We study an aggregation-disaggregation model which is relevant to biological processes such as the growth of senile plaques in Alzheimer disease. In this model, during the aggregation each deposited particle has a probability of producing a new particle in its vicinity, while during disaggregation the particles are anihilated randomly. The model is held in a dynamic equilibrium by a feedback mechanism which changes the disaggregation probability in proportion to the change in the total number of particles. We also include surface diffusion which influences the morphology of growing aggregates and colonies. A colony includes the descendents of a single particle. We investigate the statistical properties of the model in two dimensions. We find that unlike the colonies, individual aggregates are fractals with a fractal dimension of D(f)=1.92+/-0.06 in the absence of surface diffusion. We show that the surface diffusion changes the fractal dimension of aggregates: at a small aggregation-disaggregation rate, D(f) is independent of the strength of the surface diffusion, D(f)=1.73+/-0.03. At larger aggregation-disaggregation rates and different strengths of surface diffusion, aggregates with fractal dimensions between D(f)=1.73 and 1.92 form. The steady-state distribution of aggregate sizes is shown to be power law if the aggregation-disaggregation process dominates over the surface diffusion. In the limit of weak aggregation-disaggregation and strong surface diffusion the size distribution is log-normal.