Phase Portrait of Lotka-Volterra Equation

​
parameters
a
1
b
1
h
1
k
1
critical points (CP)
(
x
1
,
y
1
) = {0,0}
(
x
2
,
y
2
) = {h/b, k/a}
=
{1,1}
1
2
eigenvalues (λ, μ)
at (
x
1
,
y
1
) : (​
λ
1
​,​
μ
1
​) = {-h,k} =
{-1,1}
at (
x
2
,
y
2
 : (​
λ
2
​,​
μ
2
​) = -
hk
​, 
hk
​​ =
{-,}
stability behavior
at (
x
1
,
y
1
)  saddle point
at (
x
2
,
y
2
)  center
This Demonstration shows a phase portrait of the Lotka–Volterra equations, including the critical points. The eigenvalues at the critical points are also calculated, and the stability of the system with respect to the varying parameters is characterized.

Details

The Lotka–Volterra equations are a pair of coupled differential equations
x
t
=ax-bxy
,
y
t
=-hy+kxy
describing the dynamics of predator-prey populations,
y
and
x
, respectively.

References

[1] Wikipedia. "Lotka–Volterra Equations." (Jun 20, 2017) en.wikipedia.org/wiki/Lotka-Volterra_equations.

External Links

Lotka–Volterra Equations (Wolfram MathWorld)

Permanent Citation

Wusu Ashiribo Senapon, Akanbi Moses Adebowale
​
​"Phase Portrait of Lotka-Volterra Equation"​
​http://demonstrations.wolfram.com/PhasePortraitOfLotkaVolterraEquation/​
​Wolfram Demonstrations Project​
​Published: June 21, 2017