Numerical Issues for a Non-autonomous Logistic Model

TL;DR

This paper investigates numerical issues in solving non-autonomous logistic models with time-varying growth rates, highlighting the robustness of a simple Runge-Kutta method over common black-box solvers for accurate biological simulations.

Contribution

It demonstrates that standard numerical solvers can produce unstable or inaccurate results for non-autonomous logistic equations, and shows that a manually-programmed Runge-Kutta method offers more reliable solutions.

Findings

Common solvers like lsoda and Runge-Kutta can produce unstable solutions.

A simple Runge-Kutta method accurately captures the analytical solution.

Manual implementation provides consistent and reliable simulations.

Abstract

The logistic equation has been extensively used to model biological phenomena across a variety of disciplines and has provided valuable insight into how our universe operates. Incorporating time-dependent parameters into the logistic equation allows the modeling of more complex behavior than its autonomous analog, such as a tumor's varying growth rate under treatment, or the expansion of bacterial colonies under varying resource conditions. Some of the most commonly used numerical solvers produce vastly different approximations for a non-autonomous logistic model with a periodically-varying growth rate changing signum. Incorrect, inconsistent, or even unstable approximate solutions for this non-autonomous problem can occur from some of the most frequently used numerical methods, including the lsoda, implicit backwards difference, and Runge-Kutta methods, all of which employ a black-box framework. Meanwhile, a simple, manually-programmed Runge-Kutta method is robust enough to accurately capture the analytical solution for biologically reasonable parameters and consistently produce reliable simulations. Consistency and reliability of numerical methods are fundamental for simulating non-autonomous differential equations and dynamical systems, particularly when applications are physically or biologically informed.