Practice question

A logistic population produces its greatest natural surplus at K/2. Harvesting that increment gives the theoretical maximum yield because dN/dt = rN(1 - N/K) peaks at half carrying capacity. At K the population has no net surplus, while densities much below K/2 contain too few reproductive individuals to generate the maximum increment. Graphically, logistic abundance through time is sigmoid, whereas its instantaneous increment plotted against abundance is a downward-opening parabola. Confusing these two plots leads to incorrect curve descriptions. The time trajectory approaches K, while the production curve peaks at K/2 and is zero at both model equilibria. This causal chain is what makes the keyed content ecologically meaningful rather than merely definitional. This interpretation connects individual-level processes with measurable changes in survival, reproduction, recruitment, or abundance across the population. Field evidence should therefore be compared with the model assumptions before extending the conclusion to every species, habitat, or time period.