Dynamics of a two-dimensional system of rational difference equations of Leslie--Gower type

By Senada Kalabusic, Mustafa R. S. Kulenovic, Esmir Pilav. 2011

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AbstractWe investigate global dynamics of the following systems of difference equations xn+1=α1+β1xnA1+ynyn+1=γ2ynA2+B2xn+yn,n=0,1,2,… where the parameters α1, β1, A1, γ2, A2, B2 are positive numbers, and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We show that this system has rich dynamics which depends on the region of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or non-hyperbolic equilibrium points. We give examples of a globally attractive non-hyperbolic equilibrium point and a semi-stable non-hyperbolic equilibrium point. We also give an example of two local attractors with precisely determined basins of attraction. Finally, in some regions of parameters, we give an explicit formula for the global stable manifold. Mathematics Subject Classification (2000) Primary: 39A10, 39A11 Secondary: 37E99, 37D10

Subfields: Geometry and Topology, Modeling and Simulation, Public Health, Environmental and Occupational Health

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