Rabbit wolf populations and eigenvalues

[stylophora]

Homework Statement

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.

Homework Equations

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

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?

 

[mighty2000]

Hello there,

I would suggest approaching this problem using the concept of stability in dynamical systems. In order for the populations to converge to a finite non-zero limit, the system must be stable. This means that small perturbations in the initial conditions should not result in large changes in the populations over time.

One way to achieve stability is by designing a matrix A with negative eigenvalues, as you mentioned. However, as you correctly pointed out, this would imply that the populations are not influencing each other.

Another approach would be to design a matrix A that has a stable fixed point, where the populations of rabbits and wolves remain constant over time. This can be achieved by setting the eigenvalues to 0 and -1, which would result in a diagonal matrix with entries 1 and e^-t. This implies that the populations are influencing each other, but they are also stabilizing each other.

In summary, the key is to design a matrix A that results in a stable system, where small changes in the initial conditions do not result in large changes in the populations over time. I hope this helps guide you in the right direction. Good luck with your research!