Logistic Equation

Author

Eric W. Weisstein
May 3, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/LogisticEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LogisticEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Continuous

N'[t]==((r N[t] (K-N[t]))/K)

((N'[t])/K)==((r N[t] (1-((N[t])/K)))/K)

Letting x=(n/k)

x'(t)==r x(t) (1-x(t))

soln=x[t]/.DSolve[{x'[t]==r x[t](1-x[t]),x[0]==x0},x[t],t][[1]]

soln2=(( 1)/(1+(E^(-r t))((1/x0)-1)))

Simplify[soln-soln2]

Plots

Show[GraphicsArray[Partition[Table[
Plot[Evaluate[Table[(( x0)/(x0+(E^(-r t))(1-x0))),{x0,0,1,.05}]],{t,0,10},
DisplayFunction->Identity,
PlotLabel->TraditionalForm[HoldForm[r]==PaddedForm[r,{2,1}]],
AxesLabel->TraditionalForm/@{t,x[t]},PlotStyle->Hue/@Range[0,1,.05],PlotRange->All],
{r,-1,1.5,.5}],3]
],ImageSize->500,GraphicsSpacing->{-.07,.1}]