I need to find the positive equilibrium point of the system, Evaluate the Jacobi matrix, then determine the type and stability of the equilibrium.
The last three parts I can do (I think!), but we just began these non-linear systems and I need help getting started. Normally, I would set f(x,y)= 1-xy and g(x,y)= x-y^3 equal to zero, giving:
0=1-xy and 0=x-y^3, the latter gives x=y^3, subbing into the former gives y^4=1. I remember from precalc that gives me 4 roots: +/- 1, and +/- i. Now, the problem asks for the positive equilibrium, so I can discard the negative roots. y=1 gives x=1. If y=i, then x= -i. So that pair is not a positive equilibrium. Positive equilibrium at (x,y)=(1,1).
For the Jacobi matrix, I take the partial derivatives of f and g, and put them in their spots, which gives:
| (1-y) (1-x) |
| (1-y^3) (x-3y^2) |
plugging my equilibrium point in for x and y gives:
| 0 0 |
| 0 -2 | and taking this matrix and subtracting a (lambda)I matrix and taking the determinant, then solving for the roots gives lambda=0,2.
And now my brain hurts, because I don't know what to do with a zero root....