= Solution
Let $q_i>0$ measure the relative trapping vulnerability and use fixed species-specific catches $q_iH$. With no natural interaction, the two populations satisfy independent <constant-quota harvested logistic growth> equations,
$$
\dot N_i=r_iN_i\left(1-\frac{N_i}{K_i}\right)-q_iH,
\qquad i=1,2.
$$
This treats $H$ as a common harvest scale: a total fixed quota can equivalently be allocated in fixed fractions $q_1+q_2=1$. Catches are not reassigned after one species disappears. <Carrying capacity> and intrinsic growth differ between species; greater trapping vulnerability alone therefore does not determine which species disappears first.
The maximum natural surplus occurs at $N_i=K_i/2$. Define the critical harvesting level
$$
\boxed{H_{c,i}=\frac{r_iK_i}{4q_i}.}
$$
For $0<H<H_{c,i}$ there are two positive <equilibria>,
$$
\boxed{N_{i,\pm}=\frac{K_i}{2}
\left(1\pm\sqrt{1-H/H_{c,i}}\right).}
$$
The <derivative> of the population vector field is negative at the upper branch and positive at the lower branch. The upper branch is attracting; the lower is an unstable survival threshold. An initial abundance above the lower branch approaches the upper branch, whereas one below it reaches extinction. At $H=0$, positive populations approach $K_i$, while zero remains zero.
At $H=H_{c,i}$ the branches meet in a <saddle-node bifurcation> and
$$
\dot N_i=-\frac{r_i}{K_i}(N_i-K_i/2)^2.
$$
Initial abundances above $K_i/2$ approach it algebraically from above; those below decline to extinction. For $H>H_{c,i}$ there is no positive <equilibrium> and the <derivative> of abundance is everywhere negative. In fact $\dot N_i\le-q_i(H-H_{c,i})$, giving finite extinction time. At zero the biological model stops harvesting and keeps the population extinct rather than extending the constant-quota equation to negative abundance.
Order the thresholds as $H_{c,1}<H_{c,2}$. Below the first threshold, both species can persist if each starts above its own survival threshold; either or both can still disappear from inadequate initial abundance. Between the thresholds only species 2 can persist. Above the second, neither can persist. At each threshold use the one-sided critical behaviour just derived. Equal thresholds give simultaneous loss of both positive branches. Because the equations are independent, there are no sustained oscillations or competitive replacements in this model.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-52-harvest-equilibria.png]
{title=Stable and unstable <equilibrium> branches of two independently harvested logistic populations}
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