= Solution
A <steady state> $\mathbf c_*$ satisfies $0=A\mathbf c_*-\mathbf b$, hence
$$
A\mathbf c_*=\mathbf b.
$$
The sign is positive on the right after rearranging the supplied equation. A solution exists exactly when $\mathbf b$ lies in the <column space> of $A$, which above was found to be the zero-sum <vector subspace>. Equivalently, multiply by the <left null space> <vector> $\mathbf1^T$:
$$
\boxed{\mathbf1^T\mathbf b=b_1+b_2+b_3+b_4=0.}
$$
This condition is necessary and sufficient for an algebraic <steady state>, since both the <column space> and that zero-sum <vector subspace> have <dimension> $3$. It expresses global <conservation of mass>: in the closed <box model>, total tracer production must balance total tracer removal. With the paper's convention $-\mathbf b$, a positive $b_i$ is a sink and a negative $b_i$ is a source. If the sum is nonzero, $dM/dt=-\sum_i b_i$ and the inventory changes linearly in time, preventing a <steady state>.
When the condition holds, all algebraic <steady states> form the <affine solution space of a linear equation> $\mathbf c_p+\alpha\mathbf1$, for any particular solution $A\mathbf c_p=\mathbf b$. The initial inventory selects one member through $\mathbf1^T\mathbf c_* =\mathbf1^T\mathbf c(0)$. For a physically admissible solution, its four <concentrations> must additionally be nonnegative; the compatibility condition alone imposes no such inequalities.
Back to article page
