As the behavior of a system composed of cyclically competing species is strongly influenced by the presence of fluctuations, it is of interest to study cyclic dominance in low dimensions where these effects are the most prominent. We here discuss rock-paper-scissors games on a one-dimensional lattice where the interaction rates and the mobility can be species dependent. Allowing only single site occupation, we realize mobility by exchanging individuals of different species. When the interaction and swapping rates are symmetric, a strongly enhanced swapping rate yields an increased mixing of the species, leading to a mean-field like coexistence even in one-dimensional systems. This coexistence is transient when the rates are asymmetric, and eventually only one species will survive. Interestingly, in our spatial games the dominating species can differ from the species that would dominate in the corresponding nonspatial model. We identify different regimes in the parameter space and construct the corresponding dynamical phase diagram.
A hallmark of topological phases is the occurrence of topologically protected modes at the system`s boundary. Here we find topological phases in the antisymmetric Lotka-Volterra equation (ALVE). The ALVE is a nonlinear dynamical system and describes, e.g., the evolutionary dynamics of a rock-paper-scissors cycle. On a one-dimensional chain of rock-paper-scissor cycles, topological phases become manifest as robust polarization states. At the transition point between left and right polarization, solitonic waves are observed. This topological phase transition lies in symmetry class $D$ within the…
In the evolutionary dynamics of a rock-paper-scissor (RPS) model, the effect of natural death plays a major role in determining the fate of the system. Coexistence, being an unstable fixed point of the model becomes very sensitive towards this parameter. In order to study the effect of mobility in such a system which has explicit dependence on mortality, we perform Monte Carlo simulation on a $2$-dimensional lattice having three cyclically-competing species. The spatio-temporal dynamics has been studied along with the two-site correlation function. Spatial distribution exhibits emergence of sp…
We investigate the impact of parity on the abundance of weak species in the context of the simplest generalization of the rock-paper-scissors model to an arbitrary number of species -- we consider models with a total number of species ($N_S$) between 3 and 12, having one or more (weak) species characterized by a reduced predation probability (by a factor of ${\mathcal P}_w$ with respect to the other species). We show, using lattice based spatial stochastic simulations with random initial conditions, large enough for coexistence to prevail, that parity effects are significant. We find that the …
This work reports on two related investigations of stochastic simulations which are widely used to study biodiversity and other related issues. We first deal with the behavior of the Hamming distance under the increase of the number of species and the size of the lattice, and then investigate how the mobility of the species contributes to jeopardize biodiversity. The investigations are based on the standard rules of reproduction, mobility and predation or competition, which are described by specific rules, guided by generalization of the rock-paper-scissors game, valid in the case of three spe…
Species survival in the $(3, 1)$ May-Leonard system is determined by the mobility, with a critical mobility threshold between long-term coexistence and extinction. We show experimentally that the critical mobility threshold is determined by the topology of the graph, with the critical mobility threshold of the periodic lattice twice that of a topological 2-sphere. We use topological techniques to monitor the evolution of patterns in various graphs and estimate their mean persistence time, and show that the difference in critical mobility threshold is due to a specific pattern that can form on …
We consider a two-dimensional model of three species in rock-paper-scissors competition and study the self-organisation of the population into fascinating spiraling patterns. Within our individual-based metapopulation formulation, the population composition changes due to cyclic dominance (dominance-removal and dominance-replacement), mutations, and pair-exchange of neighboring individuals. Here, we study the influence of mobility on the emerging patterns and investigate when the pair-exchange rate is responsible for spiral waves to become elusive in stochastic lattice simulations. In particul…
This work deals with a system of three distinct species that changes in time under the presence of mobility, selection, and reproduction, as in the popular rock-paper-scissors game. The novelty of the current study is the modification of the mobility rule to the case of directional mobility, in which the species move following the direction associated to a larger (averaged) number density of selection targets in the surrounding neighborhood. Directional mobility can be used to simulate eyes that see or a nose that smells, and we show how it may contribute to reduce the probability of coexisten…
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. With the objective of reducing the impact of the use of a di…
We model a situation in which a collection of species derive their fitnesses via a rock-paper-scissors-type game; however, the precise payoffs are a function of the environment. The new aspect of our model lies in adding a feedback loop: the environment changes according to the relative fitnesses of the species; in particular, it gives a boost to the species having small populations. We cast our model in the setting of a differential game and we show that for a certain setting of parameters, this dynamics cycles. Our model is a natural one, since depletion of resources used by more frequent sp…
When faced with an imminent risk of predation, many animals react to escape consumption. Antipredator strategies are performed by individuals acting as a group to intimidate predators and minimize the damage when attacked. We study the antipredator prey response in spatial tritrophic systems with cyclic species dominance using the rock-paper-scissors game. The impact of the antipredator behavior is local, with the predation probability reducing exponentially with the number of preys in the predator's neighborhood. In contrast to the standard Lotka-Volterra implementation of the rock-paper-scis…
Rock-paper-scissors games metaphorically model cyclic dominance in ecology and microbiology. In a static environment, these models are characterized by fixation probabilities obeying two different "laws" in large and small well-mixed populations. Here, we investigate the evolution of these three-species models subject to a randomly switching carrying capacity modeling the endless change between states of resources scarcity and abundance. Focusing mainly on the zero-sum rock-paper-scissors game, equivalent to the cyclic Lotka-Volterra model, we study how the ${\it coupling}$ of demographic and …
The rock-paper-scissors game is a paradigmatic model for biodiversity, with applications ranging from microbial populations to human societies. Research has shown, however, that mobility jeopardizes biodiversity by promoting the formation of spiral waves, especially if there is no conservation law in place for the total number of competing players. Firstly, we show that even if such a conservation law applies, mobility still jeopardizes biodiversity in the spatial rock-paper-scissors game if only a small fraction of links of the square lattice is randomly rewired. Secondly, we show that zealot…
We investigate a modified spatial stochastic Lotka-Volterra formulation of the rock-paper-scissors model using off-lattice stochastic simulations. In this model one of the species moves preferentially in a specific direction -- the level of preference being controlled by a noise strength parameter $η\in [0, 1]$ ($η= 0$ and $η= 1$ corresponding to total preference and no preference, respectively) -- while the other two species have no referred direction of motion. We study the behaviour of the system starting from random initial conditions, showing that the species with asymmetric mobility has …
We investigate a tritrophic system whose cyclic dominance is modelled by the rock-paper-scissors game. We consider that organisms of one or two species are affected by movement limitations, which unbalances the cyclic spatial game. Performing stochastic simulations, we show that mobility unevenness controls the population dynamics. In the case of one slow species, the predominant species depends on the level of mobility restriction, with the slow species being preponderant if the mobility limitations are substantial. If two species face mobility limitations, our outcomes show that being higher…
We revisit the problem of the predominance of the 'weakest' species in the context of Lotka-Volterra and May-Leonard implementations of a spatial stochastic rock-paper-scissors model in which one of the species has its predation probability reduced by $0 < \mathcal{P}_w < 1$. We show that,despite the different population dynamics and spatial patterns, these two implementations lead to qualitatively similar results for the late time values of the relative abundances of the three species (as a function of $\mathcal{P}_w$), as long as the simulation lattices are sufficiently large for coexistence…
Antipredator behaviour 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 performed 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 gr…
We study a class of the stochastic May-Leonard models, with three species dominating each other in a cyclic nonhierarchical way, according to the rock-paper-scissors game. We introduce an unevenness in the system, by considering that one of the species is weaker because of a lower selection probability. The simulation results show that the pattern formation is drastically affected by the presence of the weaker species, with no spiral waves arising immediately from random initial conditions. Instead, single-species spatial domains cyclically dominate the entire territory until a region occupied…
Multiple species in the ecosystem are believed to compete cyclically for survival and thus maintain balance in nature. Stochasticity has also an inevitable role in this dynamics. Considering these attributes of nature, the stochastic dynamics of the rock-paper-scissor model based on the idea of cyclic dominance becomes an effective tool to capture different aspects of ecosystem. The evolutionary dynamics of this model crucially depends on different interactions representing different natural habits. In this framework we explore the role of mortality of individual organism in the collective sur…
In this letter, we investigate the population dynamics in a May-Leonard formulation of the rock-paper-scissors game in which one or two species, which we shall refer to as "weak", have a reduced predation or reproduction probability. We show that in a nonspatial model the stationary solution where all three species coexist is always unstable, while in a spatial stochastic model coexistence is possible for a wide parameter space. We find, that a reduced predation probability results in a significantly higher abundance of "weak" species, in models with either one or two "weak" species, as long a…
One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When…
We study a nonhierarchical tritrophic system, whose predator-prey interactions are described by the rock-paper-scissors game rules. In our stochastic simulations, individuals may move strategically towards the direction with more conspecifics to form clumps instead of moving aimlessly on the lattice. Considering that the conditioning to move gregariously depends on the organism's physical and cognitive abilities, we introduce a maximum distance an individual can perceive the environment and a minimum conditioning level to perform the gregarious movement. We investigate the pattern formation an…
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 re…
Organisms may respond to local stimuli that benefit or threaten their fitness. The adaptive movement behaviour may allow individuals to adjust their speed to maximise the chances of being in comfort zones, where death risk is minimal. We investigate spatial cyclic models where the rock-paper-scissors game rules describe the nonhierarchical dominance. We assume that organisms of one out of the species can control the mobility rate in response to the information obtained from scanning the environment. Running a series of stochastic simulations, we quantify the effects of the movement strategy on…
Antipredator behavior is present in many biological systems where individuals collectively react to an imminent attack. The antipredator response may influence spatial pattern formation and ecosystem stability but requires an organism's cost to contribute to the collective effort. We investigate a nonhierarchical tritrophic system, whose predator-prey interactions are described by the rock-paper-scissors game rules. In our spatial stochastic simulations, the radius of antipredator response defines the maximum prey group size that disturbs the predator's action, determining the individual cost …
We study a three-species cyclic game system where organisms face a contagious disease whose virulence may change by a pathogen mutation. As a responsive defence strategy, organisms' mobility is restricted to reduce disease dissemination in the system. The impact of the collective self-preservation strategy on the disease infection risk is investigated by performing stochastic simulations of the spatial version of the rock-paper-scissors game. Our outcomes show that the mobility control strategy induces plasticity in the spatial patterns with groups of organisms of the same species inhabiting s…
We study the role of the adaptive movement strategy in promoting biodiversity in cyclic models described by the rock-paper-scissors game rules. We assume that individuals of one out of the species may adjust their movement to escape hostile regions and stay longer in their comfort zones. Running a series of stochastic simulations, we calculate the alterations in the spatial patterns and population densities in scenarios where not all organisms are physically or cognitively conditioned to perform the behavioural strategy. Although the adaptive movement strategy is not profitable in terms of ter…
