Dynamical Behaviors of an Environmental Protection Expenses Model in Protected Areas with Two Delays

Abstract

This paper investigates an environmental protection expenses model, which considers the relations between the visitors to the protected areas V, the quality of the environmental resource E, and the capital stock K. In this model, the total tourism income is used partly to increase the capital stock or as the environmental protection expenses. Two time delays are introduced into the number of visitors, since the visitors need time to respond the changes of the environment, and the environment will take time to respond to the input of money. Stability crossing curves in the plane of delays (τ1,τ2) are used to obtain the stable region of equilibrium. Numerical simulations represent the mutual transformation of the supercritical bifurcation and the subcritical bifurcation. Our model shows that under some parameter conditions, the share of tourism income η is related closely to the delay τ1, while the capital stock and the environmental quality can be maintained persistently if the delay τ1 is not too large.