We analyse the asymptotic behaviour of integro-differential equations modelling $N$ populations in interaction, where interactions are modelled by non-local terms involving linear combinations of the total number of individuals in each population. These models have already been shown to be suitable for the modelling of drug resistance in cancer. They also generalise the usual Lotka-Volterra ordinary differential equations. Our aim is to give conditions under which there is persistence of all species, in the sense of global asymptotical stability of coexistence steady-states at the level of the total number of individuals in each species. Through the analysis of a Lyapunov function, our first main result gives a simple and general condition on the matrix of interactions, together with a convergence rate. The second main result establishes another type of condition in the specific case of mutualistic interactions. These conditions are compared to the well-known condition given by Goh for classical Lotka-Volterra ordinary differential equations.