Dynamics of General Class of Difference Equations and Population Model with Two Age Classes

In this paper, we study the qualitative behavior of solutions for a general class of difference equations. The criteria of local and global stability, boundedness and periodicity character (with period 2 k ) of the solution are established. Moreover, by applying our general results on a population model with two age classes, we establish the qualitative behavior of solutions of this model. To support our results, we introduce some numerical examples.

One interesting example for both facts is Riccati difference equations J n+1 = a + bJ n c + dJ n , For many results, applications and open problems on higher-order equations and difference systems, see References .
This paper is concerned with investigation of the asymptotic behavior of the solutions of a general class of difference equation where a is positive real number, the function f (u, v) : (0, ∞) 2 → [0, ∞) is continuous real function and homogenous with degree one and the initial conditions J −1 , J 0 are positive real numbers. The main reason for studying this general Equation is that its solutions have a peculiar periodicity character (with period-even) and it also involves a population model with two age classes (3), as a special case. One purpose of this paper is to further complement the results of Reference [13] for periodic solutions of the population model (2). In Section 3, we state a new necessary and sufficient condition for locally asymptotically stable of the population model (2). Also, we will confirm that the population model (2) has periodic solutions of a prime period 2k, k = 0, 1, . . . , (this means that Definition 1 is not accurate).
Furthermore, we introduce general theorems in order to study the asymptotic behavior of Equation (4). Namely, we give a complete picture regarding the local stability of equilibrium point, and we study the global stability and boundedness nature of the solutions. Also, we study the existence of periodic solutions of a prime period 2k. Moreover, we apply our results on the population model (2). Finally, we gave many numerical examples to support our results.

Stability and Boundedness of Equation (4)
In the next, we state a necessary and sufficient condition for locally asymptotically stable of equilibrium point of Equation (4). For our next considerations, we define the function Φ : An equilibrium point of Equation (6) is a point J * such that J * = Φ (J * , J * ). Then, the equilibrium point of Equation (4) is given by J * = a J * e −J * f (1,1) . Hence The linearized Equation of (4) of J * is where µ s = Φ s (J * , J * ) , s = u, v. A linear Equation will be called stable, asymptotically stable, or unstable provided that the zero equilibrium has that property. From (6), we get and In the next theorems, we study the asymptotic stability for (4).

Theorem 2.
For local stability of the equilibrium point J * = 0 of Equation (4), we have the following cases: (1) If a < 1, then J * is locally asymptotically stable and sink; (2) If a > 1, then J * is unstable and repeller; (3) If a = 1, then J * is nonhyperbolic point.
Proof. If we put J * = 0 in (9) and (10), then we have µ u = 0 and µ v = a. Thus, the roots of characteristic Equation  (1) Equilibrium point J * is locally asymptotically stable and sink if and only if (2) Equilibrium point J * is unstable saddle point if and only if (3) Equilibrium point J * is unstable and repeller if and only if α − |α| > γ, or where α = f u (1, 1) and γ = f (1, 1).
Proof. Since f homogenous with degree one, we have from Reference [19] that f u and f v homogenous with degree zero and hence and where β = f v (1, 1). Thus, the characteristic Equation of (8) is Also, from Euler's homogeneous function theorem, we have u f u + v f v = f , and hence α + β = γ (at (u, v) = (1, 1)).
In the following theorems, we study the boundedness of the solutions of Equation (4).
Proof. Assume that {J n } ∞ n=−1 is a solution of Equation (4). From (4) and f (u, v) ≥ 0, we note that Since a ≤ 1, we get J n+1 ≤ J n−1 . Thus, we can divide the sequence {J n } ∞ n=−1 to two bounded above subsequence by the initial conditions as Hence, we see that J n ≤ max {J −1 , J 0 } for all n > 0. The proof of the theorem is complete.
Theorem 5. Assume that there exists a constant δ > 0 such that f (u, v) ≥ δv. Then every solution of Equation (4) is bounded and for all n > 0.
Proof. Assume that {J n } ∞ n=−1 is a solution of Equation (4). By using the fact that ue −λu < 1/λe and f (u, v) ≥ δv, we obtain for all n > −1. Then every solution of Equation (4) is bounded. The proof of the theorem is complete.
Then every positive solution of Equation (4) converges to J * .

The Existence of Periodic Solutions
Here, we investigate the periodicity character of the solution for Equation (4).

Lemma 1. Assume that {J n } ∞
n=−1 is a solution of Equation (4). Then, Proof. Let {J n } ∞ n=−1 is a solution of Equation (4). From Equation (4), we have and so on, we find The proof of the lemma is complete.

Theorem 7.
Assume that a > 1, {J n } ∞ n=−1 is a solution of Equation (4) and there exists a couple of integers η ≥ −1 and k > 0 such that where Then {J n } ∞ n=−1 is an eventually periodic solution with period 2k.
Proof. Assume that there exists an integer number η ≥ −1 such that (25) holds. First, from (25), we have and Now, from Lemma 1, we get Using (26), we obtain J η+2k = J η . Also, from Lemma 1, we see that Since J η+2k = J η and from (27), we get Similarly, we can prove that  In the next theorem, we state a new necessary and sufficient condition for periodic solutions of a period two.

A Population Model
Difference equations have been widely used as mathematical models for describing real life situations in biology. In this section, we study the discrete model with two age classes, adults and juveniles (2) where r, κ ∈ (0, ∞). Expression exp (r − (I n + αJ n )) represents reproduction rate and is a decreasing exponential which captures the over crowding phenomenon as the population grows. To apply our results, we set system (2) as the following J n+1 = J n−1 e r−(κ J n +J n−1 ) .
For locally asymptotically stable of equilibrium point J * = r/ (κ + 1) of Equation (29), we have the next theorem.

Corollary 2.
We have the following cases:

2.
Equilibrium point J * is unstable saddle point if and only if κ > 1.
Proof. The proof is immediate (from Theorem 3) and hence is omitted. (2) is bounded and 0 < J n ≤ e r−1 ,for all n > 0.

Corollary 3. Every solution of Equation
Proof. Since f (u, v) = κu + v ≥ v for all u ∈ [0, ∞), we have that, from Theorem 5, every solution of Equation (2) is bounded and 0 < J n ≤ a/e = e r−1 , and hence the proof is complete. Now, we give the periodicity character of the solution for Equation (29).

Corollary 4.
Assume that {J n } ∞ n=−1 is a solution of Equation (29) and κ = 1. If there exists a positive integer number η such that then {J n } ∞ n=−1 converges to a prime period 2k solution.

Conclusions
Difference equations have been widely used as mathematical models for describing real life situations in biology. So, this paper is concerned with the qualitative behavior of the solution of the general class of the nonlinear difference equations which involves a population model with two age classes, as a special case. For general equation, we studied the stability (local and global), boundedness and periodicity character (with period 2k) of the solution. Moreover, by applying our general results on a population model with two age classes, adults and juveniles J n+1 = J n−1 e r−(κ J n +J n−1 ) , where expression exp (r − (I n + αJ n )) represents reproduction rate and is a decreasing exponential which captures the over crowding phenomenon as the population grows, we give a complete picture applying the local stability of equilibrium point of population model and we study the boundedness of soluations. Furthermore, we studied the existence of periodic solutions of a prime period-even of this model, as improved and complemented of results of Franke 1999 and conjecture of Kulenovic 2001. In order to support our results, we introduced some numerical examples. Further, we can try to get a necessary and sufficient condition for global stability as well as bifurcation behavior for (4) in the future work.