On the Global Asymptotic Stability and 4-Period Oscillation of the Third-Order Difference Equation

. The main objective of this paper is to study the global behavior and oscillation of the following third-order rational di ﬀ erence equation x n +1 = α x n x n − 1 x n − 2 / β x 2 n − 1 + γ x 2 n − 2 , where the initial conditions x − 2 , x − 1 , x 0 are nonzero real numbers and α , β , γ are positive constants such that α ≤ β + γ . Visual examples supporting solutions are given at the end of the study. The ﬁ gures are found with the help of MATLAB.


Introduction
Considering every field of biology such as physiology, genetics, development, ecology, or evolution, these fields cannot be examined without considering the time.Life occurs over time.It is not surprising that the mathematical modeling of equations created in biology is defined by temporal processes.Physiological events, such as hair growth, occur continuously over time.Processes such as population growth in populations occur more discretely over time.
Difference equations are known as mathematical expressions that are mostly used to describe a process that develops in discrete time.That is why difference equations are of great importance in applications.Since annual plants complete their life cycle in one year, they can be explained with discrete-time models.Bahar and Erdogan [1] investigated the amount of seed production required for an annual plant with seeds capable of remaining dormant underground for a maximum of 3 years.The mathematical model obtained in the study is a discrete-time 3rd-order linear difference equation: Wisnoski and Shoemaker [2] conducted a study showing that competition in the seed bank alters diversity.In their study, they referred to previous studies and supported their work by presenting mathematical models.
Many scholars are interested in rational difference equations because they are more challenging to study in terms of dynamics than linear models.Actually, the fact that difference equations are present in several biological models with a wide range of applications makes them important to investigate.The Riccati difference equation is as follows: where a, b, c, d and initial condition are real numbers, describing one of the intriguing models.AlSharawi and Rhouma [3] investigated the effect of different harvesting strategies in a deterministic environment on the discrete Beverton-Holt model: which is a special case of the Riccati difference equation.
Yang [4] investigated the global asymptotic stability of the difference equation: Kulenović et al. [5] studied the behavior of rational recursive sequence: Elabbasy et al. [6] investigated and study some special cases of the difference equation: Khaliq and Elsayed [7] studied the behavior and obtained some special cases of the difference equation: See also [8][9][10][11][12][13][14][15][16][17][18][19].Our aim is to examine the global behavior of the following third-order rational difference equation that will serve as the basis for such modelling: where the initial conditions x −2 , x −1 , x 0 are nonzero real numbers and α, β, γ are positive constants such that Computational examples are given at the end of study and simulated solutions of some problems via MATLAB.We hope that the results of this study contribute to the development of the theory on the global stability of nonlinear rational differential equations.
Let us give some definitions and theorems that we need.
Definition 1 (see [18]).Let I be some interval of real numbers and let f I 3 ⟶ I 10 be a continuously differentiable function.Then, for every set of initial conditions x 0 , x −1 , x −2 ∈ I, the difference equation that is, is a solution of (11), or equivalently, x is a fixed point of f .Definition 2 (see [18]).Let x be an equilibrium point of Eq. ( 11).
(i) The equilibrium x of Eq. ( 11) is called locally stable if for every ϵ > 0, there exists δ > 0 such that for all The equilibrium x of Eq. ( 11) is called locally asymptotically stable if it is locally stable, and if there exists γ > 0 such that for all The equilibrium x of Eq. ( 11) is called global attractor if for every x 0 , x −1 , x −2 ∈ I, we have lim The equilibrium x of Eq. ( 11) is called global asymptotically stable if it is locally stable and a global attractor (v) The equilibrium x of Eq. ( 11) is called unstable if it is not stable (vi) The equilibrium x of Eq. ( 11) is called source or a repeller, if there exists r > 0 such that for all The linearized equation of ( 11) about the equilibrium point x is where The characteristic equation of ( 11) is Definition 3 (see [18]).A positive semicycle of x n ∞ n=−2 of Eq. ( 11) consists of a "string" of terms x l , x l+1 , ⋯, x m , all greater than or equal to x, with l ≥ −2 and m < ∞ and such that either l = −2 or l > −2 and x l−1 < x and either m = ∞ or m < ∞ and x m+1 < %x.

Journal of Applied Mathematics
A negative semicycle of x n ∞ n=−2 of Eq. ( 11) consists of a "string" of terms x l , x l+1 , ⋯, x m all less than x, with l ≥ −2 and m < ∞ and such that either l = −2 or l > −2 and x l−1 ≥ x and either m = ∞ or m < ∞ and x m+1 ≥ x.
Theorem 5 (see [18]).Let p, q be an interval of real numbers and assume that f p, q 3 ⟶ p, q is a continuous function satisfying the following properties: (a) f x, y, z is nondecreasing in y, z ∈ p, q for each x ∈ p, q and nonincreasing in x ∈ p, q for each y, z ∈ p, q Then, Eq. ( 11) has a unique equilibrium x ∈ p, q , and every solution of Eq. ( 11) converges to x.

Dynamics of Eq. (8)
In this section, we investigate the dynamics of (8) under the assumptions that all parameters in the equation are positive and the initial conditions are nonnegative.
Proof.It follows by Theorem 4 that Eq. ( 22) is asymptotically stable if we can get is regained.This completes the proof.

Global Asymptotic Stability of x of Eq. (8)
Theorem 7. The equilibrium point of x of Eq. ( 8) is globally asymptotically stable.
Proof.Let p, q be real numbers and assume that f p, q 3 ⟶ p, q is a function defined by f u, v, t = αuvt/ βv 2 + γ t 2 .Then, we can easily see that the function is increasing in u and decreasing in v, t.Suppose that m, M is a solution of the system Then, from Eq. ( 8), By Theorem 5, x is a global attractor of Eq. ( 8).From Theorem 6 and Definition 1, x is globally asymptotically stable of Eq. ( 8) and the proof is complete.

Computational Examples
In this section, I perform computational examples to illustrate the validity of the main results.In order to better express the numerical samples, a graph of the solutions was obtained by using MATLAB.These graphs are drawn with different parameters and different starting conditions.

Conclusion
It is very interesting for researchers to examine the dynamics of rational difference equations, especially high-period dynamics.In this article, the solutions of a third-order rational difference equation with four-period oscillations are examined.First of all, the local stability of eq. ( 8) and then the global asymptotic stability are examined.Afterwards, the oscillation of equation ( 8) was examined, and the obtained theoretical results are supported by numerical examples and graphics of solutions.