I'm going through a question on my maths biology course, and I'm confused at one part. you are given the formulas for a tumour and the drug that effects it. They are shown below:
I've also non-dimensionalised it to:
Then the question tells me there is molecule that inactivates the chemotherapeutic drug, which has the equation:
Now I've looked at the answers and it's given to be :
My question is, how do you turn that $M+2C$ equation into the $-2k_1 MC^2$ and $\mu n -k_1 MC^2$ equations?
You have a population $N$ with density/concentration $n$ that without external influences grows, exponentially $\dot n\sim \frac{\rho}{\kappa}n$ for small $n$ and linearly $\dot n\sim ρ$ for large $n$. Then there is substance $C$ with concentration $c$ that is added at a steady rate $\alpha$ and decays at a rate $\beta$. Presumably there is an reaction $N+C\xrightarrow{\gamma}K+C$ that converts population $N$ to population $K$ with rate $\gamma$ without using up substance $C$. This gives the term $-γnc$ in $\dot n$.
Now you add that the population $N$ generates a neutralizing agent $M$ at rate $\mu$, $N\xrightarrow\mu N+M$ that renders $C$ inactive $M+2C\xrightarrow{k_1} W$. To produce one particle of $W$ you need that two of $C$ and one of $M$ are closely together (and in correct relative positions etc.). That happens with a probability proportional to $mc^2$. Thus the reaction takes place with a rate $k_1mc^2$. For each successful reaction two of $C$ and one of $M$ are removed, resulting in the modified equations
\begin{align} \dot m &= μn -k_1mc^2,\\ \dot c &= α-βc-2k_1mc^2. \end{align}
One could also make the model more complicated by hypothesizing that three meeting at once in a correct configuration is rather improbable, a better model would be that there is an unstable compound $U$ of one $M$ and one $C$ with the extended the reaction model $M+C\xrightarrow{k_1}U$, $U\xrightarrow{k_2}M+C$, $U+C\xrightarrow{k_3}W$ which have reaction rates of $k_1mc$, $k_2u$, $k_3uc$ and which give the modified system incorporating the consumption and production rates of the particles \begin{align} \dot m &= μn -k_1mc+k_2u,\\ \dot u &= k_1mc-k_2u-k_3uc,\\ \dot c &= α-βc-k_1mc+k_2u-k_3uc. \end{align} etc.