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Global asymptotic stability of coexistence steady-states in integro-differential Lotka-Volterra systems

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. This model generalises the usual Lotka-Volterra ordinary differential equations. Our aim is to give conditions under which there is 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.