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The two-species population dynamics model is the simplest paradigm ofinterspecies interaction. Here, we include intraspecific competition to theLotka-Volterra model and solve it analytically. Despite being simple andthoroughly studied, this model presents a very rich behavior and somecharacteristics not so well explored, which are unveiled. The forbidden regionin the mutualism regime and the dependence on initial conditions in thecompetition regime are some examples of these characteristics. From thestability 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 theconsidered ecological regimes. The time transient allows one to defined timescales for the system evolution, which can be relevant for the study of tumorgrowth by theoretical or computer simulation models.

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