A cubic nonlinear population growth model for single species: theory, an explicit–implicit solution algorithm and applications

Advances in Difference Equations

Benjamin Wacker¹  Jan Christian Schlüter² 

[1] Department of Engineering and Natural Sciences, University of Applied Sciences Merseburg, Eberhard-Leibnitz-Straße 2, D-06217, Merseburg, Germany;Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute of Dynamics and Self-Organization, Am Fassberg 17, D-37077, Göttingen, Germany;Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute of Dynamics and Self-Organization, Am Fassberg 17, D-37077, Göttingen, Germany;

关键词: Continuous Nonlinear Differential Equation;    Discrete Difference Equation;    Global Existence;    Global Uniqueness;    Numerical Solution Algorithm;    Population Dynamics;    34A12;    34A34;    34C11;    65L05;    92D25;   

DOI  :  10.1186/s13662-021-03399-5

来源: Springer

【 摘 要 】

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.

【 授权许可】

CC BY