•The spatial size and competition strength are main factors of network complexity in ecological networks.•Finding a phase transition accompanying an expansion of a basin for coexistence on the phase ...space of spatial.•The intensification of alternative competition can yield the decrease of levels of critical mobility and require larger size to validate coexistence.
One of the common assumptions in previous spatial dynamics of cyclic competition is that, regardless of competing structure and strength among species, the spatial size of a network is considered as large as possible to avoid finite size effect for species biodiversity. In real ecosystems, however, species richness, which can be defined by spatial size and competition strength, can sensitively affect species coexistence as a competition among individuals becomes complicated. In this paper, we investigate the structural stability of coexistence of mobile species in three cyclic competition games due to network complexity in which imposes a size of a square lattice and competition strength among species. By exploiting the coexistence probability, our computations quantitatively reveal that the network complexity due to changes in the competition rate and lattice size can strongly affect the structural stability of coexistence in each model. In particular, intense intraspecific competition can yield the robust coexistence at small-sized lattices regardless of mobility, and strengthening interspecific competition simultaneously induces changes in critical mobility that hampers coexistence and in spatial size for stable coexistence. Qualitatively, we find that such structural stability of coexistence relates to the degree of stability of fixed points in deterministic systems. Our finding can be useful to gain insights into species coexistence on spatially extended systems with respect to network complexity.
•Modified RPSLS game model where certain predator-prey interactions are broken is investigated.•The lake of alternative interactions influence the coexistence of species.•High mobility always ...destroys interactions between directly related species.•Three-species cyclic interactions is crucial for biodiversity at low mobility.
The evolutionary game of cyclic competition has been extensively studied to gain insight into one of the most fundamental phenomena in nature: biodiversity. The three-species Rock-Paper-Scissors (RPS) game and its extension Rock-Paper-Scissors-Lizard-Spock (RPSLS) game are typical models in this field. However, in real ecosystems, species richness can be easily influenced by the interaction structure among species. In this paper, we investigate species coexistence in modified RPSLS games by removing certain predator-prey interactions. We find that the interaction structure impacts the evolutionary dynamics and different interaction structures allow for different states of multi-species coexistence. We also find that the competition between different three-species-cyclic interactions is crucial for the realization of different asymptotic behaviors at low mobility. Our findings may be useful to understand the subtle effects of competitive structure on species coexistence and evolutionary game outcomes.
1. Over 40 years after the introduction of the concept into ecology, intransitive (i.e. non-hierarchical) competition remains overlooked by ecological theory, despite theoretical work showing it ...could be a major driver of species coexistence. 2. This special feature presents six studies, including models, reviews, experimental studies and large-scale observational studies. Collectively, these studies help to (i) link intransitive competition with short- and long-term coexistence and with other ecological patterns, (ii) evaluate the conditions under which intransitivity is more common and (iii) determine how best to quantify the degree of intransitivity. 3. The studies in this special feature show the generality of intransitive competition in nature, explore interactions between intransitivity and other coexistence mechanisms, and illustrate the effect of environmental conditions (drought, shade, fertility) on intransitivity and coexistence. They also show which metrics best quantify intransitivity and highlight the importance of adopting a more continuous view of competition as varying from strongly transitive to strongly intransitive. The studies also examine relationships between intransitivity and functional diversity and explore the evolution of intransitivity over time. 4. Synthesis. The studies presented here advance the field by integrating intransitive competition into species coexistence and general ecological theory. We also highlight important research gaps that will hopefully inspire the next generation of studies in this topic.
In nature, different species compete among themselves for common resources and favorable habitat. Therefore, it becomes really important to determine the key factors in maintaining the bio-diversity. ...Also, some competing species follow cyclic competition in real world where the competitive dominance is characterized by a cyclic ordering. In this paper, we study the formation of a wide variety of spatiotemporal patterns including stationary, periodic, quasi-periodic and chaotic population distributions for a diffusive Lotka–Volterra type three-species cyclic competition model with two different types of cyclic ordering. For both types of cyclic ordering, the temporal dynamics of the corresponding non-spatial system show the extinction of two species through global bifurcations such as homoclinic and heteroclinic bifurcations. For the spatial system, we show that the existence of Turing patterns is possible for a particular cyclic ordering, while it is not the case for the other cyclic ordering through both the analytical and numerical methods. Further, we illustrate an interesting scenario of short-range invasion as opposed to the usual invasion phenomenon over the entire habitat. Also, our study reveals that both the stationary and dynamic population distributions can coexist in different parts of a habitat. Finally, we extend the spatial system by incorporating nonlocal intra-specific competition terms for all the three competing species. Our study shows that the introduction of nonlocality in intra-specific competitions stabilizes the system dynamics by transforming a dynamic population distribution to stationary. Surprisingly, this nonlocality-induced stationary pattern formation leads to the extinction of one species and hence, gives rise to the loss of bio-diversity for intermediate ranges of nonlocality. However, the bio-diversity can be restored for sufficiently large extent of nonlocality.
Intransitivity is a property of connected, oriented graphs representing species interactions that may drive their coexistence even in the presence of competition, the standard example being the three ...species Rock-Paper-Scissors game. We consider here a generalization with four species, the minimum number of species allowing other interactions beyond the single loop (one predator, one prey). We show that, contrary to the mean field prediction, on a square lattice the model presents a transition, as the parameter setting the rate at which one species invades another changes, from a coexistence to a state in which one species gets extinct. Such a dependence on the invasion rates shows that the interaction graph structure alone is not enough to predict the outcome of such models. In addition, different invasion rates permit to tune the level of transitiveness, indicating that for the coexistence of all species to persist, there must be a minimum amount of intransitivity.
► We model a four species cyclic game whose trophic network is fully connected, but with different invasion rates. ► The system is studied within the fully mixed (mean field) assumption and on a regular, square lattice. ► On the square lattice, as the invasion rates change, there is a transition from a coexistence state to one with extinctions. ► This transition is not observed within the mean field approach. ► These results indicate that for the coexistence state to persist, there must be a minimum amount of intransitivity.
Initial community evenness has been shown to be a key factor in preserving the functional stability of an ecosystem, but has not been accounted for in previous modelling studies. We formulate a model ...that allows the initial evenness of the community to be varied in order to investigate the consequent impact on system diversity. We consider a community of four interacting bacterial species, and present a stochastic, spatial individual-based model simulating the ecosystem dynamics. Interactions take place on a two-dimensional lattice. The model incorporates three processes: reproduction, competition and mobility. In addition to variable initial evenness, multiple competition schemes are implemented, modelling various possible communities, which results in diverse coexistence and extinction scenarios. Simulations show that long-term system behaviour is strongly dependent on initial evenness and competition structure. The system is generally unstable; higher initial evenness has a small stabilizing effect on ecosystem dynamics by extending the time until the first extinction.
•We model a four species bacterial community with fully connected competition.•The model allows initial community evenness to be varied, unlike previous studies.•Simulation outcomes depend strongly on initial evenness and competition scheme.•The system is generally unstable; extinction events are frequent.•Initial evenness has a small stabilizing effect on system dynamics.
Barreto et al. (2017) showed that the genotypic cyclic competition model with three phenotypes appearing as three morphs of male lizards' throats had the same equilibrium but a wider stability region ...as the corresponding phenotypic model. In this paper we re-examine stability conditions under the symmetric choice of parameters for the phenotypic model so we can set the same internal equilibrium densities for all three phenotypes. In this setting we compare the stability regions of cyclic allele dominance rule. Next we consider the dynamics on a two-dimensional square lattice space and then show the effect of this spatial structure on the stability of phenotypic model. We obtain the following results: (i) Cyclic allele dominance rule in a genotypic model gives a wider stable region of internal equilibrium than the allele dominance rule observed in lizards; and (ii) spatial structure drastically changes dynamical behavior, especially when all three phenotypes coexist in almost all the parameter spaces when both competition and dispersal occur locally.
Game dynamics in which three or more strategies are cyclically competitive, as represented by the rock-scissors-paper game, have attracted practical and theoretical interests. In evolutionary ...dynamics, cyclic competition results in oscillatory dynamics of densities of individual strategists. In finite-size populations, it is known that oscillations blow up until all but one strategies are eradicated if without mutation. In the present paper, we formalize replicator dynamics with players who have different adaptation rates. We show analytically and numerically that the heterogeneous adaptation rate suppresses the oscillation amplitude. In social dilemma games with cyclically competing strategies and homogeneous adaptation rates, altruistic strategies are often relatively weak and cannot survive in finite-size populations. In such situations, heterogeneous adaptation rates save coexistence of different strategies and hence promote altruism. When one strategy dominates the others without cyclic competition, fast adaptors earn more than slow adaptors. When not, mixture of fast and slow adaptors stabilizes population dynamics, and slow adaptation does not imply inefficiency for a player.
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