Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong SAR, China.
Math Biosci Eng. 2022 May 20;19(8):7543-7569. doi: 10.3934/mbe.2022355.
Stage structured models, by grouping individuals with similar demographic characteristics together, have proven useful in describing population dynamics. This manuscript starts from reviewing two widely used modeling frameworks that are in the form of integral equations and age-structured partial differential equations. Both modeling frameworks can be reduced to the same differential equation structures with/without time delays by applying Dirac and gamma distributions for the stage durations. Each framework has its advantages and inherent limitations. The net reproduction number and initial growth rate can be easily defined from the integral equation. However, it becomes challenging to integrate the density-dependent regulations on the stage distribution and survival probabilities in an integral equation, which may be suitably incorporated into partial differential equations. Further recent modeling studies, in particular those by Stephen A. Gourley and collaborators, are reviewed under the conditions of the stage duration distribution and survival probability being regulated by population density.
阶段结构模型通过将具有相似人口特征的个体分组，已被证明在描述人口动态方面非常有用。本文从回顾两种广泛使用的建模框架开始，这两种框架的形式为积分方程和年龄结构偏微分方程。通过对阶段持续时间应用 Dirac 和伽马分布，这两种建模框架都可以简化为具有/不具有时滞的相同微分方程结构。每个框架都有其优点和内在局限性。净繁殖数和初始增长率可以很容易地从积分方程中定义。然而，在积分方程中整合对阶段分布和生存概率的密度依赖性调节变得具有挑战性，这可能适合纳入偏微分方程。进一步的最近的建模研究，特别是由 Stephen A. Gourley 及其合作者进行的研究，在阶段持续时间分布和生存概率受人口密度调节的条件下进行了回顾。
