= Solution
Write $c=(1-a-b)/a$. The admissible <equilibrium points> are $O=(0,0)$, $E=(1-a,0)$ when $a\le1$, and $P=(b,c)$ when $a+b\le1$. The <Jacobian matrix> is
$$
J(x,y)=\begin{pmatrix}(1-a-2x)/a-y&-x\\y/b&x/b-1\end{pmatrix}.
$$
At $O$ the <eigenvalues> are $(1-a)/a$ and $-1$; at $E$ they are $-(1-a)/a$ and $(1-a-b)/b$. At a positive coexistence <equilibrium point>,
$$
J(P)=\begin{pmatrix}-b/a&-b\\c/b&0\end{pmatrix},\quad
\operatorname{tr}J=-b/a<0,\quad\det J=c>0.
$$
It is therefore asymptotically stable. It is a node for $b^2>4a(1-a-b)$, a focus for the reverse inequality, and has repeated real <eigenvalues> at equality. The node–focus change does not alter the attracting <equilibrium point>'s topology.
There are three open physical parameter regions:
* \b[$a>1$: extinction.] Only $O$ is admissible and it is a sink.
* \b[$0<a<1$, $a+b>1$: predator extinction.] $O$ is a saddle, and $E$ is a sink.
* \b[$a+b<1$: coexistence.] $O$ and $E$ are saddles on the boundary, while $P$ is a sink.
The axes are invariant. On $x=0$, $y$ decays exponentially; on $y=0$, prey obey the <logistic differential equation>. Positivity follows because each right-hand side has its population as a factor. Also $x$ is bounded by logistic comparison, while $W=x+by$ satisfies $\dot W=(1-a)x/a-x^2/a-by$, so all forward trajectories are bounded. The <Dulac function> $1/(xy)$ has divergence $-1/(ay)<0$ in the interior, excluding <periodic orbits> by the <Bendixson-Dulac criterion>.
In the extinction region, $\dot x\le-(a-1)x/a$, and then $\dot y/y=x/b-1$ is eventually negative. In the prey-only region, the upper bound $\limsup x\le1-a<b$ implies predator decay and convergence of interior prey to $1-a$. In the coexistence region the <Lyapunov function>
$$
\mathcal V=x-b-b\log(x/b)+b\big[y-c-c\log(y/c)\big]
$$
has $\dot{\mathcal V}=-(x-b)^2/a\le0$. Its positive-quadrant sublevel sets are compact, and the only invariant subset with $x=b$ also has $y=c$. The <LaSalle invariance principle> proves convergence of every interior trajectory to $P$. These facts justify the complete qualitative <phase portraits>, not just their local arrows.
At $a=1$, $O$ and $E$ meet in a <transcritical bifurcation>; positive prey decay algebraically. At $a+b=1$ with $a<1$, $E$ and $P$ exchange stability. On its <centre manifold>, $x=b-ay+O(y^2)$ gives $\dot y=-ay^2/b+O(y^3)$, so the physically positive side is still attracting. There is no interior <Hopf bifurcation> since the coexistence <trace> never vanishes.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-65-predator-prey.png]
{title=Three continuous predator–prey phase portraits and their parameter regions}
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