Abstract：
We introduce a model that can be used for the description of the distribution of species when there is scarcity of data, based on our previous work (Ballesteros et al. J Math Biol 85(4):31, 2022). We address challenges in modeling species that are seldom observed in nature, for example species included in The International Union for Conservation of Nature\'s Red List of Threatened Species (IUCN 2023). We introduce a general method and test it using a case study of a near threatened species of amphibians called Plectrohyla Guatemalensis (see IUCN 2023) in a region of the UNESCO natural reserve \"Tacaná Volcano\", in the border between Mexico and Guatemala. Since threatened species are difficult to find in nature, collected data can be extremely reduced. This produces a mathematical problem in the sense that the usual modeling in terms of Markov random fields representing individuals associated to locations in a grid generates artificial clusters around the observations, which are unreasonable. We propose a different approach in which our random variables describe yearly averages of expectation values of the number of individuals instead of individuals (and they take values on a compact interval). Our approach takes advantage of intuitive insights from environmental properties: in nature individuals are attracted or repulsed by specific features (Ballesteros et al. J Math Biol 85(4):31, 2022). Drawing inspiration from quantum mechanics, we incorporate quantum Hamiltonians into classical statistical mechanics (i.e. Gibbs measures or Markov random fields). The equilibrium between spreading and attractive/repulsive forces governs the behavior of the species, expressed through a global control problem involving an energy operator.
摘要：
我们介绍了一个模型，该模型可用于在数据匮乏的情况下描述物种的分布，基于我们以前的工作(Ballesteros等人。数学生物学85(4):31,2022)。我们解决了在自然界中很少观察到的物种建模方面的挑战，例如，列入国际自然保护联盟濒危物种红色名录（IUCN2023）的物种。我们介绍了一种通用方法，并使用联合国教科文组织自然保护区“塔卡纳火山”地区的两栖动物濒临灭绝物种（见IUCN2023）的案例研究对其进行了测试，在墨西哥和危地马拉之间的边界。由于受到威胁的物种很难在自然界中找到，收集的数据可以大大减少。这产生了一个数学问题，即通常的马尔可夫随机场表示与网格中的位置相关的个体的建模会在观测值周围生成人工聚类。这是不合理的。我们提出了一种不同的方法，其中随机变量描述了个体数量而不是个体数量的期望值的年平均值（并且它们在紧凑的间隔内取值）。我们的方法利用了来自环境属性的直观见解：在自然界中，个体被特定特征所吸引或排斥（Ballesteros等人。数学生物学85(4):31,2022)。从量子力学中汲取灵感,我们将量子哈密顿量纳入经典统计力学（即吉布斯测度或马尔可夫随机场）。扩散和吸引/排斥力之间的平衡决定了物种的行为，通过涉及能源运营商的全局控制问题来表示。
