

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