Practice question

In the exponential model dN/dt = rN, population size N is positive, so the sign of dN/dt is determined by r. When r exceeds zero, the instantaneous change is positive and abundance increases according to Nt = N0e^(rt). A larger r produces more rapid proportional growth and a shorter doubling time. If r equals zero, e^(rt) equals one and abundance remains constant; if r is negative, the exponential factor declines with time. This interpretation assumes that r summarizes the net per-capita effects included in the model, commonly births minus deaths in a closed population. Immigration and emigration can alter observed local abundance if they are not incorporated. Positive r also does not imply indefinite real-world increase. Density-dependent competition, predators, pathogens, or resource depletion may later reduce the effective growth rate. Within the stated exponential phase, however, r is treated as constant and positive, making increase the necessary mathematical outcome. Extinction is associated with sustained negative growth or stochastic loss, not a positive intrinsic rate.