Article Contents
Article Contents

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

• * Corresponding author: Junping Shi

Partially supported by NSF grants DMS-1331021, DMS-1853598, and Simons Foundation grant 351047

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

Mathematics Subject Classification: Primary: 15B99, 05C20; Secondary: 15B35.

 Citation:

• Table 1.  List of potential digraphs with $3$ vertices and $5$ edges

Table 2.  List of potential digraphs with 3 vertices and 6 edges

Table 3.  Nonequivalent sign patterns that are PETI (potentially exhibiting Turing Instability)

Table 4.

Table 5.

Table 6.

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