Neutral competition boosts cycles and chaos in simulated food webs

Similarity of competitors has been proposed to facilitate coexistence of species because it slows down competitive exclusion, thus making it easier for equalizing mechanisms to maintain diverse communities. On the other hand, previous studies suggest that chaotic ecosystems can have a higher biodiversity. Here, we link these two previously unrelated findings, by analysing the dynamics of food web models. We show that near-neutrality of competition of prey, in the presence of predators, increases the chance of developing chaotic dynamics. Moreover, we confirm that chaotic dynamics correlate with a higher biodiversity.


A.1 Gottwald-Melbourne 0-1 test in a nutshell
The 0-1 test for chaos is designed for distinguishing between regular and chaotic dynamics in deterministic systems. It works directly with the observed time series, so a prior knowledge of the underlying dynamics is not required (as long as we know that they are deterministic). This short section is more a motivation than a rigorous proof. A minimal, geometrical approach to the method will be outlined. For a detailed, complete explanation please refer to (24).
The main input for the test is a one-dimensional time series of observations, φ k , where the integer k represents the time index. This time series is used to build the functions of the parameter θ: The summands in equation A.1 are the horizontal and vertical components of a vector of length φ k pointing in the direction kθ. Consequently, each observation in our time series can be understood as the size of a step in the plane, being kθ its direction (see φ k e ikθ ... Table 2: Example showing a step by step geometrical construction of the elements inside the summation operator in equation (A.1). In this example we use a time series whose first elements are φ j = {2, 1, 0.5, 0.25, ...}. The parameter θ has been set to π 6 .
Adding up the elements in table 2 as indicated by equation (A.1) can be interpreted geometrically as vector addition, i.e., performing one "step" after another (see figure A.1). With this picture in mind, it is easy to understand the kind of paths that different types of time series will give rise to (see figure A.2). Constant time series generate cyclic paths (polygons) or pseudocyclic paths (polygons that do not close after a first round). Periodic or pseudoperiodic time series generate periodic or pseudoperiodic paths. Random time series generate brownian-motionlike paths. Provided that our system is deterministic, the apparent stochasticity of our path is a strong indicator of chaos. The case of an underlying chaotic time series is the only one that generates a path that doesn't stay inside a bounded domain around the starting point (compare the third panel in figure A.2 with the other two). The 0-1 test uses the mean square displacement as a measure of this drift. The system is considered to be chaotic if the square displacement keeps growing for large times. If, on the contrary, it stays bounded, the test will consider the system not chaotic.
In the current manuscript, we used the time series corresponding to a nonextinct prey as input (φ k ) to the 0-1 test. Each time series was tested for 100 different values of θ, chosen from a uniform random distribution between π 3 and 2π 3 . The median convergence was taken as statistic (this is done to exclude coincidental non-convergence if θ is close to the period of the system).

A.2 Results for species pools of different sizes
In the main body of the paper we focused our attention in families of food webs with species pools consisting of 12 prey and 8 predator species. In this section we show the results of the same analysis for food webs of different sizes.
A.2.1 Probability of chaos grouped by number of species

A.2.3 Biodiversity measurements
For each simulation, a biodiversity index was estimated as the number of prey species whose population was higher than a minimum threshold of 0.01 mg l −1 , averaged respective to time. The yellow circles represent the average prey biodiversity of those simulations who had chaotic dynamics. The red and blue circles represent the same for, respectively, cyclic and stable dynamics. The relative area of the circles represents the ratio of each kind of dynamics.

A.4 Effect of the immigration term
In figure A.9 we show that our qualitative results remain true for different values of the immigration parameter f . Figure A.9: Results for a system with 8 predators and 12 prey. The first and second columns have immigration rates f of 10 −4 and 10 −6 respectively. The upper row shows the ratio of each dynamical regime vs. competition parameter (�) for the whole set of simulations. The lower row shows the average biomasses grouped by trophic level vs. competition parameter (�). The width represents standard deviation.
The curious reader is invited to re-run our simulations using different sets of parameters. The code is available in the Zenodo repository (25).