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Basins of attraction of certain rational anticompetitive system of difference equations in the plane
Advances in Difference Equations volume 2012, Article number: 153 (2012)
Abstract
We investigate the global asymptotic behavior of solutions of the following anticompetitive system of rational difference equations:
where the parameters {\gamma}_{1}, {\beta}_{2}, {A}_{1}, {A}_{2} and {B}_{2} are positive numbers and the initial conditions ({x}_{0},{y}_{0}) are arbitrary nonnegative numbers. We find the basins of attraction of all attractors of this system, which are the equilibrium points and periodtwo solutions.
MSC:39A10, 39A11.
1 Introduction
A firstorder system of difference equations
where \mathcal{R}\subset {\mathbb{R}}^{2}, (f,g):\mathcal{R}\to \mathcal{R},f,g are continuous functions is competitive if f(x,y) is nondecreasing in x and nonincreasing in y, and g(x,y) is nonincreasing in x and nondecreasing in y.
System (1) where the functions f and g have a monotonic character opposite of the monotonic character in competitive system will be called anticompetitive.
We consider the following anticompetitive system of difference equations:
where the parameters {A}_{1}, {\gamma}_{1}, {A}_{2}, {B}_{2} and {\beta}_{2} are positive numbers and the initial conditions ({x}_{0},{y}_{0}) are arbitrary nonnegative numbers. In the classification of all linear fractional systems in [1], System (2) was mentioned as System (16, 39).
Competitive and cooperative systems of the form (1) were studied by many authors such as Clark and Kulenović [2], Clark, Kulenović and Selgrade [3], Hirsch and Smith [4], Kulenović and Ladas [5], Kulenović and Merino [6], Kulenović and Nurkanović [7, 8], GarićDemirović, Kulenović and Nurkanović [9, 10], Smith [11, 12] and others.
The study of anticompetitive systems started in [13] and has advanced since then (see [14, 15]). The principal tool of the study of anticompetitive systems is the fact that the second iterate of the map associated with an anticompetitive system is a competitive map, and so the elaborate theory for such maps developed recently in [4, 16, 17] can be applied.
The main result on the global behavior of System (2) is the following theorem.
Theorem 1 (a) If {\beta}_{2}{\gamma}_{1}\le {A}_{1}{A}_{2}, then {E}_{0}=(0,0) is a unique equilibrium, and the basin of attraction of this equilibrium is \mathcal{B}({E}_{0})=\{(x,y):x\ge 0,y\ge 0\} (see Figure 1(a)).
(b) If {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}>{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})] and {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}>0, then there exist two equilibrium points: {E}_{0} which is a repeller and {E}_{+} which is an interior saddle point, and minimal periodtwo solutions {A}_{0}=(0,\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{\gamma}_{1}{B}_{2}}) and {B}_{0}=(\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{A}_{1}{B}_{2}},0) which are locally asymptotically stable. There exists a set \mathcal{C}\subset \mathcal{R}=[0,\mathrm{\infty})\times [0,\mathrm{\infty}) such that {E}_{0}\in \mathcal{C}, and {\mathcal{W}}^{s}({E}_{+})=\mathcal{C}\setminus {E}_{0} is an invariant subset of the basin of attraction of {E}_{+}. The set \mathcal{C} is a graph of a strictly increasing continuous function of the first variable on an interval and separates \mathcal{R} into two connected and invariant components, namely
which satisfy (see Figure 1(b)):
(i) If ({x}_{0},{y}_{0})\in {\mathcal{W}}_{+}, then
and
(ii) If ({x}_{0},{y}_{0})\in {\mathcal{W}}_{}, then
and
(c) If 0<{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}={B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})], then (see Figure 1(c))
(i) There exist two equilibrium points: {E}_{0} which is a repeller and {E}_{+}\in int(\mathcal{R}) which is a nonhyperbolic, and an infinite number of minimal periodtwo solutions
for x\in [0,\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{A}_{1}{B}_{2}}], that belong to the segment of the line (15) in the first quadrant.
(ii) All minimal periodtwo solutions and the equilibrium {E}_{+} are stable but not asymptotically stable.
(iii) There exists a family of strictly increasing curves {\mathcal{C}}_{+}, {\mathcal{C}}_{{A}_{x}}, {\mathcal{C}}_{{B}_{x}} for x\in (0,\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{A}_{1}{B}_{2}}) and
that emanate from {E}_{0} and {A}_{x}\in {\mathcal{C}}_{{A}_{x}}, {B}_{x}\in {\mathcal{C}}_{{B}_{x}} for all x\in [0,\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{A}_{1}{B}_{2}}), such that the curves are pairwise disjoint, the union of all the curves equals {\mathbb{R}}_{+}^{2}. Solutions with initial points in {\mathcal{C}}_{+} converge to {E}_{+} and solutions with an initial point in {\mathcal{C}}_{{A}_{x}} have evenindexed terms converging to {A}_{x} and oddindexed terms converging to {B}_{x}; solutions with an initial point in {\mathcal{C}}_{{B}_{x}} have evenindexed terms converging to {B}_{x} and oddindexed terms converging to {A}_{x}.
(d) If 0<{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}<{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})], then System (2) has two equilibrium points: {E}_{0} which is a repeller and {E}_{+} which is locally asymptotically stable, and minimal periodtwo solutions {A}_{0} and {B}_{0} which are saddle points. The basin of attraction of the equilibrium point {E}_{+} is the set
and solutions with an initial point in \{(x,y):x=0,y>0\} have evenindexed terms converging to {A}_{0} and oddindexed terms converging to {B}_{0}, solutions with an initial point in \{(x,y):x>0,y=0\} have evenindexed terms converging to {B}_{0} and oddindexed terms converging to {A}_{0} (see Figure 1(d)).
2 Preliminaries
We now give some basic notions about systems and maps in the plane of the form (1).
Consider a map T=(f,g) on a set \mathcal{R}\subset {\mathbf{R}}^{2}, and let E\in \mathcal{R}. The point E\in \mathcal{R} is called a fixed point if T(E)=E. An isolated fixed point is a fixed point that has a neighborhood with no other fixed points in it. A fixed point E\in \mathcal{R} is an attractor if there exists a neighborhood \mathcal{U} of E such that {T}^{n}(\mathbf{x})\to E as n\to \mathrm{\infty} for \mathbf{x}\in \mathcal{U}; the basin of attraction is the set of all \mathbf{x}\in \mathcal{R} such that {T}^{n}(\mathbf{x})\to E as n\to \mathrm{\infty}. A fixed point E is a global attractor on a set \mathcal{K} if E is an attractor and \mathcal{K} is a subset of the basin of attraction of E. If T is differentiable at a fixed point E, and if the Jacobian {J}_{T}(E) has one eigenvalue with modulus less than one and a second eigenvalue with modulus greater than one, E is said to be a saddle. See [18] for additional definitions.
Here we give some basic facts about the monotone maps in the plane, see [11, 16, 17, 19]. Now, we write System (2) in the form
where the map T is given as
The map T may be viewed as a monotone map if we define a partial order on {\mathbf{R}}^{2} so that the positive cone in this new partial order is the fourth quadrant. Specifically, for \mathbf{v}=({v}_{1},{v}_{2}), \mathbf{w}=({w}_{1},{w}_{2})\in {\mathbf{R}}^{2} we say that \mathbf{v}\u2aaf\mathbf{w} if {v}_{1}\le {w}_{1} and {w}_{2}\le {v}_{2}. Two points \mathbf{v},\mathbf{w}\in {\mathbf{R}}_{+}^{2} are said to be related if \mathbf{v}\u2aaf\mathbf{w} or \mathbf{w}\u2aaf\mathbf{v}. Also, a strict inequality between points may be defined as \mathbf{v}\prec \mathbf{w} if \mathbf{v}\u2aaf\mathbf{w} and \mathbf{v}\ne \mathbf{w}. A stronger inequality may be defined as \mathbf{v}\prec \prec \mathbf{w} if {v}_{1}<{w}_{1} and {w}_{2}<{v}_{2}. A map f:int{\mathbf{R}}_{+}^{2}\to Int{\mathbf{R}}_{+}^{2} is strongly monotone if \mathbf{v}\prec \mathbf{w} implies that f(\mathbf{v})\prec \prec f(\mathbf{w}) for all \mathbf{v},\mathbf{w}\in Int{\mathbf{R}}_{+}^{2}. Clearly, being related is an invariant under iteration of a strongly monotone map. Differentiable strongly monotone maps have Jacobian with constant sign configuration
The mean value theorem and the convexity of {\mathbf{R}}_{+}^{2} may be used to show that T is monotone, as in [20].
For \mathbf{x}=({x}_{1},{x}_{2})\in {\mathbb{R}}^{2}, define {Q}_{l}(\mathbf{x}) for l=1,\dots ,4 to be the usual four quadrants based at x and numbered in a counterclockwise direction, for example, {Q}_{1}(\mathbf{x})=\{\mathbf{y}=({y}_{1},{y}_{2})\in {\mathbb{R}}^{2}:{x}_{1}\le {y}_{1},{x}_{2}\le {y}_{2}\}.
The following definition is from [11].
Definition 1 Let \mathcal{S} be a nonempty subset of {\mathbb{R}}^{2}. A competitive map T:\mathcal{S}\to \mathcal{S} is said to satisfy condition (O+) if for every x, y in \mathcal{S}, T(x){\u2aaf}_{ne}T(y) implies x{\u2aaf}_{ne}y, and T is said to satisfy condition (O−) if for every x, y in \mathcal{S}, T(x){\u2aaf}_{ne}T(y) implies y{\u2aaf}_{ne}x.
The following theorem was proved by de MottoniSchiaffino for the Poincaré map of a periodic competitive LotkaVolterra system of differential equations. Smith generalized the proof to competitive and cooperative maps [11].
Theorem 2 Let \mathcal{S} be a nonempty subset of {\mathbb{R}}^{2}. If T is a competitive map for which (O+) holds then for all x\in \mathcal{S}, \{{T}^{n}(x)\} is eventually componentwise monotone. If the orbit of x has compact closure, then it converges to a fixed point of T. If instead (O−) holds, then for all x\in \mathcal{S}, \{{T}^{2n}\} is eventually componentwise monotone. If the orbit of x has compact closure in \mathcal{S}, then its omega limit set is either a periodtwo orbit or a fixed point.
The following result is from [11], with the domain of the map specialized to be the Cartesian product of intervals of real numbers. It gives a sufficient condition for conditions (O+) and (O−).
Theorem 3 Let \mathcal{R}\subset {\mathbb{R}}^{2} be the Cartesian product of two intervals in \mathbb{R}. Let T:\mathcal{R}\to \mathcal{R} be a {C}^{1} competitive map. If T is injective and det{J}_{T}(x)>0 for all x\in \mathcal{R} then T satisfies (O+). If T is injective and det{J}_{T}(x)<0 for all x\in \mathcal{R} then T satisfies (O−).
Next two results are from [17, 21].
Theorem 4 Let T be a competitive map on a rectangular region \mathcal{R}\subset {\mathbb{R}}^{2}. Let \overline{\mathbf{x}}\in \mathcal{R} be a fixed point of T such that \mathrm{\Delta}:=\mathcal{R}\cap int({Q}_{1}(\overline{\mathbf{x}})\cup {Q}_{3}(\overline{\mathbf{x}})) is nonempty (i.e., \overline{\mathbf{x}} is not the NW or SE vertex of \mathcal{R}), and T is strongly competitive on Δ. Suppose that the following statements are true.

a.
The map T has a {C}^{1} extension to a neighborhood of \overline{\mathbf{x}}.

b.
The Jacobian matrix of T at \overline{\mathbf{x}} has real eigenvalues λ, μ such that 0<\lambda <\mu, where \lambda <1, and the eigenspace {E}^{\lambda} associated with λ is not a coordinate axis.
Then there exists a curve \mathcal{C}\subset \mathcal{R} through \overline{\mathbf{x}} that is invariant and a subset of the basin of attraction of \overline{\mathbf{x}}, such that \mathcal{C} is tangential to the eigenspace {E}^{\lambda} at \overline{\mathbf{x}}, and \mathcal{C} is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of \mathcal{C} in the interior of \mathcal{R} are either fixed points or minimal periodtwo points. In the latter case, the set of endpoints of \mathcal{C} is a minimal periodtwo orbit of T.
Theorem 5 (Kulenović & Merino)
Let {\mathcal{I}}_{1}, {\mathcal{I}}_{2} be intervals in \mathbb{R} with endpoints {a}_{1}, {a}_{2} and {b}_{1}, {b}_{2} with endpoints respectively, with {a}_{1}<{a}_{2} and {b}_{1}<{b}_{2}, where \mathrm{\infty}\le {a}_{1}<{a}_{2}\le \mathrm{\infty} and \mathrm{\infty}\le {b}_{1}<{b}_{2}\le \mathrm{\infty}. Let T be a competitive map on a rectangle {\mathcal{R}=\mathcal{I}}_{1}\times {\mathcal{I}}_{2} and \overline{\mathbf{x}}\in int(\mathcal{R}). Suppose that the following hypotheses are satisfied:

1.
T(int(\mathcal{R}))\subset int(\mathcal{R}) and T is strongly competitive on int(\mathcal{R}).

2.
The point \overline{\mathbf{x}} is the only fixed point of T in ({Q}_{1}(\overline{\mathbf{x}})\cup {Q}_{3}(\overline{\mathbf{x}}))\cap int(\mathcal{R}).

3.
The map T is continuously differentiable in a neighborhood of \overline{\mathbf{x}}.

4.
At least one of the following statements is true:

a.
T has no minimal period two orbits in ({Q}_{1}(\overline{\mathbf{x}})\cup {Q}_{3}(\overline{\mathbf{x}}))\cap int(\mathcal{R}).

b.
det{J}_{T}(\overline{\mathbf{x}})>0 and T(\mathbf{x})=\overline{\mathbf{x}} only for \mathbf{x}=\overline{\mathbf{x}}.

5.
\overline{\mathbf{x}} is a saddle point.
Then the following statements are true.

(i)
The stable manifold {\mathcal{W}}^{s}(\overline{\mathbf{x}}) is connected and it is the graph of a continuous increasing curve with endpoints in \partial \mathcal{R}. int(\mathcal{R}) is divided by the closure of {\mathcal{W}}^{s}(\overline{\mathbf{x}}) into two invariant connected regions {\mathcal{W}}_{+} (“below the stable set”), and {\mathcal{W}}_{} (“above the stable set”), where

(ii)
The unstable manifold {\mathcal{W}}^{u}(\overline{\mathbf{x}}) is connected, and it is the graph of a continuous decreasing curve.

(iii)
For every \mathbf{x}\in {\mathcal{W}}_{+}, {T}^{n}(\mathbf{x}) eventually enters the interior of the invariant set {Q}_{4}(\overline{\mathbf{x}})\cap \mathcal{R}, and for every \mathbf{x}\in {\mathcal{W}}_{}, {T}^{n}(\mathbf{x}) eventually enters the interior of the invariant set {Q}_{2}(\overline{\mathbf{x}})\cap \mathcal{R}.

(iv)
Let \mathbf{m}\in {Q}_{2}(\overline{\mathbf{x}}) and \mathbf{M}\in {Q}_{4}(\overline{\mathbf{x}}) be the endpoints of {\mathcal{W}}^{u}(\overline{\mathbf{x}}), where \mathbf{m}{\u2aaf}_{se}\overline{\mathbf{x}}{\u2aaf}_{se}\mathbf{M}. For every \mathbf{x}\in {\mathcal{W}}_{} and every \mathbf{z}\in \mathcal{R} such that \mathbf{m}{\u2aaf}_{se}z, there exists m\in \mathbb{N} such that {T}^{m}(\mathbf{x}){\u2aaf}_{se}z, and for every \mathbf{x}\in {\mathcal{W}}_{+} and every \mathbf{z}\in \mathcal{R} such that \mathbf{z}{\u2aaf}_{se}\mathbf{M}, there exists m\in \mathbb{N} such that \mathbf{M}{\u2aaf}_{se}{T}^{m}(\mathbf{x}).
3 Linearized stability analysis
Lemma 1

(i)
If {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}\le 0, then System (2) has a unique equilibrium point {E}_{0}=(0,0).

(ii)
If {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}>0, then System (2) has two equilibrium points {E}_{0} and {E}_{+}=(\overline{x},\overline{y}), \overline{x}>0, \overline{y}>0.
Proof The equilibrium point E(\overline{x},\overline{y}) of System (2) satisfies the following system of equations:
It is easy to see that {E}_{0}=(0,0) is one equilibrium point for all values of the parameters, and {E}_{+}=(\overline{x},\overline{y}) is a positive equilibrium point if {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}>0. Indeed, substituting \overline{y} from the first equation in (4) in the second equation in (4), we obtain that \overline{x} satisfies the following equation:
By using Descartes’ theorem, we have that equation (5) has one positive equilibrium if the condition
is satisfied, i.e., {\beta}_{2}{\gamma}_{1}>{A}_{1}{A}_{2}. □
Theorem 6

(i)
If {\beta}_{2}{\gamma}_{1}<{A}_{1}{A}_{2}, then {E}_{0} is locally asymptotically stable.

(ii)
If {\beta}_{2}{\gamma}_{1}={A}_{1}{A}_{2}, then {E}_{0} is nonhyperbolic.

(iii)
If {\beta}_{2}{\gamma}_{1}>{A}_{1}{A}_{2}, then {E}_{0} is a repeller.
Proof The map T associated to System (2) is of the form (3). The Jacobian matrix of T at the equilibrium E=(\overline{x},\overline{y}) is
and
The corresponding characteristic equation has the following form:
from which {\lambda}_{1,2}=\pm \sqrt{\frac{{\beta}_{2}{\gamma}_{1}}{{A}_{1}{A}_{2}}}.

(i)
If {\beta}_{2}{\gamma}_{1}<{A}_{1}{A}_{2}, then {\lambda}_{1,2}<1, i.e., {E}_{0} is locally asymptotically stable.

(ii)
If {\beta}_{2}{\gamma}_{1}={A}_{1}{A}_{2}, then {\lambda}_{1,2}=1, which implies that {E}_{0} is nonhyperbolic.

(iii)
If {\beta}_{2}{\gamma}_{1}>{A}_{1}{A}_{2}, then {\lambda}_{1,2}>1, which implies that {E}_{0} is a repeller.
□
Theorem 7

(1)
Assume that {\beta}_{2}{\gamma}_{1}>{A}_{1}{A}_{2} and
{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}>{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})].(8)
Then the positive equilibrium {E}_{+} is a saddle point.

(2)
Assume that
0<{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}={B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})].(9)
Then the positive equilibrium {E}_{+} is a nonhyperbolic point and

(3)
Assume that
0<{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}<{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})].(10)
Then the positive equilibrium {E}_{+} is locally asymptotically stable.
Proof The Jacobian matrix of T at the equilibrium {E}_{+}=(\overline{x},\overline{y}) is of the form (7) and the corresponding characteristic equation has the following form:
where
Hence, for {E}_{+}=(\overline{x},\overline{y}), we have p<0, q<0, so {p}^{2}4q>0. Since
we obtain
Similarly,
where
Now, for the positive equilibrium, it holds
If {A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})\ge 0, then \varphi (x)>0 for all x>0, which implies that {E}_{+} is a saddle point. If {A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})<0, then \varphi (x)=0 for {x}_{\pm}={A}_{1}\pm \sqrt{{\gamma}_{1}({A}_{1}{B}_{2}{A}_{2})} ({x}_{}<0, {x}_{+}>0).
Now we have three cases: {x}_{+}<\overline{x}, {x}_{+}=\overline{x} or \overline{x}<{x}_{+}. Functions f(x) and \varphi (x) are increasing for x>0.

(1)
If {x}_{+}<\overline{x}, then 0=\varphi ({x}_{+})<\varphi (\overline{x}), i.e., 1+p+q<0 and f({x}_{+})<f(\overline{x})=0. So,
from which it follows
i.e.,
Now we have
so we can see that the conditions (8) and (6) are sufficient for {E}_{+}=(\overline{x},\overline{y}) to be a saddle point.

(2)
If {x}_{+}=\overline{x}, then 0=\varphi ({x}_{+})=\varphi (\overline{x}), hence 1+p+q=0, i.e.,
f({x}_{+})=f(\overline{x})=f({A}_{1}+\sqrt{{\gamma}_{1}({A}_{1}{B}_{2}{A}_{2})})=0,
from which
If conditions (12) and (6) are satisfied, then
holds, i.e., {E}_{+}=(\overline{x},\overline{y}) is a nonhyperbolic point of the form

(3)
If \overline{x}<{x}_{+}, then \varphi (\overline{x})<\varphi ({x}_{+})=0 and
0=f(\overline{x})<f({x}_{+})=f({A}_{1}+\sqrt{{\gamma}_{1}({A}_{1}{B}_{2}{A}_{2})}),
from which
Hence, if conditions (13) and (6) are satisfied, then
holds, so {E}_{+} is a locally asymptotically stable. □
4 Periodic character of solutions
In this section, we give the existence and local stability of periodtwo solutions.
Lemma 2 Assume that {\beta}_{2}{\gamma}_{1}>{A}_{1}{A}_{2}. Then System (2) has the following minimal periodtwo solutions:
If
then System (2) has an infinite number of minimal periodtwo solutions of the form
for x\in [0,\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{A}_{1}{B}_{2}}], located along the line
Proof The second iterate of T is (25). Equilibrium curves of the map {T}^{2}(x,y) are
and
We get periodtwo solutions as the intersection point of equilibrium curves (16) and (17) in the first quadrant. If x\ne 0, y=0, then System (16), (17) is reduced to the equation
and the positive solution of this equation is
If x=0, y\ne 0, then System (16), (17) is reduced to the equation
with the positive solution
On the other hand, if x>0, y>0, then we have
that is
and
Therefore, it must be ({\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2})>0 in order to get any positive solution. By eliminating the term ({\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}) from (18) and using condition (9), we get
which implies
hence
Now, by eliminating y and the term ({A}_{1}{A}_{2}{\beta}_{2}{\gamma}_{1}) from (19), we get the identity
If x={\gamma}_{1}{B}_{2}{A}_{1}, we have
So, periodic solutions are located along line (15) with endpoints given by (14) using conditions (9). It is easy to see that {A}_{x},{B}_{x}\in \mathcal{H} if {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}={B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})]. □
Let (x,y)\in \mathcal{H}, then the corresponding Jacobian matrix of the map {T}^{2} has the following form:
where a:={F}_{x}(x,y), b:={F}_{y}(x,y), c:={G}_{x}(x,y), d:={G}_{y}(x,y).
Lemma 3 Assume that 0<{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}={B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})]. Then the following statements are true.

(a)
The points {A}_{x},{B}_{x}\in \mathcal{H} are nonhyperbolic fixed points for the map {T}^{2}, and both of them have eigenvalues {\lambda}_{1}=1 and {\lambda}_{2}\in (0,1).

(b)
Eigenvectors corresponding to the eigenvalues {\lambda}_{1} and {\lambda}_{2} are not parallel to coordinate axes.
Proof (a) From (15) we have {y}_{\mathcal{H}}^{\mathrm{\prime}}(x)=\frac{{A}_{1}}{{\gamma}_{1}}<0. Since
by implicit differentiation of equations F(x,y)=x and G(x,y)=y at the point (x,y)\in \mathcal{H}, we obtain
Since a>0, b<0, c<0 and d>0, from (22), we get
The characteristic polynomial of the matrix (21) at the point (x,y)\in \mathcal{H} is of the form
Now, using (22) we have (1a)(1d)=bc, and since
we get {\lambda}_{1}=1, and due to Vieta’s formulas and condition (23), it follows
i.e., 0<{\lambda}_{2}<1.
(b) Eigenvectors corresponding to the eigenvalues {\lambda}_{1} and {\lambda}_{2} are {\mathbf{v}}_{1}=(1d,c) and {\mathbf{v}}_{2}=(a1,c). By condition (23) it is easy to see that these vectors are not parallel to the coordinate axes. □
Lemma 4 The periodic points {A}_{0} and {B}_{0} given by (14) are

(a)
locally asymptotically stable if {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}>{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})] and {\beta}_{2}{\gamma}_{1}>{A}_{1}{A}_{2},

(b)
nonhyperbolic if 0<{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}={B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})],

(c)
saddle points if 0<{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}<{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})].
Proof We have that
and characteristic eigenvalues are
Now,
Therefore,
On the other hand, we have
and the corresponding eigenvalues are
so it comes to the same conclusion! □
5 Global results
In this section, we present the results on the global dynamics of System (2).
Lemma 5 Every solution of System (2) satisfies

1.
{x}_{n}\le \frac{{\gamma}_{1}}{{A}_{1}}\cdot \frac{{\beta}_{2}}{{B}_{2}}, {y}_{n}\le \frac{{\beta}_{2}}{{B}_{2}}, n=2,3,\dots.

2.
If {\beta}_{2}{\gamma}_{1}<{A}_{1}{A}_{2}, then {lim}_{n\to \mathrm{\infty}}{x}_{n}=0, {lim}_{n\to \mathrm{\infty}}{y}_{n}=0.
The map T satisfies:

3.
T(\mathcal{B})\subseteq \mathcal{B}, where \mathcal{B}=[0,\frac{{\gamma}_{1}}{{A}_{1}}\cdot \frac{{\beta}_{2}}{{B}_{2}}]\times [0,\frac{{\beta}_{2}}{{B}_{2}}], that is, \mathcal{B} is an invariant box.

4.
T(\mathcal{B}) is an attracting box, that is T{([0,\mathrm{\infty})}^{2})\subseteq \mathcal{B}.
Proof From System (2), we have
for n=0,1,2,\dots , and
for n=1,2,\dots . Furthermore, we get
i.e.,
so it follows that {lim}_{n\to \mathrm{\infty}}{x}_{n}=0, {lim}_{n\to \mathrm{\infty}}{y}_{n}=0 if {\beta}_{2}{\gamma}_{1}<{A}_{1}{A}_{2}.
Proof of 3. and 4. is an immediate checking. □
Lemma 6 The map {T}^{2} is injective and det{J}_{{T}^{2}}(x,y)>0, for all x\ge 0 and y\ge 0.
Proof (i) Here we prove that map T is injective, which implies that {T}^{2} is injective. Indeed, T\left(\begin{array}{c}{x}_{1}\\ {y}_{1}\end{array}\right)=T\left(\begin{array}{c}{x}_{2}\\ {y}_{2}\end{array}\right) implies that
By solving System (24) with respect to {x}_{1}, {x}_{2} or {y}_{1}, {y}_{2}, we obtain that ({x}_{1},{y}_{1})=({x}_{2},{y}_{2}).

(ii)
The map {T}^{2}(x,y)=\left(\begin{array}{c}F(x,y)\\ G(x,y)\end{array}\right) is of the form
(25)
and
where
Now, we obtain
where
and the Jacobian matrix of {T}^{2}(x,y) is invertible for all x\ge 0 and y\ge 0. □
Corollary 1 The competitive map {T}^{2} satisfies the condition (O+). Consequently, the sequences \{{x}_{2n}\}, \{{x}_{2n+1}\}, \{{y}_{2n}\}, \{{y}_{2n+1}\} of every solution of System (2) are eventually monotone.
Proof It immediately follows from Lemma 6, Theorem 2 and 3. □
Lemma 7 Assume {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}>0. System (2) has periodtwo solutions (14) and

(a)
If ({x}_{0},{y}_{0})=(x,0), x>0, then
\underset{n\to \mathrm{\infty}}{lim}{T}^{2n}(x,0)=(\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{A}_{1}{B}_{2}},0)={B}_{0}
and

(b)
If ({x}_{0},{y}_{0})=(0,y), y>0, then
\underset{n\to \mathrm{\infty}}{lim}{T}^{2n}(0,y)=(0,\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{\gamma}_{1}{B}_{2}})={A}_{0}
and
Proof (a) For all x>0, x\ne \frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{A}_{1}{B}_{2}}, we have
and by induction,
Now, we have
and
□
Lemma 8 The map {T}^{2} associated to System (2) satisfies the following:
Proof Since {T}^{2} is injective, then {T}^{2}(x,y)=(\overline{x},\overline{y})={T}^{2}(\overline{x},\overline{y})\Rightarrow (x,y)=(\overline{x},\overline{y}). □
Proof of Theorem 1 Case 1 {\beta}_{2}{\gamma}_{1}\le {A}_{1}{A}_{2}
Equilibrium {E}_{0} is unique (see Lemma 1), and by Lemma 5, every solution of System (2) belongs to
which is an invariant box. In view of Corollary 1 and Theorem 2, every solution converges to minimal periodtwo solutions or {E}_{0}. System (2) has no minimal periodtwo solutions (Lemma 2). So, every solution of System (2) converges to {E}_{0}.
Case 2 {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}>{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})] and {\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}>0
By Lemmas 1, 2, 4 and Theorems 6 and 7, there exist two equilibrium points: {E}_{0} which is a repeller and {E}_{+} which is a saddle point, and minimal periodtwo solutions {A}_{0} and {B}_{0} which are locally asymptotically stable. Clearly {T}^{2} is strongly competitive and it is easy to check that the points {A}_{0} and {B}_{0} are locally asymptotically stable for {T}^{2} as well. System (2) can be decomposed into the system of the evenindexed and oddindexed terms as follows:
The existence of the set \mathcal{C} with the stated properties follows from Lemmas 6, 2, 7, 8, Corollary 1, Theorems 4 and 5.
Case 3 0<{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}={B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})]
Cases (i) and (ii) from (c) in Theorem 1 are the consequence of Lemmas 1, 2, 4 and Theorems 6 and 7.
Since {T}^{2} is strongly competitive and points {A}_{x} and {B}_{x}, for all x\in [0,\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{A}_{1}{B}_{2}}), are nonhyperbolic points of the map {T}^{2}, by Lemmas 1, 6, 2, 3, 7, Corollary 1, Theorems 2, 5, 6 and 7, it follows that all conditions of Theorem 4 are satisfied for the map {T}^{2} with \mathcal{R}=[0,\mathrm{\infty})\times [0,\mathrm{\infty}). By Lemma 7, it is clear that
Case 4 0<{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}<{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}{A}_{1}{B}_{2})]
Lemma 2 implies that System (2) has minimal periodtwo solutions (14). Furthermore, Corollary 1 and Theorem 2 imply that all solutions of System (2) converge to an equilibrium or minimal periodtwo solutions, and since, by Theorem 6, {E}_{0} is a repeller, all solutions converge to {E}_{+} (which is, in view of Theorem 7, locally asymptotically stable) or minimal periodtwo solutions (14). The points {A}_{0} and {B}_{0} are saddle points of the strongly competitive map {T}^{2}; and by Lemma 7, the stable manifold of {A}_{0} (under {T}^{2}) is
and the stable manifold of {B}_{0} (under {T}^{2}) is
and each of these stable manifolds is unique. This implies that the basin of attraction of the equilibrium point {E}_{+} is the set
and Lemma 7 completes the conclusion (d) of Theorem 1. □
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The authors are very grateful to Professor M.R.S. Kulenović for his valuable suggestions. They thank also the referees for their useful comments.
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Moranjkić, S., Nurkanović, Z. Basins of attraction of certain rational anticompetitive system of difference equations in the plane. Adv Differ Equ 2012, 153 (2012). https://doi.org/10.1186/168718472012153
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DOI: https://doi.org/10.1186/168718472012153