Local Hopf Bifurcation in a Competitive Model of Market Structure with Consumptive Delays

A competitive model of market structure with consumptive delays is considered. The local stability of the positive equilibrium and the existence of local Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equation.The explicit formulas determining the stability and other properties of bifurcating periodic solutions are derived by using normal form theory and center manifold argument. Finally, numerical simulations are given to support the analytical results.


Introduction
The Lotka-Volterra predator-prey model is proposed by Lotka and Volterra to describe the dynamics between populations in ecology.And it has been extensively studied by many authors [1][2][3][4][5][6][7].Recently, the Lotka-Volterra predator-prey model has been used in economics by many scholars [8][9][10][11].In [8], Brander and Taylor proposed a general equilibrium model of renewable resource and population dynamics related to the Lotka-Volterra predator-prey model.They applied the model to the rise and fall of Easter Island and showed that plausible parameter values generate a "feast and famine" pattern of cyclical adjustment in population and resource stocks.In [9], Delfino and Simmons investigated the links between the health structure of the population and the productive system of an economy which is subject to infectious disease by combining the Lotka-Volterra model with Solow-Swan growth model.They analyzed the local dynamics and found that the epidemiological-economic stationary state is locally stable and an attractor for a wide range of initial conditions.In [10], Farmer derived a simple nonequilibrium model for price formation.He applied the model to several commonly used trading strategies and discussed how the model can be used to understand the long term evolution of financial markets.In [11], Kong analyzed the evolution of market structure with Lotka-Volterra model.And a model of market structure was established by simulating the relations of product in market: where () denotes the output of the product  at time .() denotes the output of the product  at time . 1 and  2 denote the growth rates of the products  and , respectively. 1 and  2 denote the production scale of the products  and , respectively. 12 and  21 are the competition rates between the products  and .In system (1),  and  are products of the same type but produced by different manufacturers.
Kong [11] obtained the conditions for the market structure coming into being by analyzing the stability of system (1).It is well known that time delays are universal in the market structure due to the consumption, competition, or other reasons.Therefore, it is necessary to incorporate time delays into system (1).Based on this consideration, we consider the following model with time delays: where  1 and  2 are the consumption delays of the product  and , respectively.All the parameters in system (2) have the same meanings as in ( 1) and all of them are assumed to be positive.This paper is organized as follows.In Section 2, we analyze the local stability of the positive equilibrium and the existence of the local Hopf bifurcation.In Section 3, we determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by using normal form theory and center manifold argument.To support the analytical results, numerical simulations are included at last.
Case 2 ( 1 > 0,  2 = 0).Equation ( 7) can be transformed into the following form: Let  =  1 ( 1 > 0) be a root of (10); then we get From (11), we can have It is easy to know that (12) has only one positive real root Next, we verify the transversality condition.Differentiating (10) with respect to  1 , we have Thus, From ( 13) and ( 16), it is easy to verify that Re [/ 1 ] −1  1 = 10 > 0. Thus, the transversality condition is satisfied.In conclusion, we have the following results.
Case 3 ( 1 = 0,  2 > 0).Equation ( 7) becomes Let  =  2 ( 2 > 0) be a root of (17); then we get which follows that Similar to Case 1, we can know that (19) has only one positive root Similar to Case 2, we can get Re That is, the transversality condition is satisfied.Therefore, we have the following results.

Numerical Simulation
In this section, some numerical simulations were given to support the analytical results obtained in the previous Obviously, conditions ( 1 ) and ( 2 ) hold.By a simple calculation, we obtain that (48) has a unique positive equilibrium  * (0.7894, 1.0526).Then we have  +  = 0.9471 > 0. Namely, condition ( 3 ) holds.

Conclusion
A competitive model of market structure with consumptive delays is studied in this paper.By analyzing the distribution of the roots of the associated characteristic equation, we obtained the conditions for the local stability of the model and the existence of the Hopf bifurcation.The main results are given in Theorems 1-4, which show that the consumptive delays play important roles in the model.It is proved that when some conditions are satisfied, then Hopf bifurcation occurs when the delay passes through the corresponding critical value.In reality, the occurrence of Hopf bifurcation means that the coexistence of the two products in system (2) changes from the positive equilibrium to a limit cycle, which is not welcome in reality.Furthermore, the explicit formulas determining the stability and the direction of the bifurcating periodic solutions are given by using the normal form theory and center manifold argument.The main results are given in Theorem 5. From the view of economy, if the bifurcating periodic solutions are stable, the two products of the same type may coexist in an oscillatory mode.This is valuable from the view of economics.However, our study is restricted only to the theoretical analysis of such economic phenomena.It may be helpful for field investigation or experimental studies on the real situation.
4, suppose the following.( 7 ) : equation (23) has at least finite positive roots, which are denoted by  1 ,  2 , . . .,   .The corresponding critical value of time delay for every   is