Asymptotic behavior of quasi-competitive species defined by Lotka-Volterra dynamics (Q1913900)

The paper studies the dynamical behaviour of three-dimensional Lotka-Volterra systems \[ dx_i (t)/dt = x_i \left( r_i + \sum^3_{j = 1} a_{ij} x_j \right)\;(i = 1,2,3) \] satisfying \(r_1 = r_2 = r_3 > 0\); \(a_{11}, a_{22}, a_{33} < 0\), \(a_{ij}/a_{jj} + a_{ji}/a_{ii} = \beta > 0\) \((i \neq j\); \(i,j = 1,2,3)\). Dynamical behaviour includes, depending on the parameter values, globally stable coexistence equilibrium, periodic oscillations of the three species, extinction of two species, and extinction of the three species through nonperiodic oscillations (May-type trajectories).