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.‎