2008/12/17 by Sukanya Basu, Basu, Sukanya, Orlando Merino +1
Mathematics · Medicine · #39A05 #39A11 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Differential Equations Analysis #math.DS #msc:39A05 #msc:39A11
paper · pdf · doi:10.48550/arxiv.0812.3318
13 pages
arxiv created 2008/12/17 · openalex publication_date 2008/12/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 % xn+1 · = · \fracb1 xn1+xn+c1 yn +h1 yn+1 · = · \fracb2 yn1+yn+c2 xn +h2 n=0,1,..., (x0,y0) ∈ [0,∞)× [0,∞) 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.