Dynamic Analysis for Bertrand Competition Model with Exponential Form ∗

. This paper will consider a nonliear system of diﬀerence equations which describes a qualitative study of Bertrand oligopoly games with two boundedly rational players. With nonlinear demand function of exponential form, the local stability of equilibria and the global convergence of positive solutions for the dynamical system are analyzed.


Introduction
Market economy is fundamentally a dynamic system, which can usually be described mathematically by difference equations. In the dynamic study of economics, a couple of economic models represented by difference equations are investigated, such as the classical cobweb model describing the variation of the supply and demand, the Cournot models of oligopoly, and so on [2], [1], [6], [11]. It is well known that Bertrand duopoly competition game is one of the basic oligopoly games with two players. In this paper we consider a Bertrand duopoly game with an industry where two firms produce heterogeneous products. Price strategic interactions appear because the total demand of the good depends on the price of the industry according to a given demand function. In the classical study of Bertrand game, the demand function is usually linear or quadratic. However the demand function of exponential form can describe the bounded rationality duopoly game more realistic [11] and at the same time the dynamic will be more complicated. We assume in this paper the demand function where a is a parameter of maximum demand in the market and p i , i = 1, 2 denotes the price of the good produced by firm i. Assume the cost function C i (Q) = c i Q, i = 1, 2, (1.2) where c i is the marginal cost of the ith firm. Then the profit resulting from the above Bertrand duopoly game is given by The classical oligopoly games, and the associated notion of Nash equilibrium, are based on quite demanding notion of rationality. However, since the available information in the oligopoly market is incomplete, the rational players make their price decisions on a local estimate of the expected marginal profit ∂Πi ∂pi . Hence, the dynamical equation of the bounded rationality player i has the form where ν i , i = 1, 2 is a positive parameter which represents the relative speed of the price adjustment by producer i. Therefore, using (1.1)-(1.3), the discrete dynamical system becomes an iterated two-dimensional mapping which has the form We can rewrite this system in the new form In this paper we study the boundedness and the global asymptotic behavior of the positive solutions of the system of difference equation where α i , β i ∈ (0, ∞) with α i > β i ,i = 1, 2, and the initial values x 0 , y 0 are positive numbers.
In [5], by using the inverse demand function of exponential form, MF Elettreby and H El-Metwally studied a Cournot competition model described by difference systems of exponential form and they obtained the local stability of the equilibrium point and the global convergence of positive solutions. In [4], HA El-Metwally and AA Elsadany investigated the chaotic behavior of a duopoly Cournot game model of difference systems of exponential form. As for the studies of the behavior of positive solutions for difference equations of exponential form, we refer the readers to [3], [8], [9], [10] and the references therein. The next theorem will be a useful tool later in Section 3. Theorem 1. [7] Suppose T = (f, g) be a monotone map on a closed and bounded rectangular region S ⊂ R 2 . If T has a unique fixed point E = (x,ȳ) in S, then E is a global attractor of T on S.
The proof is completed.
The equilibria points E 1 and E 2 of system (1.4) are saddle points.
Proof. The Jacobian matrix of system (1.4) about the equilibrium point E 1 = (0, α 2 /β 2 ) has the form Therefore the eigenvalues of J(E 1 ) are given by λ 1 = 1 + α 1 e −α2/β2 and λ 2 = 1 − α 2 e −α2/β2 . Then Thus it follows that the equilibrium point E 1 of system (1.1) is a saddle point. Similarly, one can easily prove that the equilibrium point E 2 of system (1.4) is also a saddle point. This completes the proof.
Proof. The Jacobian matrix of system (1.4) about the equilibrium point .
Therefore the eigenvalues of J(E 3 ) are given by It is well known that the equilibrium point E The proof is completed.

Global Stability Analysis of (1.4)
In this section we first concern with the boundedness properties of the positive solutions for system (1.4). Under appropriate conditions, we give some bounded results related to system (1.4).
of system (1.4) with x 0 > 0 and y 0 > 0, satisfies that x n > 0 and y n > 0 for all n > 0.
Proof. Let H i (x, y), i = 1, 2 be continuous functions defined by Then system (1.4) can be rewritten in the form is a solution of system (1.4) with positive initial values. Then it suffices to show that H i (x, y), i = 1, 2 are positive for all x 0 > 0 and y 0 > 0. Observe that Therefore H i have no positive critical points. Let a and b be arbitrary positive numbers and consider the domain Then for i = 1, 2, we see that Using elementary differential calculus, we obtain that the absolute minimum of > 0 for all (x, y) ∈ D. Since a and b are arbitrary positive numbers, we can conclude that H i (x, y) > 0 for i = 1, 2 and for all (x, y) ∈ (0, ∞) 2 .
Proof. Let n 0 ≥ 0 be such that x n0 ∈ (0, α1 β1 ]. It follows from system (1.4) that . So x n ≤ α1 β1 for all n ≥ n 0 . Similar method can be applied for y n . This completes the proof.
Theorem 4. Assume that α i < 1. Then every solution {(x n , y n )} ∞ n=0 of system (1.4) satisfies for any n 0 satisfying x n0 ∈ (0, α1 β1 ], y n0 ∈ (0, α2 β2 ]. Proof. Let n ≥ 0 be such that x n ∈ (0, α1 β1 ], y n ∈ (0, α2 β2 ]. It follows from system (1.4) that Since α 1 < 1, we have By induction, for fixed n 0 satisfying x n0 ∈ (0, α 1 /β 1 ], y n0 ∈ (0, α 2 /β 2 ], we obtain Similarly, we could prove for any n 0 satisfying x n0 ∈ (0, α1 β1 ], y n0 ∈ (0, α2 β2 ]. The following corollary is coming immediately from Theorem 4. Proof. Let {(x n , y n )} ∞ n=0 be a solution of system (1.4). In the first case, if x 0 ≤x < α1 β1 , Then the sequence {x n } ∞ n=0 is increasing and since it was shown that it is bounded above by α1 β1 , then it converges to the unique positive equilibrium pointx. The second case is x 0 ≥x, we will show that there exists a positive integer N such that x N ≤x. Notice that in this case Repeating this step confirms that {x n } is a decreasing sequence. Thus either there exists N ∈ N such that x N <x or x n is a bounded decreasing sequence which has a limit d 1 >x. In whichever cases, we can assume similarly that y n has a limit d 2 . Set f (x, y) = (β 1 x − α 1 )xe −(x+y) . By the continuity of f and convergence of (x n , y n ), we have there exists n 1 ∈ N such that for any n ≥ n 1 , f (x n , y n ) ≥ δ, where δ = f (d1,d2) 2 > 0. Therefore, for any n ≥ n 1 , So there exists a positive integer N such that x N <x.
Similarly, it can be shown that the sequence {y n } ∞ n=0 converges to the unique positive equilibrium pointȳ. Thus {(x n , y n )} ∞ n=0 converges to (x,ȳ).

Remark
In this last section, just from the mathematics point of view, we give a remark to discuss the case α i ≤ β i , i = 1, 2. We will see the unique positive equilibrium point (x,ȳ) of system (1.4) is a global attractor of all positive solutions of system (1.4).
Then the sequence {x n } ∞ n=0 is increasing and since it was shown that it is bounded above, then it converges to the unique positive equilibrium pointx. Similarly, it is easy to show that the sequence {y n } ∞ n=0 is increasing and since it was shown that it is bounded above, then it converges to the unique positive equilibrium pointȳ. Thus {(x n , y n )} ∞ n=0 converges to (x,ȳ).  Figure 2 shows the stability of equilibrium of (1.4).