Predominance of the weakest species in Lotka-Volterra and May-Leonard formulations of the rock-paper-scissors model

Understanding the role of neutral species by means of high-order interaction in the rock-paper-scissors dynamics

The existence of neutral species carries profound ecological implications that warrant further investigation. In this paper, we study the impact of neutral species on biodiversity in a spatial tritrophic system of cyclic competition, in which the neutral species are identified as the fourth species that may affect the competition process of the other three species under the rock-paper-scissors (RPS) rule. Extensive simulations showed that neutral species can promote coexistence in a high mobility regime within the system. When coexistence occurs, we found that the state can be maintained by two mechanisms: Species can either (i) adhere to traditional RPS rule or (ii) form patches to resist invasion. Our findings might aid in understanding the impact of neutral species on biodiversity in ecosystems.

Response of a three-species cyclic ecosystem to a short-lived elevation of death rate

A balanced ecosystem with coexisting constituent species is often perturbed by different natural events that persist only for a finite duration of time. What becomes important is whether, in the aftermath, the ecosystem recovers its balance or not. Here we study the fate of an ecosystem by monitoring the dynamics of a particular species that encounters a sudden increase in death rate. For exploration of the fate of the species, we use Monte-Carlo simulation on a three-species cyclic rock-paper-scissor model. The density of the affected (by perturbation) species is found to drop exponentially immediately after the pulse is applied. In spite of showing this exponential decay as a short-time behavior, there exists a region in parameter space where this species surprisingly remains as a single survivor, wiping out the other two which had not been directly affected by the perturbation. Numerical simulations using stochastic differential equations of the species give consistency to our results.

Emerging solutions from the battle of defensive alliances

Competing strategies in an evolutionary game model, or species in a biosystem, can easily form a larger unit which protects them from the invasion of an external actor. Such a defensive alliance may have two, three, four or even more members. But how effective can be such formation against an alternative group composed by other competitors? To address this question we study a minimal model where a two-member and a four-member alliances fight in a symmetric and balanced way. By presenting representative phase diagrams, we systematically explore the whole parameter range which characterizes the inner dynamics of the alliances and the intensity of their interactions. The group formed by a pair, who can exchange their neighboring positions, prevail in the majority of the parameter region. The rival quartet can only win if their inner cyclic invasion rate is significant while the mixing rate of the pair is extremely low. At specific parameter values, when neither of the alliances is strong enough, new four-member solutions emerge where a rock-paper-scissors-like trio is extended by the other member of the pair. These new solutions coexist hence all six competitors can survive. The evolutionary process is accompanied by serious finite-size effects which can be mitigated by appropriately chosen prepared initial states.

How local antipredator response unbalances the rock-paper-scissors model

Antipredator behavior is a self-preservation strategy present in many biological systems, where individuals join the effort in a collective reaction to avoid being caught by an approaching predator. We study a nonhierarchical tritrophic system, whose predator-prey interactions are described by the rock-paper-scissors game rules. We perform a set of spatial stochastic simulations where organisms of one out of the species can resist predation in a collective strategy. The drop in predation capacity is local, which means that each predator faces a particular opposition depending on the prey group size surrounding it. Considering that the interference in a predator action depends on the prey's physical and cognitive ability, we explore the role of a conditioning factor that indicates the fraction of the species apt to perform the antipredator strategy. Because of the local unbalancing of the cyclic predator-prey interactions, departed spatial domains mainly occupied by a single species emerge. Unlike the rock-paper-scissors model with a weak species because of a nonlocal reason, our findings show that if the predation probability of one species is reduced because individuals face local antipredator response, the species does not predominate. Instead, the local unbalancing of the rock-paper-scissors model results in the prevalence of the weak species' prey. Finally, the outcomes show that local unevenness may jeopardize biodiversity, with the coexistence being more threatened for high mobility.

Effects of species vigilance on coexistence in evolutionary dynamics of spatial rock-paper-scissors game

Recognizing surrounding situations, such as enemy attacks, which can be realized by predator-prey relationships, is one of the common behaviors of the population in ecosystems. In this paper, we explore the relationship between such species' behavior and biodiversity in the spatial rock-paper-scissors game by employing the ecological concept "vigilance." In order to describe the vigilance process, we adopt a multiplex structure where two distinct layers describe virtual and physical interactions. By investigating the process of evolution in species, we also found that species with different vigilance go together. In addition, by utilizing the dynamic time warping method, we found that species with the same vigilance have consistent behavior, but species with different vigilance have diverse behavior. Our findings may lead to broader interpretations of mechanisms promoting biodiversity via vigilance in species ecosystems.

Enhancing coexistence of mobile species in the cyclic competition system by wildlife refuge

We investigate evolving dynamics of cyclically competing species on spatially extended systems with considering a specific region, which is called the "wildlife refuge," one of the institutional ways to preserve species biodiversity. Through Monte-Carlo simulations, we found that the refuge can play not groundbreaking but an important role in species survival. Species coexistence is maintained at a moderate mobility regime, which traditionally leads to the collapse of coexistence, and eventually, the extinction is postponed depending on the competition rate rather than the portion of the refuge. Incorporating the extinction probability and Fourier transform supported our results in both stochastic and analogous ways. Our findings may provide valuable evidence to assist fields of ecological/biological sciences in understanding the presence and construction of refuges for wildlife with associated effects on species biodiversity.

Population persistence under two conservation measures: Paradox of habitat protection in a patchy environment

Anthropogenic modification of natural habitats is a growing threat to biodiversity and ecosystem services. The protection of biospecies has become increasingly important. Here, we pay attention to a single species as a conservation target. The species has three processes: reproduction, death and movement. Two different measures of habitat protection are introduced. One is partial protection in a single habitat (patch); the mortality rate of the species is reduced inside a rectangular area. The other is patch protection in a two-patch system, where only the mortality rate in a particular patch is reduced. For the one-patch system, we carry out computer simulations of a stochastic cellular automaton for a "contact process". Individual movements follow random walking. For the two-patch system, we assume an individual migrates into the empty cell in the destination patch. The reaction-diffusion equation (RDE) is derived, whereby the recently developed "swapping migration" is used. It is found that both measures are mostly effective for population persistence. However, comparing the results of the two measures revealed different behaviors. ⅰ) In the case of the one-patch system, the steady-state densities in protected areas are always higher than those in wild areas. However, in the two-patch system, we have found a paradox: the densities in protected areas can be lower than those in wild areas. ⅱ) In the two-patch system, we have found another paradox: the total density in both patches can be lower, even though the proportion of the protected area is larger. Both paradoxes clearly occur for the RDE with swapping migration.

Lotka-Volterra versus May-Leonard formulations of the spatial stochastic rock-paper-scissors model: The missing link

The rock-paper-scissors (RPS) model successfully reproduces some of the main features of simple cyclic predator-prey systems with interspecific competition observed in nature. Still, lattice-based simulations of the spatial stochastic RPS model are known to give rise to significantly different results, depending on whether the three-state Lotka-Volterra or the four-state May-Leonard formulation is employed. This is true independently of the values of the model parameters and of the use of either a von Neumann or a Moore neighborhood. In this paper, we introduce a simple modification to the standard spatial stochastic RPS model in which the range of the search of the nearest neighbor may be extended up to a maximum Euclidean radius R. We show that, with this adjustment, the Lotka-Volterra and May-Leonard formulations can be designed to produce similar results, both in terms of dynamical properties and spatial features, by means of an appropriate parameter choice. In particular, we show that this modified spatial stochastic RPS model naturally leads to the emergence of spiral patterns in both its three- and four-state formulations.

Survival of the weakest in non-transitive asymmetric interactions among strains of E. coli

Hierarchical organization in ecology, whereby interactions are nested in a manner that leads to a dominant species, naturally result in the exclusion of all but the dominant competitor. Alternatively, non-hierarchical competitive dynamics, such as cyclical interactions, can sustain biodiversity. Here, we designed a simple microbial community with three strains of E. coli that cyclically interact through (i) the inhibition of protein production, (ii) the digestion of genomic DNA, and (iii) the disruption of the cell membrane. We find that intrinsic differences in these three major mechanisms of bacterial warfare lead to an unbalanced community that is dominated by the weakest strain. We also use a computational model to describe how the relative toxin strengths, initial fractional occupancies, and spatial patterns affect the maintenance of biodiversity. The engineering of active warfare between microbial species establishes a framework for exploration of the underlying principles that drive complex ecological interactions.

The maintenance of ecological diversity depends on the strength and direction of competitive interactions, but these interactions are difficult to study in microbial communities. Here the authors use engineered E. coli strains to show that competitively weak strains can persist when pairwise interactions are asymmetrical.

Independent evolution of cutaneous lymphoma subclones in different microenvironments of the skin

Mycosis fungoides (MF) is the most common cutaneous T-cell lymphoma. Lesions of MF are formed by hematogenous seeding the skin with polyclonal (clonotypically diverse) neoplastic T-cells which accumulate numerous mutations and display a high degree of mutational, intratumoral heterogeneity (ITH). A characteristic but poorly studied feature of MF is epidermotropism, the tendency to infiltrate skin epithelial layer (epidermis) in addition to the vascularized dermis. By sequencing the exomes of the microdissected clusters of lymphoma cells from the epidermis and the dermis, we found that those microenvironments comprised different malignant clonotypes. Subclonal structure witnessed the independent mutational evolution in the epidermis and dermis. Thus, the epidermal involvement in MF could not be explained by gradual infiltration from the dermis but was caused by a separate seeding process followed by a quasi-neutral, branched evolution. In conclusion, tissue microenvironments shape the subclonal architecture in MF leading to “ecological heterogeneity” which contributes to the total ITH. Since ITH adversely affects cancer prognosis, targeting the microenvironment may present therapeutic opportunities in MF and other cancers.

Mortality makes coexistence vulnerable in evolutionary game of rock-paper-scissors

Multiple species in the ecosystem are believed to compete cyclically for maintaining balance in nature. The evolutionary dynamics of cyclic interaction crucially depends on different interactions representing different natural habits. Based on a rock-paper-scissors model of cyclic competition, we explore the role of mortality of individual organisms in the collective survival of a species. For this purpose a parameter called "natural death" is introduced. It is meant for bringing about the decease of an individual irrespective of any intra- and interspecific interaction. We perform a Monte Carlo simulation followed by a stability analysis of different fixed points of defined rate equations and observe that the natural death rate is surprisingly one of the most significant factors in deciding whether an ecosystem would come up with a coexistence or a single-species survival.

Performance of weak species in the simplest generalization of the rock-paper-scissors model to four species

We investigate the problem of the predominance and survival of "weak" species in the context of the simplest generalization of the spatial stochastic rock-paper-scissors model to four species by considering models in which one, two, or three species have a reduced predation probability. We show, using lattice based spatial stochastic simulations with random initial conditions, that if only one of the four species has its probability reduced, then the most abundant species is the prey of the "weakest" (assuming that the simulations are large enough for coexistence to prevail). Also, among the remaining cases, we present examples in which "weak" and "strong" species have similar average abundances and others in which either of them dominates-the most abundant species being always a prey of a weak species with which it maintains a unidirectional predator-prey interaction. However, in contrast to the three-species model, we find no systematic difference in the global performance of weak and strong species, and we conjecture that a similar result will hold if the number of species is further increased. We also determine the probability of single species survival and coexistence as a function of the lattice size, discussing its dependence on initial conditions and on the change to the dynamics of the model which results from the extinction of one of the species.