Structured interactions as a stabilizing mechanism for competitive ecological communities

Violeta Calleja-Solanas, Nagi Khalil, Jesús Gómez-Gardeñes, Emilio Hernández-García, and Sandro Meloni

Phys. Rev. E 106, 064307 – Published 12 December 2022

Abstract

How large ecosystems can create and maintain the remarkable biodiversity we see in nature is probably one of the biggest open questions in science, attracting attention from different fields, from theoretical ecology to mathematics and physics. In this context, modeling the stable coexistence of species competing for limited resources is a particularly challenging task. From a mathematical point of view, coexistence in competitive dynamics can be achieved when dominance among species forms intransitive loops. However, these relationships usually lead to species' relative abundances neutrally cycling without converging to a stable equilibrium. Although in recent years several mechanisms have been proposed, models able to explain species coexistence in competitive communities are still limited. Here we identify locality in the interactions as one of the simplest mechanisms leading to stable species coexistence. We consider a simplified ecosystem where individuals of each species lay on a spatial network and interactions are possible only between nodes within a certain distance. Varying such distance allows to interpolate between local and global competition. Our results demonstrate, within the scope of our model, that species coexist reaching a stable equilibrium when two conditions are met: individuals are embedded in space and can only interact with other individuals within a short distance. On the contrary, when one of these ingredients is missing, large oscillations and neutral cycles emerge.

Received 28 September 2022
Accepted 21 November 2022

DOI:https://doi.org/10.1103/PhysRevE.106.064307

Physics Subject Headings (PhySH)

Collective behavior in networks Complex systems Patterns in complex systems

Interdisciplinary PhysicsNetworks

Authors & Affiliations

Violeta Calleja-Solanas ¹, Nagi Khalil ², Jesús Gómez-Gardeñes ^3,4,5, Emilio Hernández-García ¹, and Sandro Meloni ^1,*

¹Institute for Cross-Disciplinary Physics and Complex Systems (IFISC), CSIC-UIB, 07122 Palma de Mallorca, Spain
²Complex Systems Group & GISC, Universidad Rey Juan Carlos, Móstoles 28933, Madrid, Spain
³GOTHAM Lab., Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, 50018 Zaragoza, Spain
⁴Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain
⁵Center for Computational Social Science (CCSS), University of Kobe, 657-8501 Kobe, Japan

^*sandro@ifisc.uib-csic.es

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Vol. 106, Iss. 6 — December 2022

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Images

Figure 1
Schematic representation of the interaction networks and competitive dynamics. (a) Diagram of the model. A random plant is selected to die with a probability $1 / N$ , leaving a vacant fertile region (i.e., an empty node). Two (highlighted in the middle panel) of the three neighbors are selected at random. Finally, the winner is chosen according to the probabilities of the species dominance matrix $H$ , and its descendant sprouts in the vacant node. (b)–(d) Illustration of the three spatial interaction networks considered. The neighborhood of the black node is depicted (in green) for different interaction ranges. (b) A 2D lattice with a regular distribution of individuals. The left side of the panel depicts the neighborhood for the smallest possible interaction range while the right side highlights the neighborhood when the smallest interaction range has been increased by one unit. (c) In a random geometric graph, the coordinates of the individuals are uniformly set at random in the unit square and two nodes are connected if their Euclidean distance is less or equal than $R_{RGG} = r_{small}$ (left side of the panel) or $R_{RGG} = r_{large}$ (right side of the panel). (d) Erdős-Rényi graphs have no spatial structure. Each pair of nodes connects with probability $p$ independently of their distance. The left and right sides of the panel illustrate the same spatial arrangement as in (c), but the neighbors of the black node are determined at random by the linking probabilities $p = 0.2$ and 0.4, respectively.
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Figure 2
(a)–(c) Temporal evolution of the species relative abundances for a 3-species system, comparing different interaction schemes. In each panel, the ecosystem is represented by a RGG of $N = 10^{4}$ nodes. Relative abundances $x_{1}, x_{2}, x_{3}$ are plotted after the transient has vanished. (a) All-to-all interactions: the range is set to cover the entire plane ( $R_{RRG} = R_{max} = \sqrt{2}$ ), hence, individuals can compete for any vacant node. (b) Long-range interactions: we set $R_{RGG} = 0.15$ leading to an average degree $〈 k 〉 ≃ 706$ . (c) Short-range interaction: $R_{RGG} = 0.03$ and $〈 k 〉 ≃ 28$ . (d)–(f) Trajectories in the phase space represented on the standard 2-simplex (the portion of the $x_{1} + x_{2} + x_{3} = 1$ plane in which $x_{1}, x_{2}, x_{3} \geq 0$ ). The plots show a view perpendicular to the simplex, and correspond to the time evolution of the left panels. The color code represents time evolution. With all-to-all and long-range interactions [(d) and (e)], abundances oscillate in large cycles around what seems to be an equilibrium point (represented by a black cross). With short-range interactions (f), abundances remain confined in a small region around the equilibrium.
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Figure 3
(a) Average area inside the trajectory on the 2-simplex of the $(x_{1}, x_{2}, x_{3})$ point of a 3-species community [see Figs 2] as a function of average degree $〈 k 〉$ , for different networks. The total number of individuals is $N = 10^{4}$ , and the same dominance matrix $H$ is used for all networks. The points represent the mean area obtained over 50 realizations, each simulated in different networks. Areas have been calculated excluding the $5 %$ of out-layer points in the trajectory. Shaded areas indicate $95 %$ confidence interval.
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Figure 4
Spatial organization of a 3-species community in a 2D lattice of $N = 10^{4}$ for short- and long-range interactions with the dominance matrix given in Eq. (1). Individuals of each species are depicted in a different color. (a) Short-range interactions: when a plant dies only the eight closest neighbors at distance one compete for a vacant node [see the left network from Fig. 1]. (b) Long-range interactions: the 360 individuals at distance less or equal to 9 from a vacant node participate in the competition. Videos for the two ranges of interactions are available in the Supplemental Material [33].
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Figure 5
(a) Time evolution of the recovery from a $90 %$ pulse perturbation in a 3-species community for the dominance matrix $H$ of Eq. (1). The relative abundance of one species (blue) is artificially modified from its equilibrium value to be the $90 %$ of the whole population, whereas other species' relative abundances (in gray) are proportionally decreased. The simulation is performed in a RGG of $10^{4}$ individuals and $R_{RGG} = 0.03$ . The red line represents the fit of the local maxima of the relative abundance (blue crosses) to the function $a e^{- α} + b$ with $α = 0.018$ , $a = 0.53$ and $b = 0.38$ . (b) For the same setting than in (a), we have varied the interaction range to obtain how the extinction probability varies with the average degree. Each bar corresponds to the mean over 50 different networks with $95 %$ confidence intervals shown as error bars.
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Figure 6
Scaling of fluctuations, measured by the coefficient of variation of species 1 ( $σ_{1} / 〈 x_{1} 〉$ ) with the system size $N$ for a RGG with 3 species. Each point is the result of 10 different realizations where the variance and mean relative abundance of species 1 have been calculated over at least $Δ t = 10^{8}$ time steps after the transient. Short-range interactions correspond to an average degree $〈 k 〉 = 15 \pm 2$ , and we find a decrease of the relative fluctuations with system size as $σ_{1} / 〈 x_{1} 〉 \sim N^{- 0.47}$ , consistent with a scenario of uncorrelated domains. For a situation of long-range interactions we set $〈 k 〉 = 980 \pm 190$ , giving a scaling of the relative fluctuations as $N^{- 0.14}$ .
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