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

## TL;DR

This paper introduces a new model to study how different protocell species compete for nutrients, revealing complex ecological behaviors like competitive exclusion and keystone species.

## Contribution

The paper presents an extended discrete-time model that introduces a new ecological anomaly and allows for varied reproduction and appearance times among species.

## Key findings

- The model demonstrates complex phenomena such as competitive exclusion and keystone species in a three-species protocell community.
- A new ecological anomaly was discovered, where species survival correlates with decreasing nutrient rates in unexpected ways.
- The golden ratio appears in the equilibrium state when only the generalist protocell species survives.

## Abstract

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.

The online version contains supplementary material available at 10.1007/s11538-025-01488-0.