Effect of Resource Dynamics on Species Packing in Diverse Ecosystems

The competitive exclusion principle asserts that coexisting species must occupy distinct ecological niches (i.e., the number of surviving species cannot exceed the number of resources). An open question is to understand if and how different resource dynamics affect this bound. Here, we analyze a generalized consumer resource model with externally supplied resources and show that—in contrast to self-renewing resources—species can occupy only half of all available environmental niches. This motivates us to construct a new schema for classifying ecosystems based on species packing properties.

Figure 1

Schematic description for two types of resources. (a) Self-renewing resources (e.g., plants), which are replenished through organic reproduction, and (b) externally supplied resources (e.g., nutrients that sustain gut microbiota), which are replenished by a constant flux from some external source, and diluted at a constant rate. (c) The supply rate as a function of resource abundance for both choices, with $\kappa ={\omega }_{\alpha }=K_{\alpha }=1$.

Figure 2

Comparison between cavity solutions (see text for definition) and simulations for the fraction of surviving species ${\varphi }_N=(S^{*}/S)$ and the first and second moments of the species and resources distributions as a function of ${\sigma }_c$. The error bar shows the standard deviation from 1000 numerical simulations with $M=S=100$, and all other parameters are defined in the SM [22]. Simulations were run using the cvxpy package [31].

Figure 3

Comparison of the species packing ratio $S^{*}/M$ at various ${\sigma }_c$ and $K$ for self-renewing and externally supplied resource dynamics. The simulations represent averages from 1000 independent realizations with the system size $M=100$, $S=500$ (parameters in SM [22]).

Figure 4

Species packing bounds in the presence of metabolic trade-offs. (a) The species packing ratio $S^{*}/M$ as a function of ${\sigma }_m/{\sigma }_c$, where ${\sigma }_m$ is the standard deviation of the $\delta m_i$ and ${\sigma }_c/\sqrt{M}$ is the standard deviation of $C_{i\alpha }$. Simulations are for binary consumer preference matrix $C_{i\alpha }$ drawn from a Bernoulli distribution with probability $p$. (b) $m_i$ versus ${\sum }_{\alpha }{C}_{i\alpha }$ for $p=0.1$ and ${\sigma }_m/{\sigma }_c={10}^{-0.5}$. See SM for all parameters [22].