We explore the stability of equilibrium solution(s) of a simple model of microvascular blood flow in a two-node network. The model takes the form of convection equations for red blood cell concentration, and contains two important rheological effects—the Fåhræus–Lindqvist effect, which governs viscosity of blood flow in a single vessel, and the plasma skimming effect, which describes the separation of red blood cells at diverging nodes. We show that stability is governed by a linear system of integral equations, and we study the roots of the associated characteristic equation in detail. We demonstrate using a combination of analytical and numerical techniques that it is the relative strength of the Fåhræus–Lindqvist effect and the plasma skimming effect which determines the existence of a set of network parameter values which lead to a Hopf bifurcation of the equilibrium solution. We confirm these predictions with direct numerical simulation and suggest several areas for future research and application.