Cross-diffusion driven instabilities have gained a considerable attention in the field of population dynamics, mainly due to their ability to predict some important features in the study of spatial distribution of species in ecological systems. This paper is concerned with some mathematical and numerical aspects of a particular reaction-diffusion system with cross-diffusion, modeling the effect of allelopathy on two plankton species. Based on a stability analysis and a series of numerical simulations performed with a finite volume scheme, we show that the cross-diffusion coefficient plays a important role on the pattern selection.