How this calculator works
The logistic equation was written down by Pierre Francois Verhulst in 1838 to describe a population that grows quickly when it is small but slows as it uses up food, space or nesting sites. The growth rate r sets how fast the population would grow with unlimited resources, and the carrying capacity K is the population at which births and deaths balance. The rate of change is rN(1 - N/K): when N is small the bracket is close to one and growth is close to exponential, and as N approaches K the bracket falls to zero. The largest possible yearly increase is rK/4, at exactly half of the carrying capacity. The calculator rearranges the same equation for whichever quantity you do not know. Solving for N gives a quadratic with two answers, because a small population growing quickly and a large one growing slowly can add the same number of individuals; both are reported, and the smaller one is placed in the main output.
Formula
dN/dt = r x N x (1 - N/K). Solving for K: K = N / (1 - change / (rN)). Solving for r: r = change / (N x (1 - N/K)). Solving for N: set (r/K)N^2 - rN + change = 0, so N = K/2 +/- (K/2) x sqrt(1 - 4 x change / (rK)). The largest annual increase is rK/4.
Worked example
A population of 500 with r = 0.2 a year and K = 1,000. The annual change is 0.2 x 500 x (1 - 0.5) = 50 individuals a year. At half of K that is the largest growth the model allows: 0.2 x 1,000 / 4 = 50. If instead the population added 40 a year at r = 0.2 with K = 1,000, the small solution is 1,000/2 - (1,000/2) x sqrt(1 - 4 x 40 / (0.2 x 1,000)) = 500 - 500 x 0.447 = 276 individuals, and the second solution is 724.
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Frequently asked questions
What is carrying capacity in plain terms?
It is the number of individuals an area can support indefinitely without the resource base degrading. It is not a hard ceiling that nature enforces cleanly: populations can overshoot it, damage the habitat, and then fall back below it.
Why are there two populations for one change?
The logistic curve is symmetric around half of K. A population below half grows slowly because there are few individuals, and one above half grows slowly because it is close to the limit. Both can add the same number in a year. The smaller value is reported first.
Is r a birth rate?
No. It is the net intrinsic rate of increase: births minus deaths under unlimited resources. A birth rate of 0.3 a year with a death rate of 0.1 gives r = 0.2. Time units must match the time unit of the change.
Does the model fit real populations?
It is a good first approximation for many single-species populations in a stable environment. It ignores age structure, migration, predation, disease and time lags, all of which can produce cycles or chaos. Use it to understand the shape of growth, not to forecast a real species precisely.