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- Open Access

# Basins of attraction of certain rational anti-competitive system of difference equations in the plane

- Samra Moranjkić
^{1}and - Zehra Nurkanović
^{1}Email author

**2012**:153

https://doi.org/10.1186/1687-1847-2012-153

© Moranjkić and Nurkanović; licensee Springer 2012

**Received: **20 March 2012

**Accepted: **23 August 2012

**Published: **4 September 2012

## Abstract

We investigate the global asymptotic behavior of solutions of the following anti-competitive 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 period-two solutions.

**MSC:**39A10, 39A11.

## Keywords

- difference equations
- anti-competitive
- map
- stability
- stable manifold

## 1 Introduction

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 non-decreasing in *x* and non-increasing in *y*, and $g(x,y)$ is non-increasing in *x* and non-decreasing 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 *anti-competitive.*

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], Garić-Demirović, Kulenović and Nurkanović [9, 10], Smith [11, 12] and others.

The study of anti-competitive systems started in [13] and has advanced since then (see [14, 15]). The principal tool of the study of anti-competitive systems is the fact that the second iterate of the map associated with an anti-competitive 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 period*-

*two 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 non*-

*hyperbolic*,

*and an infinite number of minimal period*-

*two 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 period*-*two 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 even*-*indexed terms converging to* ${A}_{x}$ *and odd*-*indexed terms converging to* ${B}_{x}$; *solutions with an initial point in* ${\mathcal{C}}_{{B}_{x}}$ *have even*-*indexed terms converging to* ${B}_{x}$ *and odd*-*indexed 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 period*-

*two 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 even*-*indexed terms converging to* ${A}_{0}$ *and odd*-*indexed terms converging to* ${B}_{0}$, *solutions with an initial point in* $\{(x,y):x>0,y=0\}$ *have even*-*indexed terms converging to* ${B}_{0}$ *and odd*-*indexed 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.

*T*is given as

*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 Mottoni-Schiaffino for the Poincaré map of a periodic competitive Lotka-Volterra 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 period*-*two 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 period*-*two points*. *In the latter case*, *the set of endpoints of* $\mathcal{C}$ *is a minimal period*-*two 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:

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 non*-*hyperbolic*. - (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

- (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 non-hyperbolic.

- (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 non*-

*hyperbolic 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:

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 ${x}_{\pm}=-{A}_{1}\pm \sqrt{{\gamma}_{1}({A}_{1}{B}_{2}-{A}_{2})}$ (${x}_{-}<0$, ${x}_{+}>0$).

- (1)

*i.e.*,

- (2)If ${x}_{+}=\overline{x}$, then $0=\varphi ({x}_{+})=\varphi (\overline{x})$, hence $1+p+q=0$,
*i.e.*,$f({x}_{+})=f(\overline{x})=f(-{A}_{1}+\sqrt{{\gamma}_{1}({A}_{1}{B}_{2}-{A}_{2})})=0,$

*i.e.*, ${E}_{+}=(\overline{x},\overline{y})$ is a non-hyperbolic point of the form

- (3)If $\overline{x}<{x}_{+}$, then $\varphi (\overline{x})<\varphi ({x}_{+})=0$ and$0=f(\overline{x})<f({x}_{+})=f(-{A}_{1}+\sqrt{{\gamma}_{1}({A}_{1}{B}_{2}-{A}_{2})}),$

holds, so ${E}_{+}$ is a locally asymptotically stable. □

## 4 Periodic character of solutions

In this section, we give the existence and local stability of period-two solutions.

**Lemma 2**

*Assume that*${\beta}_{2}{\gamma}_{1}>{A}_{1}{A}_{2}$.

*Then System*(2)

*has the following minimal period*-

*two solutions*:

*If*

*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

*y*and the term $({A}_{1}{A}_{2}-{\beta}_{2}{\gamma}_{1})$ from (19), we get the identity

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 ${\beta}_{2}{\gamma}_{1}-{A}_{1}{A}_{2}=-{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}-{A}_{1}{B}_{2})]$. □

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 non*-*hyperbolic 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

*i.e.*, $0<{\lambda}_{2}<1$.

(*b*) Eigenvectors corresponding to the eigenvalues ${\lambda}_{1}$ and ${\lambda}_{2}$ are ${\mathbf{v}}_{1}=(1-d,c)$ and ${\mathbf{v}}_{2}=(a-1,c)$. 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)
*non*-*hyperbolic 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

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*$T{([0,\mathrm{\infty})}^{2})\subseteq \mathcal{B}$.

*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

- (ii)

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 period*-

*two 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

□

**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 ${\beta}_{2}{\gamma}_{1}\le {A}_{1}{A}_{2}$

which is an invariant box. In view of Corollary 1 and Theorem 2, every solution converges to minimal period-two solutions or ${E}_{0}$. System (2) has no minimal period-two 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 ${\beta}_{2}{\gamma}_{1}-{A}_{1}{A}_{2}>0$

*.*System (2) can be decomposed into the system of the even-indexed and odd-indexed 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.

Case 4 $0<{\beta}_{2}{\gamma}_{1}-{A}_{1}{A}_{2}<-{B}_{2}[{A}_{1}^{2}+{\gamma}_{1}({A}_{2}-{A}_{1}{B}_{2})]$

and Lemma 7 completes the conclusion (d) of Theorem 1. □

## Declarations

### Acknowledgements

The authors are very grateful to Professor M.R.S. Kulenović for his valuable suggestions. They thank also the referees for their useful comments.

## Authors’ Affiliations

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