Model exponential and logistic population growth. Adjust birth rate, death rate, and carrying capacity with animated population curves.
This simulator draws two population curves side by side: an exponential curve driven only by net growth rate, and a logistic curve that bends toward a carrying capacity. You set a starting population, birth rate, death rate, and, for the logistic model, a carrying capacity, and the tool animates both curves as time runs. The input that trips people up most is the death rate: growth rate is birth minus death, so a species with 4% births and 3% deaths grows at 1%, not 4%. Zeroing the death rate quietly doubles any projection.
A concrete input and expected output from the current implementation.
Input
Starting population 100; birth rate 3% per year; death rate 1% per year; carrying capacity 1000; run 50 years
Expected output
Exponential: 272 at year 50, climbing at about 5.4 per year. Logistic: 232 at year 50, climbing at about 3.6 per year. Both curves start together at 100 and the gap between them grows every year.
Net growth is r = 0.03 - 0.01 = 0.02 per year, so the exponential model gives 100 x e^(0.02 x 50) = 100 x e^1, about 272. The logistic model applies the density factor (1 - N/K) at every moment, giving 1000 / (1 + 9 x e^(-1)), about 232, with a slope already below the exponential curve.
N(t) = N0 x e^(rt) with r = birth - death. Positive r means unbounded compounding: the population doubles every ln(2)/r years, about 35 years at 2% per year. Nothing in the equation slows growth, so over long runs the curve passes any finite cap.
N(t) = K / (1 + ((K - N0)/N0) x e^(-rt)). The carrying capacity K caps the curve; growth peaks at the inflection point N = K/2 and tapers toward zero as N nears K, producing the S-shape. Start above K and the curve instead declines toward K.
Net growth r = 0, so both curves stay flat at the starting population forever. The equilibrium is fragile: a one-point change in either rate tips the curves into steady growth or decline. This is also the only setting where the exponential and logistic curves coincide.
The logistic model applies density pressure from the first moment: its growth rate is rN(1 - N/K), always smaller than the exponential rate rN while N is below K. With identical inputs the logistic curve therefore trails the exponential one at every instant, and the gap only widens.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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