paper-scissors game. In this game, rock loses to paper, paper to scissors, but scissors lose to rock, just as the wild-type bacteria lose to the antibiotic-producing bacteria, which lose to the antibiotic-resistant bacteria, which in turn lose to the wild-type bacteria. Assume that in a mixed community of bacteria, a fraction $P(t)$ are antibiotic producing, a fraction $R(t)$ are wild type (i.e., don't produce antibiotics and aren't resistant to it either), and a fraction $S(t)$ are antibiotic resistant. Since all bacteria are one of the three types, $P(t)+R(t)+S(t)=1$. The growth rate of each population depends on their interactions with other bacteria. For example, if an antibiotic-resistant bacterium encounters mainly antibiotic-producing bacterium, then it will reproduce faster since it has a competitive advantage over those bacteria; if, on the other hand, it encounters wild-type bacteria, it will be outcompeted.

(a) Explain how the model

$\frac{d S}{d t}=S(P-R)$

is consistent with the biological model described above. Write down similar equations for $\frac{d R}{d t}$ and $\frac{d P}{d t}$.

(b) Show that $\frac{d R}{d t}+\frac{d P}{d t}+\frac{d S}{d t}=0$. Why is this to be expected?

(c) We have a system of three differential equations, but our methods are designed for systems of two differential equations. However, since we know that $R+P+S=1$, we can eliminate $R$ using $R=1-P-S$. Write down two differential equations for $d P / d t$ and $d S / d t$ that do not include $R$.

(d) Assume that there are no antibiotic-resistant bacteria present (i.e., $S(t)=0$ ). Write down a differential equation for $d P / d t$. Find the equilibria for this differential equation and describe how $P(t)$ will change with time.

(e) Now consider the case where both $P$ and $S$ can be non-zero. Find all possible equilibria of the system, and classify them by linearization.

(f) One of the equilibria that you found in (d) is predicted to be a linear center. In such circumstances we cannot trust linearization. However, just as for the predator-prey system, we can write solutions as level curves of a function. Show using the chain rule that the function

$F(P, S)=P S(1-P-S)$

is constant on solution curves.

(g) Use your graphical calculator to draw the level curves of $F$. Show that, based on these level curves, we expect $P(t), S(t)$, and $R(t)$ to all oscillate in time, if all three species of bacteria are present in a habitat.