Clustering of diffusing organisms

Recently, a Brownian bugs model has been proposed as a possible explanation for the clustering of organisms such as planktons [W.R. Young, A.J. Roberts, and G. Stuhne, Nature (London) 412, 328 (2001)]. In this model, diffusing organisms die and give birth with equal probabilities and are independent of each other. The clustering phenomenon is due to the discreteness of birth and death events and cannot be explained in the framework of the continuous model of population dynamics. We show by exact calculation that at dimensions smaller than or equal to 2, this simple mechanism is enough to provoke aggregations of the organisms. At dimensions higher than 2, the clustering phenomenon disappears, but reproductive fluctuations can still be large compared to diffusive ones.