  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.
