Practice question

Exponential growth is formulated for continuous time, so it describes the instantaneous rate at which population size changes. Its differential equation is dN/dt = rN, where r is the instantaneous per-capita rate of increase. Because the absolute rate is proportional to current abundance, a larger population adds more individuals per unit time even when r remains constant. Integrating the equation gives Nt = N0e^(rt). Geometric growth instead advances through discrete intervals using a finite multiplier λ; it is appropriate for seasonal breeding or nonoverlapping generations. Linear growth would add the same absolute number in every interval, which is biologically different from adding a constant proportion. Stabilized growth implies zero net change or density regulation around an equilibrium. The word “instantaneous” is therefore decisive: derivatives describe change at an arbitrarily small moment, whereas finite ratios describe change between censuses. In practice, exponential growth is an ideal approximation valid when density dependence and resource limitation are negligible and the demographic rates summarized by r remain approximately constant.