Metabolic Trade-Offs Promote Diversity in a Model Ecosystem

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Metabolic Trade-Offs Promote Diversity in a Model Ecosystem

Anna Posfai, Thibaud Taillefumier, and Ned S. Wingreen

Phys. Rev. Lett. 118, 028103 – Published 12 January 2017

Abstract

In nature, a large number of species can coexist on a small number of shared resources; however, resource-competition models predict that the number of species in steady coexistence cannot exceed the number of resources. Motivated by recent studies of phytoplankton, we introduce trade-offs into a resource-competition model and find that an unlimited number of species can coexist. Our model spontaneously reproduces several notable features of natural ecosystems, including keystone species and population dynamics and abundances characteristic of neutral theory, despite an underlying non-neutral competition for resources.

Received 20 September 2016

DOI:https://doi.org/10.1103/PhysRevLett.118.028103

Physics Subject Headings (PhySH)

Biological self-organization Collective behavior Ecological pattern formation Marine ecology

Physics of Living Systems

Authors & Affiliations

Anna Posfai¹, Thibaud Taillefumier², and Ned S. Wingreen^1,3

¹Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey 08544, USA
²Department of Mathematics and Department of Neuroscience, The University of Texas at Austin, Austin, Texas 78712, USA
³Department of Molecular Biology, Princeton University, Princeton, New Jersey 08544, USA

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Issue

Vol. 118, Iss. 2 — 13 January 2017

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Images

Figure 1
Schematic of model with two species competing for three resources. Resources are steadily and homogeneously supplied to the environment with rates $\vec{s} = (s_{1}, s_{2}, s_{3})$ . Different “species,” i.e., different metabolic strategies, are defined by their specific distributions of enzymes for resource utilization. Since the total number of enzymes a species produces is subject to the budget constraint $\sum_{i = 1}^{p} α_{σ i} = E$ , each species can be represented by a point on the triangle in the space of resource utilization rates $(α_{1}, α_{2}, α_{3})$ . To indicate the nutrient supply in the triangle, we show ${\vec{s}}_{α} = (E / S) \vec{s}$ with a black diamond.
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Figure 2
Outcomes of a three-species competition, i.e., three distinct metabolic strategies compete for three steadily supplied resources. (a)–(c) Three illustrative examples of competition among the same three species, represented by the three colored points in all panels. Simulated population dynamics, starting from equal populations, is shown for three different steady nutrient supplies: the appropriately normalized supply rate of nutrients is indicated by the black diamonds. (d) The left triangle shows which of the three species survive for different values of the nutrient supply ${\vec{s}}_{α}$ : the red/green/blue species drives both others to extinction if ${\vec{s}}_{α}$ lies in the red/green/blue regions; the red & green/green & blue/blue & red species eradicate the third species if ${\vec{s}}_{α}$ lies in the orange/cyan/purple region; all three species coexist if ${\vec{s}}_{α}$ lies in the gray region. The right triangle shows the colormap of the corresponding steady-state concentrations of resources: the color is a mixture of red, green, and blue with proportions $c_{1}^{*}$ : $c_{2}^{*}$ : $c_{3}^{*}$ . If ${\vec{s}}_{α}$ lies outside the convex hull of the species, the steady-state nutrient concentrations mirror the proportions in which they are supplied. However, if ${\vec{s}}_{α}$ lies within the convex hull of the species, the steady-state concentrations always reach $c_{1}^{*} = c_{2}^{*} = c_{3}^{*}$ .
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Figure 3
Outcomes of many species competing for three steadily supplied resources. (a) The triangle shows the competing species. The black diamond is the (appropriately normalized) supply rate of nutrients. The graph shows simulated population dynamics starting from equal populations of each species. The populations of all species except one decay to extinction. (b) Same as (a) but with one additional species (orange). All species now coexist, as the black diamond now lies within the convex hull of the metabolic strategies.
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Figure 4
Comparison of resource-competition model to neutral model. (a) Rank-abundance curves for the resource competition birth-death-immigration process (red) and the neutral birth-death-immigration process (blue), for a total population of 100, individuals competing for three resources (equally supplied for our model) and species immigration probabilities $ν = 0.001$ (solid curves), $ν = 0.01$ (dashed curves), and $ν = 0.1$ (dotted curves). The curves indicate the mean population size of the largest (rank 1), second largest (rank 2), etc., populations during simulations of $2^{6} \times 1 0^{7}$ , $2^{5} \times 1 0^{7}$ , and $2^{4} \times 1 0^{7}$ time steps, respectively. The mean Shannon entropies of the distributions for the resource-competition and neutral models, in increasing order of $ν$ , are $H_{R C (N)} = 0.33 (0.22), 1.6 (1.4), 4.04 (4.04)$ ; $H = - \sum_{σ} p_{σ} \log_{2} (p_{σ})$ , where $p_{σ}$ is the probability of an individual belonging to species $σ$ . (b) The triangles show colormaps of the lifetime of species as fractions of the total simulation time, from top to bottom for $ν = 0.001$ , 0.01, 0.1.
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