The problem of the random zoo
Countless studies in theoretical community ecology have shown that coexistence of large communities is difficult to achieve. Take the multi-species Lotka-Volterra model, and choose parameters such that n species can coexist (where n is large). The choice of parameters becomes more and more restrictive as n grows. This means that for coexistence, parameters have to be “just right”. However, this is problematic: when we say that a certain population has an intrinsic growth rate of √2 = 1.4142... , what we really mean is that intrinsic growth rates vary in time, space, and among individuals, such that the mean of their distribution is close to 1.4—fine-tuning and biology are at odds, and we cannot reach robust conclusions whenever slight changes in parameters result in major changes to dynamics and outcomes.
In “Coexistence of many species in random ecosystems”, we take a different approach, and consider the fact that the extant communities we observe in nature are a subset of a much larger species pool (a “meta-community”) that has been pruned by population dynamics—the illusion of fine tuning comes from the fact that what we observe is the coexisting part of the community.
In this context, one can ask a simple question: if we start with n species, and run our model, how large will the final community be? To perform this calculation, we need to specify how are we going to choose the dynamical model and its parameters. For simplicity, we consider Lotka-Volterra dynamics, and sample parameters independently from a distribution. That is, species have not had time to co-evolve or co-adapt: take a large zoo, and open all the cages; come back after a few decades—how many species will you find?
Under certain parameterizations and technical assumptions, the problem of the random zoo can be solved analytically: for Lotka-Volterra dynamics, and intrinsic growth rates and interactions sampled independently from distributions that are symmetric about zero, the probability of finding k species when starting from n is simply the Binomial distribution with parameters n and ½. Therefore, in this setting we would end up with about half of the species coexisting, irrespective of the choice of n (and of the choice of the distribution of the interaction coefficients, provided that it is symmetric about zero).
This problem is strongly connected with that of “feasibility”: given a system of linear equations with random coefficients, with which probability will the solution of the system be totally positive? (In ecology, we cannot deal with negative densities or negative number of individuals). I had expected to find a rich mathematical literature on this topic, given that many quantities in science only make sense when they are positive (e.g., concentrations, probabilities, information), but I was mistaken. In fact, all I have found was an entertaining article by Kent Morrison. I thought Kent’s results could be further extended, and in fact Carlos and Kent were able to prove a more general theorem; José and Jacopo (after some heroic calculations presented in the supplement) were able to extend the results to the case of parameters with mean nonzero.
Overall, this is a quaint problem that could have been posed and solved decades ago. However, ecology is experiencing a renewed interest in the issue of assembly and dynamics of large communities, a problem that could yield precious insights for ecology and evolutionary biology.