Boundary equilibria of multiplicative resource prices

In multiplicative resource-price dynamics, a zero price remains zero even if demand exceeds supply there. For one route using two resources, unit weight and supplies $q_j(u)=u$, both $(0,1)$ and $(1,0)$ are equilibria, while the unique positive equilibrium is $(1/\sqrt{2},1/\sqrt{2})$. Thus uniqueness on the positive price domain cannot be extended to every nonnegative price trajectory without an extra boundary rule.