Thieme Horst R
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA.
Math Biosci Eng. 2019 Nov 15;17(2):1168-1217. doi: 10.3934/mbe.2020061.
If the individual state space of a structured population is given by a metric space , measures on the -algebra of Borel subsets of offer a modeling tool with a natural interpretation: () is the number of individuals with structural characteristics in the set . A discrete-time population model is given by a on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. Under suitable assumptions, has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a . For a semelparous population, it can be identified with the . A spectral radius can be defined by the usual Gelfand formula.We investigate in how far it serves as a threshold parameter between population extinction and population persistence. The variation norm on the space of measures is too strong to give the basic turnover operator enough compactness that its spectral radius is an eigenvalue associated with a positive eigenmeasure. A suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm), which, as a trade-off, makes the basic turnover operator only continuous on the cone of nonnegative measures but not on the whole space of real measures.
如果结构化种群的个体状态空间由度量空间给出，那么在的波莱尔子集的 - 代数上的测度提供了一种具有自然解释的建模工具：() 是集合中具有结构特征的个体数量。离散时间种群模型由一个作用在有限非负波莱尔测度锥上的给出，它将给定年份的结构种群分布映射到下一年的分布。在适当的假设下，在零测度（灭绝不动点）处有一阶近似，它是实测度有序向量空间上的正线性算子，并且可以解释为一个。对于一次性繁殖种群，它可以与相识别。谱半径可以通过通常的盖尔范德公式定义。我们研究它在多大程度上作为种群灭绝和种群持续存在之间的阈值参数。测度空间上的变差范数太强，以至于无法赋予基本周转算子足够的紧性，使其谱半径是与正特征测度相关联的特征值。一个合适的替代是平坦范数（也称为（对偶）有界利普希茨范数），作为一种权衡，它使基本周转算子仅在非负测度锥上连续，而不在整个实测度空间上连续。
