Basins of attraction of period-two solutions of monotone difference equations
- Arzu Bilgin^{1},
- Mustafa RS Kulenović^{1}Email author and
- Esmir Pilav^{2}
https://doi.org/10.1186/s13662-016-0801-y
© Bilgin et al. 2016
Received: 3 February 2016
Accepted: 5 March 2016
Published: 14 March 2016
Abstract
Keywords
attractivity basin difference equation invariant manifolds period-two solutionsMSC
39A10 39A20 37B25 37D101 Introduction
The first theorem, which has also been very useful in applications to mathematical biology, was really motivated by a problem in [1].
Theorem 1
- (i)
\(f(x,y)\) is non-decreasing 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)
Theorem 2
- (a)
\(f(x,y)\) is non-decreasing in each of its arguments;
- (b)
equation (1) has a unique equilibrium \(\bar{x}\in[a, b]\).
Theorems 1 and 2 were extended to a kth order difference equation where the right hand side is a non-decreasing function of all its variables; see [1, 2].
The following result has been obtained in [3].
Theorem 3
- (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
In [5] the authors consider the difference equation (1) with several equilibrium points under the condition of the nonexistence of minimal period-two 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 period-two solutions which are in North-East ordering. More precisely, we will give sufficient conditions for the precise description of the basins of attraction of different equilibrium points and period-two solutions. The results can be immediately extended to the case of any number of the equilibrium points and the period-two 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, Beverton-Holt or sigmoid Beverton-Holt. 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 period-two 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 non-hyperbolic 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.
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 order-preserving maps on \(\mathbb{R}^{n}\) and it also holds in ordered Banach spaces [10].
Theorem 4
- 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 non-negative 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 \(\mathcal {I}\times\mathcal{I}\).
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, two-dimensional 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 period-two 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 period-two solutions. As is well known [12, 13] the period-two solutions are the points in the South-East 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 period-two 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
- (i)
If there are no minimal period-two 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 period-two 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 \(E_{\mathrm{SW}}\in\mathcal{C}_{1}^{+}\), \(E_{\mathrm{SW}}\in\mathcal {C}_{1}^{-}\), \(E_{\mathrm{SW}}\in\mathcal{C}_{2}^{+}\), \(E_{\mathrm{SW}}\in\mathcal{C}_{2}^{-}\).
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 non-existence of period-two solutions.
Theorem 5
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 Perron-Frobenius 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.
Theorem 6
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 period-two 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. □
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 period-two 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 North-East 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 North-East 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
Now, we consider the dynamical scenario when there exist three minimal period-two solutions.
Theorem 8
Proof
Now, we consider two dynamical scenarios when there exists a minimal period-two solution \(\{\Phi,\Psi\}\) which is a non-hyperbolic 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
Proof
Since \(\{\Phi,\Psi\}\) is a non-hyperbolic period-two 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.
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
4 Examples
4.1 Example 1: \(x_{n+1}=\frac{Ax^{2}_{n}}{1+x^{2}_{n}}+\frac {Bx^{2}_{n-1}}{1+x^{2}_{n-1}}\)
In this part we apply Theorems 5-10 to describe the global dynamics of the difference equation in the title.
4.1.1 The equilibrium points
Lemma 3
- (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 \(x_{\mathrm{NE}}=\frac{1}{2} (A+B+\sqrt{(A+B)^{2}-4} )\); \(x=(A+B)/2\).
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 non-hyperbolic if \(A+B=2\).
Proof
Lemma 6
- (i)
a saddle point if \(2 A (A+B)+(A-B) \sqrt{(A+B)^{2}-4}>0\) and \(A+B>2\);
- (ii)
a repeller if \(2 A (A+B)+(A-B) \sqrt{(A+B)^{2}-4}<0\) and \(A+B>2\);
- (iii)
a non-hyperbolic if \(A+B=2\) or \(2 A (A+B)+(A-B) \sqrt{(A+B)^{2}-4}=0\).
Proof
4.1.2 Period-two solutions
Lemma 7
- (a)
Consider equation (12). Then all its real roots are positive numbers. Furthermore, equation (8) has up to three minimal period-two 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 period-two 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 period-two solution.
- (d)
If \(\Delta_{i}>0\) for all \(1\leq i \leq j-1\) 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 period-two 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 period-two 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 period-two 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 period-two solutions.
- (f.1)
Proof
The proof of (a) follows from Descartes’ rule of signs.
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 period-two 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
- (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 period-two 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 period-two 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 period-two 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 period-two 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 non-hyperbolic and \(\{\phi_{2},\psi_{2}\}\) is a saddle point or \(\{\phi_{1},\psi_{1}\}\) is a saddle point and \(\{\phi _{2},\psi_{2}\}\) is non-hyperbolic. If period-two solution which is non-hyperbolic 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
(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 period-two 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 period-two 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 period-two 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 period-two 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.
4.2 Example 2: \(x_{n+1}=Ax_{n}+\frac{Bx^{2}_{n-1}}{1+x^{2}_{n-1}}\)
In this part we apply Theorems 5-10 to describe the global dynamics of difference equation in the title.
4.2.1 The equilibrium points
Lemma 9
- (i)
If \(A\geq1\) or \(B^{2}-4 (A-1)^{2}<0\) then equation (14) has a unique equilibrium point \(x_{0}=0\).
- (ii)
If \(A<1\) and \(B^{2}-4 (A-1)^{2}=0\) then equation (14) has two equilibrium points \(x_{0}=0\), \(x_{\mathrm{SW}}=\frac{B}{2(1- A)}\).
- (iii)
If \(A<1\) and \(B^{2}-4 (A-1)^{2}>0\) then equation (14) has three equilibrium points \(x_{0}=0\), \(x_{\mathrm{SW}}=\frac{B-\sqrt{B^{2}-4 (A-1)^{2}}}{2(1- A)}\), and \(x_{\mathrm{NE}}=\frac{B+\sqrt{B^{2}-4 (A-1)^{2}}}{2(1- A)}\).
Lemma 10
Assume that \(A,B>0\). Then the equilibrium point \(x_{0}\) of equation (14) is locally asymptotically stable if \(A<1\), non-hyperbolic if \(A=1\), and a saddle point if \(A>1\).
Lemma 11
Assume that \(A<1\). If \(B^{2}-4 (A-1)^{2}>0\) then the equilibrium point \(x_{\mathrm{NE}}\) of equation (14) is locally asymptotically stable and non-hyperbolic if \(B^{2}-4 (A-1)^{2}=0\).
Proof
Lemma 12
- (i)
a saddle point if \((A-1) \sqrt{B^{2}-4 (A-1)^{2}}+2 A B>0\) and \(B^{2}-4 (A-1)^{2}>0\);
- (ii)
a repeller if \((A-1) \sqrt{B^{2}-4 (A-1)^{2}}+2 A B<\) and \(B^{2}-4 (A-1)^{2}>0\);
- (iii)
a non-hyperbolic point if \(B^{2}-4 (A-1)^{2}=0\).
Proof
4.2.2 Period-two solutions
Next, we investigate the existence and stability of the positive minimal period-two solutions of equation (14).
The proof of the following lemma is similar to the proof of Lemma 7, so we skip it.
Lemma 13
- (a)
Consider equation (18). Then all its real roots are positive numbers. Furthermore, equation (14) has up to three minimal period-two 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 period-two 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 period-two solutions.
- (d)
If \(\Delta_{i}>0\) for all \(1\leq i \leq j-1\) 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 period-two 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 period-two 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 period-two 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 period-two 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 \(\operatorname{int}(Q_{2}(E_{\mathrm{SW}})\cup Q_{4}(E_{\mathrm{SW}}))\).
Lemma 14
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 \(v_{n}=A v_{n-1}\) for \(n>1\). □
Theorem 12
- (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 period-two 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 period-two 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 period-two 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 period-two 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 non-hyperbolic and \(\{\phi_{2},\psi_{2}\}\) is a saddle point or \(\{\phi _{1},\psi_{1}\}\) is a saddle point and \(\{\phi_{2},\psi_{2}\}\) is non-hyperbolic. If a period-two solution which is non-hyperbolic 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
All figures are created by Dynamica 4 [26].
Declarations
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 05-39-3935-1/15.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Hautus, MLJ, Bolis, TS: Solution to Problem E2721. Am. Math. Mon. 86, 865-866 (1979) MathSciNetView ArticleGoogle Scholar
- Kulenović, MRS, Ladas, G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC, Boca Raton (2001) View ArticleMATHGoogle Scholar
- Amleh, AM, Camouzis, E, Ladas, G: On the dynamics of a rational difference equation, part I. Int. J. Difference Equ. 3, 1-35 (2008) MathSciNetMATHGoogle Scholar
- Camouzis, E, Ladas, G: Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures. Advances in Discrete Mathematics and Applications, vol. 5. Chapman & Hall/CRC, Boca Raton (2008) MATHGoogle Scholar
- Brett, A, Kulenović, MRS: Basins of attraction of equilibrium points of monotone difference equations. Sarajevo J. Math. 5, 211-233 (2009) MathSciNetMATHGoogle Scholar
- Kulenović, MRS, Yakubu, A-A: Compensatory versus overcompensatory dynamics in density-dependent Leslie models. J. Differ. Equ. Appl. 10, 1251-1265 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Kulenović, MRS, Moranjkić, S, Nurkanović, Z: Global dynamics and bifurcation of a perturbed sigmoid Beverton-Holt difference equation. Math. Methods Appl. Sci. (2016). doi:https://doi.org/10.1002/mma.3722 Google Scholar
- Robinson, C: Stability, Symbolic Dynamics, and Chaos. CRC Press, Boca Raton (1995) MATHGoogle Scholar
- Kulenović, MRS, Merino, O: Global bifurcation for competitive systems in the plane. Discrete Contin. Dyn. Syst., Ser. B 12, 133-149 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Hess, P: Periodic-Parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics Series, vol. 247. Longman, Harlow (1991); copublished in the United States with Wiley, New York MATHGoogle Scholar
- de Mottoni, P, Schiaffino, A: Competition systems with periodic coefficients: a geometric approach. J. Math. Biol. 11, 319-335 (1981) MathSciNetView ArticleMATHGoogle Scholar
- Hirsch, M, Smith, H: Monotone dynamical systems. In: Handbook of Differential Equations: Ordinary Differential Equations, vol. II, pp. 239-357. Elsevier, Amsterdam (2005) Google Scholar
- Hirsch, M, Smith, HL: Monotone maps: a review. J. Differ. Equ. Appl. 11, 379-398 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Smith, HL: Planar competitive and cooperative difference equations. J. Differ. Equ. Appl. 3, 335-357 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Takáč, P: Asymptotic behavior of strongly monotone time-periodic dynamical processes with symmetry. J. Differ. Equ. 100(2), 355-378 (1992) MathSciNetView ArticleMATHGoogle Scholar
- Takáč, P: Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups. J. Reine Angew. Math. 423, 101-173 (1992) MathSciNetMATHGoogle Scholar
- Brett, A, Kulenović, MRS: Two species competitive model with the Allee effect. Adv. Differ. Equ. 2014, 307 (2014) MathSciNetView ArticleGoogle Scholar
- Kulenović, MRS, Merino, O: Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete Contin. Dyn. Syst., Ser. B 6, 1141-1156 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Kulenović, MRS, Merino, O: Invariant manifolds for competitive discrete systems in the plane. Int. J. Bifurc. Chaos Appl. Sci. Eng. 20(8), 2471-2486 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Smith, HL: Periodic solutions of periodic competitive and cooperative systems. SIAM J. Math. Anal. 17, 1289-1318 (1986) MathSciNetView ArticleMATHGoogle Scholar
- Hadžiabdić, V, Kulenović, MRS, Pilav, E: Dynamics of a two-dimensional competitive system of rational difference equations with quadratic terms. Adv. Differ. Equ. 2013, 301 (2014) MathSciNetGoogle Scholar
- Janson, S: Resultant and discriminant of polynomials. http://www2.math.uu.se/~svante/papers/ (2010)
- Yang, L, Hou, X, Zeng, Z: Complete discrimination system for polynomials. Sci. China Ser. E 39(6), 628-646 (1996) MathSciNetMATHGoogle Scholar
- Basu, S, Merino, O: On the behavior of solutions of a system of difference equations. Commun. Appl. Nonlinear Anal. 16(1), 89-101 (2009) MathSciNetMATHGoogle Scholar
- Lakshmikantham, V, Trigiante, D: Theory of Difference Equations: Numerical Methods and Applications, 2nd edn. Monographs and Textbooks in Pure and Applied Mathematics, vol. 251. Dekker, New York (2002) View ArticleMATHGoogle Scholar
- Kulenović, MRS, Merino, O: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman & Hall/CRC, Boca Raton (2002) View ArticleMATHGoogle Scholar