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

Kalabušić et al. Advances in Difference Equations 2011, 2011:29 http://www.advancesindifferenceequations.com/content/2011/1/29 RESEARCH Open Access Dynamics of a two-dimensional system of rational difference equations of Leslie–Gower type S Kalabušić1, MRS Kulenović2* and E Pilav1 * Correspondence: 2 Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA Full list of author information is available at the end of the article Abstract We investigate global dynamics of the following systems of difference equations ⎧ ⎪ ⎨ xn+1 = α1 + β1 xn A1 + y n , γ 2 yn ⎪ ⎩ yn+1 = A2 + B 2 x n + y n n = 0, 1, 2, . . . where the parameters a1, b1, A1, g2, 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 Keywords: Basin of attraction, Competitive map, Global stable manifold, Monotonicity, Period-two solution 1 Introduction In this paper, we study the global dynamics of the following rational system of difference equations ⎧ ⎪ ⎨ xn+1 = α1 + β1 xn A1 + y n , γ 2 yn ⎪ ⎩ yn+1 = A2 + B 2 x n + y n n = 0, 1, 2, . . . (1) where the parameters a1, b1, A1, g2, A2, B2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers. System (1) was mentioned in [1] as one of three systems of Open Problem 3, which asked for a description of the global dynamics of some rational systems of difference equations. In notation used to label systems of linear fractional difference equations © 2011 Kalabušićć et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Kalabušić et al. Advances in Difference Equations 2011, 2011:29 http://www.advancesindifferenceequations.com/content/2011/1/29 Page 2 of 29 used in [1], System (1) is referred to as (29, 38). This system is dual to the system where the roles of xn and yn are interchanged, which is labeled as (29, 38) in [1], and so all results proven here extend to the latter system. In this paper, we provide a precise description of the global dynamics of the System (1). We show that System (1) may have between zero and three equilibrium points, which may have different local character. If System (1) has one equilibrium point, then this point is either locally asymptotically stable or saddle point or non-hyperbolic equilibrium point. If System (1) has two equilibrium points, then they are either locally asymptotically stable and nonhyperbolic, or locally asymptotically stable and saddle point. If System (1) has three equilibrium points, then two of equilibrium points are locally asymptotically stable and the third point, which is between these two points in southeast ordering defined below, is a saddle point. The major problem for global dynamics of the System (1) is determining the basins of attraction of different equilibrium points. The difficulty in analyzing the behavior of all solutions of the System (1) lies in the fact that there are many regions of parameters where this system possesses different equilibrium points with different local character and that in several cases, the equilibrium point is nonhyperbolic. However, all these cases can be handled by using recent results from [2]. System (1) is a competitive system, and our results are based on recent results about competitive systems in the plane, see [2,3]. System (1) can be used as a mathematical model for competition in population dynamics. In fact, second equation in (1) is of Leslie-Gower type, and first equation can be considered to be of Leslie-Gower type with stocking which is represented with the term a1, see [4-6]. In the next section, we present some general results about competitive systems in the plane. Section 3 contains some basic facts such as the non-existence of period-two solution of System (1). Section 4 analyzes local stability which is fairly complicated for this system. Finally, Section 5 gives global dynamics for all values of parameters. 2 Preliminaries A first-order system of difference equations  xn+1 = f (xn , yn ) , yn+1 = g(xn , yn ) n = 0, 1, 2, . . . (2) where S ⊂ ℝ2, (f, g): S ® S , f, g are continuous functions is competitive if f(x, y) is non-decreasing in x and non-increasing in y, and g(x, y) is non-increasing in x and non-decreasing in y. If both f and g are non-decreasing in x and y, the System (2) is cooperative. Competitive and cooperative maps are defined similarly. Strongly competitive systems of difference equations or strongly competitive maps are those for which the functions f and g are coordinate-wise strictly monotone. Competitive and cooperative systems have been investigated by many authors, see [2,3,5-19]. Special attention to discrete competitive and cooperative systems in the plane was given in [2,3,5-7,10,12,17,20]. One of the reasons for paying special attention to two-dimensional discrete competitive and cooperative systems is their applicability and the fact that many examples of mathematical models in biology and economy which involve competition or cooperation are models which involve two species. Another reason is that the theory of two-dimensional discrete competitive and cooperative systems is very well developed, unlike such theory for three and higher Kalabušić et al. Advances in Difference Equations 2011, 2011:29 http://www.advancesindifferenceequations.com/content/2011/1/29 dimensional systems. Part of the reason for this situation is de Mottoni and Schiaffino theorem given below, which provides relatively simple scenarios for possible behavior of many two-dimensional discrete competitive and cooperative systems. However, this does not mean that one cannot encounter chaos in such systems as has been shown by Smith, see [17]. If v = (u, v) Î ℝ2, we denote with Ql (v), ℓ Î {1, 2, 3, 4}, the four quadrants in ℝ2 relative to v, i.e., Q1 (v) = {(x, y) ℝ2: x ≥ u, y ≥ v}, Q2 (v) = {(x, y) Î ℝ2: x ≤ u, y ≥ v}, and so on. Define the South-East partial order ≼se on ℝ2 by (x, y) ≼se (s, t) if and only if x ≤ s and y ≥ t. Similarly, we define the (...truncated)