One of the key challenges in synthetic biology is devising robust signaling primitives for engineered microbial consortia. In such systems, a fundamental signal amplification problem is the majority consensus problem: given a system with two input species with initial difference of $\Delta$ in population sizes, what is the probability that the system reaches a state in which only the initial majority species is present? In this work, we consider a discrete and stochastic version of competitive Lotka--Volterra dynamics, a standard model of microbial community dynamics. We identify new threshold properties for majority consensus under different types of interference competition: - We show that under so-called self-destructive interference competition between the two input species, majority consensus can be reached with high probability if the initial difference satisfies $\Delta \in \Omega(\log^2 n)$, where $n$ is the initial population size. This gives an exponential improvement compared to the previously known bound of $\Omega(\sqrt{n \log n})$ by Cho et al. [Distributed Computing, 2021] given for a special case of the competitive Lotka--Volterra model. In contrast, we show that an initial gap of $\Delta \in \Omega(\sqrt{\log n})$ is necessary. - On the other hand, we prove that under non-self-destructive interference competition, an initial gap of $\Omega(\sqrt{n})$ is necessary to succeed with high probability and that a $\Omega(\sqrt{n \log n})$ gap is sufficient. This shows a strong qualitative gap between the performance of self-destructive and non-self-destructive interference competition. Moreover, we show that if in addition the populations exhibit interference competition between the individuals of the same species, then majority consensus cannot always be solved with high probability, no matter what the difference in the initial population counts.

翻译：合成生物学的一个关键挑战是为工程微生物群落设计稳健的信号传递基元。在此类系统中，一个基本的信号放大问题是多数共识问题：给定一个包含两种输入物种的系统，其初始种群规模差异为Δ，系统达到仅初始多数物种存在的状态的概率是多少？本文考虑离散随机版本的竞争性Lotka—Volterra动力学（微生物群落动力学的标准模型），并识别出不同干扰竞争类型下多数共识的新阈值性质：——我们证明，在两种输入物种间的所谓自毁干扰竞争下，若初始差异满足Δ ∈ Ω(log² n)（其中n为初始种群规模），则能以高概率达成多数共识。这相较于Cho等人[《分布式计算》，2021]针对竞争性Lotka—Volterra模型特例给出的已知界Ω(√(n log n))呈指数级改进。相反，我们证明Δ ∈ Ω(√(log n))的初始缺口是必要的。——另一方面，我们证明非自毁干扰竞争下，需初始缺口Δ ∈ Ω(√n)才能以高概率成功，且Δ ∈ Ω(√(n log n))的缺口已足够。这表明自毁与非自毁干扰竞争的性能存在显著定性差异。此外，我们证明若种群同时存在同种个体间的干扰竞争，则无论初始种群计数差异多大，多数共识均无法始终以高概率实现。