Kicsiny Richárd, Bódai Tamás, Székely László, Varga Zoltán
Department of Mathematics and Modelling, Institute of Mathematics and Basic Science, Hungarian University of Agriculture and Life Sciences, Páter K. u. 1., Gödöllő, 2100, Hungary.
Department of Applied Statistics, Institute of Mathematics and Basic Science, Hungarian University of Agriculture and Life Sciences, Páter K. u. 1., Gödöllő, 2100, Hungary.
Bull Math Biol. 2025 Jul 18;87(8):111. doi: 10.1007/s11538-025-01488-0.
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.
对原始细胞（作为基本的类生命有机体）的简单群落行为进行建模至关重要，因为对它们的更好理解可能有助于描述更复杂的（人工和真实的）生态系统。在本文中，我们将最近开发的离散时间动态种群模型（称为初步模型）扩展为一个更通用、完全重新制定的版本，用于描述三种原始细胞物种（一种通才物种和两种专性物种）群落中的竞争。通才物种的优势在于它产生的营养种类比专性物种更多。与初步模型不同，扩展模型中三种物种的繁殖时间和（首次）出现时间可以各不相同。目的是实现一个已经展示出复杂种群现象的最基本/基础模型，如竞争排斥、关键物种以及关于某些物种的生存与环境中某些营养物质减少速率之间联系的一个有趣的“异常”现象。虽然我们可以用一个三种物种的模型实现这个目标，但在最简单的层面上，该模型未来可以很容易地扩展到更多物种。上述“异常”是一个新发现，因为在初步模型中未观察到。对仅通才物种存活时的一个特定平衡点进行了精确分析，有趣的是，不同年龄原始细胞的密度出现了黄金比例。在未来的工作中，扩展模型可以作为一个有用的工具，以其纯粹/抽象的形式研究生态系统中的进一步现象。
