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

• Published in: Advances in difference equations Vol. 2011; no. 1; pp. 1 - 29
• Main Authors: Kalabušić, S, Kulenović, MRS, Pilav, E
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 23.08.2011; Springer Nature B.V; BioMed Central Ltd; SpringerOpen
• Subjects: Analysis; Asymptotic properties; Attraction; Basin of attraction; Basins; Competitive map; Difference and Functional Equations; Difference equations; Dynamical systems; Dynamics; Functional Analysis; Global stable manifold; Manifolds; Mathematical analysis; Mathematics; Mathematics and Statistics; Monotonicity; Ordinary Differential Equations; Partial Differential Equations; Period-two solution; Basin of attraction; Global stable manifold; Competitive map; Period-two solution; Monotonicity
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/1687-1847-2011-29

We investigate global dynamics of the following systems of difference equations x n + 1 = α 1 + β 1 x n A 1 + y n y n + 1 = γ 2 y n A 2 + B 2 x n + y n , n = 0 , 1 , 2 , … where the parameters α 1 , β 1 , A 1 , γ 2 , A 2 , B 2 are positive numbers, and the initial conditions x 0 and y 0 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