= Reservoir-fed magma intrusion
Let $V_T$ be total magma volume shared between a reservoir and an advancing <toughness-controlled magma intrusion>, $p_r=E(V_T-V)$, and conduit conductance $\beta>0$. With $A=\sqrt{8\pi/3}B\kappa_f^{3/2}$,
$$
\dot V=\beta[E(V_T-V)-A/\sqrt V].
$$
Equilibria satisfy $\sqrt V(V_T-V)=A/E$. Two positive equilibria exist when $A/E<2V_T^{3/2}/(3\sqrt3)$; the smaller is unstable and the larger is stable under this volume equation. An advancing solution between them approaches the larger. The singular pressure as $V\to0$ prevents this late-time model from nucleating an intrusion by itself.
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