Fig 1.
Diauxic resource competition can violate the competitive exclusion principle due to community-driven oscillations.
(A) Definition of our model of resource competition with sequential utilization, also known as diauxie. (i) We assume a model of boom–and–bust cycles. Throughout all figures, circles with tails represent species and polygons represent resources. Species are inoculated into a well-mixed environment with finite quantities of resources available and fully deplete those resources as their populations grow. Species then experience a mortality before an instantaneous influx of resources and the start of the next growth cycle (Methods). (ii) Species are defined by their resource preferences orders (Pref.) and growth rates (G.R.) for each resources (Methods), as is illustrated for two example species. (iii) Simulated growth (top) and resource consumption (bottom) dynamics for those two species over four growth cycles. The resources depletion times determine each species’ overall growth on each cycle and are indicated with vertical lines. Species A and B coexist because, while A initially outpaces B, after R1 is depleted A’s growth slows while B continues growing at its maximum growth rate. (B) When species compete for only one or two resources, the competitive exclusion principle, which predicts that only as many species as resources can survive, is obeyed. Species’ population fractions at the end of each growth cycle for two example communities are shown (Section A in S1 Text for parameters). (C) However, when three resources are supplied, diauxie can produce community-driven oscillations and competitive exclusion violations. Shown is an example of five species competing for three resources with all five stably coexisting (Section A in S1 Text for parameters and Fig 2 for further exploration of this example).
Fig 2.
Oscillations allow for highly diverse communities due to an emergent temporal-niche structure with more distinct niches than resources.
(A) Population fractions at the end of each growth cycle for the example of five species coexisting on three resources presented in Fig 1C (Section A in S1 Text for parameters). (B) Growth and resource-consumption dynamics on three different growth cycles. Top row shows population growth over the course of each growth cycle, while middle row shows decaying resource concentrations. Because each resource is depleted at a different time, species grow first in the three-resource environment, then in a two-resource environment, and finally in a single-resource environment. These sequentially realized environments are “temporal niches”. Fluctuating population sizes cause resources to be depleted in different orders on different growth cycles, so which temporal niches occur also varies, as is highlighted in the bottom row. (C) Resource depletion times (with tdep i being the time spent in a cycle until resource Ri is depleted) across the entire period of the oscillation show a total of five temporal niches. Lines show the resource depletion times and shaded regions highlight the temporal niches. (D) Species’ growth rates by resource (left) and in each temporal niche (right). Species only have one growth rate per resource, but their differing resource preferences produce different combinations of growth rates in each temporal niche such that each niche becomes a distinct growth phase with independent dynamics.
Fig 3.
The maximum number of coexisting species scales exponentially with the number of resources.
(A) The number of possible temporal niches is the number of binary combinations of whether or not each resource is still present () minus the combination corresponding to all resources being depleted. Shown are all the possible temporal niches for the cases of three, four, and five resources. (B) There can be at most one surviving species for each temporal niche, creating an upper bound on the maximum number of coexisting species that rises exponentially with the number of resources, as shown here in red. For comparison, one species per resource (the traditional competitive exclusion principle) is shown as a dashed gray line. Oscillations or chaos are necessary for more species than resources to coexist and require at least three species and three resources (Section C in S1 Text), so the bound of one species per resource applies if there are only one or two resources (blue line).
Fig 4.
With environmental fluctuations, communities of random species frequently violate competitive exclusion.
(A) Top row illustrates our implementation of extrinsic environmental fluctuations by randomly sampling the resource supply fractions on each growth cycle while keeping total supply constant (Methods). Plot shows population fractions at the end of each growth cycle for an example in which 5,000 random species were inoculated and simulated for 106 growth cycles with a uniform-randomly sampled resource supply on each day (σRS = 0.236). (B) Illustration of our resource supply sampling distributions as a function of the fluctuation magnitude σRS on simplex diagrams in which an equal supply of all resources maps to the middle of the triangle and a single resource being supplied maps to a corner (Section A in S1 Text, Methods). For σRS = 0.09, 0.15, and 0.236, one thousand randomly sampled resource supplies are shown. (C) We sampled 100 communities of 5000 species competing for three resources with the metabolic constraint that the L2-norm of species’ growth rates equal 1 hr-1, simulated these communities at 9 different fluctuation magnitudes, and tallied the survivors (Methods). Points represent the number of coexisting species for each community at each fluctuation magnitude, the dashed line shows the competitive exclusion principle (one species for each of the three resources), and the solid line and shaded region provide the mean number of survivors and standard error. With even the smallest level of fluctuation 13/100 of communities violated competitive exclusion, and with fluctuations of σRS = 0.09 or larger competitive exclusion was usually violated, with a peak frequency of violations of 83/100 at σRS = 0.15. (D) We next simulated 25 random species pools for each of two through seven resources using a uniform-random resource supply (σRS = 0.236 for the three-resource case, Methods). Points represent the number of survivors in the simulations. Shown for comparison are the competitive exclusion principle (one species per resource) and the exponential upper bound (Fig 3B). Inset shows the same data on a log-log scale with a monomial best fit. The monomial was a better fit than both linear and exponential models (Section A in S1 Text).
Fig 5.
Temporal niches explain the composition of highly diverse communities in fluctuating environments.
(A) We extended the resource supply sampling distributions to now have a maximum value σRS = 0.471, corresponding to a single randomly supplied resource on each growth cycle. Resource supply distributions are shown in the left column on the same simplex diagrams as in Fig 4B. We then simulated the 100 random communities of 5000 species from Fig 4 at 8 additional magnitudes for 17 total fluctuation magnitudes from σRS = 0 to σRS = 0.471. To develop a prediction of optimal strategies, we began by looking at the time spent in each temporal niche (middle column) and the time species spent growing on each of their resource preferences (right column). As the environmental fluctuation magnitude increased so did the time species spent growing on their second and third preferences, which would favor resource generalists. (B) Because species had been sampled with a metabolic constraint on their growth rates (), we could predict the optimal strategies (Section B in S1 Text). We compared these optimal strategies against the survivors in simulations using a simplex plot which is nonlinear in gμi but in which complete specialization on a single resource still maps to the corners and equal investment in all resources still maps to the center (Methods, Section A in S1 Text). At all fluctuation magnitudes, survivor growth rates are tightly clustered around the predicted optimal strategies.