Permanence versus impermanence for the Lotka–Volterra model of three competing species with seasonal succession

In this paper, we investigate the permanence versus impermanence of the Lotka–Volterra model of three competing species with seasonal succession, which exhibits that populations experience an externally periodically forced environment. By employing the carrying simplex theory for discrete competitive dynamical systems, we present a comprehensive classification of permanence and impermanence based on inequalities involving the model parameters. Furthermore, by constructing numerical examples, we show that the permanence cannot guarantee the global asymptotic stability of the positive harmonic periodic solution in this model, where there may exist a family of attracting quasi-periodic solutions in certain permanent classes.