  An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system
on the one-dimensional torus, arising in population dynamics, is proposed and
analyzed. The kernels are assumed to be in detailed balance and satisfy a weak
cross-diffusion condition. The latter condition allows for negative
off-diagonal coefficients and for kernels defined by an indicator function. The
scheme preserves the nonnegativity of the densities, conservation of mass, and
production of the Boltzmann and Rao entropies. The key idea is to ``translate''
the entropy calculations for the continuous equations to the finite-volume
scheme, in particular to design discretizations of the mobilities, which
guarantee a discrete chain rule even in the presence of nonlocal terms. Based
on this idea, the existence of finite-volume solutions and the convergence of
the scheme are proven. As a by-product, we deduce the existence of weak
solutions to the continuous cross-diffusion system. Finally, we present some
numerical experiments illustrating the behavior of the solutions to the
nonlocal and associated local models.
