Majority consensus thresholds in competitive Lotka--Volterra populations

Abstract: 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.