Extended Discrete-Time Population Model to Describe the Competition of Nutrient-Producing Protocells

Modeling the behavior of simple communities of protocells (as basic life-like organisms) is of vital importance since their better understanding may help to describe more complex (artificial and real) ecological systems. In this paper, we extend a recently developed discrete-time dynamic population model (called preliminary model) to a more general, completely reformulated version for describing the competition in a community of three protocell species (one generalist and two specialists). The advantage for the generalist is that it produces more kinds of nutrients than the specialists. In contrast to the preliminary model, the reproduction times and the times of (first) appearance of the three species can be all different in the extended model. The aim is to achieve the most basic/fundamental model that already displays complex population phenomena, like competitive exclusion, keystone species and an interesting "anomaly" regarding the connection between the survival of certain species and the decreasing rates of certain nutrients in the environment. Although, we could achieve this aim with a three-species model, at the simplest level, the model can be easily extended for more species in the future. The mentioned "anomaly" is a new discovery as it was not observed in the preliminary model. A particular equilibrium, when only the generalist survives, is exactly analyzed, where, interestingly, the golden ratio arises regarding the densities of the protocells of different ages. In future works, the extended model may serve as a useful tool for studying further phenomena in ecosystems, in their pure/abstract form.