Modeling and dynamic behavior of a discontinuous tourism-based social-ecological dynamical system

Abstract

In this paper we study a four-dimensional tourism-based social-ecological system including two different types of tourists, i.e., eco-tourists and mass-tourists. First we develop the mathematical model of this system, in order to make it more realistic. In fact, the negative effects of eco-tourists on the environment can not be always ignored. Therefore, we consider a discontinuous function for describing the damage induced by tourist activities to ecosystem quality. By introducing a discontinuous model in this way, we can provide a tool for better investigating the behavior of tourism phenomenon. We analyse the dynamic of the obtained discontinuous system, by using the theory of discontinuous dynamical systems. Moreover, in order to discuss about profitability, compatibility and sustainability of the obtained system, stability regions for that will be found. Furthermore by verifying tangent points of the discontinuous system, more dynamic features of eco and mas-tourists will be shown. In this regard, we determine some regions for existence of the collision of two tangent points for the system. Some numerical simulations are carried out to demonstrate our theoretical results. Here our numerical and theoretical achievements can provide useful information for analysing the tourism industry.