Transitions and heteroclinic cycles in the general Gierer-Meinhardt equation and cardiovascular calcification model. (English) Zbl 1197.35136
Summary: The article is concerned with transitions and pattern selection analysis of the inhibitor-activator system proposed in connection with recent studies of cardiovascular calcification patterns. Explicit criteria are derived to enable us to distinguish between stable and metastable patterns. By deriving a reduced system of equations, the existence of certain complicated structures is discussed; in particular, heteroclinic cycles are identified and their properties are studied. It is also discussed that the change of boundary conditions can affect the transitions of the system. In this connection, we will also study asymptotic behavior of patterns after transitions and will compare the results with numerical simulations.
MSC:
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K58 | Semilinear parabolic equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
References:
| [1] | Abedin, M.; Tintut, Y.; Demer, L., Mesenchymal stem cells and the artery wall, Circulation Research, 95, 671-676 (2004) |
| [2] | Tintut, Y.; Alfonso, Z.; Saini, T.; Radcliff, K.; Watson, K.; Bostrom, K.; Demer, L., Multilineage potential of cells from the artery wall, Circulation, 108, 20, 2505-2510 (2003) |
| [3] | Abedin, M.; Tintut, Y.; Demer, L., Vascular calcification mechanisms and clinical ramifications, Arteriosclerosis, Thrombosis, and Vascular Biology, 24, 1161-1170 (2004) |
| [4] | Gierer, A.; Meinhardt, H., A theory of biological pattern formation, Biological Cybernetics, 12, 30-39 (1972) · Zbl 1434.92013 |
| [5] | Garfinkel, A.; Tintut, Y.; Petrasek, D.; Bostrom, K.; Demer, L., Pattern formation by vascular mesenchymal cells, Proceedings of the National Academy of Sciences, 101, 9247-9250 (2004) |
| [6] | Koch, A.; Meinhardt, H., Biological pattern formation: from basic mechanisms to complex structures, Reviews of Modern Physics, 66, 1481-1507 (1994) |
| [7] | Haken, H.; Olbrich, H., Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis, Journal of Mathematical Biology, 6, 317-331 (1978) · Zbl 0393.92003 |
| [8] | Iron, D.; Ward, M.; Wei, J., The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D: Nonlinear Phenomena, 150, 25-62 (2001) · Zbl 0983.35020 |
| [9] | Kolokolnikov, T.; Sun, W.; Ward, M.; Wei, J., The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM Journal on Applied Dynamical Systems, 5, 313-363 (2006) · Zbl 1210.35016 |
| [10] | J. Wei, M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in \(r^1\).; J. Wei, M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in \(r^1\). · Zbl 1195.35033 |
| [11] | Yochelis, A.; Garfinkel, A., Front motion and localized states in an asymmetric bistable activator-inhibitor system with saturation, Physical Review E, 77, 35204 (2008) |
| [12] | Yochelis, A.; Tintut, Y.; Demer, L. L.; Garfinkel, A., The formation of labyrinths, spots and stripe patterns in a biochemical approach to cardiovascular calcification, New Journal of Physics, 10, 055002 (2008) |
| [13] | Zhu, M.; Murray, J., Parameter domains for generating spatial pattern: a comparison of reaction-diffusion and cell-chemotaxis models, International Journal of Bifurcation and Chaos, 5, 1503-1524 (1995) · Zbl 0889.35051 |
| [14] | Arcuri, P.; Murray, J., Pattern sensitivity to boundary and initial conditions in reaction-diffusion models, Journal of Mathematical Biology, 24, 141-165 (1986) · Zbl 0595.92001 |
| [15] | Ma, T.; Wang, S., Bifurcation Theory And Applications (2005), World Scientific · Zbl 1085.35001 |
| [16] | Smoller, J., Shock Waves and Reaction-Diffusion Equations (1994), Springer · Zbl 0807.35002 |
| [17] | Ma, T.; Wang, S., Dynamic phase transitions for ferromagnetic systems, Journal of Mathematics and Physics, 49, 053506 (2008) · Zbl 1152.81546 |
| [18] | Knobloch, E., Spatially localized structures in dissipative systems: open problems, Nonlinearity, 21, 45-60 (2008) · Zbl 1139.35002 |
| [19] | Henry, D., (Geometric Theory of Semilinear Parabolic Equations. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840 (1984), Springer-Verlag: Springer-Verlag New York) |
| [20] | Chow, S.; Li, C.; Wang, D., Normal Forms and Bifurcation of Planar Vector Fields (1994), Cambridge University Press · Zbl 0804.34041 |
| [21] | Golubitsky, M.; Stewart, I., The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space (2003), Birkhauser |
| [22] | Kaper, H. G.; Wang, S.; Yari, M., Dynamical transitions of Turing patterns, Nonlinearity, 22, 601-626 (2009) · Zbl 1157.92003 |
| [23] | Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1997), Springer · Zbl 0871.35001 |
| [24] | Crawford, J.; Gollub, J.; Lane, D., Hidden symmetries of parametrically forced waves, Nonlinearity, 6, 119-164 (1993) · Zbl 0773.58017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
