Oscillations and limit cycles in Lotka–Volterra systems with delays

In this paper, competitive Lotka–Volterra systems are studied that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotically constant average. Algebraic conditions are found to rule out non-vanishing oscillations for such systems and heteroclinic limit cycles for autonomous systems. As a supplement to these results, simple sufficient conditions are provided for certain components of all solutions to vanish and a criterion is given for partial permanence. An outstanding feature of all these results is that the conditions are irrelevant of the size and distribution of the delays.