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

Abstract

<p style='text-indent:20px;'>It is shown that for any positive integer <inline-formula><tex-math id="M1">\begin{document}$ n \ge 3 $\end{document}</tex-math></inline-formula>, there is a stable irreducible <inline-formula><tex-math id="M2">\begin{document}$ n\times n $\end{document}</tex-math></inline-formula> matrix <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ 2n+1-\lfloor\frac{n}{3}\rfloor $\end{document}</tex-math></inline-formula> nonzero entries exhibiting Turing instability. Moreover, when <inline-formula><tex-math id="M5">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula>, the result is best possible, i.e., every <inline-formula><tex-math id="M6">\begin{document}$ 3\times 3 $\end{document}</tex-math></inline-formula> stable matrix with five or fewer nonzero entries will not exhibit Turing instability. Furthermore, we determine all possible <inline-formula><tex-math id="M7">\begin{document}$ 3\times 3 $\end{document}</tex-math></inline-formula> irreducible sign pattern matrices with 6 nonzero entries which can be realized by a matrix <inline-formula><tex-math id="M8">\begin{document}$ A $\end{document}</tex-math></inline-formula> that exhibits Turing instability.</p>