Multiple Limit Cycles for Three Dimensional Lotka-Volterra Equations

A 3D competitive Lotka-Volterra equation with two limit cycles is constructed. Keywords-Lotka-Volterra equations, Competitive systems, Limit cycles, Hopf bifurcation. INTRODUCTION It is a classical result (due to Moisseev 1939 and/ or Bautin 1954, see [l, p. 213, Section 12, Example 71 or [2, 18.21) that 2D Lotka-Volterra equations cannot have limit cycles: if there is a periodic orbit, then the interior fixed point is a center (i.e., surrounded by a continuum of periodic orbits). Hence, a center is a codimension one phenomenon for 2D Lotka-Volterra equations, like for linear equations. On the other hand, 3D Lotka-Volterra equations allow already complicated dynamics (see [3-51): The period doubling route to chaos and many other phenomena known from the iteration of the quadratic map have been observed by numerical simulations. For 3D competitive systems, the dynamical possibilities are more restricted: According to Hirsch [6, Theorem 1.71, there is an invariant manifold (called the carrying simplex) that is homeomorphic to the twodimensional simplex and that attracts all orbits except the origin. Therefore in 3D competitive systems, the Poincare-Bendixson theorem holds. Based on this, M. L. Zeeman [7] has given a classification of all possible stable phase portraits of 3D competitive LotkaVolterra equations, thus extending a related classification of the game dynamical equation [8]. However, the question of how many limit cycles can surround the interior fixed point was left open and is still open. Up to now only examples with at most one limit cycle seem to have appeared in the literature. In this paper, we will give an example where the (locally stable) interior equilibrium is surrounded by (at least) two limit cycles. The idea for constructing such an example with multiple limit cycles is as follows: We consider a competitive LV-system which is permanent (i.e., the boundary of lR: is repelling) and where the unique interior fixed point has a pair of purely imaginary eigenvalues, but is repelling on its center manifold (which is part of the carrying simplex). This implies the existence of an Research partially supported by FWF of Austria and NSERC of Canada.