Practice question

With a positive constant r, exponential growth follows Nt = N0e^(rt). The exponential term increases slowly at first and then ever more rapidly because each new individual contributes to subsequent reproduction. On a graph with arithmetic population size on the vertical axis and time on the horizontal axis, this compounding produces a J-shaped curve. A sigmoid or S-shaped curve instead emerges when density dependence lowers per-capita growth as N approaches carrying capacity. Exponential growth contains no such stabilizing feedback, so it neither levels off nor approaches an equilibrium within the model. A decline would require r below zero, while stabilization requires r equal to zero or balancing regulation. Natural populations can display a J-shaped phase after entering a resource-rich environment, but unlimited continuation is unrealistic. Resource depletion, waste accumulation, predation, disease, or competition eventually alters demographic rates, causing deceleration or collapse. Thus the J shape represents the mathematical consequence of constant proportional increase, not a guarantee that a real population can grow without bound.