Stability Analysis of a Nonlinear Difference Equation

The local and global behavior of the positive solutions of the difference equation

where the parameters  and  are positive numbers and the initial conditions are arbitrary non-negative real numbers.This equation may be viewed as a model in Mathematical Biology, where  is the immigration rate and  the population growth rate.
In [2] was investigated the globally asymptotically stability of the difference equation where the parameters  ,  and  and the initial con- ditions are arbitrary positive numbers.In Section 2, the local asymptotic stability of the equilibrium point of Equation (1.4) was investigated by using the Linearized Stability Theorem.A suitable Lyapunov function for the analysis of the global asymptotic stability behavior was used, like the idea in [8,9].Furthermore, the characterization of the stability was examined that depends on the conditions of the coefficients (see [10]).In Section 3, the semi-cycle of positive solutions was analyzed.All this results will be shown theoretical and by simulations at the end of the paper.

Local and Global Asymptotic Stability Analysis
In this section, we discuss the local and global asymptotic stability of the unique positive equilibrium point of Equation (1.4) by using the theorems in [4,8-10].
The equilibrium points of Equation (1.4) are the solutions of the equation for 0 y  and we obtain respectively, y   It follows that Equation (2.1) has exactly one solution y .From this result Equation (1.4) has a unique equilib- rium y .
The linearized equation and the characteristic equation associated with Equation (1.4) about the equilibrium y is which gives that every positive solution of the Equation (1.4) is bounded.Thus 1) is true.
2) Assume that 0 y  .Then then the positive equilibrium point of Equation (1.4) is locally asymptotically stable.
Proof.From the Linearized Stability Theorem, we can write The inequality (2.8) can be shown under two cases; From 2), we get e .
By 1), we will have which always holds and since    , we can also obtain (2.11) Considering both (2.9) and (2.11), if Rewriting (2.12), we get In view of (2.12) and (2.13), we obtain Theorem 2.3.Let the conditions in Theorem 2.2 hold and assume that 1 y and 2 y are the equilibrium points of Equation (1.4), which parameters have the conditions . If the parameter  decreases, then the local stability of the positive equilibrium point (2.16) From (2.16), computations will give us where we get the positive equilibrium point Let us write y y y y y y Considering the conditions in Theorem 2.2, if furthermore (2.22) holds, than the stability of 2 y is weaker than 1 .
y By computing (2.22), we obtain e e e e 0.
This inequality can be also written in the form

y y y y y y y y y
From (2.20) we get (2.25) which always holds.This completes the proof.Proof.We consider a Lyapunov function   V n de- fined by The change along the solutions of Equation (1.4) is From (2.28) we can write It can be compute that by using the hypothesis we have

The Semi-Cycle and Oscillation
In this section, we consider the semi-cycle and oscillation f the positive solutions of Equation (1.4).o Copyright © 2013 SciRes.IJMNTA , all greater than or equal to the equilibrium x  , with and and such that either or and all less than the equilibrium x  , with and and such that either or and Then every oscillatory solution of Equation (1.4) has semi-cycle of length at most two.
Proof.By the Theorem B, we can write Equation (1.4) such as The first derivative of (3.1) with respect to x and y are respectively.These derivatives are less than zero if x y Copyright © 2013 SciRes.IJMNTA -Metwally et al.[1] studied the global stability of the

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 ,  and  are positive num- bers and the initial conditions are arbitrary non-negative numbers.In [3] the boundedness and the global asymptotic behavior of the difference equation where  and  are positive real numbers, and the initial conditions 1 0 are arbitrary numbers.Similar studies can be shown in [4-7].The aim in this paper is to study the local and global behavior of the positive solutions of the difference equa- equilibrium point of Equation (1.4) is globally asymptotically stable.
which is the condition for the global asymptotic stability of the positive equilibrium point of Equation (1.4).