Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability

It is shown that for any positive integer \begin{document}$ n \ge 3 $\end{document} , there is a stable irreducible \begin{document}$ n\times n $\end{document} matrix \begin{document}$ A $\end{document} with \begin{document}$ 2n+1-\lfloor\frac{n}{3}\rfloor $\end{document} nonzero entries exhibiting Turing instability. Moreover, when \begin{document}$ n = 3 $\end{document} , the result is best possible, i.e., every \begin{document}$ 3\times 3 $\end{document} stable matrix with five or fewer nonzero entries will not exhibit Turing instability. Furthermore, we determine all possible \begin{document}$ 3\times 3 $\end{document} irreducible sign pattern matrices with 6 nonzero entries which can be realized by a matrix \begin{document}$ A $\end{document} that exhibits Turing instability.