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Abstract

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.