Dynamical analysis of a prey-predator-tourist model: Environmental preferences and optimal fee control

This paper examines the economic and ecological dynamics that arise in a natural park from the interaction between the tourists that visit the park and the species living there. We consider two species whose interaction is determined by Lotka–Volterra prey–predator equations. The tourists’ decision to visit the park is affected by the entrance fee as well as by the possibility of observing the two species, mediated by their preferences towards the prey and predator populations. A third nonlinear equation governs such dynamics. Tourism has conflicting effects: a higher number of visitors can be detrimental to the habitat and its species, but promoting ecotourism while preserving environmental sustainability and equilibrium is also in the mission of protected areas. We analyse the impact on the stability of the equilibrium of different levels of the entrance fee and tourists’ preferences for the two species. It is shown that, under specific conditions, an instability may arise leading to species loss and/or no tourists choosing to visit the park. Local and global sensitivity analyses of the equilibrium coordinates with respect to the model inputs highlight the major effects of the entrance fee. Interestingly, the preference for the preys (not the predators) is the crucial parameter when optimizing a fitness utility function for the park in a static setting. Finally, it is shown how to implement an optimal fee-policy control to steer the system towards its stable equilibrium following a path that also maximizes the discounted cumulated utility.