We examine the kinetics of the following autocatalytic chemical reaction $$\text{A}+\text{B}\longrightarrow 2\,\text{B}$$ where the reaction rate is proportional to the product of the concentrations of A and B, say $$\frac{\dd[\text{A}]}{\dd t}=-k\, [\text{A}]\, [\text{B}]$$ for some rate constant $k$. The rate of formation of B satisfies the differential equation $$\frac{\dd[\text{B}]}{\dd t}=k\, [\text{A}]\, [\text{B}]$$ Because of the stoichiometry of the reaction, we have that the sum of the concentrations of A and B is constant, say $$[\text{A}]+[\text{B}]=K$$ for some constant $K$. Then we can rewrite the equation for the formation rate as $$\frac{\dd[\text{B}]}{\dd t}=k\, [\text{B}]\bigl(K-[\text{B}]\bigr)$$ The concentration of B is a solution of a logistic differential equation and is explicitly described by a logistic function. This differential equation can be solved exactly. Below is a simulation of the kinetic model. Play with parameter choices and check that the concentration of B is upper bounded and is reached even when there is only a little bit of B is 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)