  We consider a fitness-driven model of dispersal of $N$ interacting
populations, which was previously studied merely in the case $N=1$. Based on
some optimal transport distance recently introduced, we identify the model as a
gradient flow in the metric space of Radon measures. We prove existence of
global non-negative weak solutions to the corresponding system of parabolic
PDEs, which involves degenerate cross-diffusion. Under some additional
hypotheses and using a new multicomponent Poincar\'e-Beckner functional
inequality, we show that the solutions converge exponentially to an ideal free
distribution in the long time regime.
