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doi: 10.3934/dcdss.2021128
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Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability

Christopher Logan Hambric 1, , Chi-Kwong Li 2, , Diane Christine Pelejo 2, and  Junping Shi 2,,

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

Received  March 2021 Revised  July 2021 Early access October 2021

Fund Project: 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.

Keywords: Stable matrices, minimum number of non-zero-entries, Turing instability, directed graphs, sign patterns.
Mathematics Subject Classification: Primary: 15B99, 05C20; Secondary: 15B35.
Citation: Christopher Logan Hambric, Chi-Kwong Li, Diane Christine Pelejo, Junping Shi. Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021128
References:
[1]

A. Anma, K. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247.  doi: 10.2996/kmj/1341401049.

[2]

R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, vol. 39 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9781107325708.

[3]

M. Cavers, Polynomial stability and potentially stable patterns, Linear Algebra Appl., 613 (2021), 87-114.  doi: 10.1016/j.laa.2020.12.015.

[4]

X. Diego, L. Marcon, P. Müller and and J. Sharpe, Key features of Turing systems are determined purely by network topology, Physical Review X, 8 (2018), 021071.  doi: 10.1103/PhysRevX.8.021071.

[5]

D. A. Grundy, D. D. Olesky and P. van den Driessche, Constructions for potentially stable sign patterns, Linear Algebra Appl., 436 (2012), 4473-4488.  doi: 10.1016/j.laa.2011.08.011.

[6]

C. L. Hambric, C.-K. Li, D. C. Pelejo and J. Shi, Minimum number of non-zero-entries in a $7\times 7$ stable matrix, Linear Algebra Appl., 572 (2019), 135-152.  doi: 10.1016/j.laa.2019.03.002.

[7]

C. R. Johnson, J. S. Maybee, D. D. Olesky and P. van den Driessche, Nested sequences of principal minors and potential stability, Linear Algebra Appl., 262 (1997), 243-257.  doi: 10.1016/S0024-3795(97)80034-1.

[8]

A. N. Landge, B. M. Jordan, X. Diego and P. Müller, Pattern formation mechanisms of self-organizing reaction-diffusion systems, Developmental Biology, 460 (2020), 2-11.  doi: 10.1016/j.ydbio.2019.10.031.

[9]

A. Liénard and M. Chipart, Sur le signe de la partie réelle des racines dúne équation algébrique, J. Math. Pures Appl., 10 (1914), 291-346. 

[10]

P. K. Maini, K. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems, Journal of the Chemical Society, Faraday Transactions, 93 (1997), 3601-3610.  doi: 10.1039/a702602a.

[11]

L. Marcon, X. Diego, J. Sharpe and P. Müller, High-throughput mathematical analysis identifies Turing networks for patterning with equally diffusing signals, Elife, 5 (2016), e14022. doi: 10.7554/eLife.14022.

[12]

J. Maybee and J. Quirk, Qualitative problems in matrix theory, SIAM Rev., 11 (1969), 30-51.  doi: 10.1137/1011004.

[13]

M. Mincheva and M. R. Roussel, A graph-theoretic method for detecting potential Turing bifurcations, The Journal of Chemical Physics, 125 (2006), 204102.  doi: 10.1063/1.2397073.

[14]

J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, vol. 18 of Interdisciplinary Applied Mathematics, 3rd edition, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.

[15]

J. Raspopovic, L. Marcon, L. Russo and J. Sharpe, Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients, Science, 345 (2014), 566-570.  doi: 10.1126/science.1252960.

[16]

R. A. Satnoianu, M. Menzinger and P. K. Maini, Turing instabilities in general systems, J. Math. Biol., 41 (2000), 493-512.  doi: 10.1007/s002850000056.

[17]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[18]

P. van den Driessche, Sign pattern matrices, in Combinatorial Matrix Theory, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, (2018), 47–82. doi: 10.1007/978-3-319-70953-6_2.

[19]

L. Wang and M. Y. Li, Diffusion-driven instability in reaction-diffusion systems, J. Math. Anal. Appl., 254 (2001), 138-153.  doi: 10.1006/jmaa.2000.7220.

[20]

K. A. J. White and C. A. Gilligan, Spatial heterogeneity in three species, plant–parasite–hyperparasite, systems, Philos. Trans. Roy. Soc. London Ser. B, 353 (1998), 543-557.  doi: 10.1098/rstb.1998.0226.

show all references

References:
[1]

A. Anma, K. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247.  doi: 10.2996/kmj/1341401049.

[2]

R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, vol. 39 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9781107325708.

[3]

M. Cavers, Polynomial stability and potentially stable patterns, Linear Algebra Appl., 613 (2021), 87-114.  doi: 10.1016/j.laa.2020.12.015.

[4]

X. Diego, L. Marcon, P. Müller and and J. Sharpe, Key features of Turing systems are determined purely by network topology, Physical Review X, 8 (2018), 021071.  doi: 10.1103/PhysRevX.8.021071.

[5]

D. A. Grundy, D. D. Olesky and P. van den Driessche, Constructions for potentially stable sign patterns, Linear Algebra Appl., 436 (2012), 4473-4488.  doi: 10.1016/j.laa.2011.08.011.

[6]

C. L. Hambric, C.-K. Li, D. C. Pelejo and J. Shi, Minimum number of non-zero-entries in a $7\times 7$ stable matrix, Linear Algebra Appl., 572 (2019), 135-152.  doi: 10.1016/j.laa.2019.03.002.

[7]

C. R. Johnson, J. S. Maybee, D. D. Olesky and P. van den Driessche, Nested sequences of principal minors and potential stability, Linear Algebra Appl., 262 (1997), 243-257.  doi: 10.1016/S0024-3795(97)80034-1.

[8]

A. N. Landge, B. M. Jordan, X. Diego and P. Müller, Pattern formation mechanisms of self-organizing reaction-diffusion systems, Developmental Biology, 460 (2020), 2-11.  doi: 10.1016/j.ydbio.2019.10.031.

[9]

A. Liénard and M. Chipart, Sur le signe de la partie réelle des racines dúne équation algébrique, J. Math. Pures Appl., 10 (1914), 291-346. 

[10]

P. K. Maini, K. J. Painter and H. N. P. Chau, Spatial pattern formation in chemical and biological systems, Journal of the Chemical Society, Faraday Transactions, 93 (1997), 3601-3610.  doi: 10.1039/a702602a.

[11]

L. Marcon, X. Diego, J. Sharpe and P. Müller, High-throughput mathematical analysis identifies Turing networks for patterning with equally diffusing signals, Elife, 5 (2016), e14022. doi: 10.7554/eLife.14022.

[12]

J. Maybee and J. Quirk, Qualitative problems in matrix theory, SIAM Rev., 11 (1969), 30-51.  doi: 10.1137/1011004.

[13]

M. Mincheva and M. R. Roussel, A graph-theoretic method for detecting potential Turing bifurcations, The Journal of Chemical Physics, 125 (2006), 204102.  doi: 10.1063/1.2397073.

[14]

J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, vol. 18 of Interdisciplinary Applied Mathematics, 3rd edition, Springer-Verlag, New York, 2003. doi: 10.1007/b98869.

[15]

J. Raspopovic, L. Marcon, L. Russo and J. Sharpe, Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients, Science, 345 (2014), 566-570.  doi: 10.1126/science.1252960.

[16]

R. A. Satnoianu, M. Menzinger and P. K. Maini, Turing instabilities in general systems, J. Math. Biol., 41 (2000), 493-512.  doi: 10.1007/s002850000056.

[17]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[18]

P. van den Driessche, Sign pattern matrices, in Combinatorial Matrix Theory, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, (2018), 47–82. doi: 10.1007/978-3-319-70953-6_2.

[19]

L. Wang and M. Y. Li, Diffusion-driven instability in reaction-diffusion systems, J. Math. Anal. Appl., 254 (2001), 138-153.  doi: 10.1006/jmaa.2000.7220.

[20]

K. A. J. White and C. A. Gilligan, Spatial heterogeneity in three species, plant–parasite–hyperparasite, systems, Philos. Trans. Roy. Soc. London Ser. B, 353 (1998), 543-557.  doi: 10.1098/rstb.1998.0226.

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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Table 5.   
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Table 6.   
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