Spatial Ecology of Territorial Populations

Spatial ecology of territorial populations Benjamin G. Weinera, Anna Posfaib, and Ned S. Wingreenc,1 aDepartment of Physics, Princeton University, Princeton, NJ 08544; bSimons Center for Quantitative Biology, Cold Spring Harbor Laboratory, Cold Spring Harbor, NY 11724; and cLewis–Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544 Edited by Nigel Goldenfeld, University of Illinois at Urbana–Champaign, Urbana, IL, and approved July 30, 2019 (received for review July 9, 2019) Many ecosystems, from vegetation to biofilms, are composed of and penalize niche overlap (25), but did not otherwise struc- territorial populations that compete for both nutrients and phys- ture the spatial interactions. All these models allow coexistence ical space. What are the implications of such spatial organization when the combination of spatial segregation and local interac- for biodiversity? To address this question, we developed and ana- tions weakens interspecific competition relative to intraspecifc lyzed a model of territorial resource competition. In the model, competition. However, it remains unclear how such interactions all species obey trade-offs inspired by biophysical constraints on relate to concrete biophysical processes. metabolism; the species occupy nonoverlapping territories, while Here, we study biodiversity in a model where species interact nutrients diffuse in space. We find that the nutrient diffusion through spatial resource competition. We specifically consider time is an important control parameter for both biodiversity and surface-associated populations which exclude each other as they the timescale of population dynamics. Interestingly, fast nutri- compete for territory. This is an appropriate description for ent diffusion allows the populations of some species to fluctuate biofilms, vegetation, and marine ecosystems like mussels (28) or to zero, leading to extinctions. Moreover, territorial competition coral (29), in contrast with models that represent populations as spontaneously gives rise to both multistability and the Allee overlapping densities and better describe motile or planktonic effect (in which a minimum population is required for survival), so populations (9, 30). The well-mixed environment is an explicit that small perturbations can have major ecological effects. While limit of our model, so we are able to isolate the unique effects of the assumption of trade-offs allows for the coexistence of more spatial structure. species than the number of nutrients—thus violating the prin- We find that, contrary to expectations, introducing population ciple of competitive exclusion—overall biodiversity is curbed by territories into a model with metabolic trade-offs reduces bio- the domination of “oligotroph” species. Importantly, in contrast diversity relative to the well-mixed case. Extinctions occur over to well-mixed models, spatial structure renders diversity robust to a new timescale inversely related to the nutrient mixing time. inequalities in metabolic trade-offs. Our results suggest that terri- Spatial structure also leads to the emergence of multiple steady torial ecosystems can display high biodiversity and rich dynamics states and the Allee effect, so that small perturbations may have simply due to competition for resources in a spatial community. drastic consequences. Finally, we find that overall biodiversity is curbed by the domination of “oligotroph” species but is robust to spatial ecology j biodiversity j trade-offs j microbial ecology j modeling inequalities in metabolic trade-offs. Results iving things exist not in isolation but in communities, many Lof which are strikingly diverse. Tropical rainforests can have Model. We developed a model of territorial populations compet- more than 300 tree species in a single hectare (1), and it has ing for diffusing resources to clarify the relationship between been estimated that 1 g of soil contains 2,000–30,000+ dis- tinct microbial genomes (2, 3). Understanding the relationship Significance between biodiversity and the environment remains a major chal- lenge, particularly in light of the competitive exclusion principle: All organisms live in spatial communities. In many cases, such In simple models of resource competition, no more species can as vegetation or bacterial biofilms, dense surface-bound pop- coexist indefinitely than the number of limiting resources (4, 5). ulations compete for both resources and physical space. How In modern niche theory, competitive exclusion is circumvented do these territorial interactions impact ecosystem behavior by mechanisms which reduce niche overlaps and/or intrinsic fit- and biodiversity? We study a theoretical model of terri- ness differences (6, 7), suggesting that trade-offs may play an torial resource competition with trade-offs and show that important role in the maintenance of biodiversity. Intriguingly, many features of real ecosystems emerge naturally, including diversity beyond the competitive-exclusion limit was recently slow population dynamics that render community composi- demonstrated in a resource-competition model with a well-mixed tion susceptible to demographic and other noise. We also environment and exact metabolic trade-offs (8). However, many observe alternate steady states, including the Allee effect in ecosystems are spatially structured, and metabolic trade-offs are which survival requires a minimum population. Importantly, unlikely to be exact. While some spatial structure is externally we demonstrate that biodiversity occurs robustly and can imposed, it also arises from the capacity of organisms to shape arise in territorial communities simply due to competition for their environment. How does self-generated spatial structure, resources. along with realistic metabolic constraints, impact diversity? Various studies have clarified how intrinsic environmental het- Author contributions: B.G.W., A.P., and N.S.W. designed research; B.G.W., A.P., and N.S.W. erogeneity (e.g., an external resource gradient) fosters biodiver- performed research; B.G.W. analyzed data; and B.G.W. and N.S.W. wrote the paper.y sity by creating spatial niches (9–13). Others have demonstrated The authors declare no conflict of interest.y that migration between low-diversity local environments can lead This article is a PNAS Direct Submission.y to “metacommunities” with high global diversity (14–19). But This open access article is distributed under Creative Commons Attribution License 4.0 how is diversity impacted by local spatial structure? Recent (CC BY).y models suggest that spatial environments without intrinsic het- Data deposition: The code used in this paper has been deposited in GitHub (https:// erogeneity can support higher diversity than the well-mixed case github.com/BenjaminWeiner/ecology-territorial-populations).y (20–25), although the effect depends on the interactions and 1 To whom correspondence may be addressed. Email: [email protected] details of spatial structure (26, 27). In these models, competi- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. tion follows phenomenological interaction rules. In some cases, 1073/pnas.1911570116/-/DCSupplemental.y trade-offs have been invoked to limit fitness differences (21) Published online August 21, 2019. 17874–17879 j PNAS j September 3, 2019 j vol. 116 j no. 36 www.pnas.org/cgi/doi/10.1073/pnas.1911570116 Downloaded by guest on September 27, 2021 spatial structure, metabolic trade-offs, and biodiversity. The regime where population growth is nutrient-limited, so the rate model is spatially explicit and relates the mechanistic dynam- of uptake of each nutrient is linear in its concentration. Thus, ics of competition to parameters with clear biological meaning. within each region occupied by a single species σ, the nutrient Crucially, competing populations are not interpenetrating, so concentrations cσi obey populations are competing for both nutrients and territory. 2 m p @cσi @ cσi Specifically, we consider species competing for nutri- = S − α c + D , [1] ents in a 1-dimensional space of size L with periodic boundary @t i σi σi @x 2 conditions (a ring). The rate of supply of nutrients is speci- P where D is the diffusion coefficient for all nutrients. As nutri- fied by the supply vector S~ = (S1, S2 ::: Sp ) such that Si = S, i ent processing is generally much faster than growth, we assume S where is the total nutrient supply rate in units of concentra- a separation of timescales, such that nutrient concentrations tion/time. The nutrient supply is spatially uniform, so there is no equilibrate before populations change. Then, @c = 0, and external environmental heterogeneity. Each species σ 2 [1 ::: m] @t is defined by its metabolic strategy ~ασ = (ασ1, ασ2 : : : ασp ), r r Si ασi ασi which specifies the proportion of its metabolic resources (e.g., cσi (x) = + Aσi exp x + Bσi exp −x : enzymes) it allocates to the consumption of each nutrient. ασi D D Metabolic trade-offs are implemented via a constraint on the [2] P The constants of integration Aσi and Bσi are fixed by the phys- enzyme budget—namely, i ασi = E for all species (except where noted). Metabolic strategies and the supply can be rep- ical requirement that ci (x) be continuous and differentiable at resented as points on a simplex of dimension p − 1 (see Figs. 1A the population boundaries. Fig. 1D shows the concentrations of and 2 A, Inset, for example). Each species occupies a segment the 3 nutrients after the populations shown in Fig. 1

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