Extinction or coexistence in periodic Kolmogorov systems of competitive type

We study a periodic Kolmogorov system describing two species nonlinear competition. We discuss coexistence and extinction of one or both species, and describe the domain of attraction of nontrivial periodic solutions in the axes, under conditions that generalise Gopalsamy conditions. Finally, we apply our results to a model of microbial growth and to a model of phyto-plankton competition under the e_ect of toxins.