Consider a dynamical system of the form

$$\begin{aligned} &\dot{x}=x(1-y+a x) \\ &\dot{y}=r y(-1+x-b y) \end{aligned}$$

on $\Lambda=\{(x, y): x>0$ and $y>0\}$, where $a, b$ and $r$ are real constants and $r>0$.

(a) For $a=b=0$, by considering a function of the form $V(x, y)=f(x)+g(y)$, show that all trajectories in $\Lambda$ are either periodic orbits or a fixed point.

(b) Using the same $V$, show that no periodic orbits in $\Lambda$ persist for small $a$ and $b$ if $a b<0$.

[Hint: for $a=b=0$ on the periodic orbits with period $T$, show that $\int_{0}^{T}(1-x) d t=0$ and hence that $\int_{0}^{T} x(1-x) d t=\int_{0}^{T}\left[-(1-x)^{2}+(1-x)\right] d t<0$.]

(c) By considering Dulac's criterion with $\phi=1 /(x y)$, show that there are no periodic orbits in $\Lambda$ if $a b<0$.

(d) Purely by consideration of the existence of fixed points in $\Lambda$ and their Poincaré indices, determine those $(a, b)$ for which the possibility of periodic orbits can be excluded.

(e) Combining the results above, sketch the $a-b$ plane showing where periodic orbits in $\Lambda$ might still be possible.