How do exponential and logistic growth models differ?

A growing population can add more individuals each year simply because more individuals are reproducing. Exponential growth captures this effect by holding the net per-capita growth rate constant. With a positive rate, the same proportional increase acts on a larger starting population, even though births and deaths both contribute to the net change.

That assumption can become unsuitable when crowding changes conditions. Food or space per individual may fall, competition may increase, or infection may spread more readily. Birth rates can decrease or death rates increase. The logistic model represents such density dependence by reducing net per-capita growth as population size rises.

Its equation is dN/dt = rN(1 − N/K). Here N is population size, r is the low-density growth parameter, and K is a fixed carrying-capacity parameter. At N = K/2, the density factor halves the modeled per-capita rate. At N = K, net growth is zero: this need not mean births and deaths have stopped, only that their net effect balances.

The illustration starts both models at ten units, with r = ln(2) and K = 80. The exponential population doubles each time unit and reaches 80 at time three. Logistic growth is already reduced below the exponential rate at the starting size, so its value at time three is about 42.67. The curves are continuous mathematical sizes rather than a rule adding fixed numbers each round.

A habitat's resources, weather and age structure can change, so a permanently fixed carrying capacity is an assumption rather than a universal feature. Delayed responses can also produce behavior this equation misses. Logistic growth makes density-dependent slowing interpretable, allowing a useful comparison with unlimited proportional growth without turning either curve into a population forecast.