It is proposed in this paper that when two processes compete with each other, one tends to create long-range correlations, whereas the other tends to destroy it; the limiting object usually consists of many length scales. In particular, it might have power-law correlation functions and spatial 1/f spectra. The basic features of this competing dynamical process with spatial degrees of freedom are captured by the model expansion-modification system [W. Li, Europhys. Lett. 10, 395 (1989)]. Many detailed properties of the system, both numerical and analytical, are discussed here. The implications of the proposed mechanism for power-law correlation functions to the evolution of mass distribution in the Universe and the evolution of prebiotic nucleotide sequences are also discussed.
