Apply and reverse a state recurrence with external change
A fortnightly shelter model records animals at two care sites
Use the state order north shelter then south shelter. State matrices are columns. In T, columns are current states and rows are next states. One application represents one fortnight
The transition occurs first. The signed external-change column is applied afterwards. Positive entries of B add to a state and negative entries remove from a state
\[T=\begin{bmatrix} 0.65 & 0.2 \\ 0.35 & 0.8 \end{bmatrix}\]
\[S_{n+1}=T S_n+B\]
After each transition, an external intake adds 6 animals to south shelter
\[B=\begin{bmatrix} 0 \\ 6 \end{bmatrix}\]
\[S_{2}=\begin{bmatrix} 80 \\ 120 \end{bmatrix}\]