{
  "id": "0812.3318",
  "version": "v1",
  "published": "2008-12-17T20:27:18.000Z",
  "updated": "2008-12-17T20:27:18.000Z",
  "title": "On the Global Behavior of Solutions to a Planar System of Difference Equations",
  "authors": [
    "Sukanya Basu",
    "Orlando Merino"
  ],
  "comment": "13 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We establish the relation between local stability of equilibria and slopes of critical curves for a specific class of difference equations. We then use this result to give global behavior results for nonnegative solutions of the system of difference equations \\begin{equation*} %\\tag{LGIN} \\begin{array}{rcl} x_{n+1} & = & \\displaystyle \\frac{b_1 x_n}{1+x_n+c_1 y_{n}} +h_1 y_{n+1} & = & \\displaystyle \\frac{b_2 y_n}{1+y_n+c_2 x_{n}} +h_2 \\end{array} \\quad n=0,1,..., \\quad (x_0,y_0) \\in [0,\\infty)\\times [0,\\infty) \\end{equation*} with positive parameters. In particular, we show that the system has between one and three equilibria, and that the number of equilibria determines global behavior as follows: if there is only one equilibrium, then it is globally asymptotically stable. If there are two equilibria, then one is a local attractor and the other one is nonhyperbolic. If there are three equilibria, then they are linearly ordered in the south-east ordering of the plane, and consist of a local attractor, a saddle point, and another local attractor. Finally, we give sufficient conditions for having a unique equilibrium.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-17T20:27:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39A05",
      "39A11"
    ],
    "keywords": [
      "difference equations",
      "planar system",
      "local attractor",
      "equilibria determines global behavior",
      "global behavior results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.3318B"
    }
  }
}