Discrete Dynamics in Nature and Society

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Stability and Bifurcation Analysis of Discrete Dynamical Systems 2020

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Volume 2020 |Article ID 1364282 | https://doi.org/10.1155/2020/1364282

Toufik Khyat, M. R. S. Kulenović, "Global Dynamics of Delayed Sigmoid Beverton–Holt Equation", Discrete Dynamics in Nature and Society, vol. 2020, Article ID 1364282, 15 pages, 2020. https://doi.org/10.1155/2020/1364282

Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

Guest Editor: Abdul Qadeer Khan
Received09 Mar 2020
Accepted16 Apr 2020
Published26 May 2020

Abstract

In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation , where is decreasing in the variable and increasing in the variable . As a case study, we use the difference equation , where the initial conditions and the parameters satisfy . In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.

1. Introduction and Preliminaries

Consider the following difference equation:where the initial conditions and the parameters satisfy that . Equation (1) is a special case of the equationwhere and the parameters satisfy , , , and . By using the theory of competitive systems, we will describe precisely the basins of attraction of the equilibrium points and period-two solutions of equation (1).

Both equations (1) and (2) are special cases ofwhere all parameters are nonnegative numbers and the initial conditions so that the solution is well defined. Many special cases of equation (3) have been investigated in [16]. In particular, a special case where was studied in [1, 2], where a variety of techniques was used to obtain some global results. Other special cases of equation (3),were considered in [4, 7, 8], respectively, where the theory of monotone maps in the plane was used to derive the global dynamics of these equations. Indeed, in [4], the coexistence of a unique locally asymptotically stable equilibrium point and a locally asymptotically stable minimal period-two solution was obtained for the first time. Equation (1), on the other hand, can have up to three fixed points and up to three period-two solutions and its dynamics is similar to the dynamics of equations (5) and (6). The possible dynamic scenarios for equation (1) will be a motivation for getting corresponding results for the general second-order difference equation in Section 2, and these general results will also imply global dynamics results for equations (4)–(6). Thus, we will obtain some general global dynamic results for general second-order difference equation,which will unify global dynamic results for equations (4)–(6) in the hyperbolic case. The nonhyperbolic cases may be different for different equations and require the specific investigation.

Equation (1) contains an interesting special case when ,which is a well-known delayed sigmoid Beverton–Holt equation of which interesting dynamics is given in [8]. Thus, equation (1) can be considered as a perturbation of equation (8). Some interesting results about boundedness, stability, and attractivity for related ordinary differential equations with or without delay can be found in [9, 10].

The organization of the paper is as follows. The rest of this section contains some necessary notions of competitive systems in the plane. Section 2 contains several global dynamic scenarios for two-dimensional competitive systems and maps in the absence of minimal period-two solutions and the applications of these results to a general second-order difference equation with certain monotonic properties of transition function. Finally, Section 3 gives global dynamic results for equation (1). As we will show, the global dynamics of equation (1) can be described as a sequence of global period-doubling bifurcations with two parameters and . Three regions for parameter are , and . In the critical region , the critical subregions for the bifurcation parameter are and . In the critical region , the critical subregions for the bifurcation parameter are and . In the critical region , the critical subregions for the bifurcation parameter are and (see Figures 13). The dynamics of equation (1) is different from the dynamics of equation (6) which is another perturbation of equation (8), considered in [8]. The perturbation term in the case of equation (1) is (depends on population size at -th generation only) while in the case of equation (6) is (depends on population sizes at -th and -th generations) and as expected the number of bifurcations for equation (1) is smaller than the number for equation (6).

A unique new feature of the proofs of presented results is the use of center manifold theory for proving the global stability result in the nonhyperbolic case of equation (1) as well as the use of concavity properties of invariant manifolds in treating some nonhyperbolic cases of equation (1).

There is an extensive literature on the dynamics of monotone maps (competitive and cooperative) that can be found in [1113]. In this paper, we will use some theorems from [1416] used in several papers such as [46, 8, 17, 18] that will be important in establishing the global dynamics results for equation (1).

We will use Theorem 5 from [19] which states that, for every solution of equation (7), the subsequences and of even and odd terms are eventually monotonic, provided that a function is nonincreasing in the first variable and nondecreasing in the second variable. In view of Theorem 5 from [19], determining the basins of attraction of an equilibrium, or a period-two solution or of the point on the boundary where equation (7) may not be defined, becomes major objective of the study of global dynamics.

Remark 1. The connection between the theory of monotone maps and the asymptotic behavior of equation (7) is the consequence of the fact that if is strongly decreasing in the first argument and strongly increasing in the second argument, then the second iterate of a map associated to equation (7) is a strictly competitive map on , see [15] and Remarks 2 and 3 in [3].
The fixed point of a planar competitive or cooperative map is said to be nonhyperbolic if the Jacobian matrix has at least one eigenvalue on the unit circle , that is, if . If one eigenvalue is inside the unit circle , the fixed point is nonhyperbolic of stable type, and if the other eigenvalue is outside of the unit circle , the fixed point is nonhyperbolic of unstable type. If both eigenvalues lie on the unit circle, the fixed point is nonhyperbolic of resonance type of either (1, 1), (1, −1), (−1, 1), or (−1, −1) depending on the values of the eigenvalues. It should be noticed that the eigenvalues of the Jacobian matrix of planar competitive or cooperative map are real numbers with a largest eigenvalue which is positive.

2. Main Results

We start with specific global dynamic scenarios for competitive system (9) that will be applied to equation (7).

Theorem 1. Considering the competitive map generated by the system,on a set with a nonempty interior.

(a)Assume that has seven fixed points such that five belongs to the west and south boundaries of the region and two fixed points are interior points. Moreover, assuming that and belong to the west boundary, is the south-west corner of the region , and and are on the south boundary of such that . Moreover, assume that and that and . Finally, assume that are locally asymptotically stable, is a repeller, and are saddle points. If is either a saddle point or a nonhyperbolic point of stable type and has no period-two solutions, then all solutions which start between the stable manifolds and converge to and all solutions which start between the stable manifolds and converge to and all solutions which start between the stable manifolds and converge to .(b)Assume that has exactly six fixed points , where the points have the same configuration and points have the same local character as in part (a), while is a nonhyperbolic point of unstable type. If has no period-two solutions, then there exist two increasing continuous curves and , emanating from such that all solutions which start between and converge to . Furthermore, all solutions which start between the stable manifolds and converge to , and all solutions which start above converge to and all solutions which start below converge to .(c)Assume that has exactly nine fixed points , where the points have the same configuration and the same local character as in (a). Assuming that the fixed points are saddle points such that and . Assume that has no period-two solutions, then all solutions which start between the stable manifolds and converge to , all solutions which start above converges to , and all solutions which start below converge to . Finally, all solutions which start between the stable manifolds and converge to .

Proof. (a)The existence of the global stable and unstable manifolds of the saddle fixed points is guaranteed by Theorems 1–5 in [15]. In any case, all global stable manifolds and have an endpoint at , and has another endpoint at .In view of Theorem 4 in [15], every solution which starts between the stable manifolds and eventually enters . If the initial point is in , one can find the points on the -axis and on the -axis such that . This will imply that , and so converges to as both do.If the initial point is above , one can find the point on the -axis and a point such that . This will imply that , and so eventually. Now, in view of Corollary 1 in [15], as .In a similar way the case when the initial point is below can be handled.(b)The existence of the global stable and unstable manifolds of the saddle fixed points is guaranteed by Theorems 1–5 in [15]. Both global stable manifolds and have an endpoint at . The existence of curves and follows from Theorem 2 in [16]. The proof that the region between the stable manifolds and eventually enters and so it converges to is the same as in part (a).In a similar way as in the proof of part (a), we can show that if the initial point is above it will eventually enter and so it will converge to . In a similar way, we can show that if the initial point is below , it will eventually enter and so it will converge to .Finally, if the initial point is between and , then one can find the point and a point such that . This will imply that , and so as .(c)The proof that the region between the stable manifolds and is the basin of attraction of is same as in part (a) and will be omitted. The proof that all solutions which start above converges to and all solutions which start below converge to is same as in part (a) and so will be omitted.If the initial point is between and , then one can find the point and a point such that . This will imply that , and so as as . Now, in view of Corollary 2 in [14], as .

Theorem 2. Consider equation (7) and assume that is decreasing in the first and increasing in the second variable on the set .(a)Assume that equation (7) has three equilibrium points , where is locally asymptotically stable, is a repeller, and is either a saddle point or a nonhyperbolic point of stable type. Furthermore, assume that equation (7) has two minimal period-two solutions such that and . If is a saddle point and is locally asymptotically stable, then every solution with the initial point between the stable manifolds and converges to while every solution which starts off converges to the periodic solution .(b)Assume that equation (7) has two equilibrium points , where is locally asymptotically stable and is a nonhyperbolic point of unstable type. Furthermore, assume that equation (7) has two minimal period-two solutions such that and . If is a saddle point and is locally asymptotically stable, then every solution which starts between the stable manifolds and converges to while every solution which starts above and below converges to the periodic solution . Finally, every solution starting between and converges to .(c)Assume that equation (7) has three equilibrium points , where and are locally asymptotically stable and is a repeller. Furthermore, assume that equation (7) has three minimal period-two solutions such that and and .If and are saddle points and is locally asymptotically stable, then every solution which starts between the stable manifolds and converges to and every solution which starts between the stable manifolds and converges to while every solution which starts off in the complement of the basins of attractions of , and converges to the periodic solution .

Proof. In view of Remark 1 and Theorem 5 from [19], the second iterate of the map associated with equation (7) is strictly competitive and does not have any period-two points.(a)By noticing that the period-two points of are the fixed points of , two period-two solutions become four fixed points of . An application of Theorem 1in part (a) to completes the proof.(b)In view of Remark 1, the second iterate of the map associated with equation (7) is strictly competitive and has six equilibrium points and . An application of Theorem 1 in part (b) to yields for every between the stable manifolds and . Furthermore, we derivewhere we used a continuity of the map . Consequently, . The proof of other cases is similar.(c)By noticing that the period-two points of are the fixed points of , three period-two solutions become six fixed points of . Applying Theorem 1 in part (c) to , we complete the proof.

3. Case Study: Equation (1)

3.1. Local Stability Analysis for Equilibria

An equilibrium solution of equation (1) must satisfyi.e., . Therefore, equation (1) has the following:(1)The unique equilibrium , if or ( and )(2)Two equilibrium points and , if and (3)Three equilibrium points and otherwise

Denote ; then equation (1) has the following linearized equation:where

If , then clearly . If , then by the equilibrium equation,

Proposition 1. Given that ,(1)the equilibrium is locally asymptotically stable for all values of the parameter.(2)If and , then the positive equilibrium point is nonhyperbolic of unstable type.(3)If and , then the equilibrium point is a repeller while the stability of is subject to the following conditions:(a) is locally asymptotically stable if (b) is nonhyperbolic of stable type if (c) is a saddle point if

Proof. (1)Since for implies that the unique eigenvalue , is locally asymptotically stable for all values of , and .(2)As and , then the characteristic equation is given bywhich solutions are and . The latter shows that is nonhyperbolic of unstable type.(3)The roots of the characteristic equation are and .For ,since , one can easily check that which implies .On the other hand, one can use the fact that to show that .For ,since , one can similarly show that .Moreover, a simple algebraic verification shows the following:(i);(ii);(iii).Consequently, we conclude that is a repeller whenever it exists while(i) is locally asymptotically stable, ;(ii) is nonhyperbolic of stable type, ;(iii) is a saddle point, .

3.2. Local Stability Analysis of Minimal Period-Two Solutions

Here, we present the results about the existence and stability of minimal period-two solutions of equation (1).

Theorem 3. Given that , we have the following:(1)If , then equation (1) has no minimal period-two solutions.(2)If , then equation (1) has a minimal period-two solution:.(3)If , then equation (1) has two minimal period-two solutions:(4)If , then equation (1) has three minimal period-two solutions:

Proof. Any period-two solution satisfies the following system of equations:It follows thatIf , the first equation becomes .If , then the second equation turns into .If and , then the system is equivalent toNow, given that , we get which impliesThus, for solution of the form or , we must have and for solutions of the form , , , we must have . Consequently,(1)if , then and . It follows that equation (1) has no minimal period-two solutions.(2)If , it follows that is the unique minimal period-two solution of equation (1).(3)If , equation (1) has two minimal period-two solutions:(4)If , then equation (1) has three minimal period-two solutions: together withNow, consider the following substitution, and , which transforms equation (1) into the following two-dimensional system,to which corresponds the following map:in which second iterate is given bywhereThe map is strongly competitive and its Jacobian matrix is given byThe following theorem describes the local stability of minimal period-two solutions of equation (1) whenever they exist.

Theorem 4. (1)The minimal period-two solutions are nonhyperbolic of stable type.(2)The minimal period-two solutions are saddle points while are locally asymptotically stable.(3)The minimal period-two solutions are saddle points.

Proof. (1)The minimal period solutions exist when ; thus, the Jacobian matrix of the second iterate of the map at and is the following:with the eigenvalues and ; therefore, are nonhyperbolic of stable type.(2)Now,(a)For , the Jacobian matrix is of the formwith eigenvalues and . Clearly, are saddle points.(b)As of , the corresponding Jacobian matrix is given byBoth with eigenvalues and . Clearly, are locally asymptotically stable.(3)Now, at the interior period-two solutions , the Jacobian matrices areObserve thatMoreover,Consequently, the interior period-two solutions are saddle points whenever they exist.

Remark 2. Observing that the interior period-two solutions exist if and only if(1), since .(2)There are 3 equilibrium points where is a saddle point.

3.3. Global Dynamics of Equation (1)

The global dynamics of equation (1) is quite complicated. Thus, we provide the following three diagrams that describe all possible bifurcations produced by different values of parameters and .

Theorem 5. If , then the equilibrium is globally asymptotically stable (see Figure 4).

proof. First, observe that every solution of equation (1) is bounded, as for ,Moreover, by Theorem 5 from [19], subsequences and are eventually monotonic. Now, as for , there are no minimal period-two solutions, and we conclude that both and must converge to the unique equilibrium .

Theorem 6. Let denote the basin of attraction of the set .(1)The -axis and the -axis are invariant by .(2)If , then(a);(b);(c).(3)If , then(a);(b);(c)