Given the 3x3 linear ODE system describing the dynamic of toxic chemical exchanges in the body \begin{align*} \begin{cases} x'(t)&= -(k_{xy}+u) \cdot x(t) + k_{yx} \cdot y(t) + D\\ y'(t) &= k_{xy}\cdot x(t) -(k_{yx} + k_{yz} + \nu)\cdot y(t) + k_{zy} \cdot z(t)\\ z'(t) &= k_{yz} \cdot y(t) - k_{zy} \cdot z(t) \end{cases} \end{align*}
with $k_{xy}, k_{yx},k_{yz}, k_{zy}, \nu, u, D$ positive constant and $x,y,z$ the concentration in the blood, tissues and bones respectively.
I want to find the limit as $t \to \infty$ of $z(t)$
I can rewrite this system like \begin{align*} \begin{pmatrix} x'(t)\\ y'(t)\\ z'(t) \end{pmatrix} = \begin{pmatrix} -(k_{xy}+u) & k_{yx} & 0\\ k_{xy} & -(k_{yx}+ k_{yz}+\nu) & k_{zy}\\ 0 & k_{yz} & -k_{zy} \end{pmatrix} \begin{pmatrix} x(t)\\ y(t)\\ z(t) \end{pmatrix} + \begin{pmatrix} D \\ 0\\ 0 \end{pmatrix} \end{align*} I tried diagonalizing the matrix to find the general solution but it doesn't seem to give anything to work with. I was thinking about finding a fixed point but then I can't really say if it is stable or not.
Is there a way to find the limit without finding the general solution of the system? If not, how could I find the solution?