Global Dynamics of Some Quadratic Difference Equations

Consider the difference equation xn+1 = α + ∑k i=0 aixn−i + ∑k i=0 ∑k j=i aijxn−ixn−j β + ∑k i=0 bixn−i + ∑k i=0 ∑k j=i bijxn−ixn−j , n = 0, 1, . . . where all parameters α, β, ai, bi, aij, bij, i, j = 0, 1, . . . , k and the initial conditions xi, i ∈ {−k, . . . , 0} are nonnegative. We investigate the asymptotic behavior of the solutions of the considered equation. We give simple explicit conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation. We investigate the global dynamics of several anti-competitive systems of rational difference equations which are special cases of general linear fractional system of the form xn+1 = α1 + β1xn + γ1yn A1 +B1xn + C1yn , yn+1 = α2 + β2xn + γ2yn A2 +B2xn + C2yn , n = 0, 1, ..., where all parameters and the initial conditions x0, y0 are arbitrary nonnegative numbers such that both denominators are positive. We find the basins of attraction of all attractors of these systems. We investigate global dynamics of the equation xn+1 = xn−1 axn + cx 2 n−1 + f , n = 0, 1, 2, ..., where the parameters a, c and f are nonnegative numbers with condition a+e+f > 0 and the initial conditions x−1, x0 are arbritary nonnegative numbers such that x−1 + x0 > 0.

where all parameters α, β, a i , b i , a ij , b ij , i, j = 0, 1, . . . , k and the initial conditions We investigate the global dynamics of several anti-competitive systems of rational difference equations which are special cases of general linear fractional system of the form x n+1 = α 1 + β 1 x n + γ 1 y n A 1 + B 1 x n + C 1 y n , y n+1 = α 2 + β 2 x n + γ 2 y n A 2 + B 2 x n + C 2 y n , n = 0, 1, ..., where all parameters and the initial conditions x 0 , y 0 are arbitrary nonnegative numbers such that both denominators are positive. We find the basins of attraction of all attractors of these systems.
We investigate global dynamics of the equation I would also like to thank professor Orlando Merino and the difference equations group at URI, their support and feedback has been invaluable.

Introduction
Consider the difference equation where k ∈ {0, 1, . . .}, the parameters α, β, a i , b i , a ij , b ij , i, j = 0, 1, . . . , k and the initial conditions x i , i ∈ {−k, . . . , 0} are nonnegative and such that the denominator of Eq.(1) is always positive. The important special cases of Eq.(1) are the linear fractional equations such as well-known Riccati equation the second order linear fractional difference equation and the third order linear fractional difference equation that we get from Eq. (1) for k = 2 and a ij = b ij = 0 for all i, j. The global behavior and the exact solutions of Eq.(2) even for real parameters have been found in [18]. The global behavior of solutions of Eq.(3), in many subcases when one or more parameters are zero, was established in [18]. There is still one conjecture left whose answer will complete the global picture of the asymptotic behavior of solutions of Eq. (3). As far as the third order linear fractional difference equation is concerned there is a large number of sporadic results that are systemized in a book [6]. The characterization of the global asymptotic behavior of solutions of Eq.(1) for k = 2 seems to be much harder than for the second order equation (3). Consequently an attempt at giving the characterization of the global asymptotic behavior of solutions of Eq. (1) seems to be a formidable task at this time. However by using some known global attractivity results we can describe the global asymptotic behavior of solutions of Eq.(1) in some subspaces of the parametric space and the space of initial conditions. See [4,5,6,21] for a complete description of the behavior of some special cases of Eq.(1), in particular for the cases known as periodic trichotomies. See [14] where the difference in global behavior between the second and third order linear fractional difference equation is emphasized. The results on the global periodicity, that is the results which describe all special cases of Eq. (1) where all solutions are periodic of the same period were obtained in [1,2]. Most results in [4,5,6,10,11,21] are based on known global attractivity or global asymptotic stability results obtained in [6,17,18,19,20,22].
The special case of Eq.(1) with quadratic terms such as where a, b > 0 and the initial conditions A k-th order generalization of Eq.(4) with the same property is where a, b i > 0 and the initial conditions x −k+1 , . . . , x 0 ≥ 0, x −k+1 + . . . + x 0 > 0, when a ≤ 1.
Another special case of Eq.(1) with quadratic terms is x n+1 = Bx n x n−1 dx n + ex n−1 , n = 0, 1, . . . , where B, d, e > 0 and the initial conditions Finally, when B = d + e, then Eq. (6) has an infinite number of the equilibrium points, each with its basin of attraction, see [13].
Another interesting case of Eq.(1) with quadratic terms is the equation where a > 0 and x 0 ∈ R. When a > 2 every solution of Eq. None of these asymptotic behaviors which are present in the cases of Equations (4)- (7) are possible in the case of the linear fractional equation The following general global results will be applied to Eq.(1), see [12].
As we have observed in [12], condition (11) is actually a contraction condition in the Banach contraction principle.
In addition, we will need the following stability result from [3].
Theorem 2 Suppose that Eq.(8) can be linearized into the form wherex is an equilibrium of Eq.(8) and the functions g i : R k+1 → R. If k i=0 |g i | ≤ 1, n ≥ 0, then the equilibriumx of Eq.(8) is stable.
The next result follows from Theorems 1 and 2.
Corollary 1 Let a ∈ R. Suppose that Eq.(8) has the linearization (12), wherex is a unique nonnegative equilibrium of Eq.(8) on the interval I and the functions then the unique nonnegative equilibrium Eq.(8) is globally asymptotically stable on the interval I.
The next result is an analogue of the result obtained in [12].

Main results
In this section we investigate the stability of the unique positive equilibriumx of Eq.(1) by using Theorems 1 and 2. Observe that Eq.(1) has a zero equilibrium if and only if α = 0 and β > 0, in which case Eq.(1) becomes Equation (13) yields the nonzero equilibrium points k j=i a ij , then there is no positive equilibrium. The following result shows that there is an interval in which every solution of Eq. (13) converges to the zero equilibrium. For convenience of notation let Q denote the denominator of Eq.(1), that is Assume that there is no positive equilibrium. Then the zero equilibrium of Eq. (13) is globally asymptotically stable on the interval [0, M).
Hence x i ≤ M for i = 1, 0, . . . , −k. Then for n = 1 Again using Lemma 1 withx = 0, h i = |g i |, i = 0, . . . , k and N = 1 we get By induction we get that for n ≥ 0 and so the result follows from Corollary 1 wherex = 0 and the interval is All equilibrium solutions of Eq.(1) satisfy the equilibrium equation which can be rewritten as Equilibrium equation (14) has at least one nonnegative zero and it may have between 0 and 3 positive zeros. When α > 0 and either Descartes rule of sign implies that there is a unique positive equilibrium of Eq.(1).
Ifx > 0 is an equilibrium, then for n ≥ 0 where in view of Eqs. (14), (15) Now applying the identity we get that for n ≥ 0 Observe that for n ≥ 0 Thus for n ≥ 0 Therefore for n ≥ 0 Equation (16) is the linearized equation of Eq.(1) of the form (12) where for i = 0, . . . , k We can now obtain easy-to-check conditions which show when the positive equilibrium of Eq.(1) is globally asymptotically stable. We will then apply these conditions to varius cases of Eq.(1).
Theorem 4 Assume that Eq.(1) has a unique positive equilibriumx and there exist L ≥ 0 and U, N > 0 such that for every solution {x n } of Eq.

Proof.
As we have seen Eq.(1) can be written in the form of the linearized equation (16), where the coefficients g i are given as (17).
We have for n ≥ 0 and so for i = 0, . . . , k and n ≥ 0 Thus for n ≥ 0 By rearranging the terms we can show that for n ≥ 0 In view of (18) and (19), we obtain  will give us a better result. We present some of these cases.

Remark 1
The results on boundedness of all solutions of Eq.(1) are well known, see [6,7]. For instance, if for every i, j ∈ {0, . . . , k} such that b i > 0, b ij > 0 we have a i > 0, a ij > 0, then the uniform lower bound L for all solutions {x n } of On the other hand, if for every i, j ∈ {0, . . . , k} such that a i , a ij > 0 we have b i , b ij > 0, then the uniform lower bound L for all solutions of Eq.(1) for n ≥ 1, is The next result follows from Lemma 1 and can be used to find the part of the basin of attraction of a positive equilibrium in the case when there are several positive equilibrium points. The proof of this result is similar to the proof of Theorem 3 and it will be omitted.
The following result is a consequence of Theorem 4 in some special cases when the unique positive equilibrium satisfies some specific conditions. (2) a i ≥xb i , a ij ≥xb ij for all i, j ∈ {0, . . . , k} and α ≥ 0 and (3) a i ≤xb i , a ij ≤xb ij for all i, j ∈ {0, . . . , k} and Proof.

Now the condition (18) is simplified to
and the result follows from Theorem 4.
(3) In this case we have In view of our assumption and so the result follows from Theorem 4.
Similarly define Where Then the positive equilibriumx of Eq.(1) is globally asymptotically stable on the interval [0, ∞).
Proof. In view of the equilibrium equation (15) and by assumption (c), we have Now, the conclusion follows from Theorem 4. 2 In the case of general second order quadratic fractional equation of the form with nonnegative parameters and initial conditions such that A + B + C > 0, and ax 2 n + bx n x n−1 + cx 2 n−1 + dx n + ex n−1 + f > 0, n = 0, 1, . . ., the obtained results take the following form.
Corollary 3 Assume that Eq.(25) has the unique positive equilibriumx. If the following condition holds where L and U are lower and upper bounds of all solutions of Eq.(25) and f +L > 0, thenx is globally asymptotically stable on the interval [0, ∞).
In the special case of second order equation with quadratic terms only we obtain the following result.
Corollary 4 Consider the following equation with all positive parameters and nonnegative initial conditions such that ax 2 n + bx n x n−1 + cx 2 n−1 > 0 for all n ≥ 0. If the following condition holds then the unique equilibriumx of Eq. (27) is globally asymptotically stable on the interval [0, ∞). (18), (20), (21), (22), and (24) is replaced by equality, then the conclusions of Theorems 4, 5, 6, and Corollary 2 should be changed from global asymptotic stability to stability.

Introduction
A first order system of difference equations where is non-decreasing in x and non-increasing in y, and g (x, y) is non-increasing in x and non-decreasing in y.
System (28) where the functions f and g have monotonic character opposite of the monotonic character in competitive system is called anti-competitive, see [11,18].
In this paper, we consider the following anti-competitive systems of difference equations where all parameters are positive numbers and the initial conditions (x 0 , y 0 ) are arbitrary nonnegative numbers such that x 0 + y 0 > 0. In the classification of all linear fractional systems in [3], System (29) was mentioned as system (18,18). We also consider systems and with x 0 > 0, y 0 > 0, which were labeled as systems (18,23) and (16,23), respectively in [3]. Three systems have interesting and different dynamics. While System (29) has all bounded solutions most of solutions of Systems (30) and (31) are unbounded. Another major difference is the existence of the unique period-two solution for (29) and, in a special case, abundance of such solutions while neither (30) nor (31) has period-two solutions. We show that every solution of System (30) converge to the unique equilibrium or is approaching (0, ∞) and so System (30) gives an example of a semistable equilibrium point. Finally, we show that all solutions of (31) which start on the stable set converge to the unique equilibrium, while all solutions which start off the stable set are approaching (0, ∞) or (∞, 0).
We also get that for some special values of parameters Systems (31) and (30) can be decoupled and explicitly solved.
The study of anti-competitive systems started in [11] and has advanced since then, see [18]. The principal tool of study of anti-competitive systems is the fact that the second iterate of the map associated with anti-competitive system is competitive map and so elaborate theory for such maps developed recently in [9,14,15] can be applied.
The major result on a global behavior of System (29) is the following theorem.
Theorem 7 (a) Assume that B 1 < B 2 . Then the unique positive equilibrium (29) is locally asymptotically stable with the basin } is a saddle point and its basin of attraction is the union of coordinate axes without the origin, that is, Then every solution of System (29) converges to the period-two solution {P 1 , P 2 } or to the equilibrium E. More precisely, there The set C has the property that for Then every solution of System (29) is equal to either the period-two solution

Preliminaries
We now give some basic notions about competitive systems and maps in the plane of the form (28) where f and g are continuous functions and f (x, y) is nondecreasing in x and non-increasing in y and g(x, y) is non-increasing in x and non-decreasing in y in some domain A with non-empty interior.
has a neighborhood with no other fixed points in it. A fixed point E ∈ R is an attractor if there exists a neighborhood U of E such that T n (x) → E as n → ∞ for x ∈ U; the basin of attraction is the set of all x ∈ R such that T n (x) → E as n → ∞. A fixed point E is a global attractor on a set K if E is an attractor and K is a subset of the basin of attraction of E. If T is differentiable at a fixed point E, and if the Jacobian J T (E) has one eigenvalue with modulus less than one and a second eigenvalue with modulus greater than one, E is said to be a saddle. See [19] for additional definitions.
Definition 1 Let T = (f, g) be a continuously differentiable vector function and let U be a neighborhood of a saddle point (x,ȳ) of (28). The local stable manifold W s loc is the set The map T may be viewed as a monotone map if we define a partial order on R 2 so that the positive cone in this new partial order is the fourth (resp. first) quadrant. Define a South-east (resp. North-east) partial order se (resp. ne ) on R 2 so that the positive cone is the fourth quadrant (resp. first quadrant), that is, ) if and only if x 1 ≤ x 2 and y 1 ≥ y 2 (resp. x 1 ≤ x 2 and y 1 ≤ y 2 ). For x, y ∈ R 2 the order interval x, y is the set of all z such that x z y. The map T is called competitive (resp. cooperative) on for all v, w ∈ Int R 2 + . Clearly, being related is invariant under iteration of a strongly monotone map. Differentiable strongly monotone maps have Jacobian with constant sign configuration The mean value theorem and the convexity of R 2 + may be used to show that T is monotone, as in [6].
.., 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction, for example, We now state three results for competitive maps in the plane.
The following definition is from [20].
The following theorem was proved by de Mottoni-Schiaffino for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations.
Smith generalized the proof to competitive and cooperative maps [20,21]. the orbit of x has compact closure in S, then its omega limit set is either a periodtwo orbit or a fixed point.
The following result is from [20], with the domain of the map specialized to be the cartesian product of intervals of real numbers. It gives a sufficient condition for conditions (O+) and (O−).
Theorem 9 Let R ⊂ R 2 be the cartesian product of two intervals in R. Let The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess, see [15,16] and [8], and is helpful for determining the basins of attraction of the equilibrium points.
Corollary 5 If the nonnegative cone with respect to the partial order is a generalized quadrant in R 2 , and if the competitive map T :→ R 2 has no fixed points in u 1 , u 2 other than u 1 and u 2 , then the interior of u 1 , u 2 is either a subset of the basin of attraction of u 1 or a subset of the basin of attraction of u 2 .
Theorem 10 Let T be a competitive map on a rectangular region R ⊂ R 2 . Let x ∈ R be a fixed point of T such that ∆ : x is not the NW or SE vertex of R), and T is strongly competitive on ∆.
Suppose that the following statements are true.
a. The map T has a C 1 extension to a neighborhood of x.
b. The Jacobian matrix of T at x has real eigenvalues λ, µ such that 0 < |λ| < µ, where |λ| < 1, and the eigenspace E λ associated with λ is not a coordinate axis.
Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E λ at x, and C is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of C in the interior of R are either fixed points or minimal period-two points. In the latter case, the set of endpoints of C is a minimal period-two orbit of T .
Theorem 11 Let I 1 , I 2 be intervals in R with endpoints a 1 , a 2 and b 1, b 2 with endpoints respectively, with a 1 < a 2 and b Let T be a competitive map on an rectangle R = I 1 × I 2 and x ∈ int (R). Suppose that the following hypotheses are satisfied: 3. The map T is continuously differentiable in a neighborhood of x, and x is the saddle point.
4. At least one of the following statemets is true.
a. T has no minimal period two orbits in Then the following statements are true.
(i.) The stable manifold W s (x) is connected and it is the graph of a continuous increasing curve with endpoints in ∂R. int (R) is divided by the closure of W s (x) into two invariant connected regions W + ("below the stable set"), and W − ("above the stable set"), where (ii.)The unstable manifold W u (x) is connected and it is the graph of a continuous decreasing curve.
(iii.) For every x ∈ W + , T n (x) eventually enters the interior of the invariant set Q 4 (x) ∩ R, and for every x ∈ W − , T n (x) eventually enters the interior of the For every x ∈ W − and every z ∈ R such that m se z, there exists m ∈ N such that T m (x) se z, and for every x ∈ W + and every z ∈ R such that z se M, there exists m ∈ N such that M se T m (x). (29) The equilibrium point E(x,ȳ) of System (29) satisfies the following system of

Global Dynamics of System
It is easy to see that System (32) has unique equilibrium point E in the first quadrant, for all values of the parameters. Indeed, the positive equilibrium point is an intersection of the following two curves and It is clear that at the point of intersection E curve (33) is steeper than curve (34), that is, This inequality is equivalent to the following inequality which is always satisfied.

Linearized Stability Analysis of System (29)
In this section we present the linearized stability analysis of the equilibrium E of System (29).
Theorem 12 i) If B 1 < B 2 , then E is locally asymptotically stable.
ii) If B 1 = B 2 , then E is a non-hyperbolic equilibrium point.
iii) If B 1 > B 2 , then E is a saddle point.
Proof. The map T associated to System (29) is The Jacobian matrix of map (35) is and evaluated at the equilibrium point E = (x,ȳ) is The characteristic equation has the following form and the characteristic roots are A straight-forward calculation shows that the conditions λ 2 ∈ (−1, 0), λ 2 = −1, λ 2 ∈ (−∞, −1) respectively, are equivalent to the conditions On the other hand, by dividing two equilibrium equations (32) we obtain which implies that the condition (39) is equivalent to the condition which completes the proof of theorem. Lemma 2 System (29) has the minimal period-two solution Proof. The second iterate of the map T is given as A period-two solution of System (29) satisfies T 2 (x, y) = (x, y), which immediately leads to the following equations which have either unique solution if B 1 = B 2 or it has infinitely many solutions if In the first case, Eq.(41) gives immediately Eq.(33) and Eq.(42) gives (34), which means that in this case the only minimal period-two solution is (40).
A straight-forward calculation shows that T (P 1 ) = P 2 , T (P 2 ) = P 1 , which shows that {P 1 , P 2 } is a minimal period-two solution. Moreover, T ((a, 0) = P 2 , T (0, b) = P 1 for every a > 0, b > 0, which shows that the set B is a subset of the basin of attraction of {P 1 , P 2 }. The Jacobian matrix of T 2 is Where, The Jacobian matrix of T 2 evaluated at P 1 is and the Jacobian matrix of T 2 evaluated at P 2 is In both cases the eigenvalues of the Jacobian matrix of T 2 are λ 1 = 0, λ 2 = B 2 B 1 , which implies the result on local stability of the minimal period-two solution {P 1 , P 2 }.
Proof of Theorem 7. First, observe that the rectangle R = [0, γ 1 ] × 0, β 2 B 2 \ {0, 0} = P 2 , P 1 \ {0, 0} is an invariant and attracting set for the map T and so is for the map T 2 . More precisely, (x n , y n ) ∈ R for n ≥ 1. The map T 2 is competitive map on R.
Case (a). Assume that B 1 < B 2 . Then in view of Theorem 12 and Lemma 2 the map T 2 has three equilibrium points P 1 , P 2 and E where P 2 se E se P 1 .The equilibrium points P 1 and P 2 are saddle points and E is a local attractor. The ordered intervals P 2 , E and E, P 1 are both invariant sets of T 2 and in view of Corollary5 their interiors are attracted to E. If we take the point (x, y) ∈ R \ P 2 , E ∪ E, P 1 , we can find the points (x l , y l ) ∈ int P 2 , E and (x u , y u ) ∈ int E, P 1 , such that (x l , y l ) se (x, y) se (x u , y u ). Consequently, since T 2 is competitive T 2n ((x l , y l )) se T 2n ((x, y)) se T 2n ((x u , y u )) for n ≥ 1 and so lim n→∞ T 2n ((x, y)) = E, which by continuity of T implies that lim n→∞ T 2n+1 ((x, y)) = lim n→∞ T (T 2n ((x, y))) = T ( lim n→∞ T 2n ((x, y))) = T (E) = E, and so lim n→∞ T n ((x, y)) = E.
Case (b). Assume that B 1 > B 2 . Then in view of Theorem 12 and Lemma 2 the map T 2 has three equilibrium points P 1 and P 2 which are local attractors and E which is a saddle point. The ordered intervals P 2 , E and E, P 1 are both invariant sets for T 2 and in view of Corollary5 their interiors are attracted to P 2 and P 1 respectively. In view of Theorems 10 and 11 there is the set C with described properties. Direct calculation shows that the half-line y =ȳ x x, x > 0 is an invariant set which in view of a uniqueness of stable manifold implies that this half-line is exactly stable manifold mentioned in Theorems 10 and 11. It should be observed that because of the fact that one of the characteristic values at the equilibrium point E is 0, this equilibrium is super-attractive, that is, T (x 0 , y 0 ) = (x,ȳ), for every (x 0 , y 0 ) ∈ C.
Case (c). Assume that B 1 = B 2 . Then by dividing two equations of System (29) we obtain that the solution of (29) satisfies This means that yn xn satisfies first order difference equation u n+1 = D un , where D = which completes the proof of Case (c). (29) is an example of the homogeneous system which is a special case of a general System (28) where both functions f and g are homogeneous functions of the same degree k, that is f (tu, tv) = t k f (u, v), g(tu, tv) = t k g(u, v) for all u, v in intersection of domains of f and g and all t = 0. In that case, the ratio z n = y n /x n of every solution of (28) satisfies the first order difference equation x n+1 = α 1 + β 1 x n + γ 1 y n A 1 + B 1 x n + C 1 y n , y n+1 = α 2 + β 2 x n + γ 2 y n A 2 + B 2 x n + C 2 y n , n = 0, 1, ...,

Remark 3 System
where all parameters and the initial conditions (x 0 , y 0 ) are arbitrary nonnegative numbers such that A i + B i x 0 + C i y 0 > 0, i = 1, 2, was first used in [4] and was systematically developed in the recent paper [10]. In [10], the authors studied all possible homogeneous systems of the form (43) and they proved that every bounded solution converges to either an equilibrium solution or to period-two solution. They were able to find a part of the basin of attraction of the period-two solution but not the complete basin of attraction. In the case of system (29) the auxiliary equation

Since F is decreasing every solution of the auxiliary equation is approaching not
necessarily minimal period-two solution. Further analysis can be continued either by checking negative feedback condition for F 2 or by using Theorem 3.2 from [10].
In neither case the complete description of the basins of attraction of the equilibrium and the period-two solution is possible. We prefer our approach because it is more precise and also apply equally well to anti-competitive systems which are not homogeneous. The approach which is making use of homogeneous properties of functions is applicable also to the systems which are neither competitive nor anti-competitive.

Global Dynamics of System (30)
The equilibrium point E(x,ȳ) of System (30) satisfies the following system of It is easy to see that System (44) has the unique equilibrium point E in the first quadrant, for all values of the parameters. Indeed, the positive equilibrium point is an intersection of the following two curves and It is clear that at the point of intersection E curve (45) is steeper than curve (46), that is, This inequality is equivalent to the following inequality which in turn is equivalent to

Linearized Stability Analysis of System (30)
In this section we prove the following result

Theorem 13
The unique equilibrium E of System (30) is a saddle point.
Proof. The map S associated to System (30) is The Jacobian matrix of map (48) is and evaluated at the equilibrium point E = (x,ȳ) is The characteristic equation has the following form Set Then the necessary and sufficient condition for Eq.(51) to have one root inside the unit circle and one root outside the unit circle is |P | > |1 + Q|, P 2 − 4Q > 0, see [12,13]. The condition |P | > |1 + Q| leads to P > −1 − Q which is equivalent to which is condition (47).
The condition P 2 − 4Q > 0 becomes which is equivalent to which is clearly satisfied. 2

Global Results for System (30)
Lemma 3 System (30) has no minimal period-two solution.
Proof. The second iterate of the map S is given as S 2 (x, y) = S 1 , S 2 .
(i) We will prove that S is injective and the injectivity of S 2 will follow immediately. The condition S(x 1 , y 1 ) = S(x 2 , y 2 ) is reduced to the following two conditions which immediately implies y 1 = y 2 and so x 1 = x 2 .
(ii) A direct calculation shows that which implies our statement.
The statement on (O+) condition follows from Theorem 9.  The equilibrium point E(x,ȳ) of System (31) satisfies the following system of It is easy to see that System (54) has the unique equilibrium point E in the first quadrant, which is an intersection of two parabolas: and Proof. The map S associated to System (31) is The Jacobian matrix of map (58) has the form which evaluated at the equilibrium point E = (x,ȳ) is The characteristic equation of System (31) has the following form .
The necessary and sufficient condition for Eq.(61) to have one root inside the unit circle and one root outside the unit circle is |P | > |1+Q|, P 2 −4Q > 0, see [12,13].
In view of the fact that P > 1+Q, the condition |P | > |1+Q| leads to P > −1−Q which is equivalent to which is equivalent to the condition (57).
The condition P 2 − 4Q > 0 becomes which is equivalent to which is clearly satisfied. Proof. The second iterate of the map U is given as Where, Period-two solution satisfies U 2 (x, y) = (x, y) which reduces to the following two equations Equation (63)  This shows that System (31) has no minimal period-two solution. 2

Lemma 6
The maps U and U 2 associated with System (31) have the following properties.
(i) If A 1 β 2 = α 2 , then the maps U and U 2 are injective.
(i) We will prove that U is injective which will imply the injectivity of U 2 . The is reduced to the following two conditions This implies y 1 = y 2 and so x 1 = x 2 .
(ii) A direct calculation shows that which proves our statement.
The statement on (O+) condition follows from Theorem 9. can be decoupled and written as Proof. Using the condition A 1 β 2 = α 2 in second equation of System (31) gives and so x n+1 y n+1 = β 2 γ 1 which shows that System (31) has an invariant of the form x n y n = β 2 γ 1 , n = 1, 2, . . . . For example, the unique equilibriumx of first equation of System (64) satisfies equation while period-two solution satisfies f 2 (x) = x which becomes x 2 (A+x) 2 A 2 x+Ax 2 +C = x and so is reduced to Eq.(66). Thus every bounded solution converges to the unique equilibrium.
The result for unbounded solutions follows immediately from (65).

Introduction
In this paper, we investigate global behavior of the equation where the parameters a, c and f are nonnegative numbers with condition a + c > 0, f = 0 and the initial conditions x −1 , x 0 are arbritrary nonnegative numbers (67) is a special case of equations and x n+1 = Ax 2 n + Bx n x n−1 + Cx 2 n−1 + Dx n + Ex n−1 + F ax 2 n + bx n x n−1 + cx 2 n−1 + dx n + ex n−1 + f , n = 0, 1, 2, ....
Some special cases of Eq.(69) have been considered in the series of papers [4,5,10,12,13,24]. Some special second order quadratic fractional difference equations have appeared in analysis of competitive and anti-competitive systems of linear fractional difference equations in the plane, see [7,11,21]. In this paper we take an approach based on the theory of monotone maps developed in [16,17] and use it to describe precisely the basins of attraction of all attractors of this equation as well as all undergoing bifurcations. The special case of Eq.(69) for a = 0 is well known Thomson equation [2] used in the modeling of fish population [25].
The presence of quadratic terms in Eq.(67) effects the dynamic behavior of the corresponding linear fractional equation in an interesting way by introducing new dynamic phenomena such as the coexistence of two locally stable equilibrium points, the coexistence of locally stable equilibrium point and minimal period-two solutions and existence of so called Allee effect. The global dynamics of Eq.(67), in the case when f > 1 4c , is quite simple as the zero equilibrium is globally asymptotically stable and interesting dynamics happens when f ≤ 1 4c . The special case of Eq.(67) which plays an important role in mathematical biology is the following equation where c and f are positive numbers and the initial conditions x −1 , x 0 are arbitrary nonnegative numbers. The dynamics of (71), described in [18], is very interesting and follows from the dynamics of related equation first considered by Thomson [25] x n+1 = x 2 n cx 2 n + f n = 0, 1, . . .
and completely described in [2]. Equation (71) has either one or three period-two solutions and all its solutions are bounded and so its dynamics is considerably richer and more interesting from the modeling point of view.

Theorem 18
1. If f > 1 4c , then (71) has one equilibrium E 0 which is globally asymptotically stable with the basins of attraction 2. If f = 1 4c , then (71) has two equilibrium points E 0 , E 1 2c , 1 2c and the minimal period-two solution {P x , P y } = The basins of attraction of the equilibrium points and the period-two solution are given as 3. If f < 1 4c , then (71) has three equilibrium points , and E + and three minimal period-two solutions .
The corresponding basins of attraction are given as Thus Eq.(67) has some potential applications in mathematical modeling.
The local and global dynamics of Eq.(67) depends on the location of the parameter f . We obtain seven different dynamics scenarios for local and global dynamics of Eq.(67) vs. the parameters c−3a 4(a−c) 2 , 1 4(a+c) and 1 4c . The large role in dynamical scenarios is played by the equilibrium solutions and period-two solutions.
The global dynamics can be explained in terms of bifurcation theory for the second iterate of the corresponding map T 2 . Since both the equilibrium solutions and the period-two solutions are the equilibrium solutions of the second iterate of the corresponding map, we can express our results as a change of stability bifurcations for T 2 . Figure 1 gives a visual representation of local dynamics of Eq.(67).

Preliminaries
We use the following theorem for a general second order difference equation the following result is from [3]. Theorem 19 tells us that every bounded solution of (73) converges to either an equilibrium or a period-two solution or to the point on the boundary where equation is not defined, see [1,9]. To determine the basins of attraction of these solutions, we will use theory of monotone maps in the plane.
Consider a partial ordering on R 2 . Two points x, y ∈ R 2 are said to be related if x y or y x. Also, a strict inequality between points may be defined as x ≺ y if x y and x = y. A stronger inequality may be defined as x = (x 1 , x 2 ) y = (y 1 , y 2 ) if x y with x 1 = y 1 and x 2 = y 2 .
A map T on a nonempty set R ⊂ R 2 is a continuous function T : R → R.
The map T is monotone if x y implies T (x) T (y) for all x, y ∈ R, and it is strongly monotone on R if x ≺ y implies that T (x) T (y) for all x, y ∈ R. The map is strictly monotone on R if x ≺ y implies that T (x) ≺ T (y) for all x, y ∈ R.
Clearly, being related is invariant under iteration of a strongly monotone map.
In this paper we shall use the North-East ordering (NE) for which the positive cone is the first quadrant, i.e. this partial ordering is defined by (x 1 , y 1 ) ne (x 2 , y 2 ) if x 1 ≤ x 2 and y 1 ≤ y 2 and the South-East (SE) ordering defined as (x 1 , y 1 ) se (x 2 , y 2 ) if x 1 ≤ x 2 and y 1 ≥ y 2 .
A map T on a nonempty set R ⊂ R 2 which is monotone with respect to the North-East (NE) ordering is called cooperative and a map monotone with respect to the South-East (SE) ordering is called competitive. A map T on a nonempty set R ⊂ R 2 which second iterate T 2 is monotone with respect to the North-East (resp. South-East) ordering is called anti-cooperative (resp. anti-competitive), see [11].
If T is differentiable map on a nonempty set R, a sufficient condition for T to be strongly monotone (resp. anti-monotone) with respect to the SE ordering is that the Jacobian matrix at all points x has the sign configuration provided that R is open and convex.
For (x 1 , x 2 ) ∈ R 2 , define Q (x 1 , x 2 ) for = 1, 2, 3, 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction, for example, Basin of attraction of a fixed point (x,ȳ) of a map T , denoted as B((x,ȳ)), is defined as the set of all initial points (x 0 , y 0 ) for which the sequence of iterates T n ((x 0 , y 0 )) converges to (x,ȳ).
Similarly, we define a basin of attraction of a periodic point of period p.
The following results, from [17,16], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by H. L. Smith in [23,24].
Theorem 20 Let R be a rectangular subset of R 2 and let T be a competitive map on R. Let x ∈ R be a fixed point of T such that (Q 1 (x) ∪ Q 3 (x)) ∩ R has nonempty interior (i.e., x is not the NW or SE vertex of R).
Suppose that the following statements are true.
b. T is C 2 on a relative neighborhood of x.
c. The Jacobian matrix of T at x has real eigenvalues λ, µ such that |λ| < µ, where λ is stable and the eigenspace E λ associated with λ is not a coordinate axis.
d. Either λ ≥ 0 and or λ < 0 and Then there exists a curve C in R such that (i) C is invariant and a subset of W s (x).
(ii) the endpoints of C lie on ∂R.
(iv) C the graph of a strictly increasing continuous function of the first variable, and in all cases C is tangential to E λ at x, (vi) C separates R into two connected components, namely W − := {x ∈ R : ∃y ∈ C with x y} and W + := {x ∈ R : ∃y ∈ C with y x} (vii) W − is invariant, and dist(T n (x), Q 2 (x)) → 0 as n → ∞ for every x ∈ W − .

Theorem 21
For the curve C of Theorem 20 to have endpoints in ∂R, it is sufficient that at least one of the following conditions is satisfied.
i. The map T has no fixed points nor periodic points of minimal period two in ∆.
ii. The map T has no fixed points in ∆, det J T (x) > 0, and T (x) = x has no solutions x ∈ ∆.
iii. The map T has no points of minimal period-two in ∆, det J T (x) < 0, and T (x) = x has no solutions x ∈ ∆.
Theorem 22 (A) Assume the hypotheses of Theorem 20, and let C be the curve whose existence is guaranteed by Theorem 20. If the endpoints of C belong to ∂R, then C separates R into two connected components, namely W − := {x ∈ R \ C : ∃y ∈ C with x se y} and (75) such that the following statements are true.
(B) If, in addition to the hypotheses of part (A), x is an interior point of R and T is C 2 and strongly competitive in a neighborhood of x, then T has no periodic points in the boundary of Q 1 (x) ∪ Q 3 (x) except for x, and the following statements are true.
(iii) For every x ∈ W − there exists n 0 ∈ N such that T n (x) ∈ int Q 2 (x) for n ≥ n 0 .
(iv) For every x ∈ W + there exists n 0 ∈ N such that T n (x) ∈ int Q 4 (x) for n ≥ n 0 .
If T is a map on a set R and if x is a fixed point of T , the stable set W s (x) of x is the set {x ∈ R : T n (x) → x} and unstable set W u (x) of x is the set When T is non-invertible, the set W s (x) may not be connected and made up of infinitely many curves, or W u (x) may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map.
If the map is a diffeomorphism on R, the sets W s (x) and W u (x) are actually the global stable and unstable manifolds of x.
Theorem 23 In addition to the hypotheses of part (B) of Theorem 22, suppose that µ > 1 and that the eigenspace E µ associated with µ is not a coordinate axis.
If the curve C of Theorem 20 has endpoints in ∂R, then C is the stable set W s (x) of x, and the unstable set W u (x) of x is a curve in R that is tangential to E µ at x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of W u (x) in R are fixed points of T .

Remark 4
We say that f (u, v) is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative D 1 f negative and first partial derivative D 2 f positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of Eq.(73) follows from 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 Eq.(73) is a strictly competitive map on I × I, see [17].
Set x n−1 = u n and x n = v n in Eq.(73) to obtain the equivalent system u)). The second iterate T 2 is given by and it is strictly competitive on I × I, see [17].

Remark 5
The characteristic equation of Eq.(73) at an equilibrium point (x,x): has two real roots λ, µ which satisfy λ < 0 < µ, and |λ| < µ, whenever f is strictly decreasing in first and increasing in second variable. Thus the applicability of Theorems 20-23 depends on the existence of minimal period-two solution.
Here W + (resp. W − ) denotes the region below (resp. above) the stable manifold W s in the North-east ordering.
The next result provides better understanding of the behavior of orbits in some cases (see [11]).
Lemma 7 Let T be an anti-competitive map on a rectangular region R ⊂ R 2 and let x ∈ R be a fixed point of T . The stable set W s (x) of x in Theorem 23 satisfies:

Local stability analysis
The equilibrium points x of Eq.(67) satisfy i.e.x = 0 and/or (a + c) Consequently, i) The zero equilibrium x = 0 exists for all values of the parameters.
ii) If f > 1 4(a+c) , the zero equilibrium, E 0 , is the only solution to the equilibrium equation.
iii) If f = 1 4(a+c) , there exists the positive equilibrium point E + = 1 2(a+c) . iv) If f < 1 4(a+c) , there exists two positive equilibrium points and E L = .
a) We have that so λ 1 = −1, λ 2 = − 2a+c (a+c) < −1 therefore the equilibrium point E + is nonhyperbolic of unstable type. b) In this case we have that |q| which is always true therefore |q| > 1.
a+c , which is always true, it follows that |p| < |1 − q| , therefore E L = In regards to the left hand side, we have . Which is always true.
In regards to the right hand side, we have Which is always true.
4(a−c) 2 (a+c) = c−3a 4(a−c) 2 , which is true in this case. Therefore |p| = |1 − q|, and we have that In regards to the left hand side, which is true in this case.
In regards to the right hand side, −4(a+c)c 2 , which is always true. Therefore the equilibrium point E U is locally asymptotically stable. c) Since, p = q = 0 for x = 0, we have λ 2 = 0 which means that E 0 is locally asymptotically stable for all value of parameters. ii) If f = 1 4c , then Eq.

Proof.
Suppose that there exists a minimal period-two solution {φ, ψ, φ, ψ, ...} of Eq.(67), where φ and ψ are distinct nonnegative real numbers such that φ 2 + ψ 2 = 0. Then φ, ψ satisfy the following system: which is equivalent to the system Thus we obtain that if φ = 0, then the second equation in (79) is of the form .
iv) If φ = 0 and ψ = 0, we have the following system which implies and In light of (81) and (82) we have no interior period two solutions if a ≥ c.
By substitution x n−1 = u n , x n = v n Eq.(67) becomes the system of equations The map T corresponding to (83) is of the form (84) The second iteration of the map T is where H (u, v) = v 2 ah (u, v) 2 + cv 2 + f , and the map T 2 is competitive by Remark 5. Now we obtain that the Jacobian matrix of the map T 2 at the point (φ, ψ) is of the form ∂H(φ,ψ) ∂u ∂H(φ,ψ) ∂v = 2 + 4aφψ. 2 Theorem 26 The local character of the period-two solutions is as follows: i) The minimal period-two solutions {P x , P y } are non-hyperbolic points of stable type.
ii) The minimal period-two solutions P 1 x , P 1 y are saddle points, P 2 x , P 2 y are locally asymptotically stable.
iii) The minimal period-two solutions P 3 ∓ , P 3 ± are saddle points. Proof.
i) The minimal period-two solutions {P x , P y } we have for f = 1 4c . In that case, the Jacobian matrix of the map T 2 at the points P x and P y is of the form J T 2 (P x ) = 1 0 0 0 , J T 2 (P y ) = 0 0 0 1 with the eigenvalues λ 1 = 0 and λ 2 = 1, which means that the periodic solutions {P x , P y } are non-hyperbolic points of stable type.
ii) The Jacobian matrix of the map T 2 at the points P 1 x , P 1 y is of the form Since λ 1 = 0 < 1 and λ 2 = 1+ √ 1 − 4cf > 1, it means that periodic solutions P 1 x , P 1 y are saddle points. The Jacobian matrix of the map T 2 at the points P 2 x , P 2 y is of the form Since λ 1 = 0 and λ 2 = 1 − √ 1 − 4cf ∈ (0, 1) it follows that the periodic solutions P 2 x , P 2 y are locally asymptotically stable.
iii) The Jacobian matrix of the map T 2 at the points P 3 ∓ , P 3 ± using (85)- (88) is of the form It follows that which means that P 3 ∓ , P 3 ± are saddle points.

Global dynamics of Eq.(67) in hyperbolic case
In this section, we present global dynamics results for Eq.(67).
Equation (67) is equivalent to the system of difference equations (83), which can be decomposed into the system of the even-indexed and odd-indexed terms as follows: Theorem 28 If 1 4(b+c) < f < 1 4c then Eq.(67) has one equilibrium point E 0 (0, 0) which is locally asymptotically stable and two minimal period-two solutions: x , P 1 y which are a saddle points and P 2 x , P 2 y which are locally asymptotically stable. There exist global stable manifolds W s (P 1 x ) and W s P 1 y which are basins of attraction of the periodic solutions P 1 x , P 1 y , and the unstable manifolds have the following form W u P 1 x = {(x, 0) : x ∈ I 1 ∪ I 2 } , W u P 1 y = {(0, y) : y ∈ I 1 ∪ I 2 } .
The basin of attraction of the equilibrium point E 0 = (0, 0) is the region between the global stable sets B(E 0 ) = W − P 1 x ∩ W + P 1 y .
The basin of attraction of the minimal period-two solutions P 2 x , P 2 y is given with the following Proof. The proof follows from Theorems 4 and 5 in [17], Theorem 24. 2 The next two results have the same proof as Theorems 8 and 9 in [18] and so it will be skipped.
There exist the global stable manifolds W s (P 1 x ) and W s P 1 y which are also basins of attraction of the minimal period-two solutions P 1 x , P 1 y , while the unstable manifolds have the following form W u P 1 x = {(x, 0) : x ∈ I 1 ∪ I 2 } , W u P 1 y = {(0, y) : y ∈ I 1 ∪ I 2 } .
The basin of attraction of the equilibrium point E 0 is the region between those stable manifolds i.e.
B (E 0 ) = W − P 1 x ∩ W + P 1 y , while the basin of attraction of the minimal period-two solutions P 2 x , P 2 y is given with B P 2 x = W + P 1 x , B P 2 y = W − P 1 y .
There exist global stable manifolds W s (P 1 x ), W s P 1 y and W s P 3 ∓ , W s P 3 ± which are also basins of attraction of the minimal period-two solutions P 1 x , P 1 y and P 3 ∓ , P 3 ± respectively. The basin of attraction of the equilibrium point E 0 is the region between the stable manifolds of minimal period-two solutions P 1 x , P 1 y , while the basin of attraction of the equilibrium point E + is the region between the stable manifolds of the minimal period-two solutions P 3 ∓ , P 3 ± i.e.
The basin of attraction of the minimal period-two solutions P 2 x , P 2 y is given with B P 2 y = W − (S y ) , where S y = W s P 1 y ∪ W s P 3 ∓ .