Global Dynamics of Certain Mix Monotone Difference Equation

We investigate global dynamics of the following second order rational difference equation xn+1 = xnxn−1+αxn+βxn−1 axnxn−1+bxn−1 , where the parameters α, β, a, b are positive real numbers and initial conditions x−1 and x0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.

which represents discretization of the differential equation model in biochemical networks, see [13].
Notice that Equation (3) is an example of a rational difference equation, such that associated map is always strictly decreasing with respect to the second variable, and changes its monotonicity with respect to the first variable, i.e., can be increasing or decreasing depending on corresponding parametric space.Also, we see that Equation (3) is the special case of the linear rational difference equation (which was investigated in detail in [12]) with well known and complicated dynamics, such as Lynes' equation (see [14]).
There are not many papers that study in detail dynamics of the second order rational difference equations with quadratic terms such that associated map changes its monotonicity with respect to its variables.However, in [15] the behavior of the following rational difference equation has been investigated in great detail In both equations, (3) and ( 5), Theorems 1 and 2 were used in order to obtain the convergence results.In most cases of this paper we use the same results.However, in order to investigate the behaviour of the following four subsequences where β > b a we cannot use this method, because the associated map in this case does not have the same monotonicity with respect to its variables in invariant interval.More precisely, the corresponding map changes its monotonicity in invariant interval with respect to the first variable.Instead of that, we use the brute-force method to show that each subsequence converges to the unique equilibrium point.
In the case when associated map of Equation (1) changes its monotonicity from "decreasing-decreasing" into "increasing-decreasing", the problem of determining invariant interval appears.In all cases, we determine invariant interval and prove that the positive equilibrium of Equation (1), which is always locally asymptotically stable, is globally asymptotically stable for all values of the parameters, except in the case when α > α D (see .
The problem of determining invariant intervals in the case when the associated map changes its monotonicity with respect to its variables has been considered in [16,17].Also, see [18][19][20].Now, we state several well-known results.
Assume that f satisfies the following two conditions: Then, (6) has a unique equilibrium x ∈ [a, b] and every solution of (6) converges to x.
Then, (6) has a unique equilibrium x ∈ [a, b] and every solution of (6) converges to x. Theorem 3. [21] [Theorem 1.4] Let f be the function from (6) with is nonincreasing in u and v respectively; 3. x f (x, x) is nondecreasing in x; 4. Equation ( 6) has a unique positive equilibrium x.
Let x be a positive equilibrium of Equation (6).Then, every oscillatory solution of Equation ( 6) has semicycles of length at most two.
] is such that: f (x, y) is increasing in x for each fixed y, and f (x, y) is decreasing in y for each fixed x.
Let x be a positive equilibrium of Equation ( 6).Then, except for the first semicycle, every oscillatory solution of Equation ( 6) has semicycles of length at least two.

Linearized Stability
In this section, we prove that Equation (1) has a unique equilibrium point which is always locally asymptotically stable.
The equilibrium point x of Equation (1) satisfies Equation ( 7) has the unique positive solution where Theorem 6.The unique positive Equilibrium (8) of Equation ( 1) is always locally asymptotically stable.
Proof.The real function f (u, v) associated to Equation ( 1) is given by Derivatives with respect to u and v evaluated at the equilibrium point (8) are respectively , and q = − α x + α + β .
We have that the function f (u, v) is always decreasing with respect to the second variable and can be either decreasing or increasing with respect to the first variable, depending on the sign of nominator, that is depending on corresponding parametric space.Now, we check the conditions of Theorem 1.1.1,see [12].The condition |p| < The second inequality is equivalent to α < x + α + β, which is always true.The first inequality becomes Now, we have which is always true.
Proof.From Equation (1), we have which completes the proof.
Lemma 2. Equation (1) does not posses a minimal period-two solution.
Proof.Period-two solution (x, y), x > 0, y > 0, x = y, satisfies the following system of algebraic equations By replacing the function f , we have By subtracting these two equations, we obtain If α ≥ β, then there does not exist a minimal period-two solution.However, if α < β, we have Similarly, by adding, from (10) we obtain By using (11), we have which is a contradiction with (12).

Global Attractivity Results
In this section, we prove several global attractivity results in the corresponding parametric space.We notice that the sign of the partial derivative with respect to the first variable at the equilibrium point depends on the sign of the b − aβ.
b − aβ ≥ 0, In this case, the function f is increasing in the first variable and decreasing in the second variable.
Suppose that α > β and M = m.Then By adding Equations ( 14) and ( 15), we have from which, by using ( 16), we have Now, we see that Equation (17) has no positive solutions (m, M) if b ≤ 1.This means that system (13) has a unique solution (m, M) = (x, x) in this case.Assume that b > 1.By substituting ( 16) into (17) we obtain quadratic equation 13) has more than one positive solution.
Theorem 7. Assume that one of the following conditions hold: Then, the unique Equilibrium (8) of Equation ( 1) is globally asymptotically stable.
Proof.In this case (see the proof of Lemma 1) the invariant interval (and an attracting interval) of Equation ( 1) is then f is increasing in the first variable and decreasing in the second variable and we can apply Theorem 1. Also, we know that the equilibrium x is locally asymptotically stable, and consequently the proof will be completed by using Lemma 3 and Theorem 1.
For some numerical values of parameters we give visual evidence for Theorem 7. (See Figure 1a) b − aβ < 0, Lemma 1 implies that we have to consider the following three cases: we have the following result about global behavior of solutions of Equation ( 1).
Theorem 8. Assume that one of the following conditions hold: , Then, the unique Equilibrium (8) of Equation ( 1) is globally asymptotically stable.
In this case we have is continuous, then f (u, v) attains its extreme points at the end of closed interval or at the stationary point.Straightforward calculations show that all values We know that g( x) = 0. On the other hand, we have that which implies that f is increasing in the first variable and decreasing in the second variable.Since Equation (1) has the unique equilibrium point in invariant interval β b , bα aβ − b , we can apply Theorem 1. Also, we know that the equilibrium x is locally asymptotically stable, and consequently the proof will be completed by using Lemma 3 and Theorem 1.
For some numerical values of parameters we give visual evidence for Theorem 8. (See Figure 1b).
and the unique Equilibrium (8) of Equation ( 1) is globally asymptotically stable.
First, we prove that the invariant interval is given by bα aβ − b , β b .Since the function f (u, v) is continuous, then this function attains its extreme points at the end of closed interval or at the stationary point.Straightforward calculations show that We know that g( x) = 0. On the other hand, we have Since the function f (u, v) is decreasing in both variables, and Equation ( 1) has the unique equilibrium point in invariant interval bα aβ − b , β b , we can apply Theorem 2. System of algebraic equations It is easy to see that this system has a unique solution m = M, which completes the proof.
For some numerical values of parameters we give visual evidence for Theorem 9. (See Figure 2a).
Remark 1.Notice that we can prove Theorem 9 by using Theorem 3. In this case, we have , where β > b a .Then, Equation (1) does not posses a minimal period-four solution.
By eliminating z and t we obtain where Since aβ − b > 0, we have that Φ (x, y, a, b, β) > 0 and Ψ (x, y, a, b, β) > 0. Therefore, system (18) has a unique solution of the form which means that Equaiton (1) has no a minimal period-four solution.
, where β > b a .Then, the unique Equilibrium (8) of Equaiton ( 1) is globally asymptotically stable.Furthermore, every solution oscillates about the equilibrium point x with semicycles of length two.
Proof.Notice that which implies that the length of the semicycle is two.By using Eq.( 1), we have x n+1 +βx n ax n+1 x n +bx n .
After straightforward calculations, we obtain From (19) we have and we see that x n and x n+4 are always on the same side of the equilibrium point x.Namely, if x n < β b , then x n < x n+4 < β b , n ∈ N, and so, if x n > β b , then Therefore, every sequence {x 4k } ∞ k=1 , {x 4k+1 } ∞ k=0 , {x 4k+2 } ∞ k=0 , {x 4k+3 } ∞ k=0 is monotone and bounded.This implies that each of the sequences is convergent.Since, by Lemma 4, Equation (1) has no minimal period-four solutions, we have that lim k→∞ x 4k = lim k→∞ x 4k+1 = lim k→∞ x 4k+2 = lim k→∞ x 4k+3 = x, which implies that the equilibrium x is an attractor.It means that x is globally asymptotically stable.
For some numerical values of parameters we give visual evidence for Theorem 10. (See Figure 2b) Remark 2. Notice that in the case when these four subsequences exist, there is an invariant interval of this form b 2 , but we can not use any of Theorems 1 and 2 because the map associated with Equation (1) changes its monotonicity with respect to the first variable in this invariant interval.Remark 3. Based on our numerical simulations and Theorems 4 and 5, we believe that every solution of Equaiton (1) oscillates about the equilibrium x with semicycles of length two.(See Figures 1 and 2).Also, based on our numerical simulations, we give the following conjecture.

Theorem 1 .
[14] [Theorem 2.22] Let [a, b] be an interval, and suppose that f : [a, b] × [a, b] → [a, b] is a continuous function.Consider the difference equation

Theorem 2 .
[12] [Theorem 1.4.7]Let [a, b] be an interval, and suppose that f: [a, b] × [a, b] → [a, b]is a continuous function satisfying the following properties: a. f (x, y) is nonincreasing in each of its arguments; b