Asymptotic Behaviors in the Dynamics of Competing Species

We consider systems of interacting species obeying Lotka–Volterra equations. We show that periodic attractors may be generated from the equilibrium point in phase space by Hopf bifurcation. An exception is the case of three species systems with equal individual growth rates, where Hopf bifurcation is abnormal. We show that the dynamics of such systems is actually bidimensional, a fact which permits us to give an analytic description of the closed orbits observed by May and Leonard. We also consider the “explosive” dynamics of a system without self interactions and show that adding a constant source term in the equations of motion has a stabilizing effect, producing periodic solutions. In the limit of small external source terms we are able to describe analytically the associated cyclic evolutions. This effect of source terms on the solutions of Lotka–Volterra equations seems to be quite general, and is explained by the fact that they remove the attraction of the orbits by the coordinate planes.