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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 |
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.
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. |
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