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Full analytical solution and complete phase diagram analysis of the Verhulst-like two-species population dynamics model

The two-species population dynamics model is the simplest paradigm of interspecies interaction. Here, we include intraspecific competition to the Lotka-Volterra model and solve it analytically. Despite being simple and thoroughly studied, this model presents a very rich behavior and some characteristics not so well explored, which are unveiled. The forbidden region in the mutualism regime and the dependence on initial conditions in the competition regime are some examples of these characteristics. From the stability of the steady state solutions, three phases are obtained: (i) extinction of one species (Gause transition), (ii) their coexistence and (iii) a forbidden region. Full analytical solutions have been obtained for the considered ecological regimes. The time transient allows one to defined time scales for the system evolution, which can be relevant for the study of tumor growth by theoretical or computer simulation models.