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

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