Logistic Population Growth

Populations grow fast while there is room and slow down as they fill their environment. The result is an S-curve that levels off at the carrying capacity.

How it works

Separable ODEs

In one sentence

The logistic equation $P' = rP(1 - P/K)$ has the solution $P(t) = \dfrac{K}{1 + \frac{K - P_0}{P_0}e^{-rt}}$ : an S-curve that grows almost exponentially at first and levels off at the carrying capacity $K$ .

General formula

{\htmlData{sym=0}{\frac{dP}{dt}}} = {\htmlData{sym=2}{r}}{\htmlData{sym=1}{P}}\left({\htmlData{sym=3}{1 - \frac{P}{K}}}\right)

$rP$ is growth in proportion to the population; the factor $1 - P/K$ shrinks to zero as crowding sets in.

What the symbols mean

$\frac{dP}{dt}$
Growth rate
How many individuals the population gains per unit of time.
$P$
Population
How many individuals there are now.
$r$
Intrinsic growth rate
How fast the population grows while it is small and resources are plentiful.
$1 - \frac{P}{K}$
Room left to grow
K is the carrying capacity, the most the environment can support. This factor is the fraction still free, so growth slows to zero as P reaches K.

When $P$ is small, $1 - P/K \approx 1$ and growth looks exponential.
As $P$ approaches $K$ , the growth rate drops to zero.
Separating variables gives $P(t) = \dfrac{K}{1 + Ae^{-rt}}$ with $A = \dfrac{K - P_0}{P_0}$ .

Learn the math

Concavity and the Second Derivative TestCalc I · Lesson 31, Applications of Derivatives Partial FractionsCalc II · Lesson 4, Integration Techniques Direction FieldsDiff Eq · Lesson 3, Basic Concepts Separable EquationsDiff Eq · Lesson 5, First Order Differential Equations Bernoulli EquationsDiff Eq · Lesson 7, First Order Differential Equations Equilibrium SolutionsDiff Eq · Lesson 11, First Order Differential Equations

Learn the math

Concavity and the Second Derivative TestCalc I · Lesson 31, Applications of Derivatives

Partial FractionsCalc II · Lesson 4, Integration Techniques

Direction FieldsDiff Eq · Lesson 3, Basic Concepts

Separable EquationsDiff Eq · Lesson 5, First Order Differential Equations

Bernoulli EquationsDiff Eq · Lesson 7, First Order Differential Equations

Equilibrium SolutionsDiff Eq · Lesson 11, First Order Differential Equations