How Lévy Flights Triggered by the Presence of Defectors Affect Evolution of Cooperation in Spatial Games.

Cooperation among individuals has been key to sustaining societies. However, natural selection favors defection over cooperation. Cooperation can be favored when the mobility of individuals allows cooperators to form a cluster (or group). Mobility patterns of animals sometimes follow a Lévy flight. A Lévy flight is a kind of random walk but it is composed of many small movements with a few big movements. The role of Lévy flights for cooperation has been studied by Antonioni and Tomassini, who showed that Lévy flights promoted cooperation combined with conditional movements triggered by neighboring defectors. However, the optimal condition for neighboring defectors and how the condition changes with the intensity of Lévy flights are still unclear. Here, we developed an agent-based model in a square lattice where agents perform Lévy flights depending on the fraction of neighboring defectors. We systematically studied the relationships among three factors for cooperation: sensitivity to defectors, the intensity of Lévy flights, and population density. Results of evolutionary simulations showed that moderate sensitivity most promoted cooperation. Then, we found that the shortest movements were best for cooperation when the sensitivity to defectors was high. In contrast, when the sensitivity was low, longer movements were best for cooperation. Thus, Lévy flights, the balance between short and long jumps, promoted cooperation in any sensitivity, which was confirmed by evolutionary simulations. Finally, as the population density became larger, higher sensitivity was more beneficial for cooperation to evolve. Our study highlights that Lévy flights are an optimal searching strategy not only for foraging but also for constructing cooperative relationships with others.

Social dilemma in the excess use of antimicrobials incurring antimicrobial resistance.

The emergence of antimicrobial resistance (AMR) caused by the excess use of antimicrobials has come to be recognized as a global threat to public health. There is a 'tragedy of the commons' type social dilemma behind this excess use of antimicrobials, which should be recognized by all stakeholders. To address this global threat, we thus surveyed eight countries/areas to determine whether people recognize this dilemma and showed that although more than half of the population pays little, if any, attention to it, almost 20% recognize this social dilemma, and 15-30% of those have a positive attitude toward solving that dilemma. We suspect that increasing individual awareness of this social dilemma contributes to decreasing the frequency of AMR emergencies.

Multilayer networks with higher-order interaction reveal the impact of collective behavior on epidemic dynamics.

The ongoing COVID-19 pandemic has inflicted tremendous economic and societal losses. In the absence of pharmaceutical interventions, the population behavioral response, including situational awareness and adherence to non-pharmaceutical intervention policies, has a significant impact on contagion dynamics. Game-theoretic models have been used to reproduce the concurrent evolution of behavioral responses and disease contagion, and social networks are critical platforms on which behavior imitation between social contacts, even dispersed in distant communities, takes place. Such joint contagion dynamics has not been sufficiently explored, which poses a challenge for policies aimed at containing the infection. In this study, we present a multi-layer network model to study contagion dynamics and behavioral adaptation. It comprises two physical layers that mimic the two solitary communities, and one social layer that encapsulates the social influence of agents from these two communities. Moreover, we adopt high-order interactions in the form of simplicial complexes on the social influence layer to delineate the behavior imitation of individual agents. This model offers a novel platform to articulate the interaction between physically isolated communities and the ensuing coevolution of behavioral change and spreading dynamics. The analytical insights harnessed therefrom provide compelling guidelines on coordinated policy design to enhance the preparedness for future pandemics.

Adapting paths against zero-determinant strategies in repeated prisoner's dilemma games.

Long-term cooperation, competition, or exploitation among individuals can be modeled through repeated games. In repeated games, Press and Dyson discovered zero-determinant (ZD) strategies that enforce a special relationship between two players. This special relationship implies that a ZD player can unilaterally impose a linear payoff relationship to the opponent regardless of the opponent's strategies. A ZD player also has a property that can lead the opponent to an unconditional cooperation if the opponent tries to improve its payoff. This property has been mathematically confirmed by Chen and Zinger. Humans often underestimate a payoff obtained in the future. However, such discounting was not considered in their analysis. Here, we mathematically explored whether a ZD player can lead the opponent to an unconditional cooperation even if a discount factor is incorporated. Specifically, we represented the expected payoff with a discount factor as the form of determinants and calculated whether the values obtained by partially differentiating each factor in the strategy vector become positive. As a result, we proved that the strategy vector ends up as an unconditional cooperation even when starting from any initial strategy. This result was confirmed through numerical calculations. We extended the applicability of ZD strategies to real world problems.

Microbial mutualism promoting the coexistence of competing species: Double-layer model for two competing hosts and one microbial species.

Gause's law of competitive exclusion holds that the coexistence of competing species is extremely unlikely when niches are not differentiated. This law is supported by many mathematical studies, yet the coexistence of competing species is nearly ubiquitous in real ecosystems. We pay attention to the fact that plants and animals usually contact with microbial species as mutualistic partners. The activity spaces of host species are different from those of micro-organisms. In the present study, we apply double-layer model to the association of two competing hosts and a microorganism. Two lattices are prepared: one is for hosts, and the other is for microorganism. The basic equation obtained by mean-field theory is an extension of Lotka-Volterra competition model. Both mathematical analysis and numerical simulations reveal that a shared microbial mutualist can permit the coexistence of competing hosts. From the derived condition of coexistence, we believe the microbial mutualism promotes biodiversity in many ecological systems.

Conditions for the existence of zero-determinant strategies under observation errors in repeated games.

Repeated games are useful models to analyze long term interactions of living species and complex social phenomena. Zero-determinant (ZD) strategies in repeated games discovered by Press and Dyson in 2012 enforce a linear payoff relationship between a focal player and the opponent. This linear relationship can be set arbitrarily by a ZD player. Hence, a subclass of ZD strategies can fix the opponent's expected payoff and another subclass of the strategies can exceed the opponent for the expected payoff. Since this discovery, theories for ZD strategies are extended to cope with various natural situations. It is especially important to consider the theory of ZD strategies for repeated games with a discount factor and observation errors because it allows the theory to be applicable in the real world. Recent studies revealed their existence of ZD strategies even in repeated games with both factors. However, the conditions for the existence has not been sufficiently analyzed. Here, we mathematically analyzed the conditions in repeated games with both factors. First, we derived the thresholds of a discount factor and observation errors which ensure the existence of Equalizer and positively correlated ZD (pcZD) strategies, which are well-known subclasses of ZD strategies. We found that ZD strategies exist only when a discount factor remains high as the error rates increase. Next, we derived the conditions for the expected payoff of the opponent enforced by Equalizer as well as the conditions for the slope and base line payoff of linear lines enforced by pcZD. As a result, we found that, as error rates increase or a discount factor decreases, the conditions for the linear line that Equalizer or pcZD can enforce become strict.

Zero-determinant strategies under observation errors in repeated games.

Zero-determinant (ZD) strategies are a novel class of strategies in the repeated prisoner's dilemma (RPD) game discovered by Press and Dyson. This strategy set enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy. In the RPD game, games with discounting and observation errors represent an important generalization, because they are better able to capture real life interactions which are often noisy. However, they have not been considered in the original discovery of ZD strategies. In some preceding studies, each of them has been considered independently. Here, we analytically study the strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first reveal that the payoffs of two players can be represented by the form of determinants as shown by Press and Dyson even with the two factors. Then, we search for all possible strategies that enforce linear payoff relationships and find that both ZD strategies and unconditional strategies are the only strategy sets to satisfy the condition. We also show that neither Extortion nor Generous strategies, which are subsets of ZD strategies, exist when there are errors. Finally, we numerically derive the threshold values above which the subsets of ZD strategies exist. These results contribute to a deep understanding of ZD strategies in society.

Winning by hiding behind others: An analysis of speed skating data.

In some athletic races, such as cycling and types of speed skating races, athletes have to complete a relatively long distance at a high speed in the presence of direct opponents. To win such a race, athletes are motivated to hide behind others to suppress energy consumption before a final moment of the race. This situation seems to produce a social dilemma: players want to hide behind others, whereas if a group of players attempts to do so, they may all lose to other players that overtake them. To support that speed skaters are involved in such a social dilemma, we analyzed video footage data for 14 mass start skating races to find that skaters that hid behind others to avoid air resistance for a long time before the final lap tended to win. Furthermore, the finish rank of the skaters in mass start races was independent of the record of the same skaters in time-trial races measured in the absence of direct opponents. The results suggest that how to strategically cope with a skater's dilemma may be a key determinant for winning long-distance and high-speed races with direct opponents.

Author Correction: Grazing enhances species diversity in grassland communities.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Grazing enhances species diversity in grassland communities.

In grassland studies, an intermediate level of grazing often results in the highest species diversity. Although a few hypotheses have been proposed to explain this unimodal response of species diversity to grazing intensity, no convincing explanation has been provided. Here, we build a lattice model of a grassland community comprising multiple species with various levels of grazing. We analyze the relationship between grazing and plant diversity in grasslands under variable intensities of grazing pressure. The highest species diversity is observed at an intermediate grazing intensity. Grazers suppress domination by the most superior species in birth rate, resulting in the coexistence of inferior species. This unimodal grazing effect disappears with the introduction of a small amount of nongrazing natural mortality. Unimodal patterns of species diversity may be limited to the case where grazers are the principal source of natural mortality.

Strategies that enforce linear payoff relationships under observation errors in Repeated Prisoner's Dilemma game.

The theory of repeated games analyzes the long-term relationship of interacting players and mathematically reveals the condition of how cooperation is achieved, which is not achieved in a one-shot game. In the repeated prisoner's dilemma (RPD) game with no errors, zero-determinant (ZD) strategies allow a player to unilaterally set a linear relationship between the player's own payoff and the opponent's payoff regardless of the strategy that the opponent implements. In contrast, unconditional strategies such as ALLD and ALLC also unilaterally set a linear payoff relationship. Errors often happen between players in the real world. However, little is known about the existence of such strategies in the RPD game with errors. Here, we analytically search for all strategies that enforce a linear payoff relationship under observation errors in the RPD game. As a result, we found that, even in the case with observation errors, the only strategy sets that enforce a linear payoff relationship are either ZD strategies or unconditional strategies and that no other strategies can enforce it, which were numerically confirmed.

Mass killing by female soldier larvae is adaptive for the killed male larvae in a polyembryonic wasp.

Self-sacrifice is very rare among organisms. Here, we report a new and astonishing case of adaptive self-sacrifice in a polyembryonic parasitic wasp, Copidosoma floridanum. This wasp is unique in terms of its larval cloning and soldier larvae. Male clone larvae have been found to be killed by female soldier larvae, which suggests intersexual conflict between male and female larvae. However, we show here that mass killing is adaptive to all the killed males as well as the female soldiers that have conducted the killing because the killing increases their indirect fitness by promoting the reproduction of their clone sibs. We construct a simple model that shows that the optimal number of surviving males for both male and female larvae is very small but not zero. We then compare this prediction with the field data. These data agree quite well with the model predictions, showing an optimal killing rate of approximately 94-98% of the males in a mixed brood. The underlying mechanism of this mass kill is almost identical to the local competition for mates that occurs in other wasp species. The maternal control of the sex ratio during oviposition, which is well known in other hymenopterans, is impossible in this polyembryonic wasp. Thus, this mass kill is necessary to maximize the fitness of the female killers and male victims, which can be seen as an analogy of programmed cell death in multicellular organisms.

Regulatory mechanism predates the evolution of self-organizing capacity in simulated ant-like robots.

The evolution of complexity is one of the prime features of life on Earth. Although well accepted as the product of adaptation, the dynamics underlying the evolutionary build-up of complex adaptive systems remains poorly resolved. Using simulated robot swarms that exhibit ant-like group foraging with trail pheromones, we show that their self-organizing capacity paradoxically involves regulatory behavior that arises in advance. We focus on a traffic rule on their foraging trail as a regulatory trait. We allow the simulated robot swarms to evolve pheromone responsiveness and traffic rules simultaneously. In most cases, the traffic rule, initially arising as selectively neutral component behaviors, assists the group foraging system to bypass a fitness valley caused by overcrowding on the trail. Our study reveals a hitherto underappreciated role of regulatory mechanisms in the origin of complex adaptive systems, as well as highlights the importance of embodiment in the study of their evolution.

Metapopulation dynamics in the rock-paper-scissors game with mutation: Effects of time-varying migration paths.

Migration paths of animals are rarely the same. The paths may change according to seasonal and circadian rhythms. We study the effect of temporal migration on population dynamics of rock-paper-scissors (RPS) games with mutation by using the metapopulation dynamic model with two patches. Via mutation, an individual R changes to S with rate µ. All agents move by random walk between two patches and the RPS game is performed in each patch. The migration path between two patches is switched on or off periodically. The dynamics are represented by the reaction-diffusion equations with time-dependent diffusion coefficients in diffusively coupled reactors. We obtain the solutions of time-dependent reaction-diffusion equations numerically and analytically. The time-varying migration path induces complex behavior for the RPS dynamics, depending on the frequency of the periodical path. We find that the phase transitions occur by varying mutation rate µ. The phase transition depends highly on the frequency.

Metapopulation model of rock-scissors-paper game with subpopulation-specific victory rates stabilized by heterogeneity.

Recently, metapopulation models for rock-paper-scissors games have been presented. Each subpopulation is represented by a node on a graph. An individual is either rock (R), scissors (S) or paper (P); it randomly migrates among subpopulations. In the present paper, we assume victory rates differ in different subpopulations. To investigate the dynamic state of each subpopulation (node), we numerically obtain the solutions of reaction-diffusion equations on the graphs with two and three nodes. In the case of homogeneous victory rates, we find each subpopulation has a periodic solution with neutral stability. However, when victory rates between subpopulations are heterogeneous, the solution approaches stable focuses. The heterogeneity of victory rates promotes the coexistence of species.

Heterogeneous network promotes species coexistence: metapopulation model for rock-paper-scissors game.

Understanding mechanisms of biodiversity has been a central question in ecology. The coexistence of three species in rock-paper-scissors (RPS) systems are discussed by many authors; however, the relation between coexistence and network structure is rarely discussed. Here we present a metapopulation model for RPS game. The total population is assumed to consist of three subpopulations (nodes). Each individual migrates by random walk; the destination of migration is randomly determined. From reaction-migration equations, we obtain the population dynamics. It is found that the dynamic highly depends on network structures. When a network is homogeneous, the dynamics are neutrally stable: each node has a periodic solution, and the oscillations synchronize in all nodes. However, when a network is heterogeneous, the dynamics approach stable focus and all nodes reach equilibriums with different densities. Hence, the heterogeneity of the network promotes biodiversity.

Asymptotic stability of a modified Lotka-Volterra model with small immigrations.

Predator-prey systems have been studied intensively for over a hundred years. These studies have demonstrated that the dynamics of Lotka-Volterra (LV) systems are not stable, that is, exhibiting either cyclic oscillation or divergent extinction of one species. Stochastic versions of the deterministic cyclic oscillations also exhibit divergent extinction. Thus, we have no solution for asymptotic stability in predator-prey systems, unlike most natural predator-prey interactions that sometimes exhibit stable and persistent coexistence. Here, we demonstrate that adding a small immigration into the prey or predator population can stabilize the LV system. Although LV systems have been studied intensively, there is no study on the non-linear modifications that we have tested. We also checked the effect of the inclusion of non-linear interaction term to the stability of the LV system. Our results show that small immigrations invoke stable convergence in the LV system with three types of functional responses. This means that natural predator-prey populations can be stabilized by a small number of sporadic immigrants.

Epidemics of random walkers in metapopulation model for complete, cycle, and star graphs.

We present the metapopulation dynamic model for epidemic spreading of random walkers between subpopulations. A subpopulation is represented by a node on a graph. Each agent or individual is either susceptible (S) or infected (I). All agents move by random walk on the graph; namely, each agent randomly determines the destination of migration. The reaction-diffusion equations are presented as ordinary differential equations, not partial differential equations. To evaluate the risk of each subpopulation (node), we obtain the solutions of reaction-diffusion equations analytically and numerically for small, complete, cycle and star graphs. If a graph is homogeneous, or if every node has the same degree, then the solution never changes for any nodes. However, when a graph is heterogeneous, the infection density in equilibrium differs entirely among nodes. For example, on star graphs, the hub seems to be a supply source of disease because the infection density at the hub is much higher than that at the other nodes. On every graph, the epidemic thresholds are identical for all nodes.

Metapopulation model for rock-paper-scissors game: Mutation affects paradoxical impacts.

The rock-paper-scissors (RPS) game is known as one of the simplest cyclic dominance models. This game is key to understanding biodiversity. Three species, rock (R), paper (P) and scissors (S), can coexist in nature. In the present paper, we first present a metapopulation model for RPS game with mutation. Only mutation from R to S is allowed. The total population consists of spatially separated patches, and the mutation occurs in particular patches. We present reaction-diffusion equations which have two terms: reaction and migration terms. The former represents the RPS game with mutation, while the latter corresponds to random walk. The basic equations are solved analytically and numerically. It is found that the mutation induces one of three phases: the stable coexistence of three species, the stable phase of two species, and a single-species phase. The phase transitions among three phases occur by varying the mutation rate. We find the conditions for coexistence are largely changed depending on metapopulation models. We also find that the mutation induces different paradoxes in different patches.

Multi-species coexistence in Lotka-Volterra competitive systems with crowding effects.

Classical Lotka-Volterra (LV) competition equation has shown that coexistence of competitive species is only possible when intraspecific competition is stronger than interspecific competition, i.e., the species inhibit their own growth more than the growth of the other species. Note that density effect is assumed to be linear in a classical LV equation. In contrast, in wild populations we can observed that mortality rate often increases when population density is very high, known as crowding effects. Under this perspective, the aggregation models of competitive species have been developed, adding the additional reduction in growth rates at high population densities. This study shows that the coexistence of a few species is promoted. However, an unsolved question is the coexistence of many competitive species often observed in natural communities. Here, we build an LV competition equation with a nonlinear crowding effect. Our results show that under a weak crowding effect, stable coexistence of many species becomes plausible, unlike the previous aggregation model. An analysis indicates that increased mortality rate under high density works as elevated intraspecific competition leading to the coexistence. This may be another mechanism for the coexistence of many competitive species leading high species diversity in nature.