Current regulator static analysis
simulate this circuit &ndash; Schematic created using CircuitLab

There really can write many equations for a circuit.

For Q1's base, we can write equations below, it's a bit different from yours:

$$
V_{be} = (I_{D}-I_{B})R_{s}-I_{B}R_{b}\quad(1)\\
I_{D} = \frac{V_{be}+I_{B}(R_{b}+R_{s})}{R_{s}}=\frac{V_{be}}{R_{s}}+I_{B}(\frac{R_{b}}{R_{s}}+1)\quad(2)\\
I_{B} = ((I_{D}-I_{B})R_{s}-V_{be}) / R_{b}\quad(3)
$$

For M1 we can write, assume M1 work in saturated region

$$
V_{GS}=V_{2}-I_{B}\beta R-(I_{D}-I_{B})R_{s} &gt; V_{TN}\quad(4)\\
I_{D}=K_{n}(V_{GS}-V_{TN})^{2}\quad(5)\\
V_{1}-V_{D1}-(V_{be}+I_{B}R_{b}) &gt; V_{GS} - V_{TN}\quad(6)
$$

Substitute (3) into (4), then differentiate it by \$I_{D}\$, \$R_{b}\$ can reduce the \$V_{GS}\$ change rate caused by \$I_{D}\$ change.

$$
\frac{dV_{GS}}{dI_{D}}=\frac{R_{s}^2-RR_{s}\beta}{R_{b}}-R_{s}
$$

Then you can do more analysis based on these equations, it's really tedious work!!!! If you really want to know how one component's change will affect your circuit behavior, you can go for help from simulators, such as sensitivity analysis in PSpice.