Abstract
We investigate the local stability and the global asymptotic stability of the difference equation , with nonnegative parameters and initial conditions such that , for all . We obtain the local stability of the equilibrium for all values of parameters and give some global asymptotic stability results for some values of the parameters. We also obtain global dynamics in the special case, where , in which case we show that such equation exhibits a global period doubling bifurcation.
1. Introduction
In this paper, we investigate local stability and global dynamics of the following difference equation: where and where the initial conditions and are arbitrary nonnegative real numbers such that , for all . Special cases of (1), such as the cases when or or or , were considered in the monograph [1] and in several papers [2–5]. Equation (1) is the special case of a general second order quadratic fractional equation of the form with nonnegative parameters and initial conditions such that , , and , . Several global asymptotic results for some special cases of (3) were obtained in [6–9].
The change of variable transforms (1) to the difference equation where we assume that and that the nonnegative initial conditions and are such that , for all . Thus, the results of this paper extend to (4).
First systematic study of global dynamics of a special quadratic fractional case of (3) where was performed in [2, 3]. Dynamics of some related quadratic fractional difference equations was considered in the papers [6–9]. In this paper, we will perform the local stability analysis of the unique equilibrium which is quite elaborate and we will give the necessary and sufficient conditions for the equilibrium to be locally asymptotically stable, a saddle point, or a nonhyperbolic equilibrium. The local stability analysis indicates that some possible dynamics scenarios for (1) include period-doubling bifurcations and global attractivity of the equilibrium. Another possible scenario includes global periodicity; that is, the possibility that all solutions are periodic of the same period. This means that the techniques we used in [5, 6, 10–14] are applicable. We will also obtain the global asymptotic stability results for (1). As we have seen in [1], an efficient way of studying the dynamics of (1) is considering the dynamics of special cases of (1) which are obtained when one or more coefficients are set to be zero. It is interesting to notice that sometimes the most complex dynamics can be one of the very special cases of general equation, see [1]. Based on our results in [1], it is difficult to prove global asymptotic stability results of the unique equilibrium even for linear fractional difference equations where there are still two remaining cases needed to prove the general conjecture that local stability of the unique equilibrium implies the global stability. This task is even more challenging in the case of quadratic fractional difference equation such as (3).
Some interesting special cases of (1) which were thoroughly studied in [1] are the following equations.(1)Beverton-Holt difference equation, when : which represents the basic discrete model in population dynamics; see [15].(2)Riccati difference equation, when : (3)Difference equation studied in [1, 4, 16], when : which represents the discretization of the differential equation model in biochemical networks; see [17].
The paper is organized as follows. The next section gives the preliminary results on global stability of general quadratic fractional difference equation (3) which are applicable to some special cases of (1). This section contains also a global period-doubling bifurcation result from [12], which is applicable to one special case of (1), for which we obtain the global dynamics. The third section gives local stability result for (1) for all parametric values. The fourth section gives global asymptotic stability results for several special cases of (1). Finally, the fifth section gives global dynamics for difference equation where all parameters are positive, which exhibits the global period-doubling bifurcation. We pose a conjecture about global asymptotic stability of the equilibrium solutions of (8) in certain region of parameters.
2. Preliminaries
The global attractivity results obtained specifically for complicated cases of (3) are the following theorems [18].
Theorem 1. Assume that (3) has the unique equilibrium . If the following condition holds: where and are lower and upper bounds of all solutions of (3) and , then is globally asymptotically stable.
Theorem 2. Assume that (3) has the unique equilibrium , where and are lower and upper bounds of specific solution of (3) and . If the following condition holds: then is globally asymptotically stable on the interval .
In the case of (1), Theorems 1 and 2 give the following special results.
Corollary 3. If the following condition holds: where and are lower and upper bounds of all solutions of (1), then is globally asymptotically stable.
Corollary 4. If the following condition holds: where and are lower and upper bounds of specific solution of (1), then the unique equilibrium is globally asymptotically stable on the interval .
Corollary 3 can be used efficiently to obtain global stability results for the special cases of (1), in particular, for some equations of types , , and , where type means that special case of (1) has terms in the numerator and terms in the denominator. See Section 4.
In this paper, we present the local stability analysis for the unique equilibrium of (1) and then we apply Corollaries 3 and 4 to some special cases of (1) to obtain global asymptotic stability results for those equations. The obtained results will give the regions of parametric space where the unique positive equilibrium of (1) is globally asymptotically stable. In the coming manuscript we will give more precise dynamics in some special cases of (1) such as the case where , where the theory of monotone maps will give the global dynamics.
Our results on period-doubling bifurcation will be based on the following theorem for a general second order difference equation in [19]:
Theorem 5. Let be a set of real numbers and let be a function which is nonincreasing in the first variable and nondecreasing in the second variable. Then, for every solution of the equation the subsequences and 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.
The consequence of Theorem 5 is that every bounded solution of (14) converges to either equilibrium or period-two solution or to the point on the boundary, and the most important question becomes determining the basins of attraction of these solutions as well as the unbounded solutions.
The next result is the global period-doubling bifurcation result from [5] which also gives precise description of the basins of attraction of equilibrium points and periodic solutions of the considered equation.
Theorem 6. Let be a connected subset of , and, for each , let be an interval. Let be a continuous function. Consider a family of difference equations
with being continuous on , for . Suppose that, for each , the following statements are true.(a1), and is strictly decreasing in and strictly increasing in in .(a2)There exists a family of isolated equilibria that vary continuously in .(a3) is continuously differentiable on a neighborhood of , with partial derivatives and being continuous in .(a4)Let and be the roots of the characteristic equation of (15) at , ordered so that :if , then ,if , then ,if , then .(a5)If , is the unique equilibrium of (15) in and there are no prime period-two solutions in the rectangle of initial conditions . If , there exist no prime period-two solutions in the region of initial conditions , where denotes the first (., third) quadrant with respect to the point .(a6)For , statement of Theorem in [12] is true for the map defined on , where .
Then the following statements are true.(i)For such that , the equilibrium is a global attractor on .(ii)For such that , the stable set of the equilibrium is a curve in which is the graph of a continuous and increasing function that passes through and that has endpoints in . Furthermore,if is compact or if there exists a compact set such that , then there exist prime period-two solutions and ;if prime period two solutions and exist in and , respectively, and if they are the only prime period-two solutions there, then every solution with initial condition in the complement of the stable set of the equilibrium is attracted to one of the period-two solutions; that is, whenever , either and or and ;if there are no prime period-two solutions, then every solution of equation with initial condition in is such that the subsequence eventually leaves any given compact subset of , and every solution of with initial condition in is such that the subsequence eventually leaves any given compact subset of .
3. Linearized Stability Analysis
In this section we present the local stability of the unique positive equilibrium of (1).
3.1. Equilibrium Points
In view of the above restriction on the initial conditions of (1), the equilibrium points of (1) are the positive solutions of the equation: or, equivalently, When the unique positive equilibrium of (17) is given by When the unique positive equilibrium of (17) is given by Finally, when then the only equilibrium point of (17) is the positive solution of the quadratic equation (17).
In summary, it is interesting to observe that when (1) has a positive equilibrium , then is unique and satisfies (17). This observation simplifies the investigation of the local stability of the positive equilibrium of (1).
3.2. Local Stability of the Positive Equilibrium
Now we investigate the stability of the positive equilibrium of (1). Set and observe that If denotes an equilibrium point of (1), then the linearized equation associated with (1) about the equilibrium point is where
Theorem 7. Assume that Then, the unique equilibrium point of (1) is(i)locally asymptotically stable, if ;(ii)a saddle point, if ;(iii)a nonhyperbolic point, if or .
Proof. It is easy to see that Then, the proof follows from Theorem in [1] and the fact that
Theorem 8. Assume that Then, the unique equilibrium point of (1) is locally asymptotically stable.
Proof. A straightforward calculation gives Then, the proof follows from Theorem in [1] and the fact that
We have previously mentioned that if then the unique positive equilibrium of (17) is the positive solution of (17).
By using the identity we can see that Let Set It is clear that and that while for some , . A straightforward computation gives us that
Lemma 9. Let and be the partial derivatives given by (38) and (39). Assume that Then, holds for all values of parameters.
Proof. The inequality is equivalent to which is equivalent to Since we have that is always true.
Lemma 10. Let and be partial derivatives given by (38) and (39). Assume that Then, , if and only if
Proof. (i)Let , then, from (40), .(ii)Assume that and . Then is equivalent to . One can see that
if and only if
since
In view of (42), we have that , if and only if
which is equivalent to
in view of
The conditions in (57) are equivalent to
and from which the proof follows.(iii)Assume that and . Then is equivalent to . It is easy to see that
if and only if
which implies that , if and only if
which is equivalent to
since
Since
we have that (64) is equivalent to
and from which the proof follows.(iv)If and , then it is enough to see that
which is equivalent in this case to
Lemma 11. Let be the partial derivative given by (39). Assume that Then, holds for all values of parameters.
Proof. Inequality is equivalent to
If , then, from (41), we have that .
Let .
Relation (71) is equivalent to
or
It can be shown that
Since
we have that , if and only if , from which it follows that (73) is equivalent to
In view of (76), we have that (72) and (73) are equivalent to .
So, holds for all values of parameters.
Thus, we proved the following result.
Theorem 12. Assume that Then, the unique equilibrium point of (1) is(i)a locally asymptotically stable point, if and only if any of the following holds:(ii)a saddle point, if and only if the following holds: (iii)a nonhyperbolic point, if and only if the following holds:
Proof. The proof follows from Theorem in [1] and Lemmas 9, 10, and 11.
4. Global Asymptotic Stability Results
In this section we give the following global asymptotic stability result for some special cases of (1).
Theorem 13.
Consider (1), where and all other coefficients are positive, subject to the condition
where and . Then is globally asymptotically stable.
Consider (1), where all coefficients are positive, subject to the condition (1) where and . Then is globally asymptotically stable.
Proof. In view of Corollary 3, we need to find the lower and upper bounds for all solutions of (1), for .(i)In this case, the lower and upper bounds for all solutions of (1), for , are derived as (ii)In this case, the lower and upper bounds for all solutions of (1), for , are derived as
A consequence of Theorem 13 is the following result.
Corollary 14. Consider (1), where and all other coefficients are positive, subject to the condition where . Then, is globally asymptotically stable.
By using similar method as in the proof of Theorem 13, one can prove the following result.
Theorem 15. Consider (1), where and all other coefficients are positive, subject to the condition where . Then is globally asymptotically stable.
Proof. Now, we have
Remark 16. Equation (1), where and all other coefficients are positive reduces to well-known equation (7) which was studied in great details in [1, 4] and for which we have shown that the unique equilibrium is globally asymptotically stable, if and only if it is locally asymptotically stable; that is, if and only if the condition of Theorem 7 holds. This result is certainly better than the global asymptotic result we derive from Corollaries 3 and 4. However, the results where local stability implies global stability are very rare and it seems to be limited only to second order linear fractional difference equations; see [16, 19].
Remark 17. Equation (1), where either or and all other coefficients are positive can be treated with Corollary 4 and global asymptotic stability of the equilibrium (whenever it exists) follows from condition (12) in the interval when , that is, when . Similarly, (1), where exactly one of the coefficients , or is zero, and all other coefficients are positive can be treated with Corollary 4 and global asymptotic stability of the equilibrium follows from condition (12) in the interval when , that is, when . In this case can be replaced by .
5. Equation
In this section we present the global dynamics of (8), which exhibits a global period-doubling bifurcation introduced in [5].
5.1. Local Stability Analysis
Equation (8), by the change of variables reduces to the equation where we left the old labels and for the parameters and for variable. Equation (90) has the unique positive equilibrium given by The partial derivatives associated to (90) at the equilibrium are The characteristic equation associated to (90) at the equilibrium point is which implies By applying the linearized stability Theorem 12, we obtain the following result.
Theorem 18. The unique positive equilibrium point of (90) is(i)locally asymptotically stable, when (ii)a saddle point, when (iii)a nonhyperbolic point, when
Lemma 19. If then (90) possesses the unique minimal period-two solution where the values and are the (positive and distinct) solutions of the quadratic equation where
Proof. Assume that is a minimal period-two solution of (90). Then
which is equivalent to
which is true, if and only if ,
By subtracting (103) and (104), we get
By dividing (103) by and (104) by and subtracting them and by using the fact that , we get
If we set
where , then and are positive and different solutions of the quadratic equation
In addition to condition , it is necessary that .
From (105) and (106), we obtain the system
We obtain that solutions () of system (109) are
Since , there is only one positive solution of system (109). We have that , , and , if and only if
Therefore, there is only one minimal period-two solution of (90) which is solution of equation
where
It is easy to see that
So, . It can be shown that . Namely, if we consider as a function of , then
always, because
5.2. Global Results and Basins of Attraction
In this section, we present global dynamics results for (90), in case .
Lemma 20. Every solution of (90) is bounded from above and from below by positive constants.
Proof. Indeed
Theorem 21. The following statements are true.(i)If , then (90) has a unique equilibrium point , which is locally asymptotically stable and has no minimal period-two solution. Furthermore, the unique positive equilibrium point is globally asymptotically stable.(ii)If , then (90) has a unique equilibrium point which is a saddle point and has the minimal period-two solution . Global stable manifold , which is continuous increasing curve, divides the first quadrant such that the following holds:every initial point in is attracted to ;if , then the subsequence of even-indexed terms is attracted to and the subsequence of odd-indexed terms is attracted to ;if , then the subsequence of even-indexed terms is attracted to and the subsequence of odd-indexed terms is attracted to .(iii)If , then (90) has a unique equilibrium point which is nonhyperbolic and has no minimal period-two solution. The equilibrium point is a global attractor.
Proof. Let (i)If , then appropriate function of (90) is nonincreasing in the first variable and nondecreasing in the second variable. From Theorem 18, (90) has a unique equilibrium point , which is locally asymptotically stable and, from Lemma 19, (90) has no minimal period-two solution. All conditions of Theorem 6 are satisfied from which follows that a unique equilibrium point is a global attractor. So, is globally asymptotically stable.(ii)From Theorem 18, (90) has a unique equilibrium point , which is a saddle point. From Lemma 19, (90) has a unique minimal period-two solution . The map has neither fixed points nor periodic points of minimal period-two in . All conditions of Theorem 6 are satisfied from which follows that every solution with initial condition in the complement of the stable set of the equilibrium is attracted to one of the period-two solutions. That is, whenever , either and or and . It is not hard to see that the map is anticompetitive (which means that is competitive) and is strongly competitive from which we have that conclusions , , and hold.(iii)The proof follows from Theorem 5 and Lemmas 19 and 20.
For graphical illustration of Theorem 21, see Figure 1.
(a)
(b)
(c)
Based on our simulations, we pose the following conjecture.
Conjecture 22. The equilibrium of (90) is globally asymptotically stable for .
Clearly, if , then and so every solution is equal to the equilibrium solution. If , then the function is increasing in and decreasing in . In addition, the interval is an invariant interval for . In order to apply Theorem of [1], we need to show that the only solution of the system in is . This system reduces to . The polynomial with is increasing on , if , which holds when . Thus Conjecture 22 is true for .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors are grateful to the anonymous referee for number of suggestions that have improved the quality of exposition of the results.