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.

中文翻译：

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猎物-捕食者-旅游模型的动力学分析：环境偏好和最优费用控制


本文研究了自然公园中由参观公园的游客与生活在那里的物种之间的互动产生的经济和生态动态。我们考虑了两个物种，它们的相互作用是由 Lotka-Volterra 猎物-捕食者方程决定的。游客参观公园的决定受到入场费以及观察这两个物种的可能性的影响，这与他们对猎物和捕食者种群的偏好有关。第三个非线性方程控制着这种动力学。旅游业具有相互矛盾的效果：游客数量增加可能对栖息地及其物种有害，但在保持环境可持续性和平衡的同时促进生态旅游也是保护区的使命。我们分析了不同水平的入场费和游客对这两个物种的偏好对平衡稳定性的影响。结果表明，在特定条件下，可能会出现不稳定因素，导致物种丧失和/或没有游客选择参观公园。相对于模型输入的均衡坐标的局部和全局敏感性分析突出了入场费的主要影响。有趣的是，在静态设置中优化公园的健身效用函数时，对猎物（而不是捕食者）的偏好是关键参数。最后，展示了如何实施最佳费用政策控制，以引导系统遵循一条同时最大化贴现累积效用的路径走向稳定的均衡。