Bifurcation analysis in the diffusive Lotka–Volterra system: An application to market economy q A.W. Wijeratne a,b , Fengqi Yi a , Junjie Wei a, * a Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China b Department of Agri-Business Management, Sabaragamuwa University of Sri Lanka, Belihuloya 70140, Sri Lanka Accepted 13 August 2007 Abstract A diffusive Lotka–Volterra system is formulated in this paper that represents the dynamics of market share at duo- poly. A case in Sri Lankan mobile telecom market was considered that conceptualized the model in interest. Detailed Hopf bifurcation, transcritical and pitchfork bifurcation analysis were performed. The distribution of roots of the char- acteristic equation suggests that a stable coexistence equilibrium can be achieved by increasing the innovation while minimizing competition by each competitor while regulating existing policies and introducing new ones for product differentiation and value addition. The avenue is open for future research that may use real time information in order to formulate mathematically sound tools for decision making in competitive business environments. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Diffusion is the process by which an innovation is communicated through certain channels over time among the members of a social system [18]. In economic context, the goal of a diffusion model is to explain the process of an inno- vation spreading among individuals. The commonly used model that describe the market diffusion or the product growth forecasting is the Bass model [1] where many variants have been developed that address issues in a range of fields [8,10,16]. Furthermore, it is observed that the dynamical behavior of the general Bass model is very similar to the Lotka–Volterra (LV) population growth model with no competition [21]. However, when a competition is present, the approach in market share attraction models has been one of the better alternatives used in a variety of contexts to describe the behavior of competitors in a market (see [2] and reference therein). The first differential equation of predator–prey type was formulated by Alfred Lotka and Vito Volterra early in the last century, when attempts were first made to find ecological laws of the nature. Since then, the study of the LV system has attracted attention of great number of investigators (see [13,14,20] and references therein). Usually, the LV equation represent a simple non-linear model for the dynamic interaction between two species in which one species benefits at the 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.043 q This research is supported by the National Natural Science Foundation of China and Specialized Research Fund for the Doctoral Program of Higher Education. * Corresponding author. E-mail address: weijj@hit.edu.cn (J. Wei). Chaos, Solitons and Fractals 40 (2009) 902–911 www.elsevier.com/locate/chaos