Competitive networks revisited: Coexistence and chaos arising from nonhierarchical competition with multiple limiting factors
Ecologists have long sought to explain species coexistence with classical niche theory or neutral theory (i.e., species equivalence). However, field evidence supporting either theory is slim, prompting examination of alternative mechanisms. In the competitive network theory of species diversity (Allesina and Levine, PNAS 2011), species engaged in nonhierarchical competition over multiple limiting resources experience stable fluctuations in population levels. Game theory has been used to show that up to half of the species pool will coexist at equilibrium, with more species coexisting when subject to higher numbers of limiting factors, assuming species’ competitive fitness with respect to one limiting factor is independent of its fitness with respect to others. These results derived from the averaged result of fixed tournaments in which the competitive abilities of each species for each limiting factor were ranked relative to the other species and in which pairwise competitive dominance was determinisitically assigned based on the number of factors for which a competitor was superior.
We contend that, rather than treating the outcome of each pairwise competitive interaction in an environment as deterministic, it is more realistic to treat it as a random event governed by binomial probability equal to the fraction of limiting factors for which a competitor is superior. Hence, we repeated Allesina and Levine’s competitive network tournaments using a Monte Carlo simulation approach rather than the game-theoretic equilibrium solution. We tested the null hypothesis that the outcome would not differ from the aggregate solution of the fixed tournaments.
The outcome of the randomized tournament simulations contrasted with those of the fixed tournaments, leading us to reject the null hypothesis. Across all simulations, levels of coexistence were highest with just two limiting factors, but even then constituted a smaller percentage (10-30%) of the initial species pool, with diminishing returns as the size of the species pool grew. With more limiting factors, just 2.1-2.7 species coexisted at equilibrium (for initial pools of 10-30 species), again with only small gains as the size of the species pool grew. Hence, competitive network theory does not appear to be a mechanism promoting coexistence of many species. Interestingly, though, introducing the outcome of pairwise competitive interactions as a random variable within tournaments expanded the range of population dynamics over time, beyond stable oscillatory single-period fluctuations, to multiple-period oscillations and even apparently chaotic behavior, with the latter types emerging only under scenarios with larger numbers of limiting resources.