Abstract
Simple birth–death phenomena are frequently examined in mathematical modeling and probability theory courses since they serve as an excellent foundation for stochastic modeling. Such mechanisms are inherent stochastic extensions of the deterministic population paradigm for population expansion of a particular species in a habitat with constant resource availability and many other organisms. Most animal behavior research differentiates such circumstances into two different events when it comes to two-choice scenarios. On the other hand, in this kind of research, the reward serves a significant role, because, depending on the chosen side and food placement, such situations may be divided into four groups. This article presents a novel stochastic equation that may be used to describe the vast majority of models discussed in the current studies. It is noteworthy that they are connected to the symmetry of the progression of a solution of stochastic equations. The techniques of fixed point theory are employed to explore the existence, uniqueness, and stability of solutions to the proposed functional equation. Additionally, some examples are offered to emphasize the significance of our findings.