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.