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Linearization of the Lorenz model of thermosolutal convection at conduction yields $(\lambda +\sigma )(\lambda +1)(\lambda +\tau )-\sigma r(\lambda +\tau )+\sigma r_s(\lambda +1)=0$. A Hopf bifurcation requires the cubic coefficient identity $a_3=a_1{a}_2$ and positive frequency squared. Its threshold is the stated $r_H$, with ${\omega }^2=\sigma r_s(1-\tau )/(\sigma +1)-{\tau }^2>0$. If $r_s={\beta }^2\tau$ and $\delta =\sigma /(1+\sigma )$, then $r_H=1+\tau /\delta +{\beta }^2\tau \delta +O({\tau }^2)$. Comparing with the steady fold of a thermosolutal Lorenz model gives a leading nonnegative difference $\tau (\beta \delta -1{)}^2/\delta$ for $\beta \ge 0$.

Set the time derivatives to zero. The scalar equations giveFor the nonzero convection branch, division of the velocity equation by $a$ givesIts zero-amplitude limit is $r_e=1+r_s/\tau$. Differentiation yieldsFor $0<\tau <1$, the sharp condition for a subcritical bifurcation at the steady onset is $r_s>{\tau }^3/(1-{\tau }^2)$. The stronger bound printed in the question is sufficient, since $(1-\tau {)}^2<1-{\tau }^2$. Under it the branch initially runs toward smaller $r$, and the unique saddle-node bifurcation hasIt is a genuine positive-amplitude minimum: $r^{\prime \prime }(x)=2r_s\tau (1-{\tau }^2)/({\tau }^2+x{)}^3>0$. Put $h=\sqrt{r_s\tau }$ and $j=\sqrt{1-{\tau }^2}$. At the minimum, $1+x_{\ast }=j(j+h)$ and ${\tau }^2+x_{\ast }=hj$, soThese results describe the steady fold of a thermosolutal Lorenz model. They do not by themselves assert stability against every oscillatory disturbance.