= Solution
For $a=b=0$, take the <Logarithmic first integral of the Lotka-Volterra equations>
$$
\boxed{V(x,y)=r(x-\log x-1)+y-\log y-1.}
$$
Along a trajectory,
$$
\begin{aligned}
\dot V
&=r\left(1-\frac1x\right)x(1-y)
+\left(1-\frac1y\right)ry(x-1)\\
&=r(x-1)(1-y)+r(y-1)(x-1)=0.
\end{aligned}
$$
Thus $V$ is a <first integral>. The scalar function $h(s)=s-\log s-1$ is nonnegative for $s>0$, vanishes only at $s=1$, and tends to infinity as $s\downarrow0$ or $s\to\infty$. Moreover,
$$
\nabla^2V=
\begin{pmatrix}r/x^2&0\\0&1/y^2\end{pmatrix}
$$
is positive definite, so $V$ is <strictly convex>. It follows that $(1,1)$ is its unique critical point and every level set $V=c>0$ is a compact smooth simple closed curve in the <positive quadrant>. The vector field is nonzero on such a level curve, so uniqueness of solutions makes each connected level curve one <periodic orbit>. The zero level consists of the fixed point $(1,1)$. \b[Hence every trajectory in $\Lambda$ is either periodic or the fixed point $(1,1)$.]
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