= Solution
Put $P(N)=(a_1-N)(a_2-N)(a_3-N)$. The harvested <vector field> is $N[P(N)-E]$. Define the extinction threshold $E_T=P(0)=a_1a_2a_3$, and let $N_M\in(a_2,a_3)$ be the positive local maximum of $P$, with $E_M=P(N_M)>0$. Writing $s_1=a_1+a_2+a_3$, $s_2=a_1a_2+a_1a_3+a_2a_3$ gives
$$
N_M=\frac{s_1+\sqrt{s_1^2-3s_2}}3.
$$
Positive <equilibria> are intersections of $P(N)$ with the horizontal line $E$. A lower stable branch in $(0,a_1)$ exists exactly for $0<E<E_T$. A pair in $(a_2,a_3)$ exists exactly for $0<E<E_M$: its smaller member has $P'>0$ and is unstable; its larger member has $P'<0$ and is stable. At a positive root the linear <eigenvalue> is $NP'(N)$, which proves these claims. Extinction has <eigenvalue> $E_T-E$, becoming stable for $E>E_T$.
The <harvested bistable population thresholds> determine the behaviour. If $E_M<E_T$, the upper stable population and its unstable separator first disappear in a <saddle-node bifurcation>. Populations then tend to the remaining lower stable state until that branch reaches extinction at $E_T$; beyond it extinction attracts every positive initial state. If $E_T<E_M$, the lower stable branch reaches extinction first. For $E_T<E<E_M$, extinction and the upper positive state are both stable, separated by the middle unstable root. Increasing effort through $E_M$ then destroys the remaining positive <attractor>, causing a sudden collapse to extinction. If $E_T=E_M$, both events occur at the same effort at different populations.
At $E=E_T$, near zero the leading term is $P'(0)N^2<0$, so extinction attracts from the physical right side algebraically, despite its zero linear <eigenvalue>. At $E=E_M$, the double positive root is semistable: locally $\dot N\simeq-\tfrac12N_M|P''(N_M)|(N-N_M)^2$, attracting from above and repelling from below. These critical cases matter for exact initial data. The figure shows examples of both strict threshold orderings.
\Image[/past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2010-population-harvest.png]
{title=Two possible orderings of the extinction and saddle-node harvesting thresholds}
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