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 = x(i)(1 - x(i) - beta(i)x(i-1) - alpha(i)x(i+1)), x(i)(0) > 0, i = 1, 2, 3 with x(0) = x(3), x(4) = x(1) under the assumption 0 < alpha(i) < 1 < beta(i), i = 1, 2, 3. Let A(i) = 1 - alpha(i) and B-i = beta(i) - 1, i = 1, 2, 3. The linear stability analysis shows that the interior equilibrium P = (p(1), p(2), p(3)) is asymptotically stable if A(1)A(2)A(3) > B1B2B3 and P is a saddle point with one-dimensional stale manifold Gamma if A(1)A(2)A(3) < B1B2B3. Hopf bifurcation occurs when A(1)A(2)A(3) = B1B2B3. For the case A(1)A(2)A(3) not equal B1B2B3 we eliminate the possibility of the existence of periodic solutions by applying the Stokes theorem. Then, from the Poincare-Bendixson theorem for three-dimensional competitive systems, we show that (i) if A(1)A(2)A(3) > B1B2B3 then P is global asymnptotically stable Int(R-+(3)), (ii) if A(1)A(2)A(3) < B1B2B3 then for each initial condition x(0) is not an element of Gamma, the solution phi(t, x(0)) cyclically oscillates around the boundary of the coordinate planes as the trajectory of the symmetric May-Leonard model does, and (iii) if A(1)A(2)A(3) = B1B2B3 then there exists a family of neutrally stable periodic orbits.