Sustainability analysis of a discontinuous tourism-based social-ecological dynamical system

‎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 to support more efficiently sustainable management of tourist destination‎. ‎We analyse the dynamic of the obtained discontinuous system‎, ‎by using the theory of discontinuous dynamical systems‎. ‎Moreover‎, ‎in order to discus about profitability‎, ‎compatibility and sustainability of the obtained system‎, ‎stability regions for that will be found‎. ‎Some numerical simulations are carried out to demonstrate our theoretical results‎.