We model the increase in temperature in compost piles or landfill sites due to micro-organisms undergoing exothermic reactions. The model incorporates the heat release due to biological activity within the pile and the heat release due to the oxidation of cellulosic materials. The heat release rate due to biological activity is modelled by a function which is a monotonic increasing function of temperature over a particular range and followed by a monotone decreasing function of temperature. This functionality represents the fact that micro-organisms die or become dormant at high temperatures. The heat release due to the oxidation reaction is modelled by the usual manner using Arrhenius kinetics. The bifurcation behaviour is investigated for two-dimensional slab geometries to determine the critical sizes of the compost piles.