Consider the discrete predator-prey model for two populations $N_{t}, P_{t}$ of prey and predators, respectively:

$$\left.\begin{array}{rl} N_{t+T} & =r N_{t} \exp \left(-a P_{t}\right) \\ P_{t+T} & =s N_{t}\left(1-b \exp \left(-a P_{t}\right)\right) \end{array}\right\},$$

where $r, s, a, b$ are constants, all assumed to be positive.

(a) Give plausible explanations of the meanings of $T, r, s, a, b$.

(b) Nondimensionalize equations $(*)$ to show that with appropriate rescaling they may be reduced to the form

$$\left.\begin{array}{l} n_{t+1}=r n_{t} \exp \left(-p_{t}\right) \\ p_{t+1}=n_{t}\left(1-b \exp \left(-p_{t}\right)\right) \end{array}\right\}$$

(c) Now assume that $b<1, r>1$. Show that the origin is unstable, and that there is a nontrivial fixed point $(n, p)=\left(n_{c}(b, r), p_{c}(b, r)\right)$. Investigate the stability of this point by writing $\left(n_{t}, p_{t}\right)=\left(n_{c}+n_{t}^{\prime}, p_{c}+p_{t}^{\prime}\right)$ and linearizing. Express the linearized equations as a second order recurrence relation for $n_{t}^{\prime}$, and hence show that $n_{t}^{\prime}$ satisfies an equation of the form

$$n_{t}^{\prime}=A \lambda_{1}^{t}+B \lambda_{2}^{t}$$

where the quantities $\lambda_{1,2}$ satisfy $\lambda_{1}+\lambda_{2}=1+b n_{c} / r, \lambda_{1} \lambda_{2}=n_{c}$ and $A, B$ are constants. Give a similar expression for $p_{t}^{\prime}$ for the same values of $A, B$.

Show that when $r$ is just greater than unity the $\lambda_{i}(i=1,2)$ are real and both less than unity, while if $n_{c}$ is just greater than unity then the $\lambda_{i}$ are complex with modulus greater than one. Show also that $n_{c}$ increases monotonically with $r$ and that if the roots are real neither of them can be unity.

Deduce that the fixed point is stable for sufficiently small $r$ but loses stability for a value of $r$ that depends on $b$ but is certainly less than $e=\exp (1)$. Give an equation that determines the value of $r$ where stability is lost, and an equation that gives the argument of the eigenvalue at this point. Sketch the behaviour of the moduli of the eigenvalues as functions of $r$.