Numerical Issues for a Non-autonomous Logistic Model

Authors

Marina Mancuso, Los Alamos National LaboratoryFollow
Kaitlyn M. Martinez, Los Alamos National LaboratoryFollow
Carrie Manore, Los Alamos National LaboratoryFollow
Fabio Milner, Arizona State UniversityFollow

Publication Date

6-20-2024

Keywords

numerical solvers, numerical stability, nonautonomous model, logistic growth

Disciplines

Mathematics | Non-linear Dynamics | Numerical Analysis and Computation | Ordinary Differential Equations and Applied Dynamics | Physical Sciences and Mathematics | Science and Mathematics Education

Abstract

The user-friendly aspects of standardized, built-in numerical solvers in
computational software aid in the simulations of many problems solved using
differential equations. The tendency to trust output from built-in numerical
solvers may stem from their ease-of-use or the user’s unfamiliarity with the
inner workings of the numerical methods. Here, we show a case where the
most frequently used and trusted built-in numerical methods in Python’s
SciPy library produce incorrect, inconsistent, and even unstable approxima-
tions for a the non-autonomous logistic equation, which is used to model
biological phenomena across a variety of disciplines. Some of the most com-
monly used built-in numerical solvers, such as the lsoda, implicit backwards
difference, and Runge-Kutta methods employ a black-box framework and
produce vastly different approximations for the non-autonomous logistic
model with a periodically-varying growth rate changing signum. Meanwhile,
a simple, manually-programmed method can accurately capture the analyti-
cal solution for biologically reasonable parameters and consistently produce
reliable simulations across solutions with differing qualitative behavior. Con-
sistency and reliability of numerical methods are fundamental for simulating
non-autonomous differential equations and dynamical systems, particularly
when applications are physically or biologically informed. This provides an
opportunity for differential equations educators to show students an example
where output from a hand-coded solver may provide more value over the
available standardized, built-in solvers.

Recommended Citation

Mancuso, Marina; Martinez, Kaitlyn M.; Manore, Carrie; and Milner, Fabio (2024) "Numerical Issues for a Non-autonomous Logistic Model," CODEE Journal: Vol. 18, Article 5.
Available at: https://scholarship.claremont.edu/codee/vol18/iss1/5

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.