Integrability and global dynamics of the May-Leonard model

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• 作者：Gamaliel Bl¨¦ ; V¨ªctor Castellanos ; Jaume Llibre ; Ingrid Quilant¨¢n
• 关键词：May&ndash ; Leonard model ; Lotka&ndash ; Volterra systems ; First integrals ; Global dynamics ; Poincaré ; compactification
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2013
• 出版时间：February, 2013
• 年：2013
• 卷：14
• 期：1
• 页码：280-293
• 全文大小：614 K

文摘

We study when the celebrated May-Leonard model in , describing the competition between three species and depending on two positive parameters and , is completely integrable; i.e. when or . For these values of the parameters we shall describe its global dynamics in the compactification of the positive octant, i.e. adding its infinity.

If and (otherwise the dynamics is very easy) the global dynamics was partially known, and roughly speaking there are invariant topological half-cones by the flow of the system. These half-cones have a vertex at the origin of coordinates and surround the bisectrix , and foliate the positive octant. The orbits of each half-cone are attracted to a unique periodic orbit of the half-cone, which lives on the plane .

If then we consider two cases. First, if then the unique positive equilibrium point attracts all the orbits of the interior of the positive octant. If then there are three equilibria in the boundary of the positive octant, which attract almost all the orbits of the interior of the octant, we describe completely their bassins of attractions.