Abstract
In this paper we analyze the global asymptotic behavior of the asymmetric May-Leonard model of three competing species: dx <sub>i</sub> /dt = x <sub>i</sub> (1-x <sub>i</sub> -β <sub>i</sub> x <sub>i-1</sub> -α <sub>i</sub> x <sub>i+1</sub> ), x <sub>i</sub> (0)>0, i = 1, 2, 3 with x <sub>0</sub> = x <sub>3</sub> , x <sub>4</sub> = x <sub>1</sub> under the assumption 0<α <sub>i</sub> <1<β <sub>i</sub> , i = 1, 2, 3. Let A <sub>i</sub> = 1-α <sub>i</sub> and B <sub>i</sub> = β <sub>i</sub> -1, i = 1, 2, 3. The linear stability analysis shows that the interior equilibrium P = (p <sub>1</sub> , p <sub>2</sub> , p <sub>3</sub> ) is asymptotically stable if A <sub>1</sub> A <sub>2</sub> A <sub>3</sub> >B <sub>1</sub> B <sub>2</sub> B <sub>3</sub> and P is a saddle point with one-dimensional stable manifold Γ if A <sub>1</sub> A <sub>2</sub> A <sub>3</sub> <B <sub>1</sub> B <sub>2</sub> B <sub>3</sub> . Hopf bifurcation occurs when A <sub>1</sub> A <sub>2</sub> A <sub>3</sub> = B <sub>1</sub> B <sub>2</sub> B <sub>3</sub> . For the case A <sub>1</sub> A <sub>2</sub> A <sub>3</sub> ≠B <sub>1</sub> B <sub>2</sub> B <sub>3</sub> 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 <sub>1</sub> A <sub>2</sub> A <sub>3</sub> >B <sub>1</sub> B <sub>2</sub> B <sub>3</sub> then P is global asymptotically stable in Int(R <sub>+</sub> <sup>3</sup> ), (ii) if A <sub>1</sub> A <sub>2</sub> A <sub>3</sub> <B <sub>1</sub> B <sub>2</sub> B <sub>3</sub> then for each initial condition x <sub>0</sub> is not a membere of thee set Γ, the solution φ(t, x <sub>0</sub> ) cyclically oscillates around the boundary of the coordinate planes as the trajectory of the symmetric May-Leonard model does, and (iii) if A <sub>1</sub> A <sub>2</sub> A <sub>3</sub> = B <sub>1</sub> B <sub>2</sub> B <sub>3</sub> then there exists a family of neutrally stable periodic orbits.