The paper investigates the majority consensus problem in discrete, stochastic Lotka-Volterra models of competitive microbial populations. It analyzes how the probability of reaching majority consensus, where one species dominates the other, depends on the initial gap between the two species and the type of interference competition between them.
The key findings are:
For self-destructive interference competition, where interactions between individuals of different species lead to the death of both, the majority consensus threshold lies in a polylogarithmic range between Ω(√log n) and O(log^2 n). This is an exponential improvement over the previously known bound of Ω(√n log n).
For non-self-destructive interference competition, where interactions between individuals of different species lead to the death of only one individual, the majority consensus threshold lies in a polynomial range between Ω(√n) and O(√n log n). This shows an exponential separation between the two types of competitive dynamics.
The paper also investigates the impact of intraspecific competition, where individuals of the same species compete with each other. It shows that in the presence of strong intraspecific competition, majority consensus may not be solvable with high probability regardless of the initial gap.
The analysis uses a new "asynchronous pseudo-coupling" technique to bound the stochastic noise arising from birth, death, and competition events in the two-species Lotka-Volterra process. This technique is more general than previous coupling approaches and may be applicable to analyzing other stochastic population models beyond Lotka-Volterra dynamics.
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arxiv.org
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