On the long time behavior of non-autonomous Lotka–Volterra models with diffusion via the sub-supertrajectory method

In this paper we study in detail the geometrical structure of global pullback and forwards attractors associated to non-autonomous Lotka–Volterra systems in all the three cases of competition, symbiosis or prey–predator. In particular, under some conditions on the parameters, we prove the existence of a unique nondegenerate global solution for these models, which attracts any other complete bounded trajectory. Thus, we generalize the existence of a unique strictly positive stable (stationary) solution from the autonomous case and we extend to Lotka–Volterra systems the result for scalar logistic equations. To this end we present the sub-supertrajectory tool as a generalization of the now classical sub-supersolution method. In particular, we also conclude pullback and forwards permanence for the above models.