= Solution
The term $p$ represents prey reproduction and $-p^2/a$ limits growth by available food, giving carrying capacity $a$ in the absence of hunters. The loss $-ph/a$ represents encounters between prey and hunters. For hunters, $hp/(8b)$ is prey-supported growth and $-h/8$ is mortality; their per-capita growth changes sign at prey population $b$. The coordinate axes are invariant, and positive initial populations remain positive.
For $a=1$, the physical <equilibrium points> are the origin, $(1,0)$, and $(b,1-b)$ when $b<1$. The <Jacobian matrix> is
$$
J(p,h)=\begin{pmatrix}
1-2p-h&-p\\h/(8b)&(p/b-1)/8
\end{pmatrix}.
$$
At $(0,0)$ its <eigenvalues> are $1$ and $-1/8$, so the origin is always a <saddle equilibrium>, attracting along the hunter-only axis and repelling along the prey-only axis. At $(1,0)$ the <eigenvalues> are
$$
-1,\qquad\frac{1-b}{8b}.
$$
It is a <saddle equilibrium> for $0<b<1/2$ and an asymptotically <stable node> for $b>1$.
For $0<b<1/2$, the coexistence <Jacobian matrix> is
$$
J_* =\begin{pmatrix}-b&-b\\(1-b)/(8b)&0\end{pmatrix},
\qquad\lambda^2+b\lambda+\frac{1-b}{8}=0.
$$
Its <trace> is negative and <determinant> positive. Its discriminant $b^2-(1-b)/2=(2b-1)(b+1)/2$ is negative, so \b[$(b,1-b)$ is an asymptotically <stable focus>]. Interior trajectories approach it in damped, counterclockwise oscillations in the $(p,h)$ plane. Both populations ultimately coexist.
To justify the global interior fate for this <logistic predator-prey model>, let $h_*=1-b$ and use the <Lyapunov function>
$$
V=p-b-b\log(p/b)+8b\bigl[h-h_*-h_*\log(h/h_*)\bigr].
$$
It is nonnegative and its sublevel sets are compact inside the positive quadrant. Differentiation gives the explicit cancellation
$$
\dot V=(p-b)(1-p-h)+(h-h_*)(p-b)=-(p-b)^2.
$$
The only invariant subset on which this vanishes is $p=b,h=h_*$: remaining at $p=b$ requires $\dot p=b(1-b-h)=0$. <LaSalle's invariance principle> therefore proves that every strictly positive initial state converges to this coexistence point.
For $b>1$, the formal coexistence point has negative $h$ and is outside the physical state space. The only physical <equilibria> are the <saddle equilibrium> at the origin and the stable prey-only node. Since $\dot p\le p(1-p)$, positive prey has $\limsup p\le1$. Eventually $p<b$ with a fixed margin, so hunter population decays exponentially. When $h$ is sufficiently small, comparison with $\dot p=p(1-\varepsilon-p)$ gives $\liminf p\ge1-\varepsilon$ for every $\varepsilon>0$. Thus \b[$h\to0$ and $p\to1$] for positive initial prey: hunters die out and prey reaches carrying capacity.
On the boundary, hunter-only initial data approach the origin; prey-only positive data approach $(1,0)$ in either parameter range. The sketches show representative values in the two requested ranges. Dashed and dotted lines are the prey and hunter <nullclines>; filled dots are attractors and open dots are <saddle equilibria>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/ia/paper-2-population-portraits.png]
{title=Predator-prey phase portraits showing stable coexistence for b=0.25 and hunter extinction for b=1.5}
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