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

Christopher Logan Hambric; Chi-Kwong Li; Diane Christine Pelejo; Junping Shi

doi:10.3934/dcdss.2021128

Article Contents

2022, Volume 15, Issue 9: 2497-2511. Doi: 10.3934/dcdss.2021128

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

1.
Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA
2.
Department of Mathematics, William & Mary, Williamsburg, VA 23187-8795, USA

^* Corresponding author: Junping Shi
^* Corresponding author: Junping Shi

Received: March 2021

Revised: July 2021

Early access: October 29, 2021

Published online: October 29, 2021

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

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Abstract

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.

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

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

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

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

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Table 4.

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