This thesis deals with a detailed linear analysis for a two-component reaction-diffusion system with constant diffusion coefficients. A comprehensive linear stability analysis results in three types of instabilities: (1) stationary periodic instability, (2) oscillatory uniform and (3) stationary uniform. The first instability involves pattern formation and the other two do not. Precise parameter regimes are identified for each. Travelling wave analysis is performed for the system and a detailed and comprehensive analysis is undertaken of a linear mechanism governing the development and propagation of nonlinear patterns. This analysis focuses on a linear selection mechanism that gives some insights into the selected speed of invasion of an unstable state by a stable one, as described both by a fixed form of travelling wave and by a modulated travelling wave.