My instructions were to find equilibrium values (the picture I added is only showing E0, I was hoping if I got it figured out I could do the others rather than someone try to do all of them for me), which my professor said to set the equations equal to zero and solve, form a jacobian matrix using all partials, take the jacobian using equilibrium values to find eigenvalues. I've done none of these things in any math class and my professor said he knew that, but I feel confident I can do the jacobian matrices if someone would explain to me what I do with these initial equations to get them into a jacobian. Thank you in advance, especially if you explain rather than just post an answer because I want to learn what is happening 🙂
Best Answer
The Jacobian is the just the matrix of partial derivatives. You can compute it row-by-row. For example the first equation is: $$f_1(T,D,C) = \lambda -\mu T - \beta T C,$$
which has partial derivatives, \begin{align} \frac{d f_1}{dT} &= -\mu -\beta C \\ \frac{d f_1}{d D} &= 0 \\ \frac{d f_1}{d C} &= -\beta T. \end{align}
This gives us the first row of the Jacobian matrix: $$J(T,D,C) = \begin{bmatrix} -\mu -\beta C & 0 & -\beta T \\ &\text{second row of Jacobian} \\ &\text{third row...} \end{bmatrix}$$.
It is a matrix that depends on the parameters $T,D,C$. So you can plug in the specific values $(T,D,C) = (T_0, 0, 0)$ to get: $$J(T_0,0,0) = \begin{bmatrix} -\mu -\beta \cdot 0 & 0 & -\beta T_0 \\ &\dots \\ &\dots \end{bmatrix}$$.
Hopefully you can do the rest of the problem, seeing how it is done for the first row.
Jacobians are a very interesting concept - if you have time you should consider looking more into them to understand the meaning, which can be obscured if you focus too much on computation. Heres a picture I made showing what the Jacobian matrix means geometrically for a system with two equations and two variables: