Practice question

When the per-capita birth rate exceeds the per-capita death rate, their difference r = b − d is greater than zero. In the exponential model dN/dt = rN, both N and r are then positive, making the population’s instantaneous change positive. The integrated trajectory, Nt = N0e^(rt), rises because e^(rt) exceeds one for positive time. The magnitude of r determines how rapidly abundance increases, not merely the direction of change. If birth and death rates were equal, r would be zero and expected abundance would remain constant; if deaths exceeded births, r would be negative and abundance would decline. This conclusion assumes a closed population or that migration has been omitted deliberately. In an open population, sufficiently strong emigration could cause local decline even when births exceed deaths, while immigration could offset a negative birth–death balance. The sign of r therefore expresses net demographic performance from the rates included in its definition. Calling it “constant” describes a model assumption and does not identify whether its numerical value is positive, zero, or negative.