Practice question

The absolute increase dN/dt under logistic growth is greatest when N equals half the carrying capacity. This follows from the product N(1 - N/K): one factor increases with N while the other decreases, and their product peaks at K/2. At very low N there are too few reproducers; at K, density limitation reduces net increase to zero. 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. The decisive distinction is therefore between a descriptive label and the demographic mechanism that generates it. Interpreting the example at the appropriate population scale keeps the causal mechanism distinct from a simple correlation or an absolute rule. Ecological predictions remain conditional on the stated environment, because changing resources, mortality, or interactions can alter the observed demographic pattern.