Global Asymptotic Stability of a Rational System

and Applied Analysis 3 2. Linearized Stability In this section, we will make some conclusions about linearized stability. Consider the map T on R2 associated with the system (1), that is, T (x, y) = ( f 1 (x, y) f 2 (x, y) ) = ( β 1 x

and the second component {  } is constant and equal to  2 for  ≥ 1.
If the initial value is given by  0 > 0, then by simple iteration, it is easy to find that is the solution of (3).
In [10], the author proposed the following conjecture.

Conjecture 2. Assume that
Show that the unique positive equilibrium (, ) of the system (1) is globally asymptotically stable.
Inspired by Conjecture 2, we investigate the global behavior of the system (1).To start our discussion, some basic results should be presented which will be useful in the sequel.
Consider the system where  = (, ) : A vital tool for dealing with the linearized stability of (8) is the following well-known result which we incorporate in the following lemma (see, e.g., [11,12]).
The following well-known comparison result will be used in estimating the value of a solution of the system (1).
To prepare for our major investigation, we consider the following equation: and the following lemma should be mentioned which is from [12].Lemma 6.Let [, ] be an interval of real numbers and assume that  : [, ] 2 → [, ] is a continuous function satisfying the following properties: (i) (, ) is nondecreasing in each of its arguments; (ii) the function (, ) =  has a unique positive solution.

Global Attractivity
In this section, we will commence global asymptotic stability analysis.Let (  ,   ) be a solution of the system (1), then it is easy to obtain the following result from the second equation of the system (1).Theorem 8. (i) Assume that  2 >  2 .Then every solution (  ,   ) of the system (1) satisfies  2 ≤   ≤  2 for  ≥ 1.
Theorem 10.Assume that  2 <  2 ≤  1 −  1 .Then every solution of the system (1) Proof.Using Theorem 8, we get that when  2 <  2 <  1 −  1 , and when since the only equilibrium of the system ( 1) is (0,  2 ) when Further, using the boundedness of   , we have The proof is complete.
For the case where  1 ≤  1 +  2 , the authors had obtained that the unique positive equilibrium (0,  2 ) is a global attractor of all solutions of the system (1) in [10], see Theorem 1 (ii).Moreover, in view of Theorem 7 (i), we may formulate the result in the following theorem.
Theorem 11.Assume that  1 −  1 <  2 <  2 .Then the unique equilibrium (0,  2 ) of the system (1) is globally asymptotically stable.Now, we pay attention to dealing with the global attractivity of the unique positive equilibrium (, ), namely, (6), under the condition that  2 <  2 .In this case, (, ) exists if and only if (7) holds.To obtain the global attractivity of (, ), the following useful lemma should first be established.
Lemma 12. Every positive solution of (30) converges to the unique positive equilibrium .
The proof is complete.
Proof.In view of Theorem 7, it is sufficient to show that (, ) is a global attractor of all positive solutions of the system (1).In this case,   ≥  2 > 0 holds for  ≥ 1 and thus the system (1) yields for  ≥ 1.Let   =   /  , V  =   , then the system (1) becomes Further, the system (33) may reduce to the following second-order difference equation: Clearly, zero is always the equilibrium of (34) and when (7) holds, (34) also possesses a unique positive equilibrium Notice that  2 ≤ V  =   ≤  2 for  ≥ 1, and we get and thus ,  ≥ 1. (37) Hence by Lemma 12, we get that every positive solution of the following difference equation converges to its unique positive equilibrium ȗ =  2 ( Similarly, by Lemma 12, we know that every positive solution of the following difference equation converges to its unique positive equilibrium û Applying Lemma 4 and (37), we find that every solution of (34) with initial value ȗ 1 = û1 =  1 = ( 1 / 1 ) > 0 satisfies ȗ  ≤   ≤ û , for  ≥ 1. (40) Hence for 0 <  < ȗ /2, there exists an integer  such that for  > , Moreover, Let  = ȗ /2,  = û + ( ȗ /2), then  >  > 0 and every solution of (34) eventually enters the invariant interval and thus the result follows.
The proof is complete.