# Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

Authors: Toufik Khyat; M. R. S. Kulenović
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## 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 equationxn+1=fxn,xn−1,n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f,n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f>0. 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.


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## Body

## 1. Introduction and Preliminaries

Consider the following difference equation:(1)xn+1=xn−12cxn−12+dxn+f,n=0,1,…,where the initial conditions x−1,x0≥0 and the parameters satisfy that c,f≥0,d>0. Equation (1) is a special case of the equation(2)xn+1=Cxn−12+Dxn+Fcxn−12+dxn+f,n=0,1,…,where x−1,x0≥0 and the parameters satisfy C,D,F,c,d,f≥0, C+D+F>0, c+d+f>0,c+D>0, and C+d>0. 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 of(3)xn+1=Axn2+Bxnxn−1+Cxn−12+Dxn+Exn−1+Faxn2+bxnxn−1+cxn−12+dxn+exn−1+f,n=0,1,…,where all parameters are nonnegative numbers and the initial conditions x−1,x0≥0 so that the solution is well defined. Many special cases of equation (3) have been investigated in [1–6]. In particular, a special case where A=C=D=a=c=d=0 was studied in [1, 2], where a variety of techniques was used to obtain some global results. Other special cases of equation (3),(4)xn+1=xn−12axn2+bxnxn−1+cxn−12,n=0,1,…,(5)xn+1=xn−12axn2+cxn−12+f,n=0,1,…,(6)xn+1=xn−12bxnxn−1+cxn−12+f,n=0,1,…,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,(7)xn+1=fxn,xn−1,x−1,x0∈I,n=0,1,…,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 d=0,(8)xn+1=xn−12cxn−12+f,n=0,1,…,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. Section2 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 d and 4fc. Three regions for parameter d are 0,1/3,1/3,1, and 1,∞. In the critical region d∈0,1/3, the critical subregions for the bifurcation parameter 4fc are 0,d−12−4d2,d−12−4d2,d−12,d−12,1 and 1,∞. In the critical region d∈1/3,1, the critical subregions for the bifurcation parameter 4fc are 0,d−12,d−12,1 and 1,∞. In the critical region d∈1,∞, the critical subregions for the bifurcation parameter 4fc are 0,1 and 1,∞ (see Figures 1–3). 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 dxn (depends on population size at n-th generation only) while in the case of equation (6) is bxnxn−1 (depends on population sizes at n−1-th and n-th generations) and as expected the number of bifurcations for equation (1) is smaller than the number for equation (6).Figure 1
Bifurcation diagram of global dynamics of equation (1) in the parametric region d<1/3 for different values of the bifurcation parameter 4fc.Figure 2
Bifurcation diagram of global dynamics of equation (1) in the parametric region 1/3≤d<1 for different values of the bifurcation parameter 4fc.Figure 3
Bifurcation diagram of global dynamics of equation (1) in the parametric region 1≤d for different values of the bifurcation parameter 4fc.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 [11–13]. In this paper, we will use some theorems from [14–16] used in several papers such as [4–6, 8, 17, 18] that will be important in establishing the global dynamics results for equation (1).We will use Theorem5 from [19] which states that, for every solution xnn=−1∞ of equation (7), the subsequences x2nn=0∞ and x2n−1n=0∞ of even and odd terms are eventually monotonic, provided that a function f 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 f 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 I×I, see [15] and Remarks 2 and 3 in [3].
The fixed pointx¯,y¯ of a planar competitive or cooperative map T is said to be nonhyperbolic if the Jacobian matrix has at least one eigenvalue on the unit circle λ=1, that is, if λ=±1. If one eigenvalue is inside the unit circle λ<1, the fixed point is nonhyperbolic of stable type, and if the other eigenvalue is outside of the unit circle λ>1, 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 mapT generated by the system,(9)xn=1=fxn,yn,yn+1=gxn,yn,n=0,1,…,x−1,x0∈R,on a set ℛ with a nonempty interior.(a)
Assume thatT has seven fixed points E1,…,E7 such that five belongs to the west and south boundaries of the region ℛ and two fixed points are interior points. Moreover, assuming that E1 and E2 belong to the west boundary, E3 is the south-west corner of the region ℛ, and E4 and E5 are on the south boundary of ℛ such that E1⪯seE2⪯seE3⪯seE4⪯seE5. Moreover, assume that E6⪯neE7 and that E6∉E2,E4 and E7∈E1,E5. Finally, assume that E1,E3,E5 are locally asymptotically stable, E6 is a repeller, and E2,E4 are saddle points. If E7 is either a saddle point or a nonhyperbolic point of stable type and T has no period-two solutions, then all solutions which start between the stable manifolds WsE2 and WsE4 converge to E3 and all solutions which start between the stable manifolds WsE2 and WsE7 converge to E1 and all solutions which start between the stable manifolds WsE4 and WsE7 converge to E5.(b)
Assume thatT has exactly six fixed points E1,…,E6, where the points have the same configuration and points E1,…,E5 have the same local character as in part (a), while E6 is a nonhyperbolic point of unstable type. If T has no period-two solutions, then there exist two increasing continuous curves C1 and C2, C2⪯seC1 emanating from E6 such that all solutions which start between C1 and C2 converge to E6. Furthermore, all solutions which start between the stable manifolds WsE2 and WsE4 converge to E1, and all solutions which start above WsE2∪C2 converge to E1 and all solutions which start below WsE4∪C1 converge to E5.(c)
Assume thatT has exactly nine fixed points E1,…,E9, where the points E1,…,E7 have the same configuration and the same local character as in (a). Assuming that the fixed points E8,E9 are saddle points such that E8∈E1,E7 and E9∈E7,E5. Assume that T has no period-two solutions, then all solutions which start between the stable manifolds WsE2 and WsE4 converge to E3, all solutions which start above WsE2∪WsE8 converges to E1, and all solutions which start below WsE4∪WsE9 converge to E5. Finally, all solutions which start between the stable manifolds WsE8 and WsE9 converge to E7.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 WsE2,WsE4 and WsE7 have an endpoint at E6, and WsE7 has another endpoint at ∞,∞.In view of Theorem 4 in [15], every solution which starts between the stable manifolds WsE2 and WsE4 eventually enters intE2,E4. If the initial point x0,y0 is in intE2,E4, one can find the points xl,yl on the y-axis and xu,yu on the x-axis such that xl,yl⪯sex0,y0⪯sexu,yu. This will imply that Tnxl,yl⪯seTnx0,y0⪯seTnxu,yu, and so Tnx0,y0 converges to E3 as both Tnxl,yl,Tnxu,yu do.If the initial pointx0,y0 is above WsE2∪WsE7, one can find the point xl,yl on the y-axis and a point xu,yu∈WsE2∪WsE7 such that xl,yl⪯sex0,y0⪯sexu,yu. This will imply that Tnxl,yl⪯seTnx0,y0⪯seTnxu,yu, and so Tnx0,y0∈intE1,E7 eventually. Now, in view of Corollary 1 in [15], Tnx0,y0⟶E1 as n⟶∞.In a similar way the case when the initial pointx0,y0 is below WsE4∪WsE7 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 WsE2 and WsE4 have an endpoint at E6. The existence of curves C1 and C2 follows from Theorem 2 in [16]. The proof that the region between the stable manifolds WsE2 and WsE4 eventually enters intE2,E4 and so it converges to E3 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 pointx0,y0 is above WsE2∪C2 it will eventually enter intE1,E6 and so it will converge to E1. In a similar way, we can show that if the initial point x0,y0 is below WsE4∪C1, it will eventually enter intE6,E5 and so it will converge to E5.Finally, if the initial pointx0,y0 is between C1 and C2, then one can find the point xl,yl∈C2 and a point xu,yu∈C1 such that xl,yl⪯sex0,y0⪯sexu,yu. This will imply that Tnxl,yl⪯seTnx0,y0⪯seTnxu,yu, and so Tnx0,y0⟶E6 as Tnxu,yu⟶E6,Tnxl,yl⟶E6.(c)
The proof that the region between the stable manifoldsWsE2 and WsE4 is the basin of attraction of E3 is same as in part (a) and will be omitted. The proof that all solutions which start above WsE2∪WsE8 converges to E1 and all solutions which start below WsE4∪WsE9 converge to E5 is same as in part (a) and so will be omitted.If the initial pointx0,y0 is between WsE8 and WsE9, then one can find the point xl,yl∈WsE8 and a point xu,yu∈WsE9 such that xl,yl⪯sex0,y0⪯sexu,yu. This will imply that Tnxl,yl⪯seTnx0,y0⪯seTnxu,yu, and so Tnx0,y0∈intE8,E9 as Tnxu,yu⟶E9,Tnxl,yl⟶E8 as n⟶∞. Now, in view of Corollary 2 in [14], Tnx0,y0⟶E7 as n⟶∞.Theorem 2.
Consider equation (7) and assume that f is decreasing in the first and increasing in the second variable on the set a,b2.(a)
Assume that equation (7) has three equilibrium points E0⪯neE−⪯neE+, where E0 is locally asymptotically stable, E− is a repeller, and E+ is either a saddle point or a nonhyperbolic point of stable type. Furthermore, assume that equation (7) has two minimal period-two solutions P1,Q1,P2,Q2 such that P2⪯seP1⪯seE0⪯seQ1⪯seQ2 and E−,E+∈P2,Q2∖P1,Q1. If P1,Q1 is a saddle point and P2,Q2 is locally asymptotically stable, then every solution with the initial point between the stable manifolds WsP1 and WsQ1 converges to E0 while every solution which starts off WsP1∪WsQ1∪WsE+ converges to the periodic solution P2,Q2.(b)
Assume that equation (7) has two equilibrium points E0⪯neE, where E0 is locally asymptotically stable and E+ is a nonhyperbolic point of unstable type. Furthermore, assume that equation (7) has two minimal period-two solutions P1,Q1,P2,Q2 such that P2⪯seP1⪯seE0⪯seQ1⪯seQ2 and E−,E+∈P2,Q2∖P1,Q1. If P1,Q1 is a saddle point and P2,Q2 is locally asymptotically stable, then every solution which starts between the stable manifolds WsP1 and WsQ1 converges to E0 while every solution which starts above WsP1∪C2 and below WsQ1∪C1 converges to the periodic solution P2,Q2. Finally, every solution starting between C1 and C2 converges to E.(c)
Assume that equation (7) has three equilibrium points E0⪯neE−⪯neE+, where E0 and E+ are locally asymptotically stable and E− is a repeller. Furthermore, assume that equation (7) has three minimal period-two solutions P1,Q1,P2,Q2,P3,Q3 such that P2⪯seP1⪯seE0⪯seQ1⪯seQ2 and E−,E+∈P2,Q2∖P1,Q1 and P2⪯seP3⪯seE+⪯seQ3⪯seQ2.IfP1,Q1and P3,Q3 are saddle points and P2,Q2 is locally asymptotically stable, then every solution which starts between the stable manifolds WsP1 and WsQ1 converges to E0 and every solution which starts between the stable manifolds WsP3 and WsQ3 converges to E7 while every solution which starts off in the complement of the basins of attractions of E0,E7, P1,Q1 and P3,Q3 converges to the periodic solution P2,Q2.Proof.
In view of Remark1 and Theorem 5 from [19], the second iterate T2 of the map T associated with equation (7) is strictly competitive and does not have any period-two points.(a)
By noticing that the period-two points ofT are the fixed points of T2, two period-two solutions P1,Q1,P2,Q2 become four fixed points of T2. An application of Theorem 1in part (a) to T2 completes the proof.(b)
In view of Remark1, the second iterate T2 of the map T associated with equation (7) is strictly competitive and has six equilibrium points E1=P2,E2=P1,E3=E0,E4=Q,E5=Q2 and E6=E. An application of Theorem 1 in part (b) to T2 yields limn⟶∞T2nx0,y0=E0 for every x0,y0 between the stable manifolds WsP1 and WsQ1. Furthermore, we derive(10)limn⟶∞T2n+1x0,y0=limn⟶∞TT2nx0,y0=Tlimn⟶∞T2nx0,y0=TE2=E2,where we used a continuity of the mapT. Consequently, limn⟶∞Tnx0,y0=E0. The proof of other cases is similar.(c)
By noticing that the period-two points ofT are the fixed points of T2, three period-two solutions P1,Q1,P2,Q2,P3,Q3 become six fixed points of T2. Applying Theorem 1 in part (c) to T2, we complete the proof.


## 3. Case Study: Equation (1)

### 3.1. Local Stability Analysis for Equilibria

An equilibrium solution of equation (1) must satisfy(11)x¯=x¯2cx¯2+dx¯+f,i.e., x¯=0orcx¯2+d−1x¯+f=0. Therefore, equation (1) has the following:(1)
The unique equilibriumE0=0,0, if d≥1 or (d<1 and d−12−4fc<0)(2)
Two equilibrium pointsE0 and E∗=1−d/2c,1−d/2c, if d<1 and d−12−4fc=0(3)
Three equilibrium pointsE0 and E±=1−d±1−d2−4fc/2c,1−d±1−d2−4fc/2cotherwiseDenoteFu,v=v2/cv2+du+f; then equation (1) has the following linearized equation:(12)zn+1=pzn+qzn−1,where(13)p=∂F∂ux¯,x¯=dx¯2cx¯2+dx¯+f2,q=∂F∂vx¯,x¯=2x¯dx¯+fcx¯2+dx¯+f2.Ifx¯=0, then clearly p=q=0. If x¯≠0, then by the equilibrium equation,(14)p=−d,q=2d+fx¯=21−cx¯.Proposition 1.
Given thatc,d,f>0,(1)
the equilibriumE0 is locally asymptotically stable for all values of the parameter.(2)
Ifd<1 and d−12−4fc=0, then the positive equilibrium point E∗ is nonhyperbolic of unstable type.(3)
Ifd<1 and d−12−4fc>0, then the equilibrium point E− is a repeller while the stability of E+ is subject to the following conditions:(a)
E+ is locally asymptotically stable if d−12−4fc>4d2(b)
E+ is nonhyperbolic of stable type if d−12−4fc=4d2(c)
E+ is a saddle point if d−12−4fc<4d2Proof.
(1)
Sincep=q=0 for x¯=0 implies that the unique eigenvalue λ=0, E0 is locally asymptotically stable for all values of c,d, and f.(2)
Asp=−d,q=21−cx¯ and x¯∗=1−d/2c, then the characteristic equation is given by(15)λ2+dλ−d+1=0,which solutions areλ+=1 and λ−=−d+1. The latter shows that E∗ is nonhyperbolic of unstable type.(3)
The roots of the characteristic equation areλ+=−d+d2+81−cx¯/2>0 and λ−=−d−d2+81−cx¯/2<0.ForE−,sincex¯−<1−d/2c, one can easily check that d2+81−cx¯−>2+d which implies λ+>1.On the other hand, one can use the fact thatx¯−<1+d/2c to show that λ−<−1.ForE+,sincex¯+>1−d/2c, one can similarly show that λ+<1.Moreover, a simple algebraic verification shows the following:(i)
λ−<1for4d2<1−d2−4fc;(ii)
λ−=1for4d2=1−d2−4fc;(iii)
λ−>1for4d2>1−d2−4fc.Consequently, we conclude thatE− is a repeller whenever it exists while(i)
E+ is locally asymptotically stable, for4d2<1−d2−4fc;(ii)
E+ is nonhyperbolic of stable type, for4d2=1−d2−4fc;(iii)
E+ is a saddle point, for4d2>1−d2−4fc.


### 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 thatc,d,f>0, we have the following:(1)
If4fc>1, then equation (1) has no minimal period-two solutions.(2)
If4fc=1, then equation (1) has a minimal period-two solution:Px1/2c,0,Py0,1/2c.(3)
Ifd−12−4d2≤4fc<1, then equation (1) has two minimal period-two solutions:(16)Px11−1−4cf2c,0,Py10,1−1−4cf2c,Px21+1−4cf2c,0,Py20,1+1−4cf2c.(4)
If4fc<d−12−4d2, then equation (1) has three minimal period-two solutions:(17)Px1,Py1,Px2,Py2,P∓i1+d−1−d2−4d2−4fc2c,1+d+1−d2−4d2−4fc2c,P±i1+d+1−d2−4d2−4fc2c,1+d−1−d2−4d2−4fc2c.Proof.
Any period-two solutionϕ,ψ satisfies the following system of equations:(18)ϕ=ϕ2cϕ2+dψ+f,ψ=ψ2cψ2+dϕ+f.
It follows thatIfψ=0, the first equation becomes cϕ2−ϕ+f=0.Ifϕ=0, then the second equation turns into cψ2−ψ+f=0.Ifϕ≠0 and ψ≠0, then the system is equivalent to(19)cϕ2+dψ−ϕ+f=0,cψ2+dϕ−ψ+f=0,⟹cϕ2−ψ2+dψ−ϕ+ψ−ϕ=0.
Now, given thatϕ≠ψ, we get ϕ=1+d/c−ψ which implies(20)cψ2−1+dψ+1+dc+f=0.
Thus, for solution of the form0,ψ or ϕ,0, we must have 1−4fc≥0 and for solutions of the form ϕ,ψ, ϕ≠0, ψ≠0, we must have 1−d2−4d2−4fc>0. Consequently,(1)
if4fc>1, then 1−4fc<0 and 1−d2−4d2−4fc<0. It follows that equation (1) has no minimal period-two solutions.(2)
If4fc=1, it follows that Px,Py is the unique minimal period-two solution of equation (1).(3)
If1−d2−4d2≤4fc<1, equation (1) has two minimal period-two solutions:(21)Px11−1−4cf2c,0,Py10,1−1−4cf2c,Px21+1−4cf2c,0,Py20,1+1−4cf2c.(4)
If4fc<1−d2−4d2, then equation (1) has three minimal period-two solutions:
Px1,Py1,Px2,Py2 together with(22)P∓i=1+d−1−d2−4d2−4fc2c,1+d+1−d2−4d2−4fc2c,P±i=1+d+1−d2−4d2−4fc2c,1+d−1−d2−4d2−4fc2c.
Now, consider the following substitution,un=xn−1 and vn=xn, which transforms equation (1) into the following two-dimensional system,(23)un+1=vn,vn+1=un2cun2+dvn+f,to which corresponds the following map:(24)Tuv=vhu,v=vu2cu2+dv+f,in which second iterate T2 is given by(25)T2uv=Tvhu,v=hu,vhv,hu,v=Gu,vHu,v,where(26)Hu,v=v2cv2+dhu,v+f.
The mapT2 is strongly competitive and its Jacobian matrix is given by(27)JT2u,v=2uf+dvcu2+f+dv2−du2cu2+f+dv2−2duv2f+dvdu2+cu2+f+dvcv2+f2vd2cu2+2f+3dvu2/cu2+f+dv2+2fdu2/cu2+f+dv+cv2+f2.
The following theorem describes the local stability of minimal period-two solutions of equation (1) whenever they exist.Theorem 4.
(1)
The minimal period-two solutionsPx,Py are nonhyperbolic of stable type.(2)
The minimal period-two solutionsPx1,Py1 are saddle points while Px2,Py2 are locally asymptotically stable.(3)
The minimal period-two solutionsP∓i,P±i are saddle points.Proof.
(1)
The minimal period solutionsPx,Py exist when 4fc=1; thus, the Jacobian matrix of the second iterate of the map T at Px and Py is the following:(28)JT2Px=1−d00,JT2Py=0001,with the eigenvaluesλ1=0 and λ2=1; therefore, Px,Py are nonhyperbolic of stable type.(2)
Now,(a)
ForPx1,Py1, the Jacobian matrix is of the form(29)JT2Px1=1+1−4cf−d00,JT2Py1=0001+1−4cf.with eigenvaluesλ1=0 and λ2=1+1−4cf. Clearly, Px1,Py1 are saddle points.(b)
As ofPx2,Py2, the corresponding Jacobian matrix is given by(30)JT2Px2=1−1−4cf−d00,JT2Py2=0001−1−4cf.Both with eigenvaluesλ1=0 and λ2=1−1−4cf<1. Clearly, Px2,Py2 are locally asymptotically stable.(3)
Now, at the interior period-two solutionsP∓i,P±i, the Jacobian matrices are(31)JT2P∓i=−dc+−c2d3d+2+4cf−1c+1−d−d−dc+c+−c2d3d+2+4cf−1cd−1dc−−c2d3d+2+4cf−1c+1,JT2P±i=−dc−−c2d3d+2+4cf−1c+1−ddcd−1+−c2d3d+2+4cf−1cd2−dc+−c2d3d+2+4cf−1c+1.
Observe that(32)p=TrJT2P∓i=TrJT2P±i=2+dd−2,q=DetJT2P∓i=DetTrJT2P±i=4d2+cf.
Moreover,(33)p>1+q⇔d−12−4d2>4cfcondition of existence.
Consequently, the interior period-two solutionsP∓i,P±i are saddle points whenever they exist.Remark 2.
Observing that the interior period-two solutionsP∓i,P±i exist if and only if(1)
d<1/3, since d−12−4d2>4cf⟹d−12−4d2>0⟹1−3d1+d>0.(2)
There are 3 equilibrium pointsE0,E−,E+ where E+ 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 d and 4fc.Theorem 5.
If4fc>1, then the equilibrium E0 is globally asymptotically stable (see Figure 4).Figure 4
Visualization of Theorem5 for 4fc>1.proof.
First, observe that every solution of equation (1) is bounded, as for xn−1≠0,(34)xn+1=xn−12cxn−12+dxn+f=1c+dxn/xn−12+f/xn−12<1c,
Moreover, by Theorem5 from [19], subsequences x2nn=0∞ and x2n+1n=0∞ are eventually monotonic. Now, as for 4fc>1, there are no minimal period-two solutions, and we conclude that both x2n and x2n+1 must converge to the unique equilibrium x¯=0.Theorem 6.
LetℬA denote the basin of attraction of the set A.(1)
Thex-axis 0,+∞×0 and the y-axis 0×0,+∞ are invariant by T2.(2)
If4fc=1, then(a)
1/2c,+∞×0⊂ℬPx;(b)
0×1/2c,+∞⊂ℬPy;(c)
0×0,1/2c∪0,1/2c×0⊂ℬE0.(3)
If4fc<1, then(a)
1−1−4cf/2c,∞×0⊂ℬPx2;(b)
0×1−1−4cf/2c,∞⊂ℬPy2;(c)
0,1−1−4cf/2c×0∪0×0,1−1−4cf/2c⊂ℬE0.Proof.
(1)
Letα≥0, then T2α,0=α2/cα2+f,0 and T20,α=0,α2/cα2+f which implies that the sets 0,+∞×0 and 0×0,+∞ are invariant by T2.(2)
First, recalling that every solution of equation (1) is bounded and by Theorem 5 from [19], every solution must either converge to an equilibrium or a minimal period-two solution. It follows that every solution generated by T2 must converge to an equilibrium. Now, consider snn=1∞ the solution with initial point s1=x1,0∈1/2c,+∞×0, then(35)T2xn,0=xn+1,0=xn2cxn2+f,0,xn+1−xn=−xncxn2−xn+fcxn2+f.(a)
If4fc=1, one can easily show that xn is monotone decreasing; thus,(i)
ifxn>1/2c, then sn=xn,0⟶1/2c,0.(ii)
Ifxn<1/2c, then sn=xn,0⟶0,0(b)
If4fc<1, one can easily show that xn is monotone decreasing in 0,1−1−4fc/2c∪1+1−4fc/2c,+∞ and monotone increasing in 1−1−4fc/2c,1+1−4fc/2c; thus,(i)
Ifxn∈1−1−4fc/2c,+∞, then sn=xn,0⟶1+1−4fc/2c,0.(ii)
Ifxn∈0,1−1−4fc/2c, then sn=xn,0⟶0,0.
The remaining part of the proof follows similarly by considering solutions of the formsn=0,xn.Theorem 7.
If4fc=1, then equation (1) has the unique equilibrium point E0 which is locally asymptotically stable and the unique minimal period-two solution Px,Py=1/2c,0,0,1/2c which is nonhyperbolic of stable type.
There exist two invariant curvesC1 and C2 which are graphs of strictly increasing continuous functions of the first coordinate on an interval with endpoints in Px and Py, respectively. Basins of attraction of the minimal period-two solutions are(36)ℬPx=C1∪W+C1,ℬPy=C2∪W−C2,while the basin of attraction of the equilibrium point E0 is the region between curves C1 and C2, i.e.,(37)ℬE0=W−C1∩W+C2;see Figure 5.Figure 5
Visualization of global dynamics of Theorem7 for 4fc=1.Proof.
The Jacobian matrix of the second iterate of the map atPx is given by(38)JT2Px=1−d00,with two eigenvalues λ1=0 associated with the eigenvector d1 and λ2=1 which corresponds to the eigenvector 10.
Observe that the eigenvector associated withλ1 is not parallel to the x-axis and the map T2 is strongly competitive. It follows by Theorems 1–5 in [15] and Theorem 2 that there exists an invariant curve C1 through the point Px which is a subset of WsPx. Moreover, C1 is the graph of a strictly increasing continuous that separates the first quadrant into two connected subregions: an upper one W−C1 and a lower one W+C1, where(39)ℬPx=C1∪W+C1.
The Jacobian matrix ofT2 at Py is JT2Py=0001 and it has two eigenvectors that are parallel to the coordinate axis, 10,01, corresponding to λ1=0 and λ2=1, respectively. By Hartman–Grobman theorem [20], we know that there exists a C1 curve C through Py that is tangential at Py to the eigenspace associated with λ=0 such that T2C⊂C.Claim 1.
The stable manifold atPy is a linearly strongly ordered curve in the northeast ordering, given as WlocsPy=t,ϕt:0≤t≤δ for small enough δ>0.Proof of Claim 1.
First, recall that0,1/2c×0,1/2c⊂ℬE0. Now, let u0>0. Since T2 is strongly competitive, we have T2Py≪seT2u0,1/2c and that implies T2u0,1/2c∈intQ4Py. Therefore, there exists a ball BεT2u0,1/2c such that BεT2u0,1/2c⊂intQ4Py. Since the map T2 is continuous on ℝ+2, there exists a ball Bδ1u0,1/2c such that(40)T2Bδ1u0,1/2c⊂BεT2u0,1/2c⊂intQ4Py,which implies T2nu,v⟶0,0 when n⟶∞ for all points u,v∈Bδ1u0,1/2c. It follows that WlocsPy∩intQ4Py=∅. Now, observe that ϕ′0=0 as its graph must be tangential to the horizontal eigenspace. Moreover, ϕ″≥0 in a small neighborhood of t=0; otherwise,ϕ″≤0⟹ there exists δ>0 such that ϕt is decreasing in 0,δ⟹ϕ′t≤0 in 0,δ⟹ϕt≤1/2c in 0,δ which contradicts the fact that WlocsPy∩intQ4Py=∅.
Therefore, for sufficiently smallδ1, WlocsPy=t,ϕt:0≤t≤δ1 is linearly ordered in the northeast ordering and as T2 is competitive, WlocsPy∩ℝ+2 can be extended to an unbounded curve (global stable manifold) C2, see [14, 15].
Hence, the curveC2 splits the region into two connected components, an upper subregion W−C2 and a lower subregion W+C2.
Clearly,ℬPy=C2∪W−C2, and finally the basin of attraction of the zero equilibrium E0 is(41)ℬE0=W+C2∩W−C1.Theorem 8.
Ifd≥1and4fc<1 or d<1andd−12<4fc<1, then equation (1) has the unique equilibrium point E0 which is locally asymptotically stable and two minimal period-two solutions Px1,Py1 which is a saddle point and Px2,Py2 which is locally asymptotically stable.
There exist global stable manifoldsWsPx1 and WsPy1 which are basins of attractions of Px1,Py1 and the global unstable manifolds have the following form:(42)WuPx1=x,0:x∈A,WuPy1=0,y:y∈A,where A=0,1+1−4cf/2c∖1−1−4cf/2c.
The basinℬE0=0,0 is the region between the global stable sets:(43)ℬE0=W−Px1∩W+Py1.
The basin of the minimal period-two solutionsPx2,Py2 is(44)ℬPx2=W+Px1,ℬPy2=W−Py1;see Figure 6.Figure 6
Different manifolds of Theorem8.Proof.
Recall that(45)JT2Px1=1+1−4cf−d00,with eigenvalues λ1=0 with eigenvector d/1−4fc+11 and λ2=1−4fc+1 associated with the eigenvector 10. Thus, there exists a local stable manifold at Px1 that is linearly strongly ordered in the northeast ordering with Px1 as an endpoint. As T2 is competitive, the local stable manifold can be extended to a curve WsPx1 which separates the region into two connected components W+Px1 and W−Px1. On the other hand,(46)JT2Py1=0001+1−4cf,with eigenvalues λ1=0 with eigenvector 10 and λ2=1−4fc+1 associated with the eigenvector 01.
By Theorem6, we know that 0,1−1−4fc/2c×0,1−1−4fc/2c⊂ℬE0; thus, we know that the local stable manifold at Py1 is tangential to the horizontal eigenspace but cannot enter the box 0,1−1−4fc/2c×0,1−1−4fc/2c. We conclude that the local stable manifold at Py1 is a linearly strongly ordered curve (in the northeast ordering) with Py1 as an endpoint. Similarly, we conclude its extension to a global stable manifold WsPy1 which separates the region into two connected components W+Py1 and W−Py1.
Finally, by the uniqueness of the stable manifold of the saddle pointPx1, we know that no solution in W+Px1 will converge to Px1. Furthermore, all solutions are bounded and we know that by monotonicity of the map T every solution must converge to an equilibrium. It follows that ℬPx2=W+Px1 and analogously ℬPy2=W+Py1.Theorem 9.
Ifd<1 and d−12=4fc<1, then equation (1) has two equilibrium solutions:(i)
E0 is locally asymptotically stable(ii)
E∗=1−d/2c,1−d/2c is a nonhyperbolic of unstable type, and two minimal period-two solutions:(iii)
Px1,Py1=1−1−4fc/2c,0,0,1−1−4fc/2c are saddle points,(iv)
Px2,Py2=1+1−4fc/2c,0,0,1+1−4fc/2c are locally asymptotically stable.
Then,
(i)
there exist global stable manifoldsWsPx1 and WsPy1 which are the basins of attraction of the periodic solutions Px1,Py1 and which are tangential at the equilibrium point E∗.(ii)
There exists a global stable manifoldWsE∗ contained in Q1E∗ which is the basin of attraction of the equilibrium E∗;
The basin of attraction of the equilibrium pointE0 is given by(47)ℬE0=W−Px1∩W+Py1.
The basins of attraction ofPx2 and Py2 are given by(48)ℬPx2=W+Sx,whereSx=WsPx1∪WsE∗,ℬPy2=W−Sy,whereSy=WsPy1∪WsE∗;see Figure 7.Figure 7
Global manifolds and basins of Theorem9.Proof.
The existence and orientation of the global stable manifold atPx1 can be determined as in Theorem 8; however, in general, this information can be determined by studying the curvature of the local curves given by
WlocsPy1=t,ϕ1t:0≤t≤δ1 and WlocsPx1=ϕ2t,t:0≤t≤δ2 for δ1 and δ2 small enough, where if fx,y and gx,y are the coordinate functions of T2, then(49)ϕ1ft,ϕ1t=gt,ϕ1t,ϕ10=1−1−4fc2c,ϕ1′0=0,ϕ2gϕ2t,t=fϕ2t,t,ϕ20=1−1−4fc2c,ϕ2′0=0.
This is useful when the local curve has a tangent parallel to the axis at the fixed point which is the case here forpy1.
By differentiating both sides of the equation above, we get(50)ϕ1″0=gxx0,ϕ10fx0,ϕ102−gy0,ϕ10=d1−1−4fcfd+2cf−d1−4fc>0,which confirms the argument used in Theorem 8. We conclude the existence of curves C1 and C2 (global stable manifolds) which are graphs of continuous, strictly increasing functions. Furthermore, the curves cannot intersect the interior of the sets Q2E∗∩ℝ2 and Q4E∗∩ℝ2, as the monotonicity of T2 forces the latter sets to be invariant. Thus, T−2nP⟶E∗ for all P∈Cl,l=1,2, therefore C1 and C2 are also center manifolds of E∗.
On the other hand, by lettingT2x,y=fx,y,gx,y, the center manifold ϕx must satisfy [21](51)ϕfx,ϕx=gx,ϕx.
By using a Taylor expansion substitution, the center manifold can be approximated by(52)ϕx=x−cx−1−d/2c2d+2−2c2x−1−d/2c3d+23+Ox−1−d2c4.
The dynamic on the center manifoldϕx is given by the reduced difference equation un+1=fun,ϕun which has the following asymptotic representation:(53)un+1=1−d2c+un−1−d2c−2cun−1−d/2c2d+2+2c2dun−1−d/2c3d+23−⋯+Oun−1−d2c5.
Sinceu¯=1−d/2c is a semistable fixed point for the latter scalar difference equation, it follows that E∗ is a semistable fixed point for T2; furthermore, the coefficient of the lowest nonlinear term in the reduced map is negative; thus, by [22], the local basin of attraction of the equilibrium E∗ is a one-dimensional curve. We conclude that there is a unique center manifold curve U which satisfies T2U⊂U. Moreover, U is tangential to the eigenspace associated with λ=1, namely, Span11. It follows that U is contained in Q1E∗ and is linearly ordered in the northeast ordering and therefore can be extended to an unbounded curve C.
Now, for every pointq∈W−Px1∩W+Py1, there exist qx∈WsPx1 and qy∈WsPy1 such that
qy⪯seq⪯seqx which implies that T2nqy⪯seT2nq⪯seT2nqx, but we know that(54)T2nqy⟶Py1,T2nqx⟶Px1.
Consequently, there existsN such that(55)Py1⪯seT2Nq⪯sePx1.
It follows by Theorem6 that q∈ℬE0. As of the basins of attractions of Px2 and Py2, the proof is analogous to the one given in Theorem 8.Theorem 10.
Ifd<1,d−12−4d2<4fc<d−12, then equation (1) has three equilibrium points E0,E−,E+, which are, respectively, locally asymptotically stable, a repeller and a saddle point and two minimal period-two solutions:(i)
Px1,Py1 are saddle points(ii)
Px2,Py2 are locally asymptotically stable
Then,
(i)
There exist global stable manifoldsWsPx1 (the basin of attraction of Px1) and WsPy1 (the basin of attraction of Px2) which are tangential at the equilibrium point E−.(ii)
There exists a global stable manifoldWsE+ which is an unbounded curve that is a graph of an increasing function contained in Q1E+∪Q3E+ with an endpoint at E− and is the basin of attraction of the equilibrium E+.(iii)
There exist a global unstable manifoldWuE+ which is a graph of a decreasing function contained in Q2E+∪Q4E+ with endpoints Px2 and Py2.(iv)
The basin of attraction ofE0 is(56)ℬE0=W−Px1∩W+Py1.(v)
The basins of attraction ofPx2 and Py2 are given by(57)ℬPx2=W+Sx,whereSx=WsPx1∪WsE+,ℬPy2=W−Sy,whereSy=WsPy1∪WsE+;see Figure 8.Figure 8
All invariant manifolds of Theorem10.Proof.
He the existence ofWsPx1 and WsPy1 as well as the basin of attraction of E0 follows from Theorem 9. The existence of the stable WsE+ and the unstable manifold WuE+ follows from Theorems 1–5 in [15] and Theorem 2. The basins of attraction of Px2 and Py2 were discussed in Theorem 8.Theorem 11.
Ifd<1 and 0<d−12−4d2=4fc, then equation (1) has three equilibrium points E0,E−,E+, where(i)
E0 is locally asymptotically stable(ii)
E−=1−3d/2c,1−3d/2c is a repeller(iii)
E+=1+d/2c,1+d/2c is a nonhyperbolic point of stable type, and two minimal period-two solutions:(iv)
Px1,Py1 is a saddle point(v)
Px2,Py2 is locally asymptotically stable
Then,
(i)
There exist global stable manifoldsWsPx1 and WsPy1 which are the basins of attraction of the periodic solutions Px1,Py1 and which are tangential at the equilibrium point E−.(ii)
There exists a global stable manifoldWsE+ which is an unbounded curve and the graph of an increasing function contained in Q1E+∪Q3E+ with an endpoint at E− and WsE+ is the basin of attraction of the equilibrium E+.(iii)
The basin of attraction of equilibrium pointE0 is the region between those stable manifolds, i.e.,(58)ℬE0=W−Px1∩W+Py1.(iv)
The basins of attraction ofPx2 and Py2 are given by(59)ℬPx2=W+Sx,whereSx=WsPx1∪WsE+,ℬPy2=W−Sy,whereSy=WsPy1∪WsE+;see Figure 9.Figure 9
Visual representation of global dynamics of Theorem11.Proof.
The existence and orientation of the global stable manifoldWsE+ follows from Theorems 1–5 in [15] and Theorem 2. The remaining of the proof is analogous to the discussions in Theorems 8 and 9.Theorem 12.
Ifd<1 and 4fc<d−12−4fc, then equation (1) has three equilibrium points E0,E−,E+ which are, respectively, locally asymptotically stable, a repeller and locally asymptotically stable, and has three minimal period-two solutions such that(i)
Px1,Py1 are saddle points,(ii)
Px2,Py2 are locally asymptotically stable,(iii)
P∓i,P±i are saddle points.(iv)
There exist global stable manifoldsWsPx1 and WsPy1 which are the basins of attraction of the periodic solutions Px1,Py1 and which are tangential at the equilibrium point E−.(v)
The basin of attraction of equilibrium pointE0 is the region between those stable manifolds, i.e.,(60)ℬE0=W−Px1∩W+Py1.(vi)
There exists a global stable manifoldWsP∓i which is an unbounded curve and the graph of an increasing function contained in Q1P∓i∪Q3P∓i with an endpoint at E−. WsP∓i is the basin of attraction of the equilibrium P∓i.(vii)
There exists a global stable manifoldWsP±i which is an unbounded curve and the graph of an increasing function contained in Q1P±i∪Q3P±i with an endpoint at E−. WsP±i is the basin of attraction of the equilibrium P±i.(viii)
The basin of attraction of the equilibrium pointE+ is the region between those stable manifolds, i.e.,(61)ℬE+=W−P±i∩W+P∓i.(ix)
There exist a global unstable manifoldWuP∓i which is the graph of a decreasing function contained in Q2P∓i∪Q4P∓i with endpoints Py2 and E+.(x)
There exist a global unstable manifoldWuP±i which is the graph of a decreasing function contained in Q2P±i∪Q4P±i with endpoints Px2 and E+.(xi)
The basins of attraction ofPx2 and Py2 are given by(62)ℬPx2=W+Sx,whereSx=WsPx1∪WsP±i,ℬPy2=W−Sy,whereSy=WsPy1∪WsP±i;see Figure 10.Figure 10
The visual illustration of Theorem12.proof.
The existence and orientation of the stable manifoldWsP∓i follows from Theorems 1–5 in [15] and Theorem 2. Moreover, WsP∓i cannot intersect another manifold or the boundary of the region at any point as the latter sets are invariant. Thus, it must have an endpoint at E−. Similarly, the existence and orientation of the unstable manifold WuP∓i are given by Theorem 5 in [15]. On the other hand, WuP∓i∩P∓i,E+≠∅ and P∓i,E+ is invariant. Thus, WuP∓i cannot leave the latter set and must end at E+. Analogous arguments and conclusions also hold for WsP±i and WuP±i. In addition, we know that for p∓∈WsP∓i and p±∈WsP±i,(63)T2np∓⟶P∓i,T2np±⟶P±i.
Furthermore, for allp∈W−P±i∩W+P∓i, there exist p∓∈WsP∓i and. p±∈WsP±i such that(64)p∓⪯sep⪯sep±⟹T2np∓⪯seT2np⪯seT2np±,for alln≥0.
It follows that there existsN>0 such that T2Np=q∈P∓i,P±i. Thus, there exist q∓∈WuP∓i and q±∈WuP±i such that(65)q∓⪯seq⪯seq±⟹T2nq∓⪯seT2nq⪯seT2nq±,for alln≥0,where(66)T2nq∓⟶E+,T2nq±⟶E+,which implies that T2nq⟶E+⟹T2np⟶E+. We conclude that(67)ℬE+=W−P±i∩W+P∓i.
As ofWsPx1, WsPy1, ℬE0, ℬPx2 and ℬPy2, the proof is analogous to the discussion in Theorems 8 and 9.


## 3.1. Local Stability Analysis for Equilibria

An equilibrium solution of equation (1) must satisfy(11)x¯=x¯2cx¯2+dx¯+f,i.e., x¯=0orcx¯2+d−1x¯+f=0. Therefore, equation (1) has the following:(1)
The unique equilibriumE0=0,0, if d≥1 or (d<1 and d−12−4fc<0)(2)
Two equilibrium pointsE0 and E∗=1−d/2c,1−d/2c, if d<1 and d−12−4fc=0(3)
Three equilibrium pointsE0 and E±=1−d±1−d2−4fc/2c,1−d±1−d2−4fc/2cotherwiseDenoteFu,v=v2/cv2+du+f; then equation (1) has the following linearized equation:(12)zn+1=pzn+qzn−1,where(13)p=∂F∂ux¯,x¯=dx¯2cx¯2+dx¯+f2,q=∂F∂vx¯,x¯=2x¯dx¯+fcx¯2+dx¯+f2.Ifx¯=0, then clearly p=q=0. If x¯≠0, then by the equilibrium equation,(14)p=−d,q=2d+fx¯=21−cx¯.Proposition 1.
Given thatc,d,f>0,(1)
the equilibriumE0 is locally asymptotically stable for all values of the parameter.(2)
Ifd<1 and d−12−4fc=0, then the positive equilibrium point E∗ is nonhyperbolic of unstable type.(3)
Ifd<1 and d−12−4fc>0, then the equilibrium point E− is a repeller while the stability of E+ is subject to the following conditions:(a)
E+ is locally asymptotically stable if d−12−4fc>4d2(b)
E+ is nonhyperbolic of stable type if d−12−4fc=4d2(c)
E+ is a saddle point if d−12−4fc<4d2Proof.
(1)
Sincep=q=0 for x¯=0 implies that the unique eigenvalue λ=0, E0 is locally asymptotically stable for all values of c,d, and f.(2)
Asp=−d,q=21−cx¯ and x¯∗=1−d/2c, then the characteristic equation is given by(15)λ2+dλ−d+1=0,which solutions areλ+=1 and λ−=−d+1. The latter shows that E∗ is nonhyperbolic of unstable type.(3)
The roots of the characteristic equation areλ+=−d+d2+81−cx¯/2>0 and λ−=−d−d2+81−cx¯/2<0.ForE−,sincex¯−<1−d/2c, one can easily check that d2+81−cx¯−>2+d which implies λ+>1.On the other hand, one can use the fact thatx¯−<1+d/2c to show that λ−<−1.ForE+,sincex¯+>1−d/2c, one can similarly show that λ+<1.Moreover, a simple algebraic verification shows the following:(i)
λ−<1for4d2<1−d2−4fc;(ii)
λ−=1for4d2=1−d2−4fc;(iii)
λ−>1for4d2>1−d2−4fc.Consequently, we conclude thatE− is a repeller whenever it exists while(i)
E+ is locally asymptotically stable, for4d2<1−d2−4fc;(ii)
E+ is nonhyperbolic of stable type, for4d2=1−d2−4fc;(iii)
E+ is a saddle point, for4d2>1−d2−4fc.


## 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 thatc,d,f>0, we have the following:(1)
If4fc>1, then equation (1) has no minimal period-two solutions.(2)
If4fc=1, then equation (1) has a minimal period-two solution:Px1/2c,0,Py0,1/2c.(3)
Ifd−12−4d2≤4fc<1, then equation (1) has two minimal period-two solutions:(16)Px11−1−4cf2c,0,Py10,1−1−4cf2c,Px21+1−4cf2c,0,Py20,1+1−4cf2c.(4)
If4fc<d−12−4d2, then equation (1) has three minimal period-two solutions:(17)Px1,Py1,Px2,Py2,P∓i1+d−1−d2−4d2−4fc2c,1+d+1−d2−4d2−4fc2c,P±i1+d+1−d2−4d2−4fc2c,1+d−1−d2−4d2−4fc2c.Proof.
Any period-two solutionϕ,ψ satisfies the following system of equations:(18)ϕ=ϕ2cϕ2+dψ+f,ψ=ψ2cψ2+dϕ+f.
It follows thatIfψ=0, the first equation becomes cϕ2−ϕ+f=0.Ifϕ=0, then the second equation turns into cψ2−ψ+f=0.Ifϕ≠0 and ψ≠0, then the system is equivalent to(19)cϕ2+dψ−ϕ+f=0,cψ2+dϕ−ψ+f=0,⟹cϕ2−ψ2+dψ−ϕ+ψ−ϕ=0.
Now, given thatϕ≠ψ, we get ϕ=1+d/c−ψ which implies(20)cψ2−1+dψ+1+dc+f=0.
Thus, for solution of the form0,ψ or ϕ,0, we must have 1−4fc≥0 and for solutions of the form ϕ,ψ, ϕ≠0, ψ≠0, we must have 1−d2−4d2−4fc>0. Consequently,(1)
if4fc>1, then 1−4fc<0 and 1−d2−4d2−4fc<0. It follows that equation (1) has no minimal period-two solutions.(2)
If4fc=1, it follows that Px,Py is the unique minimal period-two solution of equation (1).(3)
If1−d2−4d2≤4fc<1, equation (1) has two minimal period-two solutions:(21)Px11−1−4cf2c,0,Py10,1−1−4cf2c,Px21+1−4cf2c,0,Py20,1+1−4cf2c.(4)
If4fc<1−d2−4d2, then equation (1) has three minimal period-two solutions:
Px1,Py1,Px2,Py2 together with(22)P∓i=1+d−1−d2−4d2−4fc2c,1+d+1−d2−4d2−4fc2c,P±i=1+d+1−d2−4d2−4fc2c,1+d−1−d2−4d2−4fc2c.
Now, consider the following substitution,un=xn−1 and vn=xn, which transforms equation (1) into the following two-dimensional system,(23)un+1=vn,vn+1=un2cun2+dvn+f,to which corresponds the following map:(24)Tuv=vhu,v=vu2cu2+dv+f,in which second iterate T2 is given by(25)T2uv=Tvhu,v=hu,vhv,hu,v=Gu,vHu,v,where(26)Hu,v=v2cv2+dhu,v+f.
The mapT2 is strongly competitive and its Jacobian matrix is given by(27)JT2u,v=2uf+dvcu2+f+dv2−du2cu2+f+dv2−2duv2f+dvdu2+cu2+f+dvcv2+f2vd2cu2+2f+3dvu2/cu2+f+dv2+2fdu2/cu2+f+dv+cv2+f2.
The following theorem describes the local stability of minimal period-two solutions of equation (1) whenever they exist.Theorem 4.
(1)
The minimal period-two solutionsPx,Py are nonhyperbolic of stable type.(2)
The minimal period-two solutionsPx1,Py1 are saddle points while Px2,Py2 are locally asymptotically stable.(3)
The minimal period-two solutionsP∓i,P±i are saddle points.Proof.
(1)
The minimal period solutionsPx,Py exist when 4fc=1; thus, the Jacobian matrix of the second iterate of the map T at Px and Py is the following:(28)JT2Px=1−d00,JT2Py=0001,with the eigenvaluesλ1=0 and λ2=1; therefore, Px,Py are nonhyperbolic of stable type.(2)
Now,(a)
ForPx1,Py1, the Jacobian matrix is of the form(29)JT2Px1=1+1−4cf−d00,JT2Py1=0001+1−4cf.with eigenvaluesλ1=0 and λ2=1+1−4cf. Clearly, Px1,Py1 are saddle points.(b)
As ofPx2,Py2, the corresponding Jacobian matrix is given by(30)JT2Px2=1−1−4cf−d00,JT2Py2=0001−1−4cf.Both with eigenvaluesλ1=0 and λ2=1−1−4cf<1. Clearly, Px2,Py2 are locally asymptotically stable.(3)
Now, at the interior period-two solutionsP∓i,P±i, the Jacobian matrices are(31)JT2P∓i=−dc+−c2d3d+2+4cf−1c+1−d−d−dc+c+−c2d3d+2+4cf−1cd−1dc−−c2d3d+2+4cf−1c+1,JT2P±i=−dc−−c2d3d+2+4cf−1c+1−ddcd−1+−c2d3d+2+4cf−1cd2−dc+−c2d3d+2+4cf−1c+1.
Observe that(32)p=TrJT2P∓i=TrJT2P±i=2+dd−2,q=DetJT2P∓i=DetTrJT2P±i=4d2+cf.
Moreover,(33)p>1+q⇔d−12−4d2>4cfcondition of existence.
Consequently, the interior period-two solutionsP∓i,P±i are saddle points whenever they exist.Remark 2.
Observing that the interior period-two solutionsP∓i,P±i exist if and only if(1)
d<1/3, since d−12−4d2>4cf⟹d−12−4d2>0⟹1−3d1+d>0.(2)
There are 3 equilibrium pointsE0,E−,E+ where E+ 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 d and 4fc.Theorem 5.
If4fc>1, then the equilibrium E0 is globally asymptotically stable (see Figure 4).Figure 4
Visualization of Theorem5 for 4fc>1.proof.
First, observe that every solution of equation (1) is bounded, as for xn−1≠0,(34)xn+1=xn−12cxn−12+dxn+f=1c+dxn/xn−12+f/xn−12<1c,
Moreover, by Theorem5 from [19], subsequences x2nn=0∞ and x2n+1n=0∞ are eventually monotonic. Now, as for 4fc>1, there are no minimal period-two solutions, and we conclude that both x2n and x2n+1 must converge to the unique equilibrium x¯=0.Theorem 6.
LetℬA denote the basin of attraction of the set A.(1)
Thex-axis 0,+∞×0 and the y-axis 0×0,+∞ are invariant by T2.(2)
If4fc=1, then(a)
1/2c,+∞×0⊂ℬPx;(b)
0×1/2c,+∞⊂ℬPy;(c)
0×0,1/2c∪0,1/2c×0⊂ℬE0.(3)
If4fc<1, then(a)
1−1−4cf/2c,∞×0⊂ℬPx2;(b)
0×1−1−4cf/2c,∞⊂ℬPy2;(c)
0,1−1−4cf/2c×0∪0×0,1−1−4cf/2c⊂ℬE0.Proof.
(1)
Letα≥0, then T2α,0=α2/cα2+f,0 and T20,α=0,α2/cα2+f which implies that the sets 0,+∞×0 and 0×0,+∞ are invariant by T2.(2)
First, recalling that every solution of equation (1) is bounded and by Theorem 5 from [19], every solution must either converge to an equilibrium or a minimal period-two solution. It follows that every solution generated by T2 must converge to an equilibrium. Now, consider snn=1∞ the solution with initial point s1=x1,0∈1/2c,+∞×0, then(35)T2xn,0=xn+1,0=xn2cxn2+f,0,xn+1−xn=−xncxn2−xn+fcxn2+f.(a)
If4fc=1, one can easily show that xn is monotone decreasing; thus,(i)
ifxn>1/2c, then sn=xn,0⟶1/2c,0.(ii)
Ifxn<1/2c, then sn=xn,0⟶0,0(b)
If4fc<1, one can easily show that xn is monotone decreasing in 0,1−1−4fc/2c∪1+1−4fc/2c,+∞ and monotone increasing in 1−1−4fc/2c,1+1−4fc/2c; thus,(i)
Ifxn∈1−1−4fc/2c,+∞, then sn=xn,0⟶1+1−4fc/2c,0.(ii)
Ifxn∈0,1−1−4fc/2c, then sn=xn,0⟶0,0.
The remaining part of the proof follows similarly by considering solutions of the formsn=0,xn.Theorem 7.
If4fc=1, then equation (1) has the unique equilibrium point E0 which is locally asymptotically stable and the unique minimal period-two solution Px,Py=1/2c,0,0,1/2c which is nonhyperbolic of stable type.
There exist two invariant curvesC1 and C2 which are graphs of strictly increasing continuous functions of the first coordinate on an interval with endpoints in Px and Py, respectively. Basins of attraction of the minimal period-two solutions are(36)ℬPx=C1∪W+C1,ℬPy=C2∪W−C2,while the basin of attraction of the equilibrium point E0 is the region between curves C1 and C2, i.e.,(37)ℬE0=W−C1∩W+C2;see Figure 5.Figure 5
Visualization of global dynamics of Theorem7 for 4fc=1.Proof.
The Jacobian matrix of the second iterate of the map atPx is given by(38)JT2Px=1−d00,with two eigenvalues λ1=0 associated with the eigenvector d1 and λ2=1 which corresponds to the eigenvector 10.
Observe that the eigenvector associated withλ1 is not parallel to the x-axis and the map T2 is strongly competitive. It follows by Theorems 1–5 in [15] and Theorem 2 that there exists an invariant curve C1 through the point Px which is a subset of WsPx. Moreover, C1 is the graph of a strictly increasing continuous that separates the first quadrant into two connected subregions: an upper one W−C1 and a lower one W+C1, where(39)ℬPx=C1∪W+C1.
The Jacobian matrix ofT2 at Py is JT2Py=0001 and it has two eigenvectors that are parallel to the coordinate axis, 10,01, corresponding to λ1=0 and λ2=1, respectively. By Hartman–Grobman theorem [20], we know that there exists a C1 curve C through Py that is tangential at Py to the eigenspace associated with λ=0 such that T2C⊂C.Claim 1.
The stable manifold atPy is a linearly strongly ordered curve in the northeast ordering, given as WlocsPy=t,ϕt:0≤t≤δ for small enough δ>0.Proof of Claim 1.
First, recall that0,1/2c×0,1/2c⊂ℬE0. Now, let u0>0. Since T2 is strongly competitive, we have T2Py≪seT2u0,1/2c and that implies T2u0,1/2c∈intQ4Py. Therefore, there exists a ball BεT2u0,1/2c such that BεT2u0,1/2c⊂intQ4Py. Since the map T2 is continuous on ℝ+2, there exists a ball Bδ1u0,1/2c such that(40)T2Bδ1u0,1/2c⊂BεT2u0,1/2c⊂intQ4Py,which implies T2nu,v⟶0,0 when n⟶∞ for all points u,v∈Bδ1u0,1/2c. It follows that WlocsPy∩intQ4Py=∅. Now, observe that ϕ′0=0 as its graph must be tangential to the horizontal eigenspace. Moreover, ϕ″≥0 in a small neighborhood of t=0; otherwise,ϕ″≤0⟹ there exists δ>0 such that ϕt is decreasing in 0,δ⟹ϕ′t≤0 in 0,δ⟹ϕt≤1/2c in 0,δ which contradicts the fact that WlocsPy∩intQ4Py=∅.
Therefore, for sufficiently smallδ1, WlocsPy=t,ϕt:0≤t≤δ1 is linearly ordered in the northeast ordering and as T2 is competitive, WlocsPy∩ℝ+2 can be extended to an unbounded curve (global stable manifold) C2, see [14, 15].
Hence, the curveC2 splits the region into two connected components, an upper subregion W−C2 and a lower subregion W+C2.
Clearly,ℬPy=C2∪W−C2, and finally the basin of attraction of the zero equilibrium E0 is(41)ℬE0=W+C2∩W−C1.Theorem 8.
Ifd≥1and4fc<1 or d<1andd−12<4fc<1, then equation (1) has the unique equilibrium point E0 which is locally asymptotically stable and two minimal period-two solutions Px1,Py1 which is a saddle point and Px2,Py2 which is locally asymptotically stable.
There exist global stable manifoldsWsPx1 and WsPy1 which are basins of attractions of Px1,Py1 and the global unstable manifolds have the following form:(42)WuPx1=x,0:x∈A,WuPy1=0,y:y∈A,where A=0,1+1−4cf/2c∖1−1−4cf/2c.
The basinℬE0=0,0 is the region between the global stable sets:(43)ℬE0=W−Px1∩W+Py1.
The basin of the minimal period-two solutionsPx2,Py2 is(44)ℬPx2=W+Px1,ℬPy2=W−Py1;see Figure 6.Figure 6
Different manifolds of Theorem8.Proof.
Recall that(45)JT2Px1=1+1−4cf−d00,with eigenvalues λ1=0 with eigenvector d/1−4fc+11 and λ2=1−4fc+1 associated with the eigenvector 10. Thus, there exists a local stable manifold at Px1 that is linearly strongly ordered in the northeast ordering with Px1 as an endpoint. As T2 is competitive, the local stable manifold can be extended to a curve WsPx1 which separates the region into two connected components W+Px1 and W−Px1. On the other hand,(46)JT2Py1=0001+1−4cf,with eigenvalues λ1=0 with eigenvector 10 and λ2=1−4fc+1 associated with the eigenvector 01.
By Theorem6, we know that 0,1−1−4fc/2c×0,1−1−4fc/2c⊂ℬE0; thus, we know that the local stable manifold at Py1 is tangential to the horizontal eigenspace but cannot enter the box 0,1−1−4fc/2c×0,1−1−4fc/2c. We conclude that the local stable manifold at Py1 is a linearly strongly ordered curve (in the northeast ordering) with Py1 as an endpoint. Similarly, we conclude its extension to a global stable manifold WsPy1 which separates the region into two connected components W+Py1 and W−Py1.
Finally, by the uniqueness of the stable manifold of the saddle pointPx1, we know that no solution in W+Px1 will converge to Px1. Furthermore, all solutions are bounded and we know that by monotonicity of the map T every solution must converge to an equilibrium. It follows that ℬPx2=W+Px1 and analogously ℬPy2=W+Py1.Theorem 9.
Ifd<1 and d−12=4fc<1, then equation (1) has two equilibrium solutions:(i)
E0 is locally asymptotically stable(ii)
E∗=1−d/2c,1−d/2c is a nonhyperbolic of unstable type, and two minimal period-two solutions:(iii)
Px1,Py1=1−1−4fc/2c,0,0,1−1−4fc/2c are saddle points,(iv)
Px2,Py2=1+1−4fc/2c,0,0,1+1−4fc/2c are locally asymptotically stable.
Then,
(i)
there exist global stable manifoldsWsPx1 and WsPy1 which are the basins of attraction of the periodic solutions Px1,Py1 and which are tangential at the equilibrium point E∗.(ii)
There exists a global stable manifoldWsE∗ contained in Q1E∗ which is the basin of attraction of the equilibrium E∗;
The basin of attraction of the equilibrium pointE0 is given by(47)ℬE0=W−Px1∩W+Py1.
The basins of attraction ofPx2 and Py2 are given by(48)ℬPx2=W+Sx,whereSx=WsPx1∪WsE∗,ℬPy2=W−Sy,whereSy=WsPy1∪WsE∗;see Figure 7.Figure 7
Global manifolds and basins of Theorem9.Proof.
The existence and orientation of the global stable manifold atPx1 can be determined as in Theorem 8; however, in general, this information can be determined by studying the curvature of the local curves given by
WlocsPy1=t,ϕ1t:0≤t≤δ1 and WlocsPx1=ϕ2t,t:0≤t≤δ2 for δ1 and δ2 small enough, where if fx,y and gx,y are the coordinate functions of T2, then(49)ϕ1ft,ϕ1t=gt,ϕ1t,ϕ10=1−1−4fc2c,ϕ1′0=0,ϕ2gϕ2t,t=fϕ2t,t,ϕ20=1−1−4fc2c,ϕ2′0=0.
This is useful when the local curve has a tangent parallel to the axis at the fixed point which is the case here forpy1.
By differentiating both sides of the equation above, we get(50)ϕ1″0=gxx0,ϕ10fx0,ϕ102−gy0,ϕ10=d1−1−4fcfd+2cf−d1−4fc>0,which confirms the argument used in Theorem 8. We conclude the existence of curves C1 and C2 (global stable manifolds) which are graphs of continuous, strictly increasing functions. Furthermore, the curves cannot intersect the interior of the sets Q2E∗∩ℝ2 and Q4E∗∩ℝ2, as the monotonicity of T2 forces the latter sets to be invariant. Thus, T−2nP⟶E∗ for all P∈Cl,l=1,2, therefore C1 and C2 are also center manifolds of E∗.
On the other hand, by lettingT2x,y=fx,y,gx,y, the center manifold ϕx must satisfy [21](51)ϕfx,ϕx=gx,ϕx.
By using a Taylor expansion substitution, the center manifold can be approximated by(52)ϕx=x−cx−1−d/2c2d+2−2c2x−1−d/2c3d+23+Ox−1−d2c4.
The dynamic on the center manifoldϕx is given by the reduced difference equation un+1=fun,ϕun which has the following asymptotic representation:(53)un+1=1−d2c+un−1−d2c−2cun−1−d/2c2d+2+2c2dun−1−d/2c3d+23−⋯+Oun−1−d2c5.
Sinceu¯=1−d/2c is a semistable fixed point for the latter scalar difference equation, it follows that E∗ is a semistable fixed point for T2; furthermore, the coefficient of the lowest nonlinear term in the reduced map is negative; thus, by [22], the local basin of attraction of the equilibrium E∗ is a one-dimensional curve. We conclude that there is a unique center manifold curve U which satisfies T2U⊂U. Moreover, U is tangential to the eigenspace associated with λ=1, namely, Span11. It follows that U is contained in Q1E∗ and is linearly ordered in the northeast ordering and therefore can be extended to an unbounded curve C.
Now, for every pointq∈W−Px1∩W+Py1, there exist qx∈WsPx1 and qy∈WsPy1 such that
qy⪯seq⪯seqx which implies that T2nqy⪯seT2nq⪯seT2nqx, but we know that(54)T2nqy⟶Py1,T2nqx⟶Px1.
Consequently, there existsN such that(55)Py1⪯seT2Nq⪯sePx1.
It follows by Theorem6 that q∈ℬE0. As of the basins of attractions of Px2 and Py2, the proof is analogous to the one given in Theorem 8.Theorem 10.
Ifd<1,d−12−4d2<4fc<d−12, then equation (1) has three equilibrium points E0,E−,E+, which are, respectively, locally asymptotically stable, a repeller and a saddle point and two minimal period-two solutions:(i)
Px1,Py1 are saddle points(ii)
Px2,Py2 are locally asymptotically stable
Then,
(i)
There exist global stable manifoldsWsPx1 (the basin of attraction of Px1) and WsPy1 (the basin of attraction of Px2) which are tangential at the equilibrium point E−.(ii)
There exists a global stable manifoldWsE+ which is an unbounded curve that is a graph of an increasing function contained in Q1E+∪Q3E+ with an endpoint at E− and is the basin of attraction of the equilibrium E+.(iii)
There exist a global unstable manifoldWuE+ which is a graph of a decreasing function contained in Q2E+∪Q4E+ with endpoints Px2 and Py2.(iv)
The basin of attraction ofE0 is(56)ℬE0=W−Px1∩W+Py1.(v)
The basins of attraction ofPx2 and Py2 are given by(57)ℬPx2=W+Sx,whereSx=WsPx1∪WsE+,ℬPy2=W−Sy,whereSy=WsPy1∪WsE+;see Figure 8.Figure 8
All invariant manifolds of Theorem10.Proof.
He the existence ofWsPx1 and WsPy1 as well as the basin of attraction of E0 follows from Theorem 9. The existence of the stable WsE+ and the unstable manifold WuE+ follows from Theorems 1–5 in [15] and Theorem 2. The basins of attraction of Px2 and Py2 were discussed in Theorem 8.Theorem 11.
Ifd<1 and 0<d−12−4d2=4fc, then equation (1) has three equilibrium points E0,E−,E+, where(i)
E0 is locally asymptotically stable(ii)
E−=1−3d/2c,1−3d/2c is a repeller(iii)
E+=1+d/2c,1+d/2c is a nonhyperbolic point of stable type, and two minimal period-two solutions:(iv)
Px1,Py1 is a saddle point(v)
Px2,Py2 is locally asymptotically stable
Then,
(i)
There exist global stable manifoldsWsPx1 and WsPy1 which are the basins of attraction of the periodic solutions Px1,Py1 and which are tangential at the equilibrium point E−.(ii)
There exists a global stable manifoldWsE+ which is an unbounded curve and the graph of an increasing function contained in Q1E+∪Q3E+ with an endpoint at E− and WsE+ is the basin of attraction of the equilibrium E+.(iii)
The basin of attraction of equilibrium pointE0 is the region between those stable manifolds, i.e.,(58)ℬE0=W−Px1∩W+Py1.(iv)
The basins of attraction ofPx2 and Py2 are given by(59)ℬPx2=W+Sx,whereSx=WsPx1∪WsE+,ℬPy2=W−Sy,whereSy=WsPy1∪WsE+;see Figure 9.Figure 9
Visual representation of global dynamics of Theorem11.Proof.
The existence and orientation of the global stable manifoldWsE+ follows from Theorems 1–5 in [15] and Theorem 2. The remaining of the proof is analogous to the discussions in Theorems 8 and 9.Theorem 12.
Ifd<1 and 4fc<d−12−4fc, then equation (1) has three equilibrium points E0,E−,E+ which are, respectively, locally asymptotically stable, a repeller and locally asymptotically stable, and has three minimal period-two solutions such that(i)
Px1,Py1 are saddle points,(ii)
Px2,Py2 are locally asymptotically stable,(iii)
P∓i,P±i are saddle points.(iv)
There exist global stable manifoldsWsPx1 and WsPy1 which are the basins of attraction of the periodic solutions Px1,Py1 and which are tangential at the equilibrium point E−.(v)
The basin of attraction of equilibrium pointE0 is the region between those stable manifolds, i.e.,(60)ℬE0=W−Px1∩W+Py1.(vi)
There exists a global stable manifoldWsP∓i which is an unbounded curve and the graph of an increasing function contained in Q1P∓i∪Q3P∓i with an endpoint at E−. WsP∓i is the basin of attraction of the equilibrium P∓i.(vii)
There exists a global stable manifoldWsP±i which is an unbounded curve and the graph of an increasing function contained in Q1P±i∪Q3P±i with an endpoint at E−. WsP±i is the basin of attraction of the equilibrium P±i.(viii)
The basin of attraction of the equilibrium pointE+ is the region between those stable manifolds, i.e.,(61)ℬE+=W−P±i∩W+P∓i.(ix)
There exist a global unstable manifoldWuP∓i which is the graph of a decreasing function contained in Q2P∓i∪Q4P∓i with endpoints Py2 and E+.(x)
There exist a global unstable manifoldWuP±i which is the graph of a decreasing function contained in Q2P±i∪Q4P±i with endpoints Px2 and E+.(xi)
The basins of attraction ofPx2 and Py2 are given by(62)ℬPx2=W+Sx,whereSx=WsPx1∪WsP±i,ℬPy2=W−Sy,whereSy=WsPy1∪WsP±i;see Figure 10.Figure 10
The visual illustration of Theorem12.proof.
The existence and orientation of the stable manifoldWsP∓i follows from Theorems 1–5 in [15] and Theorem 2. Moreover, WsP∓i cannot intersect another manifold or the boundary of the region at any point as the latter sets are invariant. Thus, it must have an endpoint at E−. Similarly, the existence and orientation of the unstable manifold WuP∓i are given by Theorem 5 in [15]. On the other hand, WuP∓i∩P∓i,E+≠∅ and P∓i,E+ is invariant. Thus, WuP∓i cannot leave the latter set and must end at E+. Analogous arguments and conclusions also hold for WsP±i and WuP±i. In addition, we know that for p∓∈WsP∓i and p±∈WsP±i,(63)T2np∓⟶P∓i,T2np±⟶P±i.
Furthermore, for allp∈W−P±i∩W+P∓i, there exist p∓∈WsP∓i and. p±∈WsP±i such that(64)p∓⪯sep⪯sep±⟹T2np∓⪯seT2np⪯seT2np±,for alln≥0.
It follows that there existsN>0 such that T2Np=q∈P∓i,P±i. Thus, there exist q∓∈WuP∓i and q±∈WuP±i such that(65)q∓⪯seq⪯seq±⟹T2nq∓⪯seT2nq⪯seT2nq±,for alln≥0,where(66)T2nq∓⟶E+,T2nq±⟶E+,which implies that T2nq⟶E+⟹T2np⟶E+. We conclude that(67)ℬE+=W−P±i∩W+P∓i.
As ofWsPx1, WsPy1, ℬE0, ℬPx2 and ℬPy2, the proof is analogous to the discussion in Theorems 8 and 9.



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