On the Asymmetric May-Leonard Model of Three Competing Species

In this paper we analyze the global asymptotic behavior of the asymmetric May– Leonard model of three competing species: dxi dt = xi(1−xi−βixi−1−αixi+1), xi(0) > 0, i = 1, 2, 3 with x0 = x3, x4 = x1 under the assumption 0 < αi < 1 < βi, i = 1, 2, 3. Let Ai = 1−αi and Bi = βi − 1, i = 1, 2, 3. The linear stability analysis shows that the interior equilibrium P = (p1, p2, p3) is asymptotically stable if A1A2A3 > B1B2B3 and P is a saddle point with one-dimensional stable manifold Γ if A1A2A3 < B1B2B3. Hopf bifurcation occurs when A1A2A3 = B1B2B3. For the case A1A2A3 6= B1B2B3 we eliminate the possibility of the existence of periodic solutions by applying the Stokes theorem. Then, from the Poincaré–Bendixson theorem for three-dimensional competitive systems, we show that (i) if A1A2A3 > B1B2B3 then P is global asymptotically stable in Int(R+), (ii) if A1A2A3 < B1B2B3 then for each initial condition x0 6∈ Γ, the solution φ(t, x0) cyclically oscillates around the boundary of the coordinate planes as the trajectory of the symmetric May– Leonard model does, and (iii) if A1A2A3 = B1B2B3 then there exists a family of neutrally stable periodic orbits.