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Let $q_i>0$ measure the relative trapping vulnerability and use fixed species-specific catches $q_{i}H$. With no natural interaction, the two populations satisfy independent constant-quota harvested logistic growth equations,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 levelFor $0<H<H_{c,i}$ there are two positive equilibria,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 andInitial 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.