This thesis is devoted to the study of two types of degenerate parabolic equations: a dryland vegetation model developed by E. Gilad et al. [1, 2] and a mechanochemical model first introduced by J. D. Murray et al. [3] in the context of somitogenesis.

The former models vegetation patches in a region with limited water resources. Its construction is based on various feedback mechanisms between biomass and water, resulting in a system of three nonlinear parabolic partial differential equations, one of which is degenerate parabolic of the family of porous media equation. In the first part of the thesis, we prove the existence and uniqueness of weak solutions and the existence of maximal attractors. Our approach illustrates two different methods of regularization and makes use of techniques including the Galerkin methods, truncation, maximum principle and compactness, regularization effect. We observe in this way various properties and regularity results of the solutions.

A mechanochemical model is concerned with morphogenetic phenomena in an embryo and encapsulates the mechanical aspects of the cells and their interaction with the surroundings. In the second part of the thesis, we apply this model to the formation of somites, spherical balls of the mesodermal cells in vertebrate embryos, which align alongside of the neural tube, and study the effect of the cells' contractile forces exerted onto the ECM. Our approach involves the linear stability analysis and a study of asymptotic behavior of the cell density based on a priori estimates. The full model considered in 2 dimensional space is numerically simulated to show that the traction force of the cells alone can generate a pattern.

[1] E. Gilad, J. Von Hardenberg, A. Provenzale, M. Shachak, and E. Meron, Ecosystem engineers: From pattern formation to habitat creation . Physical Review Letters, 98(9):098105-1-098105-4, 2004. [2] E. Gilad, J. Von Hardenberg, A. Provenzale, M. Shachak, and E. Meron, A mathematical model of plants as ecosystem engineers. Journal of Theoretical Biology, 244:680-691, 2007. [3] G. Oster, J. D. Murray, and A. K. Harris, Mechanical aspects of mesenchymal morphogenesis. Journal of embryology and experimental morphology, 78:83-125, 1983.