Practice question

A defining assumption of exponential growth is that the instantaneous per-capita rate, (1/N)(dN/dt), equals a constant r. This means each individual contributes the same expected net amount to population growth regardless of current abundance. The total or absolute growth rate is not constant: dN/dt = rN becomes larger as N increases when r is positive. Confusing these two rates obscures why the curve accelerates. Constant per-capita performance requires births and deaths per individual to remain unchanged, which approximates an environment without density-dependent competition, disease transmission, or resource depletion. If those processes intensify with density, r effectively declines and growth becomes nonexponential. A zero per-capita rate would produce no net growth, while a decreasing rate is characteristic of regulated growth toward carrying capacity. Environmental variation can also make r fluctuate through time even at low density, so the strict model is an idealization. Its central mathematical feature remains proportionality: doubling N doubles the expected absolute increment because the same per-capita rate acts on twice as many organisms.