Stability crossing curves and Hopf bifurcation in an environmental protection expenses model with two delays

Abstract

This study is devoted to considering dynamic properties of an environmental protection expenditures model for protected areas (PAs) with two delays. It presents an improved understanding of relationship among tourism, environment, and capital stock in PAs and provides a sustainable strategy to keep the balance between them. The model considers the interactions among environmental resource E$$ E $$, visitors V$$ V $$, and capital stock K$$ K $$ in PAs. The tourism income is used partly to increase the capital stock and partly to protect the environment. In this paper, time delays τ1$$ {\tau}_1 $$ and τ2$$ {\tau}_2 $$ are introduced into the system. The technique of stability crossing curves for two delays is proposed to analyze the occurrence of Hopf bifurcation. Numerical simulations are performed for understanding the analytical results. The results show that under some parameter conditions, the share η$$ \eta $$ of total revenue generated by tourism user fees for environmental protection is related to delays τ1$$ {\tau}_1 $$ and τ2$$ {\tau}_2 $$. Moreover, the environment and the capital stock can be maintained persistently if time delay τ1$$ {\tau}_1 $$ is not too large.