# 6.5: Building Your Own Model Equation

Principles and best practices of building your own equations for a continuous-time model are very much the same as those we discussed for discrete-time models in Sections 4.5 and 4.6. The only difference is that, in differential equations, you need to describe time derivatives, i.e., instantaneous rates of change of the system’s state variables, instead of their actual values in the next time step.

Here are some modeling exercises. The ﬁrst one is exactly the same as the one in Section 4.6, except that you are now writing the model in differential equations, so that you can see the differences between the two kinds of models. The other two are on new topics, which are relevant to chemistry and social sciences. Work on these exercises to get some experience writing continuous-time models!

Exercise 6.7

Develop a continuous-time mathematical model of two species competing for the same resource, and simulate its behavior.

Exercise 6.8

Imagine two chemical species, \(S\) and \(E\), interacting in a test tube. Assume that \(E\) catalyzes the production of itself using \(S\) as a substrate in the following chemical reaction:

\[S+E \rightarrow 2E\label{6.31}\]

Develop a continuous-time mathematical model that describes the temporal changes of the concentration of S and E and simulate its behavior.

Exercise 6.9

When a new pop song is released, it sounds attractive to people and its popularity increases. After people get used to the song, however, it begins to sound boring to them, and its popularity goes down. Develop a continuous-time mathematical model that captures such rise and fall in a typical pop song’slife, and simulate its behavior.