Global Dynamics of Delayed Sigmoid Beverton–Holt Equation Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

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 x n + 1 � f ( x n ,x n − 1 ) , n � 0 , 1 , ... , where f is decreasing in the variable x n and increasing in the variable x n − 1 . As a case study, we use the diﬀerence equation x n + 1 � ( x 2 n − 1 / ( cx 2 n − 1 + dx n + f )) , n � 0 , 1 , ... , where the initial conditions x − 1 ,x 0 ≥ 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.


Introduction and Preliminaries
Consider the following difference equation: where the initial conditions x − 1 , x 0 ≥ 0 and the parameters satisfy that c, f ≥ 0, d > 0. Equation (1) is a special case of the equation where x − 1 , x 0 ≥ 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 where all parameters are nonnegative numbers and the initial conditions x − 1 , x 0 ≥ 0 so that the solution is well defined.Many special cases of equation (3) have been investigated in [1][2][3][4][5][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), , n � 0, 1, . . ., (4) , n � 0, 1, . . ., (6) 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). e 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).us, we will obtain some general global dynamic results for general second-order difference equation, x n+1 � f x n , x n− 1 , x − 1 , x 0 ∈ I, n � 0, 1, . . ., (7) which will unify global dynamic results for equations ( 4)- (6) in the hyperbolic case.e nonhyperbolic cases may be different for different equations and require the specific investigation.
e perturbation term in the case of equation ( 1) is dx n (depends on population size at n-th generation only) while in the case of equation ( 6) is bx n x n− 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).
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).
ere is an extensive literature on the dynamics of monotone maps (competitive and cooperative) that can be found in [11][12][13].In this paper, we will use some theorems from [14][15][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 eorem 5 from [19] which states that, for every solution x n   ∞ n�− 1 of equation ( 7), the subsequences x 2n   ∞ n�0 and x 2n− 1   ∞ n�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 eorem 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.
e 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].
e fixed point (x, 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.

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 T generated by the system, on a set R with a nonempty interior.
(a) Assume that T has seven fixed points E 1 , . .Finally, all solutions which start between the stable manifolds W s (E 8 ) and W s (E 9 ) converge to E 7 .Proof (a) e existence of the global stable and unstable manifolds of the saddle fixed points is guaranteed by eorems 1-5 in [15].In any case, all global stable manifolds W s (E 2 ), W s (E 4 ) and W s (E 7 ) have an endpoint at E 6 , and W s (E 7 ) has another endpoint at (∞, ∞).In view of eorem 4 in [15], every solution which starts between the stable manifolds W s (E 2 ) and one can find the points (x l , y l ) on the y-axis and (x u , y u ) on the x-axis such that (x l , y l ) ⪯ se (x 0 , y 0 ) ⪯ se (x u , y u ). is will imply that T n ((x l , y l )) ⪯ se T n ((x 0 , y 0 )) ⪯ se T n ((x u , y u )), and so T n ((x 0 , y 0 )) converges to E 3 as both T n ((x l , y l )), T n ((x u , y u )) do.If the initial point (x 0 , y 0 ) is above W s (E 2 ) ∪ W s (E 7 ), one can find the point (x l , y l ) on the y-axis and a point is will imply that T n ((x l , y l )) ⪯ se T n ((x 0 , y 0 )) ⪯ se T n ((x u , y u )), and so T n ((x 0 , y 0 )) ∈ int[E 1 , E 7 ] eventually.Now, in view of Corollary 1 in [15], T n ((x 0 , y 0 )) ⟶ E 1 as n ⟶ ∞.In a similar way the case when the initial point (x 0 , y 0 ) is below W s (E 4 ) ∪ W s (E 7 ) can be handled.(b) e existence of the global stable and unstable manifolds of the saddle fixed points is guaranteed by eorems 1-5 in [15].Both global stable manifolds W s (E 2 ) and W s (E 4 ) have an endpoint at E 6 .e existence of curves C 1 and C 2 follows from eorem 2 in [16].e proof that the region between the stable manifolds W s (E 2 ) and W s (E 4 ) eventually enters int[E 2 , E 4 ] and so it converges to E 3 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 and so it will converge to E 1 .In a similar way, we can show that if the initial point and so it will converge to E 5 .Finally, if the initial point (x 0 , y 0 ) is between C 1 and C 2 , then one can find the point (x l , y l ) ∈ C 2 and a point (x u , y u ) ∈ C 1 such that (x l , y l ) ⪯ se (x 0 , y 0 ) ⪯ se (x u , y u ). is will imply that T n ((x l , y l )) ⪯ se T n ((x 0 , y 0 )) ⪯ se T n ((x u , y u )), and so T n ((x 0 , y 0 )) ⟶ E 6 as T n ((x u , y u )) ⟶ E 6 , T n ((x l , y l )) ⟶ E 6 .(c) e proof that the region between the stable manifolds W s (E 2 ) and W s (E 4 ) is the basin of attraction of E 3 is same as in part (a) and will be omitted.e proof that all solutions which start above converge to E 5 is same as in part (a) and so will be omitted.
If the initial point (x 0 , y 0 ) is between W s (E 8 ) and W s (E 9 ), then one can find the point (x l , y l ) ∈ W s (E 8 ) and a point (x u , y u ) ∈ W s (E 9 ) such that (x l , y l ) ⪯ se (x 0 , y 0 ) ⪯ se (x u , y u ). is will imply that T n ((x l , y l )) ⪯ se T n ((x 0 , y 0 )) ⪯ se T n ((x u , y u )), and so □ Theorem 2. Consider equation (7) and assume that f is decreasing in the first and increasing in the second variable on the set (a, b) 2 .
(a) Assume that equation ( 7) has three equilibrium points , where E 0 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 is locally asymptotically stable, then every solution with the initial point between the stable manifolds W s (P 1 ) and W s (Q 1 ) converges to E 0 while every solution which starts off W s (P 1 ) ∪ W s (Q 1 ) ∪ W s (E + ) converges to the periodic solution P 2 , Q 2  .
(b) Assume that equation (7) has two equilibrium points E 0 ⪯ ne E, where E 0 is locally asymptotically stable and E + is a nonhyperbolic point of unstable type.Furthermore, assume that equation (7) has two minimal period-two solutions is locally asymptotically stable, then every solution which starts between the stable manifolds W s (P 1 ) and W s (Q 1 ) converges to E 0 while every solution which starts above W s (P 1 ) ∪ C 2 and below Finally, every solution starting between C 1 and C 2 converges to E.
(c) Assume that equation ( 7) has three equilibrium points , where E 0 and E + are locally asymptotically stable and E − is a repeller.Furthermore, assume that equation ( 7) has three minimal periodtwo solutions are saddle points and P 2 , Q 2   is locally asymptotically stable, then every solution which starts between the stable manifolds W s (P 1 ) and W s (Q 1 ) converges to E 0 and every solution which starts between the stable manifolds W s (P 3 ) and W s (Q 3 ) converges to E 7 while every solution which starts off in the complement of the basins of attractions of E 0 , E 7 , P 1 , Q 1   and P 3 , Q 3   converges to the periodic solution P 2 , Q 2  .
Proof.In view of Remark 1 and eorem 5 from [19], the second iterate T 2 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 of T are the fixed points of T 2 , two period-two solutions become four fixed points of T 2 .An application of eorem 1in part (a) to T 2 completes the proof.(b) In view of Remark 1, the second iterate T 2 of the map T associated with equation ( 7) is strictly competitive and has six equilibrium points An application of eorem 1 in part (b) to T 2 yields lim n⟶∞ T 2n ((x 0 , y 0 )) � E 0 for every (x 0 , y 0 ) between the stable manifolds W s (P 1 ) and W s (Q 1 ).Furthermore, we derive where we used a continuity of the map T. Consequently, lim n⟶∞ T n ((x 0 , y 0 )) � E 0 .e proof of other cases is similar.(c) By noticing that the period-two points of T are the fixed points of T 2 , three period-two solutions Applying eorem 1 in part (c) to T 2 , we complete the proof.

Local Stability Analysis for
Equilibria.An equilibrium solution of equation (1) must satisfy i.e., x � 0 or cx 2 + (d − 1)x + f � 0. erefore, equation (1) has the following: (1) e unique equilibrium has the following linearized equation: where If x � 0, then clearly p � q � 0. If x ≠ 0, then by the equilibrium equation, Proposition 1.Given that c, d, f > 0, (1) the equilibrium E 0 is locally asymptotically stable for all values of the parameter.

Proof
(1) Since p � q � 0 for x � 0 implies that the unique eigenvalue λ � 0, E 0 is locally asymptotically stable for all values of c, d, and f.
the characteristic equation is given by which solutions are λ + � 1 and λ − � − (d + 1).e latter shows that E * is nonhyperbolic of unstable type.
Discrete Dynamics in Nature and Society (3) e roots of the characteristic equation are On the other hand, one can use the fact that x − < (1 + d)/2c to show that λ − < − 1.For E + , since x + > (1 − d)/2c, one can similarly show that λ + < 1.Moreover, a simple algebraic verification shows the following: Consequently, we conclude that E − is a repeller whenever it exists while
(3) If (1 − d) 2 − 4d 2 ≤ 4fc < 1, equation (1) has two minimal period-two solutions: Now, consider the following substitution, u n � x n− 1 and v n � x n , which transforms equation (1) into the following two-dimensional system, to which corresponds the following map: in which second iterate T 2 is given by where e map T 2 is strongly competitive and its Jacobian matrix is given by e following theorem describes the local stability of minimal period-two solutions of equation (1) whenever they exist.

□ Theorem 4
(1) e minimal period-two solutions P x , P y   are nonhyperbolic of stable type.
(2) e minimal period-two solutions P 1 x , P

Proof
(1) e minimal period solutions P x , P y   exist when 4fc � 1; thus, the Jacobian matrix of the second iterate of the map T at P x and P y is the following: with the eigenvalues λ 1 � 0 and λ 2 � 1; therefore, P x , P y   are nonhyperbolic of stable type.(2) Now, (a) For P 1 x , P 1 y  , the Jacobian matrix is of the form with eigenvalues λ 1 � 0 and λ 2 � 1 + ������ � 1 − 4cf  .Clearly, P 1 x , P 1 y   are saddle points.(b) As of P 2 x , P 2 y  , the corresponding Jacobian matrix is given by (30) Observe that Moreover, Consequently, the interior period-two solutions P i ∓ , P i ±   are saddle points whenever they exist.

□
Remark 2. Observing that the interior period-two solutions (2) ere are 3 equilibrium points E 0 , E − , E + where E + is a saddle point.1) is quite complicated.us, we provide the following three diagrams that describe all possible bifurcations produced by different values of parameters d and 4fc.Theorem 5.If 4fc > 1, then the equilibrium E 0 is globally asymptotically stable (see Figure 4).proof.First, observe that every solution of equation ( 1) is bounded, as for x n− 1 ≠ 0, 8

Global Dynamics of Equation (1). e global dynamics of equation (
Discrete Dynamics in Nature and Society Moreover, by eorem 5 from [19], subsequences x 2n   ∞ n�0 and x 2n+1   ∞ n�0 are eventually monotonic.Now, as for 4fc > 1, there are no minimal period-two solutions, and we conclude that both x 2n and x 2n+1 must converge to the unique equilibrium x � 0. Theorem 6.Let B(A) denote the basin of attraction of the set A.
(2) First, recalling that every solution of equation ( 1) is bounded and by eorem 5 from [19], every solution must either converge to an equilibrium or a minimal period-two solution.It follows that every solution generated by T 2 must converge to an equilibrium.Now, consider s n   ∞ n�1 the solution with initial point (a) If 4fc � 1, one can easily show that x n is monotone decreasing; thus, )/2c, +∞) and monotone increasing in e remaining part of the proof follows similarly by considering solutions of the form s n � (0, x n ).□ Theorem 7. If 4fc � 1, then equation (1) has the unique equilibrium point E 0 which is locally asymptotically stable and the unique minimal period-two solution P x , P y   � (1/2c, 0), (0, 1/2c) { } which is nonhyperbolic of stable type.
ere exist two invariant curves C 1 and C 2 which are graphs of strictly increasing continuous functions of the first coordinate on an interval with endpoints in P x and P y , respectively.Basins of attraction of the minimal period-two solutions are while the basin of attraction of the equilibrium point E 0 is the region between curves C 1 and C 2 , i.e., (37) see Figure 5.
Proof. e Jacobian matrix of the second iterate of the map at P x is given by Observe that the eigenvector associated with λ 1 is not parallel to the x-axis and the map T 2 is strongly competitive.It follows by eorems 1-5 in [15] and eorem 2 that there exists an invariant curve C 1 through the point P x which is a subset of W s (P x ).Moreover, C 1 is the graph of a strictly increasing continuous that separates the first quadrant into two connected subregions: an upper one W − (C 1 ) and a lower one W + (C 1 ), where e Jacobian matrix of T 2 at P y is J T 2 (P y ) � 0 0 0 1   and it has two eigenvectors that are parallel to the coordinate axis, 1 0  , 0 1  , corresponding to λ 1 � 0 and λ 2 � 1, respectively.By Hartman-Grobman theorem [20], we know that there exists a C 1 curve C through P y that is tangential at P y to the eigenspace associated with λ � 0 such that e stable manifold at P y is a linearly strongly ordered curve in the northeast ordering, given as Proof of Claim 1. First, recall that (0, 1/2c) × (0, 1/2c) ⊂ B(E 0 ).Now, let u 0 > 0. Since T 2 is strongly competitive, we have T 2 (P y ) ≪ se T 2 (u 0 , 1/2c) and that implies T 2 (u 0 , 1/2c) ∈ int(Q 4 (P y )).erefore, there exists a ball which implies T 2n (u, v) ⟶ (0, 0) when n ⟶ ∞ for all points (u, v) ∈ B δ 1 ((u 0 , 1/2c)).It follows that W s loc (P y ) ∩ int(Q 4 (P y )) � ∅.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, is linearly ordered in the northeast ordering and as T 2 is competitive, W s loc (P y ) ∩ R 2 + can be extended to an unbounded curve (global stable manifold) C 2 , see [14,15].Hence, the curve C 2 splits the region into two connected components, an upper subregion W − (C 2 ) and a lower subregion W + (C 2 ).
Clearly, B(P y ) � C 2 ∪ W − (C 2 ), and finally the basin of attraction of the zero equilibrium E 0 is Theorem 8.If (d ≥ 1 and 4fc < 1) or (d < 1 and (d − 1) 2 < 4fc < 1), then equation ( 1) has the unique equilibrium point E 0 which is locally asymptotically stable and two minimal period-two solutions P 1 x , P 1 y   which is a saddle point and P 2 x , P 2 y   which is locally asymptotically stable.ere exist global stable manifolds W s (P 1 x ) and W s (P 1 y ) which are basins of attractions of P 1 x , P 1 y   and the global unstable manifolds have the following form: where A � (0, (1 . e basin B(E 0 ) � (0, 0) is the region between the global stable sets: e basin of the minimal period-two solutions P 2 x , P 2 y   is see Figure 6.
Proof.Recall that us, there exists a local stable manifold at P 1 x that is linearly strongly ordered in the northeast ordering with P 1 x as an endpoint.As T 2 is competitive, the local stable manifold can be extended to a curve W s (P 1 x ) which separates the region into two connected components W + (P 1 x ) and W − (P 1 x ).On the other hand, with eigenvalues λ 1 � 0 with eigenvector 1 0   and λ 2 � ������ 1 − 4fc √ + 1 associated with the eigenvector 0 1  .
By eorem 6, we know that (0, (1 ; thus, we know that the local stable manifold at P 1 y is tangential to the horizontal eigenspace but cannot enter the box (0, (1 ).We conclude that the local stable manifold at P 1 y is a linearly strongly ordered curve (in the northeast ordering) with P 1 y as an endpoint.Similarly, we conclude its extension to a global stable manifold W s (P 1 y ) which separates the region into two connected components W + (P 1 y ) and W − (P 1 y ).Finally, by the uniqueness of the stable manifold of the saddle point P 1 x , we know that no solution in W + (P 1 x ) will converge to P 1 x .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 B(P 2 x ) � W + (P 1 x ) and analogously B(P (48) see Figure 7.
Proof. e existence and orientation of the global stable manifold at P 1 x can be determined as in eorem 8; however, in general, this information can be determined by studying the curvature of the local curves given by W s loc (P 1 y ) � t, ϕ 1 (t) : 0 ≤ t ≤ δ 1   and W s loc (P 1 x ) � (ϕ 2  (t), t) : 0 ≤ t ≤ δ 2 } for δ 1 and δ 2 small enough, where if f(x, y) and g(x, y) are the coordinate functions of T 2 , then is is useful when the local curve has a tangent parallel to the axis at the fixed point which is the case here for p 1 y .By differentiating both sides of the equation above, we get which confirms the argument used in eorem 8.We conclude the existence of curves C 1 and C 2 (global stable manifolds) which are graphs of continuous, strictly increasing functions.Furthermore, the curves cannot intersect the interior of the sets Q 2 (E * ) ∩ R 2 and Q 4 (E * ) ∩ R 2 , as the monotonicity of T 2 forces the latter sets to be invariant.us, T − 2n (P) ⟶ E * for all P ∈ C l , l � 1, 2, therefore C 1 and C 2 are also center manifolds of E * .
On the other hand, by letting T 2 (x, y) � f(x, y), g(x, y), the center manifold ϕ(x) must satisfy [21] ϕ(f(x, ϕ(x))) � g(x, ϕ(x)). (51) By using a Taylor expansion substitution, the center manifold can be approximated by (52) e dynamic on the center manifold ϕ(x) is given by the reduced difference equation u n+1 � f(u n , ϕ(u n )) which has the following asymptotic representation: Since u � (1 − d)/2c is a semistable fixed point for the latter scalar difference equation, it follows that E * is a semistable fixed point for T 2 ; 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 T 2 (U) ⊂ U.Moreover, U is tangential to the eigenspace associated with ) and is linearly ordered in the northeast ordering and therefore can be extended to an unbounded curve C. Now, for every point q ∈ W − (P 1 x ) ∩ W + (P 1 y ), there exist q x ∈ W s (P 1 x ) and q y ∈ W s (P 1 y ) such that q y ⪯ se q ⪯ se q x which implies that T 2n (q y ) ⪯ se T 2n (q) ⪯ se T 2n (q x ), but we know that T 2n q y   ⟶ P 1 y , T 2n q x  ⟶ P 1 x . (54) Consequently, there exists N such that It follows by eorem 6 that q ∈ B(E 0 ).As of the basins of attractions of P 2 x and P 2 y , the proof is analogous to the one given in eorem 8. (57) see Figure 8.
Proof.He the existence of W s (P 1 x ) and W s (P 1 y ) as well as the basin of attraction of E 0 follows from eorem 9. e existence of the stable W s (E + ) and the unstable manifold W u (E + ) follows from eorems 1-5 in [15] and eorem 2.
e basins of attraction of P 2 x and P 2 y were discussed in eorem 8. (59) see Figure 9.

□
Proof. e existence and orientation of the global stable manifold W s (E + ) follows from eorems 1-5 in [15] and eorem 2. e remaining of the proof is analogous to the discussions in eorems 8 and 9. □ Theorem 12.If d < 1 and 4fc < (d − 1) 2 − 4fc, then equation (1) has three equilibrium points E 0 , E − , E + which are, respectively, locally asymptotically stable, a repeller and locally asymptotically stable, and has three minimal period-two solutions such that (i) P 1 x , P (62) see Figure 10.
proof.e existence and orientation of the stable manifold W s (P i ∓ ) follows from eorems 1-5 in [15] and eorem 2.Moreover, W s (P i ∓ ) cannot intersect another manifold or the boundary of the region at any point as the latter sets are invariant.us, it must have an endpoint at E − .Similarly, the existence and orientation of the unstable manifold W u (P i ∓ ) are given by eorem 5 in [15].On the other hand, W u (P i ∓ ) ∩ [P i ∓ , E + ] ≠ ∅ and [(P i ∓ , E + ] is invariant.us, W u (P i ∓ ) cannot leave the latter set and must end at E + .Analogous arguments and conclusions also hold for W s (P i ± ) and W u (P i ± ).In addition, we know that for p ∓ ∈ W s (P i ∓ ) and p ± ∈ W s (P i ± ), T 2n p ±  ⟶ P i ± . (63) Furthermore, for all p ∈ W − (P i ± ) ∩ W + (P i ∓ ), there exist p ∓ ∈ W s (P i ∓ ) and. p ± ∈ W s (P i ± ) such that p ∓ ⪯ se p ⪯ se p ± ⟹T 2n p ∓  ⪯ se T 2n (p) ⪯ se T 2n p ± , for all n ≥ 0. (64) It follows that there exists N > 0 such that T 2N (p) � q ∈ [P i ∓ , P i ± ]. us, there exist q ∓ ∈ W u (P i ∓ ) and q ± ∈ W u (P i ± ) such that q ∓ ⪯ se q ⪯ se q ± ⟹T 2n q ∓  ⪯ se T 2n (q) ⪯ se T 2n q ± , for all n ≥ 0, where which implies that T 2n (q) ⟶ E + ⟹T 2n (p) ⟶ E + .We conclude that As of W s (P 1 x ), W s (P 1 y ), B(E 0 ), B(P 2 x ) and B(P 2 y ), the proof is analogous to the discussion in eorems 8 and 9. □

Figure 3 :
Figure 3: Bifurcation diagram of global dynamics of equation (1) in the parametric region 1 ≤ d for different values of the bifurcation parameter 4fc.

( 2 )
If d < 1 and (d − 1) 2 − 4fc � 0, then the positive equilibrium point E * is nonhyperbolic of unstable type.(3) If d < 1 and (d − 1) 2 − 4fc > 0, then the equilibrium point E − is a repeller while the stability of E + is subject to the following conditions:
) ∪ W s (E 8 ) converges to E 1 , and all solutions which start below W s (E 4 ) ∪ W s (E 9 ) converge to E 5 .
. , E 7 such that five belongs to the west and south boundaries of the region R and two fixed points are interior points.Moreover, assuming that E 1 and E 2 belong to the west boundary, E 3 is the south-west corner of the region R, and E 4 and E 5 are on the south boundary of R such that E 1 ⪯ se E 2 ⪯ se E 3 ⪯ se E 4 ⪯ se E 5 .Moreover, assume 2 Discrete Dynamics in Nature and Society that E 6 ⪯ ne E 7 and that E 6 ∉ [E 2 , E 4 ] and E 7 ∈ [E 1 , E 5 ].Finally, assume that E 1 , E 3 , E 5 are locally asymptotically stable, E 6 is a repeller, and E 2 , E 4 are saddle points.If E 7 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 W s (E 2 ) and W s (E 4 ) converge to E 3 and all solutions which start between the stable manifolds W s (E 2 ) and W s (E 7 ) converge to E 1 and all solutions which start between the stable manifolds W s (E 4 ) and W s (E � W + S x , where S x � W s P 1 y   ∪ W s E * ; iii) ere exist a global unstable manifold W u (E + ) which is a graph of a decreasing function contained in Q 2 (E + ) ∪ Q 4 (E + ) with endpoints P 2x and P 2 y .(iv) e basin of attraction of E 0 is B E 0  � W − P 1 −and is the basin of attraction of the equilibrium E + .( Figure 8: All invariant manifolds of eorem 10.Figure 9: Visual representation of global dynamics of eorem 11. and which are tangential at the equilibrium point E − .(v)ebasin of attraction of equilibrium point E 0 is the region between those stable manifolds, i.e., B E 0  � W − P 1 (vi) ere exists a global stable manifold W s (P i ∓ ) which is an unbounded curve and the graph of an increasing function contained inQ 1 (P i ∓ ) ∪ Q 3 (P i ∓ ) with an endpoint at E − .W s (P i ∓ ) isthe basin of attraction of the equilibrium P i ∓ .(vii) ere exists a global stable manifold W s (P i ± ) which is an unbounded curve and the graph of an increasing function contained in Q 1 (P i ± ) ∪ Q 3 (P i ± ) with an endpoint at E − .W s (P i ± ) is the basin of attraction of the equilibrium P i ± .(viii) e basin of attraction of the equilibrium point E + is the region between those stable manifolds, i.e., B E +  � W − P i (ix) ere exist a global unstable manifold W u (P i ∓ ) which is the graph of a decreasing function contained in Q 2 (P i ∓ ) ∪ Q 4 (P i ∓ ) with endpoints P 2 y and E + .(x) ere exist a global unstable manifold W u (P i ± ) which is the graph of a decreasing function contained in Q 2 (P i ± ) ∪ Q 4 (P i ± ) with endpoints P 2 x and E + .(xi) e basins of attraction of P 2 x and P 2 y are given by ± are saddle points.(iv)ere exist global stable manifolds W s (P 1x ) and W s (P 1 y ) which are the basins of attraction of the x   � W + S x , where S x � W s P 1 x   ∪ W s P i ±  , B P 2 y   � W − S y  , where S y � W s P 1 y   ∪ W s P i ±  ;