On the Asymmetric May--Leonard Model of Three Competing Species

Abstract

In this paper we analyze the global asymptotic behavior of the asymmetric May--Leonard model of three competing species:\, $\frac{dx_i}{dt}=x_i(1-x_i-\beta_ix_{i-1}-\alpha_ix_{i+1})$, $x_i(0) > 0$, $i=1,2,3$ 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=(p1 , p2 , p3 ) is asymptotically stable if A1A2A3 > B1B2B3 and P is a saddle point with one-dimensional stable manifold $\Gamma$ if A1A2A3 < B1B2B3 . Hopf bifurcation occurs when A1A2A3 = B1B2B3 . For the case A1A2A3 ne ,B1B2B3 we eliminate the possibility of the existence of periodic solutions by applying the Stokes theorem. Then, from the Poincar{é}--Bendixson theorem for three-dimensional competitive systems, we show that (i) if A1A2A3 > B1B2B3 then P is global asymptotically stable in Int(R3+ ), (ii) if A1A2A3 < B1B2B3 then for each initial condition $x_0\not\in\Gamma$, the solution $\varphi(t,x_0)$ cyclically oscillates around the boundary of the coordinate planes as the trajectory of the symmetric May--Leonard model does, and (iii) if A1A2A3 = B1B2B3 then there exists a family of neutrally stable periodic orbits.