Biology and other natural sciences Example

Logistic equation

Builds on Initial value problem Exponential growth and decay

Example

Let $r > 0$ and $K > 0$ be constants. The logistic equation is $P'(t) = r\, P(t) \left(1 - \frac{P(t)}{K}\right).$ For an initial value $P(0) = P_0 > 0$ , its solution is $P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right) e^{-r t}}, \qquad t \ge 0.$ With $r = K = 1$ , the function $f(t) = 1 / \left(1 + e^{-t}\right)$ is the standard logistic function. It satisfies $f' = f(1 - f)$ and $f(0) = 1/2$ .

Remarks

The constant functions $P = 0$ and $P = K$ are the equilibrium points. For each $P_0 > 0$ , $P(t) \to K$ as $t \to \infty$ . For small $P$ , the equation is close to $P' = r P$ , which is exponential growth. If $0 < P_0 < K/2$ , the graph of $P$ is an S-shaped curve with its inflection point at the value $P = K/2$ . The equation is separable, and it is a Bernoulli equation with exponent $2$ .

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
Initial value problem definition
Exponential growth and decay example
Logistic equation example

Where this leads

8 pages build on this one.

Tag	`0CC4VC7KR3`
URI	`m2://0CC4VC7KR3`
Aliases	`verhulst-equation`
Kind	example
Writing	W2 usable
MSC	92D25
Instance of	Separable equation, Bernoulli equation
Uses	Initial value problem, Exponential growth and decay

Logistic equation

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