Ecosystems with Mutually Exclusive Interactions Self-Organize to a State of High Diversity

Abstract

Ecological systems comprise an astonishing diversity of species that cooperate or compete with each other forming complex mutual dependencies. The minimum requirements to maintain a large species diversity on long time scales are in general unknown. Using lichen communities as an example, we propose a model for the evolution of mutually excluding organisms that compete for space. We suggest that chain-like or cyclic invasions involving three or more species open for creation of spatially separated sub-populations that subsequently can lead to increased diversity. In contrast to its non-spatial counterpart, our model predicts robust co-existence of a large number of species, in accordance with observations on lichen growth. It is demonstrated that large species diversity can be obtained on evolutionary timescales, provided that interactions between species have spatial constraints. In particular, a phase transition to a sustainable state of high diversity is identified.Introduction.-Interactions between biological species may well be as old as life itself [1-3] with competition and predation as major determinants for species diversity [4,5]. Competitive exclusion [6,7] has been suggested to reduce ecosystem diversity when several species compete for the same resources. Real ecosystems, on the other hand, consist of multiple species and have a robustness that may even increase with diversity [8,9]. To obtain robustness of ecosystem diversity in theoretical models one needs first of all to limit the exponential growth by assuming a maximum carrying capacity for the population of each species [10]. Extreme version of such models [11,12] indeed predicts a sustainable but fragile coexistence of multiple species.A more robust way to maintain high diversity is to include space [13][14][15][16][17][18] e.g. in combination with hypercycles [19] or predator-prey cycles [20][21][22][23]. Studying ecosystems in marine hard-substrate environments Jackson & Buss [24] suggested that non-transitive allelopathic relationships between species could maintain species diversity on a longer timescale than pure hierarchical predation relationships. This was confirmed in a model on two-dimensional lattice where sessile species compete for space [25,26]. As in the "Buss" model [25,26] we consider a 2-dimensional lattice where the competition for resources is a zero sum game about available space. In our model, however, the focus is on the dynamic balance between an introduction of new species and exclusion of older species. With this complementary model we, for the first time show that: a) There is a sharp transition from multiple to single species as the number of interactions is increased. b) Both cycles and chain-like (hierarchical) relationships lead to spatial fragmentation of species population thus creating isolated niches for new species and increased diversity.The model is inspired by the spatial dynamics of lichen communities. Lichens have existed for as long as ∼600 million years [27] and are organisms consisting of fungi and ...

From pairwise to group interactions in games of cyclic dominance

“…In experiments with Escherichia coli, for example, it has been shown that arranging the bacteria on a Petri dish is crucial for keeping all three competing strains alive [8,36]. Accordingly, simulations of spatial rock-paper-scissors and related games of cyclic dominance have a long and fruitful history [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56], which is firmly rooted in methods of statistical physics. In general, if the mobility in the population is sufficiently small [7], the spatial rock-paper-scissors game leads to the stable coexistence of all three competing strategies, whereby the coexistence is maintained by the spontaneous formation of complex spatial patterns.…”

Characterization of phase transitions in a model ecosystem of sessile species

“…The critical value was found to be approximately γ c ≈ 0.055, for L = 192 [28,35]. For larger systems L ∈ 256, 320 we find larger critical values γ c ≈ 0.1.…”

“…The mean diversity shows a sharp transition from high-to low-values at γ ≈ 0.055 when α is close to zero and N = 192 2 [28,35] as shown in Figure 1(a). However, the fluctuation of the diversity turned out to be significant for larger α (see Fig.…”

“…For each potential interaction between the new and existing species a link is added with probability γ. Additionally the interaction network will always contain a link from the new species to the former occupant of the site at which it is introduced. This definition of α differs from our previous papers [28,35], were we chose α to be the rate of attempted introductions of a new species, which were rejected if the interaction network did not have a link from the new species to the old one. Thus, the α value used in this paper would have to be divided by γ when one compares it with previous results.…”

Labyrinthine clustering in a spatial rock-paper-scissors ecosystem

“…Investigating three strands of E. coli bacteria with cyclic interactions, it has been shown that biodiversity can not be preserved unless spacial structure is imposed by arranging the bacteria on a petri dish [7,8,26]. These results have been reproduced in Monte Carlo simulations [6,[27][28][29], but even though many different analytical approaches have been applied, exactly how spatial structure stabilizes the system is still an open problem [30][31][32][33].…”