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
{x}_{n+1}\le \frac{{\gamma}_{1}}{{A}_{1}}{y}_{n}\le \frac{{\gamma}_{1}}{{A}_{1}}\cdot \frac{{\beta}_{2}}{{B}_{2}}
for n=1,2,\dots . Furthermore, we get
{x}_{n}\le \frac{{\gamma}_{1}}{{A}_{1}}{y}_{n1}\le \frac{{\gamma}_{1}{\beta}_{2}}{{A}_{1}{A}_{2}}{x}_{n2},
i.e.,
{x}_{2n}\le {\left(\frac{{\gamma}_{1}{\beta}_{2}}{{A}_{1}{A}_{2}}\right)}^{n}{x}_{0},\phantom{\rule{2em}{0ex}}{x}_{2n+1}\le {\left(\frac{{\gamma}_{1}{\beta}_{2}}{{A}_{1}{A}_{2}}\right)}^{n}{x}_{1},
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
{A}_{1}({y}_{1}{y}_{2})={x}_{1}{y}_{2}{x}_{2}{y}_{1},\phantom{\rule{2em}{0ex}}{A}_{2}({x}_{1}{x}_{2})={x}_{2}{y}_{1}{x}_{1}{y}_{2}.
(24)
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
and
{J}_{{T}^{2}}(x,y)=\left(\begin{array}{cc}{F}_{x}& {F}_{y}\\ {G}_{x}& {G}_{y}\end{array}\right),
where
Now, we obtain
det{J}_{{T}^{2}}(x,y)={F}_{x}{G}_{y}{F}_{y}{G}_{x}=UV,
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
\underset{n\to \mathrm{\infty}}{lim}{T}^{2n+1}(x,0)=(0,\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{\gamma}_{1}{B}_{2}})={A}_{0}.

(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
\underset{n\to \mathrm{\infty}}{lim}{T}^{2n+1}(x,0)=(\frac{{\beta}_{2}{\gamma}_{1}{A}_{1}{A}_{2}}{{A}_{1}{B}_{2}},0)={B}_{0}.
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:
{T}^{2}(x,y)=(\overline{x},\overline{y})\phantom{\rule{1em}{0ex}}\mathit{\text{only for}}\phantom{\rule{0.25em}{0ex}}(x,y)=(\overline{x},\overline{y}).
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
B=\left[\begin{array}{c}0,\frac{{\beta}_{2}{\gamma}_{1}}{{A}_{1}{B}_{2}}\end{array}\right]\times \left[\begin{array}{c}0,\frac{{\beta}_{2}}{{B}_{2}}\end{array}\right],
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:
\{\begin{array}{c}{x}_{2n+1}=\frac{{\gamma}_{1}{y}_{2n}}{{A}_{1}+{x}_{2n}},\hfill \\ {x}_{2n}=\frac{{\gamma}_{1}{y}_{2n1}}{{A}_{1}+{x}_{2n1}},\hfill \\ {y}_{2n+1}=\frac{{\beta}_{2}{x}_{2n}}{{A}_{2}+{B}_{2}{x}_{2n}+{y}_{2n}},\hfill \\ {y}_{2n}=\frac{{\beta}_{2}{x}_{2n1}}{{A}_{2}+{B}_{2}{x}_{2n1}+{y}_{2n1}},\phantom{\rule{1em}{0ex}}n=1,2,\dots .\hfill \end{array}
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
{\mathcal{C}}_{{A}_{0}}=\{(x,y):x=0,y>0\}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{C}}_{{B}_{0}}=\{(x,y):x>0,y=0\}.
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
\mathcal{B}({A}_{0})=\{(x,y):x=0,y>0\}
and the stable manifold of {B}_{0} (under {T}^{2}) is
\mathcal{B}({B}_{0})=\{(x,y):x>0,y=0\}
and each of these stable manifolds is unique. This implies that the basin of attraction of the equilibrium point {E}_{+} is the set
\mathcal{B}({E}_{+})=\{(x,y):x>0,y>0\},
and Lemma 7 completes the conclusion (d) of Theorem 1. □