Mathematical Aspects of Pattern Formation in Biological Systems Introduction Existence of spikes for the Gierer Meinhardt system in one dimension The Nonlocal Eigenvalue Problem (NLEP) Stability of spikes for the Gierer Meinhardt system in one dimension Existence of spikes for the shadow Gierer Meinhardt system Existence and stability of spikes for the Gierer Meinhardt system in two dimensions The Gierer Meinhardt system with inhomogeneous coefficients Other aspects of the Gierer Meinhardt system The Gierer Meinhardt system with saturation Spikes for other two component reaction diffusion systems Reaction diffusion systems with many components Biological applications Appendix This monograph is concerned with the mathematical analysis of patterns which are encountered in biological systems. It summarises expands and relates results obtained in the field during the last fifteen years. It also links the results to biological applications and highlights their relevance to phenomena in nature. Of particular concern are large amplitude patterns far from equilibrium in biologically relevant models. The approach adopted in the monograph is based on the following paradigms: • Examine the existence of spiky steady states in reaction diffusion systems and select as observable patterns only the stable ones • Begin by exploring spatially homogeneous two component activator inhibitor systems in one or two space dimensions • Extend the studies by considering extra effects or related systems each motivated by their specific roles in developmental biology such as spatial inhomogeneities large reaction rates altered boundary conditions saturation terms convection many component systems. Mathematical Aspects of Pattern Formation in Biological Systems will be of interest to graduate students and researchers who are active in reaction diffusion systems pattern formation and mathematical biology