# Lotka-Volterra equations

• Jun 8th 2008, 04:45 AM
thermalwarrior
Lotka-Volterra equations
I have two first order differential equations of the following:

x'=pxy - qx
y'=rxy - sy

These two equations are meant to mimic two interacting populations.

Is it possible to rearrange them into the Lotka-Volterra equations to solve them for equilibrium points?

this would give:

x'=px(1 - y/Y)

y'=-ry(1 - x/X)

cheers
• Jun 8th 2008, 12:01 PM
CaptainBlack
Quote:

Originally Posted by thermalwarrior
I have two first order differential equations of the following:

x'=pxy - qx
y'=rxy - sy

x'=-qx(1-(p/q)y)
y'=-sy(1-(r/s)x)

So put Y=q/p and X=s/r for the equilibrium populations

RonL
• Jun 9th 2008, 02:36 AM
thermalwarrior
so was my rearrangement wrong?

Just im writing this out by hand showing my working and not sure if my working is correct
• Jun 9th 2008, 02:55 AM
CaptainBlack
Quote:

Originally Posted by thermalwarrior
so was my rearrangement wrong?

Just im writing this out by hand showing my working and not sure if my working is correct

You seemed to have gotten the p's and q's mixed up. But the solution I gave is the equilibrium (steady state solution) for the original coupled system you gave (its just the solution of x'=y'=0)

RonL
• Jun 9th 2008, 03:44 AM
thermalwarrior
x'=-qx(1 - Y/y)

y'=-sy(1 - X/x)

with equilibrium points at (0,0), (X,Y) ????
• Jun 9th 2008, 04:03 AM
thermalwarrior
[IMG]file:///C:/DOCUME%7E1/Paul/LOCALS%7E1/Temp/moz-screenshot-2.jpg[/IMG][IMG]file:///C:/DOCUME%7E1/Paul/LOCALS%7E1/Temp/moz-screenshot-3.jpg[/IMG][IMG]file:///C:/DOCUME%7E1/Paul/LOCALS%7E1/Temp/moz-screenshot-4.jpg[/IMG]also I have a computer question related to this question:

this is bit im having a problem with:

i need to enter into the program u(x,y).

u(x,y) = dx/dt where x(t) = [x(t) y(t)]^T

is it the equations from earlier in the question? or do I have to change the form of them???
cheers