Let I be some interval of real numbers and let \(f\in C^{1} [I\times I, I]\) be strongly increasing function. Assume that for \((x_{0},y_{0})\in I\times I\) there exists \(n_{0}\) such that \(F^{n}(x_{0},y_{0})\in[U_{1},U_{2}]^{2}\) for all \(n>n_{0}\) where \([U_{1},U_{2}]\subseteq I\) and \(\infty< U_{1}<U_{2}<\infty\) and assume that \([U_{1},U_{2}]^{2}\) is an invariant set for the map T, that is, \(T: [U_{1},U_{2}]^{2} \to[U_{1},U_{2}]^{2}\). Let \(\bar{x}_{0}, \bar{x}_{\mathrm{SW}}, \bar{x}_{\mathrm{NE}}\in I\), \(U_{1} \leq\bar{x}_{0} < \bar {x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) be three equilibrium points of the difference equation (1) where the equilibrium points \(\bar {x}_{0}\) and \(\bar{x}_{\mathrm{NE}}\) are locally asymptotically stable and \(\bar {x}_{\mathrm{SW}}\) is unstable. Then the map F has three equilibrium solutions \(E_{0}(\bar{x}_{0},\bar{x}_{0})\), \(E_{\mathrm{SW}}(\bar{x}_{\mathrm{SW}},\bar {x}_{\mathrm{SW}})\), and \(E_{\mathrm{NE}}(\bar{x}_{\mathrm{NE}},\bar{x}_{\mathrm{NE}})\) such that \(E_{0}\ll_{\mathrm{ne}} E_{\mathrm{SW}}\ll_{\mathrm{ne}} E_{\mathrm{NE}}\) where the equilibrium points \(E_{0}\) and \(E_{\mathrm{NE}}\) are locally asymptotically stable and \(E_{\mathrm{SW}}\) is unstable. By Theorem 3 all solutions converge to either equilibrium solutions or to the periodtwo solutions. As is well known [12, 13] the periodtwo solutions are the points in the SouthEast ordering, which means that they belong to \(Q_{2}(E_{\mathrm{SW}})\cup Q_{4}(E_{\mathrm{SW}})\). In the following discussion, we will assume that all periodtwo solutions belong to \(\operatorname{int}(Q_{2}(E_{\mathrm{SW}})\cup Q_{4}(E_{\mathrm{SW}}))\).
Let \(\mathcal{B}(E_{0})\) be the basin of attraction of \(E_{0}\) and \(\mathcal {B}(E_{\mathrm{NE}})\) be the basin of attraction of \(E_{\mathrm{NE}}\) with respect to the map T.
Lemma 1
Let
\(Q_{1}((x_{0},y_{0}))=\{(x,y): x\geq x_{0}\textit{ and } y\geq y_{0}\}\cap I\times I\)
and
\(Q_{3}((x_{0},y_{0}))=\{(x,y): x\leq x_{0}\textit{ and } y\leq y_{0}\}\cap I\times I\). Then the following holds:

(i)
If there are no minimal periodtwo solutions in
\(\operatorname{int}(Q_{3}(E_{\mathrm{SW}}))\)
then
\(\operatorname{int}(Q_{3}(E_{\mathrm{SW}}))\subset\mathcal{B}(E_{0})\).

(ii)
If there are no minimal periodtwo solutions in
\(\operatorname{int}(Q_{1}(E_{\mathrm{SW}}))\)
then
\(\operatorname{int}(Q_{1}(E_{\mathrm{SW}}))\subset\mathcal{B}(E_{\mathrm{NE}})\).
Proof
First, in view of Corollary 1, \(\operatorname{int} [\!\![(U_{1}, U_{1}), E_{0} ]\!\!]\subset\mathcal{B}(E_{0})\) and also \(\operatorname{int} [\!\![E_{\mathrm{NE}}, (U_{2}, U_{2}) ]\!\!]\subset\mathcal{B}(E_{\mathrm{NE}})\). By Corollary 1 we obtain \(\operatorname{int}(Q_{3}(E_{\mathrm{SW}})\cap Q_{1}(E_{0}))\subset\mathcal{B}(E_{0})\) and \(\operatorname{int}(Q_{1}(E_{\mathrm{SW}})\cap Q_{3}(E_{\mathrm{NE}}))\subset\mathcal{B}(E_{\mathrm{NE}})\). Since \((U_{1},U_{1})\preceq_{\mathrm{ne}}T(U_{1},U_{1})\preceq_{\mathrm{ne}}E_{0}\) and T is cooperative map we obtain \(T^{n}(U_{1},U_{1})\to E_{0}\) as \(n\to\infty\). For \((x_{0},y_{0})\in \operatorname{int}(Q_{3}(E_{0}))\) we have \((U_{1},U_{1})\preceq_{\mathrm{ne}}(x_{1},x_{0})\preceq _{\mathrm{ne}}E_{0}\), which implies \(T^{n}(x_{1},x_{0})\to E_{0}\) as \(n\to\infty\), i.e.
\(\operatorname{int} (Q_{3}(E_{0}))\subset\mathcal{B}(E_{0})\). Assume that \((x_{0},y_{0})\in \operatorname{int}(Q_{3}(E_{\mathrm{SW}}))\). Then there exists \((\tilde{x}_{0},\tilde {y}_{0})\in \operatorname{int}( Q_{3}(E_{0}))\) such that \((\tilde{x}_{0},\tilde{y}_{0})\preceq_{\mathrm{ne}} (x_{0},y_{0})\) and \((\tilde {x}_{1},\tilde{y}_{1})\in \operatorname{int}( Q_{3}(E_{\mathrm{SW}})\cap Q_{1}(E_{0}))\) such that \((x_{0},y_{0})\preceq_{\mathrm{ne}} (\tilde{x}_{1},\tilde{y}_{1})\). By monotonicity of T we have \(T^{n}(\tilde{x}_{0},\tilde{y}_{0})\preceq _{\mathrm{ne}}T^{n}(x_{0},y_{0})\preceq_{\mathrm{ne}} T^{n}(\tilde{x}_{1},\tilde{y}_{1})\), which implies \(T^{n}(x_{0},y_{0})\to E_{0}\) as \(n\to\infty\). This implies that \(\operatorname{int}( Q_{3}(E_{\mathrm{SW}}))\subset\mathcal{B}(E_{0})\). The proof of (ii) is similar and we skip it. □
Let \(\mathcal{C}_{1}^{+}\) denote the boundary of \(\mathcal{B}(E_{0})\) considered as a subset of \(Q_{2}(E_{\mathrm{SW}})\) (the second quadrant relative to \(E_{\mathrm{SW}}\)) and \(\mathcal {C}_{1}^{}\) denote the boundary of \(\mathcal{B}(E_{0})\) considered as a subset of \(Q_{4}(E_{\mathrm{SW}})\) (the fourth quadrant relative to \(E_{\mathrm{SW}}\)). Also, let \(\mathcal{C}_{2}^{+}\) denote the boundary of \(\mathcal{B}(E_{\mathrm{NE}})\) considered as a subset of \(Q_{2}(E_{\mathrm{SW}})\) and \(\mathcal{C}_{2}^{}\) denote the boundary of \(\mathcal{B}(E_{0})\) considered as a subset of \(Q_{4}(E_{\mathrm{SW}})\). It is easy to see that \(E_{\mathrm{SW}}\in\mathcal{C}_{1}^{+}\), \(E_{\mathrm{SW}}\in\mathcal {C}_{1}^{}\), \(E_{\mathrm{SW}}\in\mathcal{C}_{2}^{+}\), \(E_{\mathrm{SW}}\in\mathcal{C}_{2}^{}\).
The proof of the following lemma for a cooperative map is the same as the proof of Claims 1 and 2 in [21] for competitive maps, so we skip it (see Figure 1).
Lemma 2
Let
\(\mathcal{C}_{1}^{+}\)
and
\(\mathcal{C}_{1}^{}\)
be the sets defined above. Then the sets
\(\mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{}\)
and
\(\mathcal {C}_{2}^{+}\cup\mathcal{C}_{2}^{}\)
are invariant under the map
T
and they are the graphs of continuous strictly decreasing functions.
By Lemma 2 it remains to determine the behavior of the orbits of initial conditions \((x_{0},y_{0})\) such that \((\tilde {x}_{0},\tilde{y}_{0})\preceq _{\mathrm{ne}}(x_{0},y_{0})\preceq(\bar{x}_{0},\bar{y}_{0})\) for some \((\tilde {x}_{0},\tilde{y}_{0})\in\mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{}\) and \((\bar{x}_{0},\bar{y}_{0}) \in\mathcal{C}_{2}^{+}\cup\mathcal{C}_{2}^{}\).
Now, we present the global dynamics of equation (1) depending on the existence or nonexistence of periodtwo solutions.
Theorem 5
Assume that equation (1) has no minimal periodtwo solutions and has three equilibrium points
\(\bar{x}_{0} < \bar {x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\)
where the equilibrium points
\(\bar{x}_{0}\)
and
\(\bar{x}_{\mathrm{NE}}\)
are locally asymptotically stable and
\(\bar{x}_{\mathrm{SW}}\)
is a saddle point. Then there exist two continuous curves
\(\mathcal {W}^{s}(E_{\mathrm{SW}})\)
and
\(\mathcal{W}^{u}(E_{\mathrm{SW}})\), both passing through the point
\(E_{\mathrm{SW}}\), such that
\(\mathcal{W}^{s}(E_{\mathrm{SW}})\)
is a graph of decreasing function and
\(\mathcal{W}^{u}(E_{\mathrm{SW}})\)
is a graph of an increasing function. The set of initial conditions
\(Q_{1}= \{(x_{1},x_{0}):x_{1}\geq U_{1},x_{0}\geq U_{1}\}\cap I\)
is the union of three disjoint basins of attraction, namely
\(Q_{1}=\mathcal{B}(E_{0})\cup \mathcal{B}(E_{\mathrm{SW}})\cup\mathcal{B}(E_{\mathrm{NE}})\), where
\(\mathcal{B}(E_{\mathrm{SW}})=\mathcal{W}^{s}(E_{\mathrm{SW}})\)
and
$$\begin{aligned}& \mathcal{B}(E_{0}) =\bigl\{ (x_{1},x_{0})(x_{1},x_{0}) \preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}}) \textit{ for some } (x_{E_{0}},y_{E_{0}})\in\mathcal{W}^{s}(E_{\mathrm{SW}}) \bigr\} , \\& \mathcal{B}(E_{\mathrm{NE} }) =\bigl\{ (x_{1},x_{0})(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \preceq _{\mathrm{ne}}(x_{1},x_{0}) \textit{ for some } (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal {W}^{s}(E_{\mathrm{SW}}) \bigr\} . \end{aligned}$$
Thus, we have
\(\mathcal{W}^{s}(E_{\mathrm{SW}})=\mathcal{C}_{1}^{+}\cup\mathcal {C}_{1}^{}=\mathcal{C}_{2}^{+}\cup\mathcal{C}_{2}^{}\).
Proof
By assumption the map T has three equilibrium point namely \(E_{0}\), \(E_{\mathrm{SW}}\), and \(E_{\mathrm{NE}}\) such that \(E_{0}\ll_{\mathrm{ne}}E_{\mathrm{SW}}\ll_{\mathrm{ne}}E_{\mathrm{NE}}\). In this case, \(E_{0}\) and \(E_{\mathrm{NE}}\) are locally asymptotically stable and \(E_{\mathrm{SW}}\) is a saddle point. Since f is strongly increasing then the map T is strongly cooperative on \([U_{1},\infty)^{2}\). It follows from the PerronFrobenius theorem and a change of variables [14] that, at each point, the Jacobian matrix of a strongly cooperative map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively. Also, one can show that if the map is strongly cooperative then no eigenvector is aligned with a coordinate axis.
Hence, all conditions of Theorems 1 and 4 in [19] for cooperative map T are satisfied, which yields the existence of the global stable manifold \(\mathcal{W}^{s}(E_{\mathrm{SW}})\) and the global unstable manifold \(\mathcal{W}^{u}(E_{\mathrm{SW}})\), where \(\mathcal{W}^{s}(E_{\mathrm{SW}})\) is passing through the point \(E_{\mathrm{SW}}\), and it is a graph of a decreasing function. The curve \(\mathcal{W}^{u}(E_{\mathrm{SW}})\) is the graph of an increasing function. Let
$$\begin{aligned}& \mathcal{\mathcal{W}^{}}= \bigl\{ (x_{1},x_{0})(x_{1},x_{0}) \preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}}) \mbox{ for some } (x_{E_{0}},y_{E_{0}})\in\mathcal{W}^{s}(E_{\mathrm{SW}}) \bigr\} , \\& \mathcal{W^{+}}= \bigl\{ (x_{1},x_{0})(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \preceq _{\mathrm{ne}}(x_{1},x_{0}) \mbox{ for some } (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal {W}^{s}(E_{\mathrm{SW}}) \bigr\} . \end{aligned}$$
Take \((x_{0},y_{0})\in\mathcal{W}^{}\cap[U_{1},\infty)^{2}\) and \((\tilde {x}_{0},\tilde{y}_{0})\in\mathcal{W}^{+}\cap[U_{1},\infty)^{2}\). By Theorem 4 in [19] we see that there exists \(n_{0}>0\) such that \(T^{n}(x_{0},y_{0})\in \operatorname{int}(Q_{3}(E_{\mathrm{SW}}))\) and \(T^{n}(\tilde{x}_{0},\tilde {y}_{0})\in \operatorname{int} (Q_{1}(E_{\mathrm{SW}}))\) for \(n>n_{0}\). The rest of the proof follows from Lemma 1. See Figure 2(a) for its visual illustration.
□
Theorem 6
Assume that equation (1) has no minimal periodtwo solutions and there exist three equilibrium points
\(\bar {x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\)
where the equilibrium points
\(\bar {x}_{0}\)
and
\(\bar{x}_{\mathrm{NE}}\)
are locally asymptotically stable and
\(\bar {x}_{\mathrm{SW}}\)
is nonhyperbolic. Then there exist two continuous curves
\(\mathcal{C}_{1}(E_{\mathrm{SW}})\)
and
\(\mathcal{C}_{2}(E_{\mathrm{SW}})\)
passing through the point
\(E_{\mathrm{SW}}\), which are the graphs of decreasing functions. The set of the initial condition
\(Q_{1}=\{(x_{1},x_{0})\}\)
is the union of three disjoint basins of attraction, namely
\(Q_{1}=\mathcal{B}(E_{0})\cup \mathcal{B}(E_{\mathrm{SW}})\cup\mathcal{B}(E_{\mathrm{NE}})\), where
$$\begin{aligned}& \mathcal{B}(E_{0})= \bigl\{ (x_{0},y_{0})(x_{0},y_{0}) \preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}}) \textit{ for some } (x_{E_{0}},y_{E_{0}})\in\mathcal{C}_{1}(E_{\mathrm{SW}}) \bigr\} , \\& \mathcal{B}(E_{\mathrm{SW}})= \bigl\{ (x_{0},y_{0})(x_{E_{0}},y_{E_{0}}) \preceq _{\mathrm{ne}}(x_{0},y_{0})\preceq_{\mathrm{ne}}(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\textit{ for some} \\& \hphantom{\mathcal{B}(E_{\mathrm{SW}})={}}{} (x_{E_{0}},y_{E_{0}})\in \mathcal{C}_{1}(E_{\mathrm{SW}}) \textit{ and } (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \in\mathcal{C}_{2}(E_{\mathrm{SW}})\bigr\} , \\& \mathcal{B}(E_{\mathrm{NE} })= \bigl\{ (x_{0},y_{0})(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \preceq _{\mathrm{ne}}(x_{0},y_{0}) \textit{ for some } (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal {C}_{2}(E_{\mathrm{SW}}) \bigr\} . \end{aligned}$$
Proof
The characterization of \(\mathcal{B}(E_{0})\) and \(\mathcal{B}(E_{\mathrm{NE}})\) follows as in Theorem 5.
The existence of the curves \(\mathcal{C}_{1}(E_{\mathrm{SW}})\) and \(\mathcal {C}_{2}(E_{\mathrm{SW}})\) passing through the point \(E_{\mathrm{SW}}\) which are the graphs of decreasing functions follows from Lemma 2. Assume that \((x_{E_{0}},y_{E_{0}})\in\mathcal{C}_{1}(E_{\mathrm{SW}})\). Since \(\mathcal{C}_{1}(E_{\mathrm{SW}})\) is an invariant set and there are no periodtwo solutions we must have \(T^{n}(x_{E_{0}},y_{E_{0}})\to E_{\mathrm{SW}}\) as \(n\to \infty\). Similarly, we obtain \(T^{n}(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\to E_{\mathrm{SW}}\) as \(n\to\infty\) if \((x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal{C}_{2}(E_{\mathrm{SW}})\). Suppose that \((x_{E_{0}},y_{E_{0}})\preceq_{\mathrm{ne}}(x_{0},y_{0})\preceq _{\mathrm{ne}}(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\) for some \((x_{E_{0}},y_{E_{0}})\in \mathcal{C}_{1}(E_{\mathrm{SW}})\) and \((x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal {C}_{2}(E_{\mathrm{SW}})\). Then \(T^{n}(x_{E_{0}},y_{E_{0}})\preceq _{\mathrm{ne}}T^{n}(x_{0},y_{0})\preceq_{\mathrm{ne}}T^{n}(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\), which implies that \(T^{n}(x_{0},y_{0})\to E_{\mathrm{SW}}\) as \(n\to\infty\). See Figure 2(b) for the visual illustration of this result. □
Now, we consider the dynamical scenarios when there exists a minimal periodtwo solution which is a saddle point (see Figure 3(a)).
Theorem 7
Assume that equation (1) has three equilibrium points
\(U_{1} \leq\bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar {x}_{\mathrm{NE}}\)
where the equilibrium points
\(\bar{x}_{0}\)
and
\(\bar{x}_{\mathrm{NE}}\)
are locally asymptotically stable. Further, assume that there exists a minimal periodtwo solution
\(\{\Phi_{1},\Psi_{1}\}\)
which is a saddle point such that
\((\Phi_{1},\Psi_{1})\in \operatorname{int}(Q_{2}(E_{\mathrm{SW}}))\). In this case there exist four continuous curves
\(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{s}(\Psi _{1},\Phi_{1})\), \(\mathcal{W}^{u}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{u}(\Psi_{1},\Phi _{1})\), where
\(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{s}(\Psi_{1},\Phi _{1})\)
are passing through the point
\(E_{\mathrm{SW}}\)
and are graphs of decreasing functions. The curves
\(\mathcal{W}^{u}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{u}(\Psi_{1},\Phi_{1})\)
are the graphs of increasing functions and are starting at
\(E_{0}\). Every solution which starts below
\(\mathcal {W}^{s}(\Phi_{1},\Psi_{1}) \cup\mathcal{W}^{s}(\Psi_{1},\Phi_{1})\)
in the NorthEast ordering converges to
\(E_{0}\)
and every solution which starts above
\(\mathcal{W}^{s}(\Phi_{1},\Psi_{1}) \cup\mathcal{W}^{s}(\Psi_{1},\Phi _{1})\)
in the NorthEast ordering converges to
\(E_{\mathrm{NE}}\), i.e. \(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})=\mathcal{C}_{1}^{+}=\mathcal{C}_{2}^{+}\)
and
\(\mathcal{W}^{s}(\Psi_{1},\Phi_{1})=\mathcal{C}_{1}^{}=\mathcal{C}_{2}^{}\).
Proof
The map T is cooperative and all conditions of Theorems 1 and 4 in [19] are satisfied, which yields the existence of the global stable manifolds \(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{s}(\Psi _{1},\Phi_{1})\) and the global unstable manifolds \(\mathcal{W}^{u}(\Phi _{1},\Psi_{1})\), \(\mathcal{W}^{u}(\Psi_{1},\Phi_{1})\) where \(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{s}(\Psi_{1},\Phi_{1})\) are passing through the point \(E_{\mathrm{SW}}\), and they are graphs of decreasing functions. Let
$$\begin{aligned}& \mathcal{\mathcal{W}^{}}= \bigl\{ (x_{0},y_{0})(x_{0},y_{0}) \preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}}) \textit{ for some } (x_{E_{0}},y_{E_{0}})\in\mathcal{W}^{s}( \Phi_{1},\Psi_{1}) \cup\mathcal{W}^{s}( \Psi_{1},\Phi_{1})\bigr\} , \\& \mathcal{W^{+}}= \bigl\{ (x_{0},y_{0})(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \preceq _{\mathrm{ne}}(x_{0},y_{0}) \textit{ for some} \\& \hphantom{\mathcal{W^{+}}={}}{}(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal {W}^{s}( \Phi_{1},\Psi_{1}) \cup\mathcal{W}^{s}( \Psi_{1},\Phi_{1})\bigr\} . \end{aligned}$$
Take \((x_{0}, y_{0})\in\mathcal{W}^{}\cap[0,\infty)^{2}\) and \((\tilde {x}_{0},\tilde{y}_{0})\in\mathcal{W}^{+}\cap[0,\infty)^{2}\). By Theorem 4 in [19] we see that there exist \(n_{0},n_{1}>0\) such that \(T^{n}(x_{0},y_{0})\in \operatorname{int}(Q_{3}(\Phi_{1},\Psi_{1})\cap Q_{3}(\Psi_{1},\Phi_{1}))\) for \(n>n_{0}\) and \(T^{n}(\tilde{x}_{0},\tilde{y}_{0})\in \operatorname{int}(Q_{1}(\Phi_{1},\Psi _{1})\cap Q_{1}(\Psi_{1},\Phi_{1}))\) for \(n>n_{1}\). Since \(\operatorname{int}(Q_{3}(\Phi_{1},\Psi _{1})\cap Q_{3}(\Psi_{1},\Phi_{1}))\subseteq \operatorname{int}(Q_{3}(E_{\mathrm{SW}})) \subseteq\mathcal {B}(E_{0})\) and \(\operatorname{int}(Q_{1}(\Phi_{1},\Psi_{1})\cap Q_{1}(\Psi_{1},\Phi_{1})) \subseteq \operatorname{int}(Q_{1}(E_{\mathrm{SW}})) \subseteq\mathcal{B}(E_{\mathrm{NE}})\) we have \(T^{n}(x_{0},y_{0})\to E_{0}\) as \(n\to\infty\) if \((x_{0},y_{0})\in \operatorname{int}(Q_{3}(\Phi _{1},\Psi_{1})\cap Q_{3}(\Psi_{1},\Phi_{1}))\) and \(T^{n}(x_{0},y_{0})\to E_{\mathrm{NE}}\) as \(n\to\infty\) if \((x_{0},y_{0})\in \operatorname{int}(Q_{1}(\Phi_{1},\Psi_{1})\cap Q_{1}(\Psi _{1},\Phi_{1}))\). □
Now, we consider the dynamical scenario when there exist three minimal periodtwo solutions.
Theorem 8
Assume that equation (1) has three equilibrium points
\(U_{1} \leq\bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\)
where the equilibrium points
\(\bar{x}_{0}\)
and
\(\bar{x}_{\mathrm{NE}}\)
are locally asymptotically stable and
\(\bar{x}_{\mathrm{SW}}\)
is a repeller. Further, assume that there exist three minimal periodtwo solutions
\(\{\Phi_{i},\Psi_{i}\} \), such that
\((\Phi_{i},\Psi_{i})\in \operatorname{int}(Q_{2}(E_{\mathrm{SW}}))\), \(i=1,2,3\), where
\(\{ \Phi_{1},\Psi_{1}\}\)
and
\(\{\Phi_{2},\Psi_{2}\}\)
are the saddle points and
\(\{ \Phi_{3},\Psi_{3}\}\)
is locally asymptotically stable. In this case there exist four continuous curves
\(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal {W}^{s}(\Psi_{1},\Phi_{1})\), \(\mathcal{W}^{s}(\Phi_{2},\Psi_{2})\), \(\mathcal {W}^{s}(\Psi_{2},\Phi_{2})\)
where
\(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{s}(\Psi_{1},\Phi_{1})\), \(\mathcal{W}^{s}(\Phi_{2},\Psi_{2})\), \(\mathcal{W}^{s}(\Psi_{2},\Phi_{2})\)
are passing through the point
\(E_{\mathrm{SW}}\)
and are graphs of decreasing functions. Every solution which starts below
\(\mathcal{W}^{s}(\Phi_{2},\Psi_{2}) \cup\mathcal{W}^{s}(\Psi_{2},\Phi _{2})\)
in the NorthEast ordering converges to
\(E_{0}\)
and every solution which starts above
\(W^{s}(\Phi_{1},\Psi_{1}) \cup W^{s}(\Psi_{1},\Phi_{1})\)
in the NorthEast ordering converges to
\(E_{\mathrm{NE}}\). Every solution which starts above
\(\mathcal{W}^{s}(\Phi_{2},\Psi_{2}) \cup\mathcal{W}^{s}(\Psi _{2},\Phi_{2})\)
and below
\(\mathcal{W}^{s}(\Phi_{1},\Psi_{1}) \cup\mathcal {W}^{s}(\Psi_{1},\Phi_{1})\)
in the NorthEast ordering converges to
\(\{\Phi _{3},\Psi_{3}\}\). In other words, the set of the initial condition
\(Q_{1}=\{ (x_{1},x_{0})\}\)
is the union of five disjoint basins of attraction, namely
$$Q_{1}=\mathcal{B}(E_{0})\cup\mathcal{B}\bigl(\{ \Phi_{1},\Psi_{1}\}\bigr)\cup\mathcal {B}\bigl(\{ \Phi_{2},\Psi_{2}\}\bigr)\cup\mathcal{B}\bigl(\{ \Phi_{3},\Psi_{3}\}\bigr)\cup\mathcal {B}(E_{\mathrm{NE}}), $$
where
$$\begin{aligned}& \mathcal{B}\bigl(\{\Phi_{1},\Psi_{1}\}\bigr) = \mathcal{W}^{s}(\Phi_{1},\Psi_{1}) \cup \mathcal{W}^{s}(\Psi_{1},\Phi_{1}), \\& \mathcal{B}\bigl(\{\Phi_{2},\Psi_{2}\}\bigr) = \mathcal{W}^{s}(\Phi_{2},\Psi_{2}) \cup \mathcal{W}^{s}(\Psi_{2},\Phi_{2}), \\& \mathcal{B}(E_{0}) =\bigl\{ (x_{0},y_{0})(x_{0},y_{0}) \preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}}) \textit{ for some} \\& \hphantom{\mathcal{B}(E_{0}) ={}}{}(x_{E_{0}},y_{E_{0}})\in\mathcal{W}^{s}( \Phi_{2},\Psi_{2}) \cup\mathcal{W}^{s}( \Psi_{2},\Phi_{2})\bigr\} , \\& \mathcal{B}(E_{\mathrm{NE} }) =\bigl\{ (x_{0},y_{0})(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \preceq _{\mathrm{ne}}(x_{0},y_{0}) \textit{ for some} \\& \hphantom{\mathcal{B}(E_{\mathrm{NE} }) ={}}{} (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal {W}^{s}( \Phi_{1},\Psi_{1}) \cup W^{s}(\Psi_{1}, \Phi_{1})\bigr\} , \\& \mathcal{B}\bigl(\{\Phi_{3},\Psi_{3}\}\bigr) =\bigl\{ (x_{0},y_{0})(x_{E_{0}},y_{E_{0}})\preceq _{\mathrm{ne}}(x_{0},y_{0})\preceq(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \textit{ for some} \\& \hphantom{\mathcal{B}\bigl(\{\Phi_{3},\Psi_{3}\}\bigr) ={}}{}(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal{W}^{s}( \Phi_{1},\Psi_{1}) \cup W^{s}(\Psi_{1}, \Phi_{1}) \textit{ and} \\& \hphantom{\mathcal{B}\bigl(\{\Phi_{3},\Psi_{3}\}\bigr) ={}}{}(x_{E_{0}},y_{E_{0}})\in\mathcal {W}^{s}(\Phi_{2},\Psi_{2}) \cup \mathcal{W}^{s}(\Psi_{2},\Phi_{2})\bigr\} . \end{aligned}$$
Thus, we have
\(\mathcal{W}^{s}(\Phi_{2},\Psi_{2})=\mathcal{C}_{1}^{+}\), \(\mathcal{W}^{s}(\Psi_{2},\Phi_{2})=\mathcal{C}_{1}^{}\), \(\mathcal {W}^{s}(\Phi_{1},\Psi_{1})=\mathcal{C}_{2}^{+}\), and
\(\mathcal{W}^{s}(\Psi _{1},\Phi_{1})=\mathcal{C}_{2}^{}\).
Proof
All conditions of Theorems 1 and 4 in [19] for the cooperative map T are satisfied, which yields the existence of the global stable manifolds \(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{s}(\Psi_{1},\Phi _{1})\), \(\mathcal{W}^{s}(\Phi_{2},\Psi_{2})\), \(\mathcal{W}^{s}(\Psi_{2},\Phi_{2})\), which are passing through the point \(E_{\mathrm{SW}}\), and they are graphs of decreasing functions. Since T is a cooperative map it can be seen that \((\Phi_{1},\Psi_{1})\ll_{\mathrm{ne}} (\Phi _{3},\Psi_{3})\ll_{\mathrm{ne}} (\Phi_{2},\Psi_{2})\) or \((\Phi_{2},\Psi_{2})\ll_{\mathrm{ne}} (\Phi _{3},\Psi_{3})\ll_{\mathrm{ne}} (\Phi_{1},\Psi_{1})\). Assume \((\Phi_{1},\Psi_{1})\ll_{\mathrm{ne}} (\Phi_{3}, \Psi_{3})\ll_{\mathrm{ne}} (\Phi_{2},\Psi_{2})\). As in the proof of Theorem 7 one can see that
$$\begin{aligned}& \mathcal{B}(E_{0}) =\bigl\{ (x_{0},y_{0})(x_{0},y_{0}) \preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}}) \mbox{ for some} \\& \hphantom{\mathcal{B}(E_{0}) ={}}{}(x_{E_{0}},y_{E_{0}})\in\mathcal{W}^{s}( \Phi_{1},\Psi_{1}) \cup\mathcal{W}^{s}( \Psi_{1},\Phi_{1})\bigr\} , \\& \mathcal{B}(E_{\mathrm{NE} }) =\bigl\{ (x_{0},y_{0})(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \preceq _{\mathrm{ne}}(x_{0},y_{0}) \mbox{ for some} \\& \hphantom{\mathcal{B}(E_{\mathrm{NE} }) ={}}{}(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal {W}^{s}( \Phi_{2},\Psi_{2}) \cup W^{s}(\Psi_{2}, \Phi_{2})\bigr\} . \end{aligned}$$
So, we assume that \((x_{E_{0}},y_{E_{0}})\preceq _{\mathrm{ne}}(x_{0},y_{0})\preceq(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\) for some \((x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal{W}^{s}(P_{2})\) and \((x_{E_{0}},y_{E_{0}})\in\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\). By Theorem 4 in [19] we see that there exists \(n_{0}>0\) such that \(T^{2n}(x_{0},y_{0})\in \operatorname{int}(Q_{3}(\Phi_{1},\Psi_{1})\cap Q_{1}(\Phi_{2},\Psi _{2}))\) for \(n>n_{0}\). By Corollary 1 we get \([\!\![(\Phi_{2},\Psi_{2}),(\Phi_{3},\Psi_{3})]\!\!]\cup[\!\![(\Phi_{3},\Psi_{3}),(\Phi _{1},\Psi_{1})]\!\!]\subseteq\mathcal{B}(\Phi_{3},\Psi_{3})\) which implies that \(\operatorname{int}(Q_{3}(\Phi_{1},\Psi_{1})\cap Q_{1}(\Phi_{2},\Psi_{2}))=[\!\![(\Phi_{2},\Psi_{2}),(\Phi _{1},\Psi_{1})]\!\!]\subseteq\mathcal{B}(\Phi_{3},\Psi_{3})\). If \((x_{E_{0}},y_{E_{0}})\preceq _{\mathrm{ne}}(x_{0},y_{0})\preceq(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\) for some \((x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal{W}^{s}(\Phi_{2},\Psi_{2})\) and \((x_{E_{0}},y_{E_{0}})\in\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\) then there exists \(n_{0}>0\) such that \(T^{2n}(x_{0},y_{0})\in \operatorname{int}(Q_{3}(\Phi_{1},\Psi_{1})\cap Q_{1}(\Phi_{2},\Psi _{2}))\) for \(n>n_{0}\). By Corollary 1 we get \([\!\![(\Psi_{2},\Phi _{2}),(\Psi_{3},\Phi_{3})]\!\!]\cup[\!\![(\Psi_{3},\Phi_{3}),(\Psi_{1},\Phi_{1})]\!\!]\subseteq \mathcal{B}(\Phi_{3},\Psi_{3})\), which implies that \(\operatorname{int}(Q_{3}(\Psi_{1},\Phi _{1})\cap Q_{1}(\Psi_{2},\Phi_{2}))=[\!\![(\Psi_{2},\Phi_{2}),(\Psi_{1},\Phi _{1})]\!\!]\subseteq\mathcal{B}(\Phi_{3},\Psi_{3})\). This completes the proof. □
Now, we consider two dynamical scenarios when there exists a minimal periodtwo solution \(\{\Phi,\Psi\}\) which is a nonhyperbolic of stable type (i.e. if \(\mu_{1}\) and \(\mu_{2}\) are eigenvalues of \(J_{T}(\Phi,\Psi)\) then \(\mu_{1}=1\) and \(\mu_{2}<1\)).
Theorem 9
Assume that equation (1) has three equilibrium points
\(U_{1} \leq\bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\)
where the equilibrium points
\(\bar{x}_{0}\)
and
\(\bar{x}_{\mathrm{NE}}\)
are locally asymptotically stable and
\(\bar{x}_{\mathrm{SW}}\)
is a repeller or nonhyperbolic equilibrium point. Further, assume that there exist two minimal periodtwo solutions
\(\{\Phi,\Psi\}\)
and
\(\{\Phi_{1},\Psi_{1}\}\), where
\(\{\Phi,\Psi\}\)
is a nonhyperbolic periodtwo solution of the stable type and
\(\{\Phi_{1},\Psi_{1}\}\)
is a saddle point, and
\((\Phi,\Psi )\ll_{\mathrm{ne}}(\Phi_{1},\Psi_{1})\) (see Figure
4(a)). In this case there exist four continuous curves
\(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{s}(\Psi _{1},\Phi_{1})\), \(\mathcal{C}^{s}(\Phi,\Psi)\), \(\mathcal{C}^{s}(\Psi,\Phi)\)
which are passing through the point
\(E_{\mathrm{SW}}\)
and which are graphs of decreasing functions. The set
\(Q_{1}= \{(x_{1},x_{0}):x_{1}\geq U_{1},x_{0}\geq U_{1}\}\)
is the union of four disjoint basins of attraction, namely
$$Q_{1}=\mathcal{B}(E_{0})\cup\mathcal{B}\bigl(\{ \Phi_{1},\Psi_{1}\}\bigr)\cup\mathcal {B}\bigl(\{\Phi,\Psi\} \bigr)\cup\mathcal{B}(E_{\mathrm{NE}}), $$
where
$$\begin{aligned}& \mathcal{B}\bigl(\{\Phi_{1},\Psi_{1}\}\bigr) = \mathcal{W}^{s}(\Phi_{1},\Psi_{1}) \cup \mathcal{W}^{s}(\Phi_{1},\Psi_{1}), \\& \mathcal{B}(E_{0}) =\bigl\{ (x_{0},y_{0})(x_{0},y_{0}) \preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}}) \textit{ for some } (x_{E_{0}},y_{E_{0}})\in\mathcal{C}(\Phi,\Psi) \cup \mathcal{C}(\Psi,\Phi) \bigr\} , \\& \mathcal{B}(E_{\mathrm{NE} }) =\bigl\{ (x_{0},y_{0})(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \preceq _{\mathrm{ne}}(x_{0},y_{0}) \textit{ for some} \\& \hphantom{\mathcal{B}(E_{\mathrm{NE} }) ={}}{}(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal {W}^{s}( \Phi_{1},\Psi_{1}) \cup W^{s}(\Psi_{1}, \Phi_{1})\bigr\} , \\& \mathcal{B}\bigl(\{\Phi,\Psi\}\bigr) =\mathcal{C}(\Phi,\Psi) \cup\mathcal{C}( \Psi ,\Phi) \\& \hphantom{\mathcal{B}\bigl(\{\Phi,\Psi\}\bigr) ={}}{}\cup\bigl\{ (x_{0},y_{0})(x_{E_{0}},y_{E_{0}}) \preceq _{\mathrm{ne}}(x_{0},y_{0})\preceq(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\textit{ for some} \\& \hphantom{\mathcal{B}\bigl(\{\Phi,\Psi\}\bigr) ={}}{} (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in \mathcal{W}^{s}(\Phi_{1},\Psi _{1}) \cup W^{s}(\Psi_{1},\Phi_{1}) \textit{ and} \\& \hphantom{\mathcal{B}\bigl(\{\Phi,\Psi\}\bigr) ={}}{}(x_{E_{0}},y_{E_{0}})\in \mathcal{C}(\Phi,\Psi) \cup\mathcal{C}( \Psi,\Phi)\bigr\} . \end{aligned}$$
Thus, we have
\(\mathcal{C}(\Phi,\Psi)=\mathcal{C}_{1}^{+}\), \(\mathcal {C}(\Psi,\Phi)=\mathcal{C}_{1}^{}\), \(\mathcal{W}^{s}(\Phi_{1},\Psi _{1})=\mathcal{C}_{2}^{+}\), and
\(\mathcal{W}^{s}(\Psi_{1},\Phi_{1})=\mathcal{C}_{2}^{}\).
Proof
Since \(\{\Phi,\Psi\}\) is a nonhyperbolic periodtwo solution of the stable type and \(\{\Phi_{1},\Psi_{1}\}\) is a saddle point, all conditions of Theorems 1 and 4 in [19] for the cooperative map T are satisfied, which yields the existence of the global stable manifolds \(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal {W}^{s}(\Psi_{1},\Phi_{1})\) and invariant curves \(\mathcal{C}(\Phi,\Psi)\), \(\mathcal{C}(\Psi,\Phi)\) which are passing through the point \(E_{\mathrm{SW}}\), and they are graphs of decreasing functions.
Take \((x_{0},y_{0})\) such that \((x_{0},y_{0})\preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}})\) for some \((x_{E_{0}},y_{E_{0}})\in\mathcal {C}(\Phi,\Psi) \cup\mathcal{C}(\Psi,\Phi)\). In view of Theorem 4 in [19] there exists \(n_{0}>0\) such that \(T^{n}(x_{0},y_{0})\in \operatorname{int}(Q_{3}(\Phi,\Psi)\cap Q_{3}(\Psi,\Phi))\) for \(n>n_{0}\). Since \(\operatorname{int}(Q_{3}(\Phi,\Psi)\cap Q_{3}(\Psi,\Phi))\subseteq \operatorname{int}(Q_{3}(E_{\mathrm{SW}}))\) by Lemma 1 we obtain
$$ \mathcal{B}(E_{0})=\bigl\{ (x,y)(x,y)\preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}}) \mbox{ for some } (x_{E_{0}},y_{E_{0}})\in \mathcal{C}(\Phi,\Psi) \cup\mathcal{C}(\Psi,\Phi)\bigr\} . $$
If \((x_{0},y_{0})\) is such that \((x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\preceq_{\mathrm{ne}} (x_{0},y_{0})\) for some \((x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal{W}^{s}(\Phi _{1},\Psi_{1}) \cup W^{s}(\Psi_{1},\Phi_{1})\) by Theorem 4 in [19] there exists \(n_{0}>0\) such that \(T^{n}(x_{0},y_{0})\in \operatorname{int}(Q_{1}(\Phi_{1},\Psi_{1})\cap Q_{1}(\Psi_{1},\Phi_{1}))\) for \(n>n_{0}\). Since \(\operatorname{int}(Q_{1}(\Phi_{1},\Psi_{1})\cap Q_{1}(\Psi_{1},\Phi _{1}))\subseteq \operatorname{int}(Q_{1}(E_{\mathrm{SW}}))\) by Lemma 1 we obtain
$$\begin{aligned} \mathcal{B}(E_{\mathrm{NE}}) =&\bigl\{ (x_{0},y_{0})(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \preceq _{\mathrm{ne}}(x_{0},y_{0}) \mbox{ for some} \\ &(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal {W}^{s}( \Phi_{1},\Psi_{1}) \cup\mathcal{W}^{s}( \Psi_{1},\Phi_{1})\bigr\} . \end{aligned}$$
Now, we assume that \((x_{E_{0}},y_{E_{0}})\preceq _{\mathrm{ne}}(x_{0},y_{0})\preceq_{\mathrm{ne}} (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\) for some \((x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\) and \((x_{E_{0}},y_{E_{0}})\in\mathcal{C}(\Phi,\Psi)\). By Theorem 4 in [19] we see that there exists \(n_{0}>0\) such that \(T^{n}(x_{0},y_{0})\in \operatorname{int}(Q_{3}(\Phi_{1},\Psi_{1})\cap Q_{1}(\Phi,\Psi))\) for \(n>n_{0}\). By Corollary 1 we get \([\!\![(\Phi,\Psi),(\Phi_{1},\Psi_{1})]\!\!]\subseteq\mathcal{B}(\{\Phi,\Psi\})\). Similarly, if \((x_{E_{0}},y_{E_{0}})\preceq _{\mathrm{ne}}(x_{0},y_{0})\preceq_{\mathrm{ne}} (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\) for some \((x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in\mathcal{W}^{s}(\Psi_{1},\Phi_{1})\) and \((x_{E_{0}},y_{E_{0}})\in\mathcal{C}(\Psi,\Phi)\) we see that there exists \(n_{0}>0\) such that \(T^{n}(x_{0},y_{0})\in \operatorname{int}(Q_{3}(\Psi_{1},\Phi_{1})\cap Q_{1}(\Psi,\Phi))\) for \(n>n_{0}\). By Corollary 1 we get \([\!\![T(P),(\Psi_{1},\Phi_{1})]\!\!]\subseteq\mathcal{B}(\{\Phi,\Psi\})\). This implies \((x_{0},y_{0})\in\mathcal{B}(\{\Phi,\Psi\})\). □
The proof of the following result is similar to the proof of Theorem 9 and will be omitted.
Theorem 10
Assume that equation (1) has three equilibrium points
\(U_{1} \leq\bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\)
where the equilibrium points
\(\bar{x}_{0}\)
and
\(\bar{x}_{\mathrm{NE}}\)
are locally asymptotically stable and
\(\bar{x}_{\mathrm{SW}}\)
is a repeller or nonhyperbolic equilibrium point. Further, assume that there exist two minimal periodtwo solutions
\(\{\Phi,\Psi\}\)
and
\(\{\Phi_{1},\Psi_{1}\}\), where
\(\{\Phi,\Psi\}\)
is a nonhyperbolic periodtwo solution of the stable type and
\(\{\Phi_{1},\Psi_{1}\}\)
is a saddle point, and
\((\Phi_{1},\Psi _{1})\ll_{\mathrm{ne}}(\Phi,\Psi)\) (see Figure
4(b)). In this case there exist four continuous curves
\(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{s}(\Psi _{1},\Phi_{1})\), \(\mathcal{C}^{s}(\Phi,\Psi)\), \(\mathcal{C}^{s}(\Psi,\Phi)\)
where
\(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\), \(\mathcal{W}^{s}(\Psi_{1},\Phi_{1})\), \(\mathcal{C}(\Phi,\Psi)\), \(\mathcal{C}(\Psi,\Phi)\)
are passing through the point
\(E_{\mathrm{SW}}\), which are graphs of decreasing functions. The set
\(Q_{1}= \{(x_{1},x_{0}):x_{1}\geq U_{1},x_{0}\geq U_{1}\}\)
is the union of four disjoint basins of attraction, namely
$$Q_{1}=\mathcal{B}(E_{0})\cup\mathcal{B}\bigl(\{ \Phi_{1},\Psi_{1}\}\bigr)\cup\mathcal {B}\bigl(\{\Phi,\Psi\} \bigr)\cup\mathcal{B}(E_{\mathrm{NE}}), $$
where
$$\begin{aligned}& \mathcal{B}\bigl(\{\Phi_{1},\Psi_{1}\}\bigr) = \mathcal{W}^{s}(\Phi_{1},\Psi_{1}) \cup \mathcal{W}^{s}(\Phi_{1},\Psi_{1}), \\& \mathcal{B}(E_{0}) =\bigl\{ (x_{0},y_{0})(x_{0},y_{0}) \preceq _{\mathrm{ne}}(x_{E_{0}},y_{E_{0}}) \textit{ for some} \\& \hphantom{\mathcal{B}(E_{0}) ={}}{}(x_{E_{0}},y_{E_{0}})\in\mathcal{W}^{s}( \Phi_{1},\Psi_{1}) \cup W^{s}(\Psi_{1}, \Phi_{1})\bigr\} , \\& \mathcal{B}(E_{\mathrm{NE} }) =\bigl\{ (x_{0},y_{0})(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \preceq _{\mathrm{ne}}(x,y) \textit{ for some } (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}}) \in\mathcal{C}(\Phi ,\Psi) \cup\mathcal{C}(\Psi,\Phi)\bigr\} , \\& \mathcal{B}\bigl(\{\Phi,\Psi\}\bigr) =\mathcal{C}(\Phi,\Psi) \cup\mathcal{C}( \Psi ,\Phi) \\& \hphantom{\mathcal{B}\bigl(\{\Phi,\Psi\}\bigr) ={}}{}\cup\bigl\{ (x_{0},y_{0})(x_{E_{0}},y_{E_{0}}) \preceq _{\mathrm{ne}}(x_{0},y_{0})\preceq(x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\textit{ for some} \\& \hphantom{\mathcal{B}\bigl(\{\Phi,\Psi\}\bigr) ={}}{} (x_{E_{\mathrm{NE}}},y_{E_{\mathrm{NE}}})\in \mathcal{C}(\Phi,\Psi) \cup\mathcal{C}(\Psi,\Phi) \textit{ and} \\& \hphantom{\mathcal{B}\bigl(\{\Phi,\Psi\}\bigr) ={}}{} (x_{E_{0}},y_{E_{0}})\in \mathcal{W}^{s}( \Phi_{1},\Psi_{1}) \cup W^{s}(\Psi_{1}, \Phi_{1})\bigr\} . \end{aligned}$$
Thus, we have
\(\mathcal{C}(\Phi,\Psi)=\mathcal{C}_{2}^{+}\), \(\mathcal {C}(\Psi,\Phi)=\mathcal{C}_{2}^{}\), \(\mathcal{W}^{s}(\Phi_{1},\Psi _{1})=\mathcal{C}_{1}^{+}\), and
\(\mathcal{W}^{s}(\Psi_{1},\Phi_{1})=\mathcal{C}_{1}^{}\).