In this paper, we extend existing population growth models and propose a model based on a nonlinear cubic differential equation that reveals itself as a special subclass of Abel differential equations of first kind. We first summarize properties of the time-continuous problem formulation. We state the boundedness, global existence, and uniqueness of solutions for all times. Proofs of these properties are thoroughly given in the Appendix to this paper. Subsequently, we develop an explicit–implicit time-discrete numerical solution algorithm for our time-continuous population growth model and show that many properties of the time-continuous case transfer to our numerical explicit–implicit time-discrete solution scheme. We provide numerical examples to illustrate different behaviors of our proposed model. Furthermore, we compare our explicit–implicit discretization scheme to the classical Eulerian discretization. The latter violates the nonnegativity constraints on population sizes, whereas we prove and illustrate that our explicit–implicit discretization algorithm preserves this constraint. Finally, we describe a parameter estimation approach to apply our algorithm to two different real-world data sets.

在本文中，我们扩展了现有的人口增长模型，并提出了一个基于非线性三次微分方程的模型，该模型显示出自己是第一类Abel微分方程的特殊子类。我们首先总结时间连续问题公式的性质。我们陈述了解决方案在任何时候的有界性，全球存在性和唯一性。这些属性的证明已在本文的附录中彻底给出。随后，我们为我们的时间连续人口增长模型开发了显式-隐式时间离散数值解算法，并证明了时间连续案例的许多性质都转移到了我们的数字式-隐式时间离散算法中。我们提供了数值示例来说明我们提出的模型的不同行为。此外，我们将我们的显式-隐式离散化方案与经典的欧拉离散化方案进行了比较。后者违反了人口数量的非负约束，而我们证明并说明了我们的显式-隐式离散化算法保留了该约束。最后，我们描述了一种参数估计方法，可将我们的算法应用于两个不同的实际数据集。