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Basins of attraction of periodtwo solutions of monotone difference equations
Advances in Difference Equations volume 2016, Article number: 74 (2016)
Abstract
We investigate the global character of the difference equation of the form
with several periodtwo solutions, where f is increasing in all its variables. We show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or periodtwo solutions are in fact the global stable manifolds of neighboring saddle or nonhyperbolic equilibrium solutions or periodtwo solutions. As an application of our results we give the global dynamics of three feasible models in population dynamics which includes the nonlinearity of BevertonHolt and sigmoid BevertonHolt types.
1 Introduction
Let I be some interval of real numbers and let \(f\in C^{1} [I\times I, I]\) be such that \(f(I\times I)\subseteq\mathcal{K}\) where \(\mathcal {K}\subseteq I\) is a compact set. Let \(\bar{x}_{0}, \bar{x}_{\mathrm{SW}}, \bar {x}_{\mathrm{NE}}\in I\), \(\bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) be three equilibrium points of the difference equation
where f is a continuous and increasing function in both variables. There are several global attractivity results for equation (1) which give the sufficient conditions for all solutions to approach a unique equilibrium. We list three such results.
The first theorem, which has also been very useful in applications to mathematical biology, was really motivated by a problem in [1].
Theorem 1
Let \(I\subseteq{}[0,\infty)\) be some interval and assume that \(f\in C[I \times I,(0,\infty)]\) satisfies the following conditions:

(i)
\(f(x,y)\) is nondecreasing in each of its arguments;

(ii)
equation (1) has a unique positive equilibrium point x̄ and the function \(f(x, x)\) satisfies the negative feedback condition
$$ (x  \bar{x}) \bigl(f(x,x)  x\bigr) < 0 \quad \textit{for every } x\in I\{\bar {x}\}. $$(2)
Then every positive solution of equation (1) with initial conditions in I converges to x̄.
Theorem 2
Let \([a,b]\) be an interval of real numbers and assume that
is a continuous function satisfying the following properties:

(a)
\(f(x,y)\) is nondecreasing in each of its arguments;

(b)
equation (1) has a unique equilibrium \(\bar{x}\in[a, b]\).
Then every solution of equation (1) converges to x̄.
Theorems 1 and 2 were extended to a kth order difference equation where the right hand side is a nondecreasing function of all its variables; see [1, 2].
The following result has been obtained in [3].
Theorem 3
Let \(I\subseteq R\) and let \(f\in C[I \times I,I]\) be a function which increases in both variables. Then for every solution of equation (1) the subsequences \(\{ x_{2n} \}_{n=0}^{\infty}\) and \(\{ x_{2n+1} \} _{n=1}^{\infty}\) of even and odd terms of the solution do exactly one of the following:

(i)
Eventually they are both monotonically increasing.

(ii)
Eventually they are both monotonically decreasing.

(iii)
One of them is monotonically increasing and the other is monotonically decreasing.
Remark 1
Theorem 1 is actually a special case of Theorem 2. Indeed (2) implies that there exist a and \(b, a < \bar{x} < b\) such that \(f(a, a) > a\), \(f(b, b) < b\), which in view of monotonicity of f implies that
and so all conditions of Theorem 2 are satisfied. Furthermore, Theorem 2 is a special case of Theorem 3 if we additionally assume nonexistence of periodtwo solutions and singular points on the boundary of the region which may be the limiting points of the solutions of equation (1). See [4] for higher order versions of Theorems 1 and 2. None of these results provide any information as regards the basins of attraction of different equilibrium solutions or periodtwo solutions when there exist several equilibrium solutions and periodtwo solutions. Theorem 3 has been applied to many equations, see [3, 5] and references therein, but so far no example of equation (1) with a function f increasing in both variables which has stable equilibrium points and stable periodtwo solutions has been exhibited. In this paper we give several such examples for difference equations of the type
where \(f_{i}\), \(i=0,1\) are continuous and increasing functions. Many difference equations in mathematical biology are of the form (3); see [6] and references therein. As we have shown in [7] if the first order difference equation \(x_{n+1} = f_{1}(x_{n}) \) has two equilibrium points the corresponding delay difference equation \(x_{n+1} = f_{1}(x_{n1})\) has one periodtwo solution. Now if \(f_{1}\) dominates \(f_{0}\) in equation (3) then one may expect equation (3) will have periodtwo solution as well, which we show in this paper. Similarly if the difference equation \(x_{n+1} = f_{1}(x_{n}) \) has k equilibrium points the corresponding delay difference equation \(x_{n+1} = f_{1}(x_{n1})\) has \(k(k1)/2 \) periodtwo solutions, and if \(f_{1}\) dominates \(f_{0}\) in equation (3) then we show that equation (3) has a periodtwo solution as well.
In [5] the authors consider the difference equation (1) with several equilibrium points under the condition of the nonexistence of minimal periodtwo solutions and determine the basins of attraction of different equilibrium solutions. In this paper we consider equation (1) which has three equilibrium points and up to three minimal periodtwo solutions which are in NorthEast ordering. More precisely, we will give sufficient conditions for the precise description of the basins of attraction of different equilibrium points and periodtwo solutions. The results can be immediately extended to the case of any number of the equilibrium points and the periodtwo solutions by replicating our main results. An application of our results gives a precise description of the basis of attraction of all attractors of several difference equations, which are feasible models in population dynamics. Precisely, we illustrate our results with applications to three difference equations where all functions are linear, BevertonHolt or sigmoid BevertonHolt. In fact, our general results here are motivated by equations (8) and (14). Our results give first examples of difference equations with coexisting stable equilibrium solutions and stable periodtwo solutions.
2 Preliminaries
We now give some basic notions about monotone maps in the plane and connection between equation (1) and the monotone map.
Consider a map T on a nonempty set \(\mathcal{S}\subset\mathbb{R}^{2}\), and let \(\bar{\mathbf{e}} \in\mathcal{S}\). The point \(\bar{\mathbf{e}} \in\mathcal{S}\) is called a fixed point if \(T(\bar{\mathbf{e}} )=\bar{\mathbf{e}}\). An isolated fixed point is a fixed point that has a neighborhood with no other fixed points in it. A fixed point \(\bar{\mathbf{e}} \in\mathcal{S}\) is an attractor if there exists a neighborhood \(\mathcal{U}\) of \(\bar{\mathbf{e}}\) such that \(T^{n}(\mathbf{x}) \rightarrow\bar{\mathbf{e}}\) as \(n \rightarrow \infty\) for \({\mathbf{x}} \in\mathcal{U}\); the basin of attraction is the set of all \({\mathbf{x}} \in \mathcal{S}\) such that \(T^{n}(\mathbf{x}) \rightarrow\bar{\mathbf{e}}\) as \(n \rightarrow\infty\). A fixed point \(\bar{\mathbf{e}}\) is a global attractor on a set \(\mathcal{K}\) if \(\bar{\mathbf{e}}\) is an attractor and \(\mathcal{K}\) is a subset of the basin of attraction of \(\bar{\mathbf{e}}\). If T is differentiable at a fixed point \(\bar{\mathbf{e}}\), and if the Jacobian \(J_{T}(\bar{\mathbf{e}})\) has one eigenvalue with modulus less than one and a second eigenvalue with modulus greater than one, \(\bar{\mathbf{e}}\) is said to be a saddle. If one of the eigenvalues of T has absolute values 1 and a second eigenvalue has modulus greater (resp. less) than one, then \(\bar{\mathbf{e}}\) is said to be a nonhyperbolic of stable (resp. unstable) type. See [8] for additional definitions (stable and unstable manifolds, asymptotic stability).
Consider a partial ordering ⪯ on \(\mathbb{R}^{2}\). Two points \({\mathbf{v}}, {\mathbf{w}} \in\mathbb{R}^{2}\) are said to be related if \({\mathbf{v}} \preceq {\mathbf{w}}\) or \({\mathbf{w}} \preceq {\mathbf{v}}\). Also, a strict inequality between points may be defined as \({\mathbf{v}} \prec {\mathbf{w}} \) if \({\mathbf{v}} \preceq {\mathbf{w}} \) and \({\mathbf{v}} \neq {\mathbf{w}} \). A stronger inequality may be defined as \({\mathbf{v}}=(v_{1},v_{2}) \ll{\mathbf{w}}=(w_{1},w_{2})\) if \({\mathbf{v}} \preceq{\mathbf{w}}\) with \(v_{1} \neq w_{1} \) and \(v_{2} \neq w_{2} \). For u, v in \(\mathbb{R}^{2}\), the order interval \([\!\![{\mathbf{u}},{\mathbf{v}}]\!\!]\) is the set of all \({\mathbf{x}} \in\mathbb{R}^{2}\) such that \({\mathbf{u}} \preceq{\mathbf{x}} \preceq{\mathbf{v}}\).
A map T on a nonempty set \(\mathcal{S}\subset\mathbb{R}^{2}\) is a continuous function \(T:\mathcal{S} \rightarrow\mathcal{S}\). The map T is monotone if \({\mathbf{v}} \preceq{\mathbf{w}}\) implies \(T({\mathbf{v}}) \preceq T({\mathbf{w}}) \) for all \({\mathbf{v}}, {\mathbf{w}} \in\mathcal{S}\), and it is strongly monotone on \(\mathcal{S}\) if \({\mathbf{v}} \prec{\mathbf{w}}\) implies that \(T({\mathbf{v}}) \ll T({\mathbf{w}})\) for all \({\mathbf{v}}, {\mathbf{w}} \in\mathcal{S}\). The map is strictly monotone on \(\mathcal{S}\) if \({\mathbf{v}} \prec{\mathbf{w}}\) implies that \(T({\mathbf{v}}) \prec T({\mathbf{w}})\) for all \({\mathbf{v}}, {\mathbf{w}} \in\mathcal{S}\). Clearly, being related is invariant under iteration of a strongly monotone map.
Throughout this paper we shall use the NorthEast ordering for which the positive cone is the first quadrant, i.e. this partial ordering is defined by
A map T on a nonempty set \(\mathcal{S}\subset\mathbb{R}^{2}\) which is monotone with respect to the NorthEast ordering is called cooperative and a map monotone with respect to the SouthEast ordering
is called competitive.
If T is differentiable map on a nonempty set \(\mathcal{S}\), a sufficient condition for T to be strongly monotonic with respect to the NE ordering is that the Jacobian matrix at all points x has the sign configuration
provided that \(\mathcal{S}\) is open and convex.
For \({\mathbf{x}} \in\mathbb{R}^{2} \), define \(Q_{\ell}({\mathbf{x}})\) for \(\ell =1,\ldots,4\) to be the usual four quadrants based at x and numbered in a counterclockwise direction, for example, \(Q_{1}({\mathbf{x}}) = \{{\mathbf{y}} \in\mathbb{R}^{2} : x_{1} \leq y_{1}, x_{2} \leq y_{2} \} \). The (open) ball of radius r centered at x is denoted with \(\mathcal{ B}({\mathbf{x}},r)\). If \(\mathcal{K} \subset\mathbb{R}^{2}\) and \(r > 0\), write \(\mathcal{K}+\mathcal{B}(\mathbf{0},r):=\{ \mathbf{x} : \mathbf{x} = \mathbf{k}+\mathbf{y} \mbox{ for some } \mathbf{k} \in \mathcal{K} \mbox{ and } \mathbf{y} \in\mathcal{B}(\mathbf{0},r) \}\). If \(\mathbf{x} \in[\infty,\infty]^{2}\) is such that \(\mathbf{x} \preceq\mathbf{y}\) for every y in a set \(\mathcal{Y}\), we write \(\mathbf{x} \preceq\mathcal{Y}\). The inequality \(\mathcal{Y} \preceq \mathbf{x}\) is defined similarly.
The next result in [9] is stated for orderpreserving maps on \(\mathbb{R}^{n}\) and it also holds in ordered Banach spaces [10].
Theorem 4
For a nonempty set \(R\subset\mathbb{R}^{n}\) and ⪯ a partial order on \(\mathbb{R}^{n}\), let \(T:R\rightarrow R\) be an orderpreserving map, and let \(a, b \in R\) be such that \(a \prec b\) and \([\!\![a,b ]\!\!]\subset R\). If \(a \preceq T(a)\) and \(T(b) \preceq b\), then \([\!\![a,b ]\!\!]\) is an invariant set and:

i.
There exists a fixed point of T in \([\!\![a,b ]\!\!]\).

ii.
If T is strongly order preserving, then there exists a fixed point in \([\!\![a,b ]\!\!]\) which is stable relative to \([\!\![a,b ]\!\!]\).

iii.
If there is only one fixed point in \([\!\![a,b ]\!\!]\), then it is a global attractor in \([\!\![a,b ]\!\!]\) and therefore asymptotically stable relative to \([\!\![a,b ]\!\!]\).
The following result is a direct consequence of the trichotomy theorem of Dancer and Hess; see [9, 10].
Corollary 1
If the nonnegative cone of a partial ordering ⪯ is a generalized quadrant in \(\mathbb{R}^{n}\), and if T has no fixed points in \([\!\![u_{1},u_{2}]\!\!]\) other than \(u_{1}\) and \(u_{2}\), then the interior of \([\!\![u_{1},u_{2}]\!\!]\) is either a subset of the basin of attraction of \(u_{1}\) or a subset of the basin of attraction of \(u_{2}\).
We say that f is strongly increasing in both arguments if it is increasing, differentiable, and has both partial derivatives positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of equation (1) follows from the fact that if f is strongly increasing, then a map associated to equation (1) is a cooperative map on \(\mathcal{I}\times \mathcal{I}\) while the second iterate of a map associated to equation (1) is a strictly cooperative map on \(\mathcal {I}\times\mathcal{I}\).
Set \(x_{n1}= u_{n}\) and \(x_{n} = v_{n}\) in equation (1) to obtain the equivalent system
Let \(F(u,v)=(v, f(v, u))\). Then F maps \(I \times I\) into itself and is a cooperative map. The second iterate \(T:=F^{2}\) is given by
and it is clearly strictly cooperative on \(\mathcal{I}\times\mathcal{I}\).
If \(D_{1} g(u, v)\) and \(D_{2} g(u, v)\) denote the partial derivatives of a function \(g(u,v)\) with respect to u and v, the Jacobian matrix of T is
The determinant of (7) is given by
To check injectivity of T we set
which implies
and so \(f(f(v_{1}, u_{1}), v_{1}) = f(f(v_{1}, u_{1}), v_{2})\). By using the monotonicity of f we conclude that \(v_{1} = v_{2}\) which in view of \(f(v_{1}, u_{1}) = f(v_{2}, u_{2})\) gives \(u_{1} = u_{2}\).
The theory of monotone maps has been extensively developed at the level of ordered Banach spaces and applied to many types of equations such as ordinary, partial, and discrete types; see [10–16]. In particular, [12] has an extensive updated bibliography of different aspects of the theory of monotone maps. The theory of monotone discrete maps is more specialized and so one should expect stronger results in this case. An excellent review of the basic results is given in [12, 13]. In particular, twodimensional discrete maps are studied in great detail and very precise results which describe the global dynamics and the basins of attractions of equilibrium points and periodtwo solutions as well as global stable manifolds are given in [11, 14, 17–20].
3 Main results
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
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
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
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
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
where
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
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
where
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
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
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
where
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 \(\mathcal{W}^{s}(\Psi_{1},\Phi_{1})=\mathcal{C}_{1}^{}\).
4 Examples
In this section we give an application of our results in establishing global dynamics of some equations from mathematical biology. We present two cases in detail. All three models are of the types
where \(f_{i}\), \(i=1,2\) are transition functions, considered by many researchers in mathematical biology [6]. The most common transition functions in modeling are linear, BevertonHolt (\(f(u) = \frac{a u}{1+u}\)), and sigmoid BevertonHolt (\(f(u) = \frac{a u^{\sigma}}{1+u^{\sigma}}\), \(\sigma\geq1\)). The case when both transition functions are BevertonHolt functions was treated in [6] and we prove that in this case there are no periodtwo solutions and so every solution converges to an equilibrium. However, in all other cases for any other combination of transition functions the periodtwo solutions exist and play an important role in the dynamics.
4.1 Example 1: \(x_{n+1}=\frac{Ax^{2}_{n}}{1+x^{2}_{n}}+\frac {Bx^{2}_{n1}}{1+x^{2}_{n1}}\)
In this part we apply Theorems 510 to describe the global dynamics of the difference equation in the title.
4.1.1 The equilibrium points
We consider the difference equation
where \(A,B> 0\) and the initial conditions \(x_{1}\), \(x_{0}\) are nonnegative. In view of the above restriction on the initial conditions of equation (8), the equilibrium points of equation (8) are the nonnegative solutions of the equation
or equivalently
Now we have the following result.
Lemma 3
The following holds:

(i)
If \(A+B<2\) then equation (8) has a unique equilibrium point \(x_{0}=0\).

(ii)
If \(A+B=2\) then equation (8) has two equilibrium points \(x_{0}=0\) and \(x=(A+B)/2\).

(iii)
If \(A+B>2\) then equation (8) has three equilibrium points \(x_{0}=0\), \(x_{\mathrm{SW}}=\frac{1}{2} (A+B\sqrt{(A+B)^{2}4} )\), and \(x_{\mathrm{NE}}=\frac{1}{2} (A+B+\sqrt{(A+B)^{2}4} )\); \(x=(A+B)/2\).
Next, we investigate the stability of the nonnegative equilibrium points of equation (8). Set
and observe that \(f_{u}(u,v)>0\) and \(f_{v}(u,v)>0\). The next three lemmas are straightforward.
Lemma 4
Assume that \(A,B>0\). Then the equilibrium point \(x_{0}\) is locally asymptotically stable.
Lemma 5
Assume that \(A+B\geq2\). Then the equilibrium point \(x_{\mathrm{NE}}\) is locally asymptotically stable if \(A+B>2\) and nonhyperbolic if \(A+B=2\).
Proof
It can be seen that
and
Then the proof follows from Theorem 1.1.1 in [2] and the fact that
□
Lemma 6
Assume that \(A+B\geq2\). Then the equilibrium point \(x_{\mathrm{SW}}\) is:

(i)
a saddle point if \(2 A (A+B)+(AB) \sqrt{(A+B)^{2}4}>0\) and \(A+B>2\);

(ii)
a repeller if \(2 A (A+B)+(AB) \sqrt{(A+B)^{2}4}<0\) and \(A+B>2\);

(iii)
a nonhyperbolic if \(A+B=2\) or \(2 A (A+B)+(AB) \sqrt{(A+B)^{2}4}=0\).
Proof
It can be seen that
Then the proof follows from Theorem 1.1.1 in [2] and the fact that
□
4.1.2 Periodtwo solutions
Next, we investigate the existence and stability of the positive minimal periodtwo solutions of equation (8). Let \(\{\phi,\psi\} \) be a minimal periodtwo solution of equation (8). Then
which is equivalent to
which is true if and only if \(\phi\neq\psi\),
and
By eliminating ψ from (10) and (11) we obtain
and by eliminating ϕ from (10) and (11) we obtain
where
Since \((A +B) x  x^{2}1 \neq0\) for any x different from the equilibrium point, the minimal periodtwo solutions are solutions of the equation
Lemma 7
Let
Then the following holds:

(a)
Consider equation (12). Then all its real roots are positive numbers. Furthermore, equation (8) has up to three minimal periodtwo solutions.

(b)
If \(\Delta_{i}>0\), for all \(1\leq i\leq5\) and \(\Delta>0\) then equation (12) has six real roots. Consequently, equation (8) has three minimal periodtwo solutions.

(c)
If \(\Delta_{i}\leq0 \) for some \(1\leq i\leq4\) and \(\Delta_{5}>0\), \(\Delta>0\) then equation (12) has two distinct real roots and two pairs of conjugate imaginary roots. Consequently, equation (8) has one minimal periodtwo solution.

(d)
If \(\Delta_{i}>0\) for all \(1\leq i \leq j1\) and \(\Delta_{i}<0\) for all \(j\leq i \leq4\) for some \(1\leq j\leq5\) and \(\Delta_{5}<0\), \(\Delta>0\) then equation (12) has four distinct real roots and one pair of conjugate imaginary roots. Consequently, equation (8) has two minimal periodtwo solutions.

(e)
If \(\Delta_{i}\leq0\), \(\Delta_{i+1}\geq0\) for some \(1\leq i \leq 3\) and \(\Delta_{5}<0\), \(\Delta>0\) then equation (12) has three pairs of conjugate imaginary roots. Consequently, equation (8) has no minimal periodtwo solution.

(f)
Assume that \(\Delta=0\), and \(\Delta_{3}\neq0\) and \(\Delta _{5}\neq0\).

(f.1)
If (\(\Delta_{1}\leq0\) and \(\Delta_{2}\geq0\)) or (\(\Delta_{2}\leq0\) and \(\Delta_{3}> 0\)) then equation (12) has no real roots and has two distinct pairs of conjugate imaginary roots, one of them of multiplicity one and the other one of multiplicity two. Consequently, equation (8) has no minimal periodtwo solutions.

(f.2)
If \(\Delta_{1}> 0\) and \(\Delta_{2}> 0\) and \(\Delta _{3}>0\) then equation (12) has four distinct real roots, two of them are multiplicity two and other two of multiplicity one and has no conjugate imaginary roots. Consequently, equation (8) has two minimal periodtwo solutions.

(f.1)
Proof
The proof of (a) follows from Descartes’ rule of signs.
The discrimination matrix [22, 23] of \(\tilde{f}(x)\) and \(\tilde{f}'(x)\) is given by
where \(a_{6}=B^{2}\), \(a_{5}=2 B (A^{2}B^{2} )\), \(a_{4}= A^{4}2 A^{2} B^{2}+A^{2}+B^{4}+2 B^{2} \), \(a_{3}= A^{3}+2 A^{2} BA B^{2}2 B^{3}\), \(a_{2}=A^{4}A^{3} BA^{2} (B^{2}2 )+A B^{3}+B^{2}\), \(a_{1}=A^{3}A B^{2}\), and \(a_{0}=A^{2}\).
Let \(D_{k}\) denote the determinant of the submatrix of \(\operatorname{Discr}(\tilde{f})\), formed by the first 2k row and the first 2k columns, for \(k=1,\ldots,m\). By a straightforward calculation one can see that
The rest of the proof follows from Theorem 1 in [23]. □
4.1.3 The global behavior
In this section we describe the global behavior of equation (8) which has three equilibrium points \(\bar{x}_{0}, \bar{x}_{\mathrm{SW}}, \bar {x}_{\mathrm{NE}}\in I\) such that \(0 = \bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) where the equilibrium points \(\bar{x}_{0}\) and \(\bar{x}_{\mathrm{NE}}\) are locally asymptotically stable and \(\bar{x}_{\mathrm{SW}}\) is unstable. Further, \(x_{n}< A+B\) for all \(n\geq1\). One can see that all minimal periodtwo solutions of (8) belong to \(\operatorname{int}(Q_{2}(E_{\mathrm{SW}})\cup Q_{4}(E_{\mathrm{SW}}))\).
Lemma 8
If \(A+B<2\) then there exists a unique equilibrium point \(x_{0} = 0\) which is globally asymptotically stable.
Proof
The proof follows from Theorem 2 and the fact that \(x_{n}< A+B\) for \(n\geq1\). □
Theorem 11
Assume that \(A+B>2\). Then the following holds:

(i)
If \(\Delta_{i}\leq0\) and \(\Delta_{i+1}\geq0\) for some \(1\leq i \leq3\) and \(\Delta_{5}<0\), \(\Delta>0\) then equation (8) has three equilibrium points such that \(0 = \bar{x}_{0} < \bar {x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) where \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable and \(x_{\mathrm{SW}}\) is saddle point and has no periodtwo solution. The global behavior of equation (8) is described by Theorem 6. For example, this happens for \(A=0.46\) and \(B=1.98\).

(ii)
If \(\Delta_{i}\leq0 \) for some \(1\leq i\leq4\) and \(\Delta _{5}>0\), \(\Delta>0\) then equation (8) has three equilibrium points such that \(0 = \bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) where \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable and \(x_{\mathrm{SW}}\) is repeller and one periodtwo solution \(\{\phi_{1},\psi_{1}\}\) which is a saddle point. The global behavior of equation (8) is described by Theorem 7. For example, this happens for \(A=0.46\) and \(B=2.18\).

(iii)
If \(\Delta_{i}>0\), for all \(1\leq i\leq5\) and \(\Delta>0\) then equation (8) has three equilibrium points such that \(0 = \bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) where \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable and \(x_{\mathrm{SW}}\) is repeller and three minimal periodtwo solutions \(\{\phi_{1},\psi_{1}\}\), \(\{\phi_{2},\psi_{2}\}\), and \(\{\phi_{3},\psi_{3}\}\) such that \((\phi_{1},\psi_{1})\preceq_{\mathrm{ne}}(\phi _{2},\psi_{2})\preceq_{\mathrm{ne}}(\phi_{3},\psi_{3})\) where \(\{\phi_{1},\psi_{1}\}\), and \(\{\phi_{3},\psi_{3}\}\) are saddle points and \(\{\phi_{2},\psi_{2}\}\) is locally asymptotically stable. The global behavior of equation (8) is described by Theorem 8. For example, this happens for \(A=0.06\) and \(B=2.09\).

(iv)
If \(\Delta=0\), \(\Delta_{5}>0\), \(\Delta_{1}> 0\), \(\Delta_{2}> 0\), \(\Delta _{3}>0\) then equation (8) has three equilibrium points such that \(0 = \bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) where \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable and \(x_{\mathrm{SW}}\) is repeller and two minimal periodtwo solutions \(\{\phi_{1},\psi_{1}\}\) and \(\{\phi_{2},\psi_{2}\} \) such that \((\phi_{1},\psi_{1})\preceq_{\mathrm{ne}}(\phi_{2},\psi_{2})\) where either \(\{\phi_{1},\psi_{1}\}\) is nonhyperbolic and \(\{\phi_{2},\psi_{2}\}\) is a saddle point or \(\{\phi_{1},\psi_{1}\}\) is a saddle point and \(\{\phi _{2},\psi_{2}\}\) is nonhyperbolic. If periodtwo solution which is nonhyperbolic is of stable type then the global behavior of equation (8) is described by Theorems 8 and 9. For example, this happens for \(A=0.06\) and \(B=1.9998768381155188\).
Proof
(i) By assumption we have \(\Delta_{5}<0\). By Lemmas 4 and 5 the equilibrium points \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable. Since
by Lemma 6, the equilibrium point \(x_{\mathrm{SW}}\) is a saddle point. By Lemma 7 there are no minimal periodtwo solutions so the rest of the proof follows from Theorem 6.
(ii) By our assumptions we have \(\Delta_{5}>0\). By Lemmas 4 and 5 the equilibrium points \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable. In view of (13) and Lemma 6 the equilibrium point \(x_{\mathrm{SW}}\) is a repeller. Since the discriminant of polynomial \(\tilde{f}(x)\) is \(\operatorname{Dis}(\tilde {f})=D_{6}\neq0\) similarly as in Theorem 15 of [21] one can see that all periodtwo solutions are hyperbolic. In view of Lemma 2 we see that \(\mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{}\) is a totally ordered set which is invariant under T. If \((x_{0},y_{0})\in \mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{}\) then \(\{T^{n}(x_{0},y_{0})\}\) is eventually componentwise monotone. Then there exists a minimal periodtwo solution \(\{(\phi_{1},\psi_{1}),(\psi_{1},\phi_{1})\} \in\mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{} \subset Q_{2}(E_{\mathrm{SW}})\cup Q_{4}(E_{\mathrm{SW}})\) such that \(T^{n}(x_{0},y_{0})\to(\phi_{1},\psi_{1})\) as \(n\to\infty\). By Lemma 7 there exists only one minimal periodtwo solution, which implies that \(\{\phi_{1},\psi_{1}\}\) is a saddle point. The rest of the proof follows from Theorem 7.
(iii) Similarly as in (ii) one can see that \((\phi_{1},\psi_{1})\in \mathcal{C}_{1}^{+}\cup\mathcal{C}_{1}^{}\) and \((\phi_{3},\psi_{3})\in\mathcal {C}_{2}^{+}\cup\mathcal{C}_{2}^{}\) which are saddle points. By Corollary 1 there exists a minimal periodtwo solution \(\{\phi_{2},\psi_{2}\}\) such that \((\phi_{1},\psi_{1})\preceq_{\mathrm{ne}}(\phi _{2},\psi_{2})\preceq_{\mathrm{ne}}(\phi_{3},\psi_{3})\) which is locally asymptotically stable. The rest of the proof follows from Theorem 8.
(iv) Since \(\Delta_{5}>0\), by Lemmas 4 and 5 the equilibrium points \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable. Since
by Lemma 6 the equilibrium point \(x_{\mathrm{SW}}\) is a repeller. By Lemma 7 there exist two minimal periodtwo solutions. Let \(g(x,y)=f(y,x)\) and \(h(x,y)=f(f(y,x),y)\). Then periodtwo curves, that is, the curves of which the intersection is a periodtwo solution, are given by
Taking derivatives of \(g(x, y) = x\) and \(h(x, y) = y\) with respect to x we get
One can see that
where \(p(\mu)\) is the characteristic equation of the matrix
By Lemma 7, suppose that \(\{\phi_{1},\psi_{1}\}\) is a minimal periodtwo solution such that \(\{\phi_{1},\psi_{1}\}\) are roots of multiplicity two of (12) and \(\{\phi_{2},\psi_{2}\}\) are roots of multiplicity one of (12). In view of Lemma 7 in [24] we have \(y'_{\tilde{F}}(\phi_{1})y'_{\tilde{G}}(\phi_{1})=0\) and \(y'_{\tilde {F}}(\phi_{2})y'_{\tilde{G}}(\phi_{2})\neq0\), which implies that \(\{\phi _{1},\psi_{1}\}\) is nonhyperbolic. Since \(\det J_{T}(\phi_{2},\psi_{2})>0\) and \(\operatorname{tr} J_{T}(\phi_{2},\psi_{2})>0\) we see that \(\{\phi_{2},\psi_{2}\}\) is hyperbolic. Since \(E_{\mathrm{SW}}\) is a repeller it must be so that \(\{\phi _{2},\psi_{2}\}\) is a saddle point. If \(\{\phi_{1},\psi_{1}\}\) is nonhyperbolic of the stable type then the rest of the proof follows from Theorem 9. □
4.2 Example 2: \(x_{n+1}=Ax_{n}+\frac{Bx^{2}_{n1}}{1+x^{2}_{n1}}\)
In this part we apply Theorems 510 to describe the global dynamics of difference equation in the title.
4.2.1 The equilibrium points
We consider the difference equation
where \(A,B> 0\) and the initial conditions \(x_{1}\), \(x_{0}\) are nonnegative. In view of the above restriction on the initial conditions of equation (14), the equilibrium points of equation (14) are the positive solutions of the equation
or equivalently
Now we have the following result.
Lemma 9
The equilibrium points of equation (14) satisfy:

(i)
If \(A\geq1\) or \(B^{2}4 (A1)^{2}<0\) then equation (14) has a unique equilibrium point \(x_{0}=0\).

(ii)
If \(A<1\) and \(B^{2}4 (A1)^{2}=0\) then equation (14) has two equilibrium points \(x_{0}=0\), \(x_{\mathrm{SW}}=\frac{B}{2(1 A)}\).

(iii)
If \(A<1\) and \(B^{2}4 (A1)^{2}>0\) then equation (14) has three equilibrium points \(x_{0}=0\), \(x_{\mathrm{SW}}=\frac{B\sqrt{B^{2}4 (A1)^{2}}}{2(1 A)}\), and \(x_{\mathrm{NE}}=\frac{B+\sqrt{B^{2}4 (A1)^{2}}}{2(1 A)}\).
Next, we investigate the stability of the equilibrium points of equation (14). Set
and observe that \(f_{u}(u,v)>0\) and \(f_{v}(u,v)>0\). The next three lemmas are straightforward.
Lemma 10
Assume that \(A,B>0\). Then the equilibrium point \(x_{0}\) of equation (14) is locally asymptotically stable if \(A<1\), nonhyperbolic if \(A=1\), and a saddle point if \(A>1\).
Lemma 11
Assume that \(A<1\). If \(B^{2}4 (A1)^{2}>0\) then the equilibrium point \(x_{\mathrm{NE}}\) of equation (14) is locally asymptotically stable and nonhyperbolic if \(B^{2}4 (A1)^{2}=0\).
Proof
The proof follows from Theorem 1.1.1 in [2] and the fact that
where \(p=f_{u}(x_{\mathrm{NE}},x_{\mathrm{NE}})\) and \(q=f_{u}(x_{\mathrm{NE}},x_{\mathrm{NE}})\). □
Lemma 12
Assume that \(A<1\). Then the equilibrium point \(x_{\mathrm{SW}}\) is:

(i)
a saddle point if \((A1) \sqrt{B^{2}4 (A1)^{2}}+2 A B>0\) and \(B^{2}4 (A1)^{2}>0\);

(ii)
a repeller if \((A1) \sqrt{B^{2}4 (A1)^{2}}+2 A B<\) and \(B^{2}4 (A1)^{2}>0\);

(iii)
a nonhyperbolic point if \(B^{2}4 (A1)^{2}=0\).
Proof
The proof follows from Theorem 1.1.1 in [2] and the fact that
where \(p=f_{u}(x_{\mathrm{SW}},x_{\mathrm{SW}})\) and \(q=f_{u}(x_{\mathrm{SW}},x_{\mathrm{SW}})\). □
4.2.2 Periodtwo solutions
Next, we investigate the existence and stability of the positive minimal periodtwo solutions of equation (14).
Let \(\{\phi,\psi\} \) be a minimal periodtwo solution of equation (14). Then
which is true if and only if \(\phi\neq\psi\) and
By eliminating ψ from (16) and (17) we obtain
and by eliminating ϕ from (16) and (17) we obtain
where
Since \((A1) x^{2} + B x +A1 \neq0\) for any x different from the equilibrium point, the minimal periodtwo solutions are solutions of the equation
The proof of the following lemma is similar to the proof of Lemma 7, so we skip it.
Lemma 13
Let
Then the following holds:

(a)
Consider equation (18). Then all its real roots are positive numbers. Furthermore, equation (14) has up to three minimal periodtwo solutions.

(b)
If \(\Delta_{i}>0\), for all \(1\leq i\leq5\) and \(\Delta>0\) then equation (18) has six real roots. Consequently, equation (14) has three minimal periodtwo solutions.

(c)
If \(\Delta_{i}\leq0 \) for some \(1\leq i\leq4\) and \(\Delta_{5}>0\), \(\Delta>0\) then equation (18) has two distinct real roots and two pairs of conjugate imaginary roots. Consequently, equation (14) has one minimal periodtwo solutions.

(d)
If \(\Delta_{i}>0\) for all \(1\leq i \leq j1\) and \(\Delta_{i}<0\) for all \(j\leq i \leq4\) for some \(1\leq j\leq5\) and \(\Delta_{5}<0\), \(\Delta>0\) then equation (18) has four distinct real roots and one pair of conjugate imaginary roots. Consequently, equation (14) has two minimal periodtwo solutions.

(e)
If \(\Delta_{i}\leq0\), \(\Delta_{i+1}\geq0\) for some \(1\leq i \leq 3\) and \(\Delta_{5}<0\), \(\Delta>0\) then equation (18) has three pairs of conjugate imaginary roots. Consequently, equation (14) has no minimal periodtwo solution.

(f)
Assume that \(\Delta=0\), \(\Delta_{3}\neq0\), \(\Delta_{5}\neq0\).

(f.1)
If (\(\Delta_{1}\leq0 \), \(\Delta_{2}\geq0\)) or (\(\Delta_{2}\leq0\), \(\Delta_{3}> 0\)) then equation (18) has no real roots and has two distinct pairs of conjugate imaginary roots one of them is of multiplicity one and the other one of multiplicity two. Consequently, equation (14) has no minimal periodtwo solutions.

(f.2)
If \(\Delta_{1}> 0\), \(\Delta_{2}> 0\), \(\Delta_{3}>0\) then equation (18) has four distinct real roots, two of them are of multiplicity two and the other two of multiplicity one and has no conjugate imaginary roots. Consequently, equation (14) has two minimal periodtwo solutions.

(f.1)
4.2.3 The global behavior
In this section we describe the global behavior of equation (14). Equation (14) has three equilibrium points \(\bar{x}_{0}, \bar{x}_{\mathrm{SW}}, \bar{x}_{\mathrm{NE}}\in I\) such that \(0 = \bar{x}_{0} < \bar {x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) where the equilibrium points \(\bar{x}_{0}\) and \(\bar{x}_{\mathrm{NE}}\) are locally asymptotically stable and \(\bar{x}_{\mathrm{SW}}\) is unstable. One can see that all minimal period two solutions of (14) belong to \(\operatorname{int}(Q_{2}(E_{\mathrm{SW}})\cup Q_{4}(E_{\mathrm{SW}}))\).
Let \(u_{n+1}=Au_{n}+B\). By mathematical induction, it is easy to see that \(x_{n} \leq u_{n}\) for \(n>0\) if \(x_{0} \leq u_{0}\). Since
and
we obtain \(u_{n}\to\frac{B}{1A}\) as \(n\to\infty\) if \(A<1\). Thus, we conclude that the interval \([0,\frac{B}{1A}+\epsilon ]\) where \(\epsilon>0\) attracts all solutions, when \(A<1\).
Lemma 14
If \(A\geq1\) then there exists a unique equilibrium point \(x_{0} = 0\) and every solution \(\{ x_{n} \}\) satisfies
Proof
By using the difference inequalities method [25], the proof follows from the fact that \(x_{n}\geq v_{n}\) for \(n>0\) where \(x_{0}=v_{0}\) and \(v_{n}=A v_{n1}\) for \(n>1\). □
Theorem 12
Assume that \(A<1\) and \(B^{2}4 (A1)^{2}>0\). Then the following holds:

(i)
If \(\Delta_{i}\leq0\), \(\Delta_{i+1}\geq0\) for some \(1\leq i \leq 3\) and \(\Delta_{5}<0\), \(\Delta>0\) then equation (14) has three equilibrium point such that \(0 \leq\bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar {x}_{\mathrm{NE}}\) where \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable and \(x_{\mathrm{SW}}\) is a saddle point and has no periodtwo solution. The global behavior of equation (14) is described by Theorem 6. For example, this happens for \(A=0.3\) and \(B=2.0\).

(ii)
If \(\Delta_{i}\leq0 \) for some \(1\leq i\leq4\) and \(\Delta _{5}>0\), \(\Delta>0\) then equation (14) has three equilibrium points such that \(0 \leq\bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) where \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable and \(x_{\mathrm{SW}}\) is a repeller and one periodtwo solution \(\{\phi_{1},\psi_{1}\}\) which is a saddle point. The global behavior of equation (14) is described by Theorem 7. For example, this happens for \(A=0.1\) and \(B=1.9\).

(iii)
If \(\Delta_{i}>0\), for all \(1\leq i\leq5\) and \(\Delta>0\) then equation (14) has three equilibrium points such that \(0 \leq\bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) where \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable and \(x_{\mathrm{SW}}\) is a repeller and three minimal periodtwo solutions \(\{\phi_{1},\psi_{1}\}\), \(\{\phi_{2},\psi_{2}\}\), and \(\{\phi_{3},\psi_{3}\}\) such that \((\phi_{1},\psi_{1})\preceq_{\mathrm{ne}}(\phi _{2},\psi_{2})\preceq_{\mathrm{ne}}(\phi_{3},\psi_{3})\) where \(\{\phi_{1},\psi_{1}\}\), and \(\{\phi_{3},\psi_{3}\}\) are saddle points and \(\{\phi_{2},\psi_{2}\}\) is locally asymptotically stable. The global behavior of equation (14) is described by Theorem 8. For example, this happens for \(A=0.1\) and \(B=2.0\).

(iv)
If \(\Delta=0\) and \(\Delta_{5}>0\) and \(\Delta _{1}> 0\), \(\Delta_{2}> 0\), \(\Delta_{3}>0\) then equation (14) has three equilibrium points such that \(0 \leq\bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar {x}_{\mathrm{NE}}\) where \(x_{0}\) and \(x_{\mathrm{NE}}\) are locally asymptotically stable and \(x_{\mathrm{SW}}\) is a repeller and two minimal periodtwo solutions \(\{\phi _{1},\psi_{1}\}\) and \(\{\phi_{2},\psi_{2}\}\) such that \((\phi_{1},\psi_{1})\preceq _{\mathrm{ne}}(\phi_{2},\psi_{2})\) where either \(\{\phi_{1},\psi_{1}\}\) is nonhyperbolic and \(\{\phi_{2},\psi_{2}\}\) is a saddle point or \(\{\phi _{1},\psi_{1}\}\) is a saddle point and \(\{\phi_{2},\psi_{2}\}\) is nonhyperbolic. If a periodtwo solution which is nonhyperbolic is of stable type then the global behavior of equation (14) is described by Theorems 8 and 9. For example, this happens for \(A=0.1\) and \(B=1.97282\).
Remark 2
Similarly as in the previous two examples, one can see that the difference equation
has three equilibrium points \(\bar{x}_{0}, \bar{x}_{\mathrm{SW}}, \bar{x}_{\mathrm{NE}}\in [0, \infty)\) such that \(0 = \bar{x}_{0} < \bar{x}_{\mathrm{SW}}<\bar{x}_{\mathrm{NE}}\) where the equilibrium points \(\bar{x}_{0}\) and \(\bar{x}_{\mathrm{NE}}\) are locally asymptotically stable and \(\bar{x}_{\mathrm{SW}}\) is unstable and \(x_{n}< A+B\) for all \(n\geq1\). Further, one can see that all its minimal period two solutions belong to \(\operatorname{int}(Q_{2}(E_{\mathrm{SW}})\cup Q_{4}(E_{\mathrm{SW}}))\). It has up to three minimal periodtwo solutions and the global behavior can be described by Theorem 12, where the determinants \(\Delta_{i}\) will have different values which depend of the discrimination matrix.
All figures are created by Dynamica 4 [26].
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Acknowledgements
MRS Kulenović is supported in part by Maitland P Simmons Foundation. Esmir Pilav is supported in part by FMON of Bosnia and Herzegovina, number 053939351/15.
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Bilgin, A., Kulenović, M.R. & Pilav, E. Basins of attraction of periodtwo solutions of monotone difference equations. Adv Differ Equ 2016, 74 (2016). https://doi.org/10.1186/s136620160801y
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DOI: https://doi.org/10.1186/s136620160801y