# Rabbit wolf populations and eigenvalues

1. Nov 26, 2011

### stylophora

1. The problem statement, all variables and given/known data

We are initially given the system:

$$\frac{dr}{dt} = 5r -2w$$
$$\frac{dw}{dt} = r + 2w$$

Initially there are 100 rabbits and 50 wolves.

Where the above corresponds to rabbit and wolf populations over time. I solved that system of equations to find the population of rabbits and wolves. Now we want to design a matrix, A such that the populations converge to a finite non-zero limit as t goes to infinity.

2. Relevant equations

I solved the differential equations using:
$$x = YDY^{-1}x_0$$
where D is a diagonal matrix with entries:
$$e^{\lambda_kt}$$

3. The attempt at a solution

The matrix I want to design is based on the eigenvalues. I understand that any positive eigenvalue means that the system will blow up as t increases. But two negative eigenvalues means that the system converges to 0 with increasing t.

Of course I could use a simple diagonal 0 and a negative eigenvalue but this implies that the populations are not influencing each other which is not what the question seemed to imply. Can someone lead me along the right track here?