f 1 (T,D,C)=λ−μT−βTC,
f 2 (T,D,C)= βTC - αD, f 3 ( T , D , C ) = kD - γCGiven these functions, I've solved 1 of the 2 equilibria, (the simpler of the two), which is
I've included work from my presentation about it so that there is proof that I've actually tried:
Now I cannot get reasonable eigenvalues for J(E+) and I don't know how else to work it other what I've done for J(E0). If any other information is needed I will do my best to give it, these equations are based off models of human immune disease in a paper I'm proofing. I have not been formally taught anything to do with Jacobian Matrices, eigenvalues, or partial derivatives, I'm only doing this as part of augmenting my math class, so if I've made some beginner mistakes please go easy on me, and thank you for your time if you do try to explain to me what I've done wrong.
P.S: I didn't want to take up the entire screen with mathematica's results of trying to find the eigen values of J(E+), so here's the code if you're curious what wolfram alpha says it is: Eigenvalues[{-C [Beta]-[Mu],0,-T [Beta]},{C [Beta],-[Alpha],T [Beta]},{0,k,-[Gamma]}]
Edit for clarity:
The DEQ's are called T', D', and C' in the paper. I call them f1, f2, and f3 when taking the partials for the jacobian. For E0, it exists for T=T0, D=0, and C=0, so it's jacobian is missing the elements dependent on D or C.
You can convert J(E+) into an upper right triangular matrix using row operations,
And now the determinant is the product of the diagonal. Each diagonal element is one of the 3 eigenvalues for a 3x3 square matrix.