On the correlation between properties of one-dimensional mappings of control functions and chaos in a special type delay differential equation

A differential equation of a special form, which contains two control functions $f$ and $g$ and one delayed argument, is analyzed. This equation has a wide application in biology for the description of dynamic processes in population, physiological, metabolic, molecular-genetic, and other applications. Specific numerical examples show the correlation between the properties of the one-dimensional mapping, which is generated by the ratio $f /g$, and the presence of chaotic dynamics for such equation. An empirical criterion is formulated that allows one to predict the presence of a chaotic potential for a given equation by the properties of the one-dimensional mapping $f /g$.