Spatial ecology of territorial populations
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Edited by Nigel Goldenfeld, University of Illinois at Urbana–Champaign, Urbana, IL, and approved July 30, 2019 (received for review July 9, 2019)

Significance
All organisms live in spatial communities. In many cases, such as vegetation or bacterial biofilms, dense surface-bound populations compete for both resources and physical space. How do these territorial interactions impact ecosystem behavior and biodiversity? We study a theoretical model of territorial resource competition with trade-offs and show that many features of real ecosystems emerge naturally, including slow population dynamics that render community composition susceptible to demographic and other noise. We also observe alternate steady states, including the Allee effect in which survival requires a minimum population. Importantly, we demonstrate that biodiversity occurs robustly and can arise in territorial communities simply due to competition for resources.
Abstract
Many ecosystems, from vegetation to biofilms, are composed of territorial populations that compete for both nutrients and physical space. What are the implications of such spatial organization for biodiversity? To address this question, we developed and analyzed a model of territorial resource competition. In the model, all species obey trade-offs inspired by biophysical constraints on metabolism; the species occupy nonoverlapping territories, while nutrients diffuse in space. We find that the nutrient diffusion time is an important control parameter for both biodiversity and the timescale of population dynamics. Interestingly, fast nutrient diffusion allows the populations of some species to fluctuate to zero, leading to extinctions. Moreover, territorial competition spontaneously gives rise to both multistability and the Allee effect (in which a minimum population is required for survival), so that small perturbations can have major ecological effects. While the assumption of trade-offs allows for the coexistence of more species than the number of nutrients—thus violating the principle of competitive exclusion—overall biodiversity is curbed by the domination of “oligotroph” species. Importantly, in contrast to well-mixed models, spatial structure renders diversity robust to inequalities in metabolic trade-offs. Our results suggest that territorial ecosystems can display high biodiversity and rich dynamics simply due to competition for resources in a spatial community.
Living things exist not in isolation but in communities, many of which are strikingly diverse. Tropical rainforests can have more than 300 tree species in a single hectare (1), and it has been estimated that 1 g of soil contains 2,000–30,000+ distinct microbial genomes (2, 3). Understanding the relationship between biodiversity and the environment remains a major challenge, particularly in light of the competitive exclusion principle: In simple models of resource competition, no more species can coexist indefinitely than the number of limiting resources (4, 5). In modern niche theory, competitive exclusion is circumvented by mechanisms which reduce niche overlaps and/or intrinsic fitness differences (6, 7), suggesting that trade-offs may play an important role in the maintenance of biodiversity. Intriguingly, diversity beyond the competitive-exclusion limit was recently demonstrated in a resource-competition model with a well-mixed environment and exact metabolic trade-offs (8). However, many ecosystems are spatially structured, and metabolic trade-offs are unlikely to be exact. While some spatial structure is externally imposed, it also arises from the capacity of organisms to shape their environment. How does self-generated spatial structure, along with realistic metabolic constraints, impact diversity?
Various studies have clarified how intrinsic environmental heterogeneity (e.g., an external resource gradient) fosters biodiversity by creating spatial niches (9⇓⇓⇓–13). Others have demonstrated that migration between low-diversity local environments can lead to “metacommunities” with high global diversity (14⇓⇓⇓⇓–19). But how is diversity impacted by local spatial structure? Recent models suggest that spatial environments without intrinsic heterogeneity can support higher diversity than the well-mixed case (20⇓⇓⇓⇓–25), although the effect depends on the interactions and details of spatial structure (26, 27). In these models, competition follows phenomenological interaction rules. In some cases, trade-offs have been invoked to limit fitness differences (21) and penalize niche overlap (25), but did not otherwise structure the spatial interactions. All these models allow coexistence when the combination of spatial segregation and local interactions weakens interspecific competition relative to intraspecifc competition. However, it remains unclear how such interactions relate to concrete biophysical processes.
Here, we study biodiversity in a model where species interact through spatial resource competition. We specifically consider surface-associated populations which exclude each other as they compete for territory. This is an appropriate description for biofilms, vegetation, and marine ecosystems like mussels (28) or coral (29), in contrast with models that represent populations as overlapping densities and better describe motile or planktonic populations (9, 30). The well-mixed environment is an explicit limit of our model, so we are able to isolate the unique effects of spatial structure.
We find that, contrary to expectations, introducing population territories into a model with metabolic trade-offs reduces biodiversity relative to the well-mixed case. Extinctions occur over a new timescale inversely related to the nutrient mixing time. Spatial structure also leads to the emergence of multiple steady states and the Allee effect, so that small perturbations may have drastic consequences. Finally, we find that overall biodiversity is curbed by the domination of “oligotroph” species but is robust to inequalities in metabolic trade-offs.
Results
Model.
We developed a model of territorial populations competing for diffusing resources to clarify the relationship between spatial structure, metabolic trade-offs, and biodiversity. The model is spatially explicit and relates the mechanistic dynamics of competition to parameters with clear biological meaning. Crucially, competing populations are not interpenetrating, so populations are competing for both nutrients and territory.
Specifically, we consider m species competing for p nutrients in a 1-dimensional space of size L with periodic boundary conditions (a ring). The rate of supply of nutrients is specified by the supply vector
A model with spatial structure and metabolic trade-offs supports more species than expected from the principle of competitive exclusion. Example with 3 nutrients and 11 species starting with equal populations is shown. (A) Each species uptakes nutrients according to its enzyme-allocation strategy
Spatial structure reduces diversity compared to the well-mixed limit of instantaneous nutrient diffusion. (A, Upper) A well-mixed population of 10 species with equal initial populations competing for 2 nutrients. All 10 coexist at steady state. (A, Lower) Same as A, Upper, but with different initial populations. The community reaches a new steady state. (A, Upper, Inset) Strategies
While the supply of nutrients is spatially uniform, the local rate of nutrient consumption depends on the metabolic strategy of the local species, and nutrients diffuse in space. We study the regime where population growth is nutrient-limited, so the rate of uptake of each nutrient is linear in its concentration. Thus, within each region occupied by a single species σ, the nutrient concentrations
The populations change in time according to
The spatial nutrient environment influences the population dynamics via the dimensionless diffusion time
Biodiversity.
How does territorial spatial structure influence biodiversity? As an illustrative example, we consider 10 species competing for 2 resources. The simplex in Fig. 2 A, Inset shows how each of the strategies (colored dots) and the nutrient supply (diamond) divide between the 2 nutrients. In Fig. 2A, the nutrients are well-mixed (
How representative is the behavior seen in Fig. 2 A and B? In Fig. 2 C and D, we show results for many randomly generated territorial communities, confirming that the loss of biodiversity is a generic feature of spatial structure and that the nutrient diffusion time
In order to identify which features of the initial set of species determine steady-state diversity, we generated many random communities with species drawn uniformly from strategy space. Fig. 3 A and C shows the distributions of the steady-state diversity M as a function of
Steady-state diversity is governed by a simple condition: Diversity crashes if there is an “oligotroph” whose strategy satisfies
To test whether it is simply the presence/absence of oligotrophs that controls overall biodiversity, we generated random communities in an environment with an asymmetric nutrient supply (
Alternative Steady States and Slow Dynamics.
How does the outcome of spatial competition depend on initial conditions? Consider the well-mixed system in Fig. 2A. All that differs between the top and bottom subplots are the initial populations, but the same set of species has 2 very different steady states; not even the hierarchy of populations is preserved. In fact, there is an
The relationship between the steady states in the well-mixed and spatial regimes can be visualized in a simple example. Fig. 4A shows the phase behavior of 3 species competing for 2 resources. Here,
Spatial structure replaces the steady-state degeneracy of the well-mixed case with slow modes in population space. (A) Trajectories in population space for a 3-way competition at different values of
(A and B) Two species competing for 2 nutrients, with supply
What are the ecological implications of this slow relaxation to steady state? In general, diverse communities with
Although the well-mixed case has degenerate steady states, the steady-state nutrient environment is unique, and small initial population differences lead to small differences in the steady state (Fig. 4A). By contrast, spatial communities can have multiple steady-state nutrient environments, and similar populations may end in very different steady states. For example, in a competition of 2 species for 2 resources, Fig. 5 A and B shows the steady states as a function of
The Allee effect persists in more complex communities. Fig. 5C shows a 10-species competition where the brown and blue species can displace each other depending on the initial conditions, modifying the 8 other species’ fates in the process. Thus, multistability and the Allee effect emerge naturally in our territorial model, even though the species interact exclusively through competition for resources.
Unequal Enzyme Budgets.
Metabolic trade-offs are plausible because all microbes face the same biophysical constraints on metabolism and protein production, but trade-offs are unlikely to be exact in real ecosystems. How does this impact biodiversity in our model? Fig. 6A shows results for 10 species with exact trade-offs (
Territorial spatial structure renders diversity robust to variation in enzyme budgets. (A) In a community with equal enzyme budgets and
Discussion
We analyzed a model of spatial resource competition among territorial surface communities such as biofilms, vegetation, or coral. Each species has a concrete metabolic strategy subject to biophysical trade-offs. The nutrient environment has no intrinsic heterogeneity but is globally coupled via diffusion, so competitors shape it via consumption. We found that the resulting spatial structure restricts biodiversity, in stark contrast to previous models, where spatial segregation increases diversity by weakening competition. In the simplest of these cases, different resources are partitioned into different regions, providing spatial niches (9⇓⇓⇓–13). Alternatively, competitors may self-organize into patches linked by migration (14⇓–16, 18, 19) or into aggregates with local interactions (20⇓⇓⇓⇓⇓–26). External resource gradients can also increase diversity, because diffusion of dense motile populations prevents any species from monopolizing resource-rich regions (9, 10). In our model, the situation is very different. Because the external nutrient supply is uniform and the entire space is linked via diffusion, no spatial niches emerge; competitors have nowhere to hide. This is reminiscent of ecological reaction–diffusion models without external sources, where global coupling reduces diversity (26) and nonuniform steady states only become possible for unequal diffusion coefficients or complex geometries (27). Our communities are fundamentally different, however, as they occupy exclusive territories and exceed the competitive-exclusion limit despite a simple geometry and uniform diffusion coefficients.
What controls diversity in our model? The degree of nutrient mixing
Spatial structure also provides a novel mechanism for discontinuous transitions between alternative steady states. Such sudden shifts, or “catastrophes,” attract significant attention due to their implications for ecosystem resilience (31). The Allee effect occurs in a large variety of ecosystems (32) and is particularly relevant to the conservation of rare species. It is usually understood as the result of transparently cooperative processes, such as production of a public good (32), and modeled via an explicit cooperative term. In resource-competition models, multistability has been observed when species consume nutrients one at a time (33) or with unequal stoichiometries (34). Here, both the Allee effect and multistability emerge naturally from the ability of a population to render its resource environment more favorable to itself. Interestingly, the Allee-effect species are oligotrophs, underscoring the special ability these strategies have to impact their ecosystems.
It has been observed that spatial structure increases the time to reach equilibrium (35). Here, we showed precisely how a new dynamical timescale emerges from spatial structure. We found that the slow dynamics are confined to a manifold in population space. These slow modes of the population are subject to large fluctuations due to noise (e.g., demographics). Slow relaxation also means that for a rapidly changing nutrient supply, the population might never reach steady state, potentially saving some species from extinction.
Finally, we find that spatial structure allows diversity to persist with imprecise metabolic trade-offs. In the well-mixed system without noise, any deviation from exactly equal enzyme budgets leads to ecosystem collapse (8). Spatial communities, however, remain diverse with only approximate trade-offs. In fact, variation in enzyme budgets actually increases mean diversity by impairing oligotrophs. The persistence of diversity beyond competitive exclusion with inexact trade-offs makes it more credible that trade-offs play a role in maintaining the surprising diversity of real ecosystems.
Our results suggest several future research directions. A 2D extension of the model exhibits the same loss of biodiversity due to oligotrophs and uneven abundances (SI Appendix), and it will be interesting to explore 2D pattern formation in more depth. One might also consider resources that diffuse at different rates. This can lead to nonuniform steady states in reaction–diffusion systems (27). Finally, in microbial communities, gene regulation and evolution are often relevant on ecological timescales, so it would be natural to allow species to modify their strategies.
In summary, we find that spatial structure engenders more realistic communities: It curtails the unlimited diversity of the well-mixed model, but allows for coexistence beyond the competitive exclusion principle even in the absence of exact metabolic trade-offs. Our results demonstrate that mechanistic interactions, arising from biophysical constraints such as space and metabolism, can allow even simple models to capture some of the rich behaviors of real ecosystems.
Materials and Methods
The population ordinary differential equations (Eq. 3) were solved numerically by using Mathematica’s “NDSolve.” The
Details on the well-mixed model, stochastic dynamics, and figure parameters can be found in SI Appendix. Code is available on GitHub (36).
Acknowledgments
We thank S. Levin for valuable conversations and suggestions. This work was supported in part by the National Science Foundation, through Center for the Physics of Biological Function Grant PHY-1734030 (to B.G.W.); National Institutes of Health Grant R01 GM082938 (to N.S.W.); and the Princeton University Diekman Genomics Fund (A.P.).
Footnotes
- ↵1To whom correspondence may be addressed. Email: wingreen{at}princeton.edu.
Author contributions: B.G.W., A.P., and N.S.W. designed research; B.G.W., A.P., and N.S.W. performed research; B.G.W. analyzed data; and B.G.W. and N.S.W. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Data deposition: The code used in this paper has been deposited in GitHub (https://github.com/BenjaminWeiner/ecology-territorial-populations).
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1911570116/-/DCSupplemental.
- Copyright © 2019 the Author(s). Published by PNAS.
This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY).
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