Global Analysis and the Periodic Character of a Class of Difference Equations

: In biology, Difference equations is often used to understand and describe life phenomenon through mathematical models. So, In this work, we study a new class of difference equations by focusing on the periodicity character, stability (local and global) and boundedness of its solutions. Furthermore, this equation involves a May’s Host Parasitoid Model, as a special case.


Introduction
The goal of our paper is to research the dynamics of solutions of equation J n+1 = α + βJ 2 n (γ + J n ) J n−1 , n = 0, 1, ... , (1) where α, β, γ ∈ [0, ∞), β = 0 and the initial data J −1 , J 0 ∈ (0, ∞). When describing the evolution of any phenomenon as a mathematical model, difference equations often arise, frequently due to the discrete nature of time-evolving variable measurements and detached sciences. Difference equations are used in situations of real life, in various sciences (population models, genetics, psychology, economics, sociology, stochastic time series, combinatorial analysis, queuing problems, number theory, geometry, radiation quanta and electrical networks).
In fact, the nonlinear DEs have the efficiency to make a complicated behavior, regardless of their order. Among the well-known examples, the family J n+1 = y λ (J n ), λ > 0, depends on η, and its conduct changes from a bounded number of periodic solutions to chaos. Due to the many applications of differential equations, there is a growing interest in searching for various aspects in terms of dynamics and behaviors of difference equations (see ).
Our focus in this paper is on the study of qualitative behavior of solutions of the nonlinear difference equations. Furthermore, a new equation includes a May's Host Parasitoid Model, as a special case. Minutely, we discuss the local/global stability, boundedness and periodicity character of the solution. Moreover, by applying our results, we will prove the following cojecture: Conjecture 1 ([24]). Let β > 1. Show that every positive solution of May's Host Parasitoid Model (Equation (1) with α = 0 and γ = 1) is bounded.
On the Other hand, if α, β and γ are real numbers. From (2), we get where t = r/s. Then, where Similarly, from (3), we obtain with By using the fact r − st = 0, (4) and (5), we find where By simple computation, (10) shows that From definitions of A, B and C, we have Since αγt = 0 and t = 1, we obtain Now, if t ∈ R + , then the function H (t) := 1 + t 2 /t attends its minimum value on R + at t * Thus, from (7), we see that The proof is complete.
then equation (1) has a period two solutions.
Proof. Assume that equation (1) has a prime period two solution ...r, s, r, s, ..., (r = s). From (1), we get where t = r/s. Then Similarly, we obtain By using the fact r − st = 0, (8) and (9), we find Now, if t ∈ R + , then the function H (t) := t 2 + t + 1 /t attends its minimum value on R + at t * (7), we se that The proof is complete. Proof. Assume that every positive solution of Equation (1) is periodic with period p. Now, we consider the solution with J −1 = 1 and J 0 ∈ (0, ∞) .
Hence, J p−1 = 1 and J p = J 0 . From Equation (1), we have Assume that α = 0 and β > 0. If we choose J 0 < α, then which is impossible and hence α = 0. The proof of the theorem is complete.

Remark 1.
Let α = 0, it is possible that every positive solution of Equation (1) is periodic with period p. As a special case, if α = γ = 0, then we see that every positive solution of equation Also, if αγ = 0, then the only positive equilibrium is One of the fundamental objectives in the investigation of a dynamical system is to determine the behavior of its solutions near an equilibrium point. For the basic definitions of stability see [24]. To study the local stability of a positive equilibrium point, we define the function F : (0, ∞) × (0, ∞) → (0, ∞) by The partial derivatives of function F are and The equilibrium point J e is called a sink or an attracting equilibrium if every eigenvalue of Jacobian matrix of J e has absolute value less than one, see [23]. In the following theorem, by using Theorem 1.1.1 in [24], we study a locally asymptotically stable for positive equilibrium point of (1) when α, β, γ ∈ [0, ∞). Proof. By replacing both u and v with J e in Equations (11) and (12), we get ∂ ∂u F (J e , J e ) = β (γ + J e ) 2 (2γ + J e ) := µ u (13) and Then, the linearized equation associated with (1) about J e is and so β γ + J e < 1.
Moreover, we see that From (13)- (15), we obtain Hence, we have |µ u | < 1 − µ v < 2. Therefore, J e is locally asymptotically stable and sink. The proof of the theorem is complete. Proof. The proof is similar to the proof of Theorem 4 and so we omit it.
for all n > 0 and so every solution of Equation (1) is bounded.
Hence, we have From Lemma 1, we have Thus, and from 2βγ α 2 + αβ + β 2 2 < α 6 , we get Since γ > β, we find From (19) and (20), we obtain L = U. From Theorem 1.4.5 in [24], we have that all solution of (1) converges to J e . The proof of the theorem is complete.

Application and Discussion
In Equation (1), if α = 0 and γ = 1, we get the May's Host Parasitoid Model By using Theorems 1 and 5 and Lemma 2, respectively, we get the following corollaries:  Remark 2. Note that, Corollaries 1-3 gave some qualitative behaviors of the model (21). Moreover, Corollary 3 confirms the Conjecture 1.

Example 1. Let the equation
(22) Figure 1 shows the dynamics of (22) with J −1 = 1.5 and J 0 = 0.1. Let N e be the first value of n in which the solution is stable (by approximation 10 −6 ), for example, let α =