Stationary Solutions of a Mathematical Model for Formation of Coral Patterns

Abstract

A reaction-diffusion type mathematical model for growth of corals in a tank is considered. In this paper, we study stationary problem of the model subject to the homogeneous Neumann boundary conditions. We derive some existence results of the non-constant solutions of the stationary problem based on Priori estimations and Topological Degree theory. The existence of non-constant stationary solutions implies the existence of spatially variant time invariant solutions for the model.

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Somathilake, L. and Wedagedera, J. (2015) Stationary Solutions of a Mathematical Model for Formation of Coral Patterns. Applied Mathematics, 6, 1099-1106. doi: 10.4236/am.2015.66100.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Merks, R.M.H. (2003) Models of Coral Growth: Spontaneous Branching, Compactification and Laplacian Growth Assumption. Journal of Theoretical Biology, 224, 153-166.
http://dx.doi.org/10.1016/S0022-5193(03)00140-1
[2] Castro, P. and Huber, M.E. (1997) Marine Biology. WCB/McGraw-Hill, New York.
[3] Kaandorp, J.A., et al. (1996) Effect of Nutrient Diffusion and Flow on Coral Morphology. Physical Review Letters, 77, 2328-2331.
http://dx.doi.org/10.1103/PhysRevLett.77.2328
[4] Kaandorp, J.A., et al. (2005) Morphogenesis of the Branching Reef Coral Madracis Mirabilis. Proceedings of the Royal Society B, 77, 127-133.
http://dx.doi.org/10.1098/rspb.2004.2934
[5] Kaandorp, J.A., et al. (2008) Modelling Genetic Regulation of Growth and Form in a Branching Sponge. Proceedings of the Royal Society B, 275, 2569-2575.
http://dx.doi.org/10.1098/rspb.2008.0746
[6] Merks, R.M.H. (2003) Branching Growth in Stony Corals: A Modelling Approach. Ph.D. Thesis, Advanced School of Computing and Imaging, University of Amsterdam, Amsterdam.
[7] Merks, R.M.H. (2003) Diffusion-Limited Aggregation in Laminar Flows. International Journal of Modern Physics C, 14, 1171-1182.
http://dx.doi.org/10.1142/S0129183103005297
[8] Mistr, S. and Bercovici, D. (2003) A Theoretical Model of Pattern Formation in Coral Reefs. Ecosystems, 6, 61-74.
http://dx.doi.org/10.1007/s10021-002-0199-0
[9] Maxim, V.F., et al. (2010) A Comparison between Coral Colonies of the Genus Madracis and Simulated Forms. Proceedings of the Royal Society B, 277, 3555-3561.
http://dx.doi.org/10.1098/rspb.2010.0957
[10] Merks, R.M.H. (2010) Problem Solving Environment for Modelling Stony Coral Morphogenesis.
[11] Somathilake, L.W. and Wedagedera, J.R. (2012) On the Stability of a Mathematical Model for Coral Growth in a Tank. British Journal of Mathematics and Computer Science, 2, 255-280.
http://dx.doi.org/10.9734/BJMCS/2012/1387
[12] Somathilake, L.W. and Wedagedera, J.R. (2014) A Reaction-Diffusion Type Mathematical Model for Formation of Coral Patterns. Journal of National Science Foundation, 42, 341-349.
[13] Kien, B.T., et al. (2007) On the Degree Theory for General Mappings of Monotone Type. Journal of Mathematical Analysis and Applications, 340, 707-720.
http://dx.doi.org/10.1016/j.jmaa.2007.07.058
[14] Dhruba, R.A. (2007) Applications of Degree Theories to Nonlinear Operator Equations in Banach Spaces. Ph.D. Thesis, University of South Florida, Tampa.
[15] Norihiro, S. (2007) A Study on the Set of Stationary Solutions for the Gray-Scott Model. Ph.D. Thesis, Waseda University, Japan.
[16] Mawhin, J. (1999) Leray-Schauder Degree: A Half Century of Extension and Applications. Topological Methods in Nonlinear Analysis, 14, 195-228.

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