The slow dynamics and linearized stability of a two-spike quasi-equilibrium solution to a general class of reaction-diffusion (RD) system with and without sub-diffusion is analyzed. For both the case of regular and sub-diffusion, the method of matched asymptotic expansions is used to derive an ODE characterizing the spike locations in the absence of any O(1) time-scale instabilities of the two-spike quasi-equilibrium profile. These fast instabilities result from unstable eigenvalues of a certain nonlocal eigenvalue problem (NLEP) that is derived by linearizing the RD system around the two-spike quasi-equilibrium solution. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined by the roots of some simple transcendental equations. From a rigorous analysis of these transcendental equations, explicit sufficient conditions are given to predict the occurrence of either Hopf bifurcations or competition instabilities of the two-spike quasi-equilibrium solution. The theory is illustrated for several specific choices of the reaction kinetics.scale. This asymptotic limit, where only one of the two components of the RD system is localized, is called the semi-strong spike interaction limit (cf. [5]).Reaction-diffusion systems of the general class (1.1) arise in various applications. In particular, Turing [33] proposed that localized peaks in the concentration of a chemical substance, known as a morphogen, could be responsible for the process of morphogenesis, which describes the development of a complex organism from a single cell. Through the use of a linearized analysis, he showed how stable spatially complex patterns can develop from small perturbations of spatially homogeneous initial data for a coupled system of reaction-diffusion equations. Later, Gierer and Meinhardt [8] (see the more recent book [17]), have demonstrated numerically the existence of stable spatially inhomogeneous equilibrium solutions for activator-inhibitor systems of the type (1.1) with u b = 0 where the inhibitor u diffuses more slowly than the activator v. A class of GM models is g(u) = u −q , f (u) = u −s , u b = 0, where qr/(p − 1) − (s + 1) > 0 (see [12]). Another application of (1.1) is to the study of chemical patterns, involving two reactants in a gel reactor where the reactor is maintained in contact with a reservoir of one of the two chemical species. This leads to the source or feed term u b in (1.1). In this class of substrate-depletion models, a ubiquitous model is the Gray-Scott (GS) model, which was introduced for continuously stirred systems in [10]. It is given by setting u b = 1, g(u) = Au, p = r = 2, and f = −u in (1.1), where A > 0 is a bifurcation parameter.We will also consider the sub-diffusive counterpart of (1.1), formulated aswhere the anomaly exponent γ is on the range 0 < γ < 1. The regular diffusion RD system (1.1) is obtained by setting γ = 1. In (1.2), the sub-diffusive operator as applied to a function h(t) is defined by