Lotka-Volterra Model

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November 25th 2008, 06:28 PM
rainy cloud

Lotka-Volterra Model

Hi everyone,

May I get help in the following question please?

Consider the competitive Lotka-Volterra species model:
dx/dt=x(4-2x-y)
dy/dt=y(9-3x-3y)
This model has 4 equilibrium points which are:
(0,0), (0,3), (2,0) and (1,2) .

My question is:
Suppose we are given an initial condition with a mix of species x and y, what do we expect about the long term future of the species?

Thank you in advance for any help.

RainyCloud
November 26th 2008, 01:03 PM
shawsend

Hi. You have to analyze it's phase space right? Linearize it, calculate the eigenvalues, determine the type of fixed points whether they're sources, centers, saddles, sink, and then based on the initial conditions, conclude what the long term behavior is. For example, if I have initial conditions close to a sink, I'd expect the long term behavior to be drawn into the sink and reach equilibrium. You can just do a few runs: choose any old initial conditions, run it through a numerical integrator and compare the numerical results to it's phase space portrait.

I'll do the equilibrium point at the origin:

The Jacobian is $J=\left|\begin{array}{cc} 4-4x & -x \\ -3x & 9-9y\end{array}\right|$

Then $J(0,0)=\left|\begin{array}{cc} 4 & 0 \\ 0 & 9\end{array}\right|$

The eigenvalue equation then is:

$\left|\begin{array}{cc} 4-\lambda & 0 \\ 0 & 9-\lambda\end{array}\right|$

The characteristic equation then is $(4-\lambda)(9-\lambda)=0$

The eigenvalues are thus 4 and 9. Both positive. This means the origin is a source: Initial points near it will move away from it.

Now need to to the others and see how they affect the dynamcis.
November 27th 2008, 04:17 AM
rainy cloud

Quote:

Originally Posted by shawsend

Hi. You have to analyze it's phase space right? Linearize it, calculate the eigenvalues, determine the type of fixed points whether they're sources, centers, saddles, sink, and then based on the initial conditions, conclude what the long term behavior is. For example, if I have initial conditions close to a sink, I'd expect the long term behavior to be drawn into the sink and reach equilibrium. You can just do a few runs: choose any old initial conditions, run it through a numerical integrator and compare the numerical results to it's phase space portrait.

I'll do the equilibrium point at the origin:

The Jacobian is $J=\left|\begin{array}{cc} 4-4x & -x \\ -3x & 9-9y\end{array}\right|$

Then $J(0,0)=\left|\begin{array}{cc} 4 & 0 \\ 0 & 9\end{array}\right|$

The eigenvalue equation then is:

$\left|\begin{array}{cc} 4-\lambda & 0 \\ 0 & 9-\lambda\end{array}\right|$

The characteristic equation then is $(4-\lambda)(9-\lambda)=0$

The eigenvalues are thus 4 and 9. Both positive. This means the origin is a source: Initial points near it will move away from it.

Now need to to the others and see how they affect the dynamcis.

Thank you very much Shawsend for help and valuable explanations.