Lotka-Volterra Equations
Lotka-Volterra Equations
Author
Author
Eric W. Weisstein
May 18, 2007
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
Analytic
V4.2
V4.2
V5.0
V5.0
V6
V6
DSolve[{x'[t]ax[t]-bx[t]y[t],y'[t]-cy[t]+dx[t]y[t]},{x[t],y[t]},t]
InverseFunction::ifun:Inverse functions are being used. Values may be lost for multivalued inverses.
Solve::ifun:Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
Solve::tdep:The equations appear to involve the variables to be solved for in an essentially non-algebraic way.
Solve::tdep:The equations appear to involve the variables to be solved for in an essentially non-algebraic way.
y[t]-aProductLog-b,x[t]InverseFunctionK[1]&[at+C[2]]
1
b
1
a
-+
C[1]
a
dInverseFunctionK[1]&[at+C[2]]
#1
∫
1
1
K[1]1+ProductLog-
b
-C[1]+dK[1]
a
-
c
a
K[1]
a
a
-
c
a
InverseFunctionK[1]&[at+C[2]]
#1
∫
1
1
K[1]1+ProductLog-
b
-C[1]+dK[1]
a
-
c
a
K[1]
a
#1
∫
1
1
K[1]1+ProductLog-
b
-C[1]+dK[1]
a
-
c
a
K[1]
a
V7
V7
DSolve[{x'[t]ax[t]-bx[t]y[t],y'[t]-cy[t]+dx[t]y[t]},{x[t],y[t]},t]
InverseFunction::ifun:Inverse functions are being used. Values may be lost for multivalued inverses.
Solve::ifun:Inverse functions are being used by \*StyleBox[Solve, MT]\)\*StyleBox[", so some solutions may not be found; use Reduce for complete solution information.", MT]\) \*ButtonBox[, ButtonStyle->Link, ButtonFrame->None, ButtonData:>paclet:ref/message/Solve/ifun, ButtonNote -> Solve::ifun]\)
Solve::tdep:The equations appear to involve the variables to be solved for in an essentially non-algebraic way.
Solve::tdep:The equations appear to involve the variables to be solved for in an essentially non-algebraic way.
y[t]-aProductLog-b,x[t]InverseFunctionK[1]&[at+C[2]]
1
b
1
a
-+
C[1]
a
dInverseFunctionK[1]&[at+C[2]]
#1
∫
1
1
K[1]1+ProductLog-
b
-C[1]+dK[1]
a
-
c
a
K[1]
a
a
-
c
a
InverseFunctionK[1]&[at+C[2]]
#1
∫
1
1
K[1]1+ProductLog-
b
-C[1]+dK[1]
a
-
c
a
K[1]
a
#1
∫
1
1
K[1]1+ProductLog-
b
-C[1]+dK[1]
a
-
c
a
K[1]
a
V11.3
V11.3
$Version
11.3.0 for Mac OS X x86 (64-bit) (November 20, 2017)
soln=DSolve[eqn={x'[t]ax[t]-bx[t]y[t],y'[t]-cy[t]+dx[t]y[t]},{x[t],y[t]},t]
y[t]-aProductLog-b,x[t]InverseFunctionK[1]&[at+C[2]]
1
b
1
a
-+
C[1]
a
dInverseFunctionK[1]&[at+C[2]]
#1
∫
1
1
K[1]1+ProductLog-
b
-+
C[1]
a
dK[1]
a
-
c
a
K[1]
a
a
-
c
a
InverseFunctionK[1]&[at+C[2]]
#1
∫
1
1
K[1]1+ProductLog-
b
-+
C[1]
a
dK[1]
a
-
c
a
K[1]
a
#1
∫
1
1
K[1]1+ProductLog-
b
-+
C[1]
a
dK[1]
a
-
c
a
K[1]
a
Numeric
Numeric
Static Image
Static Image
Show[GraphicsArray[Plot[Evaluate[{#[[1]][t],#[[2]][t]}&[LotkaVolterra[Sequence@@#]]],{t,0,Last[#]},AxesLabelTraditionalForm/@{t,{StyleForm[x[t],FontColorRed],StyleForm[y[t],FontColorBlue]}},PlotLabelTraditionalForm[parms[#[[1]]]],PlotRangeAll,DisplayFunctionIdentity,PlotStyle{Red,Blue}]&/@{{{1.5,1,3,1},{10,5},20},{{1,1,1,1},{3,5},40}}],ImageSize500,GraphicsSpacing-.05]
⁃GraphicsArray⁃
Dynamic
Dynamic
Manipulate[Plot[Evaluate[{#[[1]][t],#[[2]][t]}&[LotkaVolterra[{a,b,c,d},{prey0,preditor0},tmax]]],{t,0,tmax},AxesLabelTraditionalForm/@{t,{StyleForm[prey[t],FontColorRed],StyleForm[predator[t],FontColorBlue]}},PlotLabelTraditionalForm[parms[{a,b,c,d}]],PlotRangeAll,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}]
