Lotka-Volterra Equations

Author

Eric W. Weisstein
May 18, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/Lotka-VolterraEquations.nb.

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Analytic

V4.2

DSolve[{x'[t]==a x[t]-b x[t]y[t],y'[t]==-c y[t]+d x[t]y[t]},{x[t],y[t]},t]

V5.0

DSolve[{x'[t]==a x[t]-b x[t]y[t],y'[t]==-c y[t]+d x[t]y[t]},{x[t],y[t]},t]

V6

DSolve[{x'[t]==a x[t]-b x[t]y[t],y'[t]==-c y[t]+d x[t]y[t]},{x[t],y[t]},t]

V7

DSolve[{x'[t]==a x[t]-b x[t]y[t],y'[t]==-c y[t]+d x[t]y[t]},{x[t],y[t]},t]

V11.3

$Version

soln=DSolve[eqn={x'[t]==a x[t]-b x[t]y[t],y'[t]==-c y[t]+d x[t]y[t]},{x[t],y[t]},t]

Numeric

Static Image

Show[GraphicsArray[
Plot[Evaluate[{#[[1]][t],#[[2]][t]}&[LotkaVolterra[Sequence@@#]]],{t,0,Last[#]},
AxesLabel->TraditionalForm/@{t,{StyleForm[x[t],FontColor->Red],
StyleForm[y[t],FontColor->Blue]}},
PlotLabel->TraditionalForm[parms[#[[1]]]],
PlotRange->All,DisplayFunction->Identity,PlotStyle->{Red,Blue}]&/@{
{{1.5,1,3,1},{10,5},20},
{{1,1,1,1},{3,5},40}
}
],ImageSize->500,GraphicsSpacing->-.05]

Dynamic

Manipulate[
Plot[Evaluate[{#[[1]][t],#[[2]][t]}&[LotkaVolterra[{a,b,c,d},{prey0,preditor0},tmax]]],{t,0,tmax},
AxesLabel->TraditionalForm/@{t,{StyleForm[prey[t],FontColor->Red],
StyleForm[predator[t],FontColor->Blue]}},
PlotLabel->TraditionalForm[parms[{a,b,c,d}]],
PlotRange->All,PlotStyle->{Red,Blue}],
{{a,1},-5,5},{{b,1},-5,5},{{c,1},-5,5},{{d,1},-5,5},{{prey0,3},0,20},{{preditor0,5},0,20},{{tmax,40},.01,50}]