We investigate the dynamics of two models of biological networks with purely suppressive interactions between the units; species interacting via niche competition and neurons via inhibitory synaptic coupling. In both of these cases, power-law scaling of the density of states with probability arises without any fine-tuning of the model parameters. These results argue against the increasingly popular notion that non-equilibrium living systems operate at special critical points, driven by there by evolution so as to enable adaptive processing of input data.