Public goods games in populations with fluctuating size

Indirect reciprocity in the public goods game with collective reputations

Indirect reciprocity unveils how social cooperation is founded upon moral systems. Within the frame of dyadic games based on individual reputations, the 'leading-eight' strategies distinguish themselves in promoting and sustaining cooperation. However, in real-world societies, there are widespread interactions at the group level, where individuals need to make a singular action choice when facing multiple individuals with different reputations. Here, through introducing the assessment of collective reputations, we develop a framework that embeds group-level reputation structure into public goods games to study the evolution of group-level indirect reciprocity. We show that changing the criteria of group assessment destabilizes the reputation dynamics of leading-eight strategies. In a particular range of social assessment criteria, all leading-eight strategies can break the social dilemma in public goods games and sustain cooperation. Specifically, there exists an optimal, moderately set assessment criterion that is most conducive to promoting cooperation. Moreover, in the evolution of assessment criteria, the preference of the leading-eight strategies for social strictness is inversely correlated with the payoff level. Our work reveals the impact of social strictness on prosocial behaviour, highlighting the importance of group-level interactions in the analysis of evolutionary games and complex social dynamics.

Cyclical oscillations and absorbing-state probabilities in optional public goods games: Interplay of reward and group size

The optional public goods game (OPGG) is a three-strategy model in which individuals can cooperate, defect, or not participate. Despite its simplicity, this model effectively captures various social dilemmas, including those involving public services, environmental sustainability, and broader societal issues. In this study, we investigate how the reward (r) and group size of potential players (S) of public goods games influence the steady-state coexistence of these strategies or the alternation of their dominance in a rock-paper-scissors dynamic. The OPGG is simulated using Monte Carlo in a nonspatial scenario, meaning there is no topology connecting the agents, allowing any player to interact with any other player. We show that under sufficiently noisy conditions, the system consistently evolves to an absorbing state, with the prevailing strategy determined by the values of r and S. In the range 2≤r≤S, the system shows multiple stable absorbing states, with groups of size S=4 exhibiting more pronounced and transient rock-paper-scissors dynamics with longer average absorbing times. We present a thorough analysis of our results in terms of the fraction of time the system spends in rock-paper-scissor cycles, the number of cycles, and the average probability that the system relaxes to each possible absorbing state, including scenarios where the system does not reach an absorbing state at all.

Extinction scenarios in evolutionary processes: a multinomial Wright-Fisher approach

We study a discrete-time multi-type Wright-Fisher population process. The mean-field dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the mean-field dynamics on it. One of the results is a limit theorem that describes sufficient conditions for an almost certain path to extinction, first eliminating the type which is the least fit at the mean-field equilibrium. The effect is explained by the metastability of the stochastic system, which under the conditions of the theorem spends almost all time before the extinction event in a neighborhood of the equilibrium. In addition to the limit theorems, we propose a maximization principle for a general deterministic replicator dynamics and study its implications for the stochastic model.

How public can public goods be? Environmental context shapes the evolutionary ecology of partially private goods

The production of costly public goods (as distinct from metabolic byproducts) has largely been understood through the realization that spatial structure can minimize losses to non-producing "cheaters" by allowing for the positive assortment of producers. In well-mixed systems, where positive assortment is not possible, the stable production of public goods has been proposed to depend on lineages that become indispensable as the sole producers of those goods while their neighbors lose production capacity through genome streamlining (the Black Queen Hypothesis). Here, we develop consumer-resource models motivated by nitrogen-fixing, siderophore-producing bacteria that consider the role of colimitation in shaping eco-evolutionary dynamics. Our models demonstrate that in well-mixed environments, single "public goods" can only be ecologically and evolutionarily stable if they are partially privatized (i.e., if producers reserve a portion of the product pool for private use). Colimitation introduces the possibility of subsidy: strains producing a fully public good can exclude non-producing strains so long as the producing strain derives sufficient benefit from the production of a second partially private good. We derive a lower bound for the degree of privatization necessary for production to be advantageous, which depends on external resource concentrations. Highly privatized, low-investment goods, in environments where the good is limiting, are especially likely to be stably produced. Coexistence emerges more rarely in our mechanistic model of the external environment than in past phenomenological approaches. Broadly, we show that the viability of production depends critically on the environmental context (i.e., external resource concentrations), with production of shared resources favored in environments where a partially-privatized resource is scarce.

Enhanced cooperation in multiplayer snowdrift games with random and dynamic groupings

An analytically tractable generalization of the N-person snowdrift (NSG) game that illustrates how cooperation can be enhanced is proposed and studied. The number of players competing within a NSG varies from one time step to another. Exact equations governing the frequency of cooperation f_{c}(r) as a function of the cost-to-benefit ratio r within an imitation strategy updating scheme are presented. For group sizes g uniformly distributed within the range g∈[1,g_{m}], an analytic formula for the critical value r_{c}(g_{m}), below which the system evolves into a totally cooperative (AllC) state, is derived. In contrast, a fixed group size NSG does not support an AllC state. The result r_{c}(g_{m}) requires the presence of sole-player groups and involves the inverse of the harmonic numbers and, more generally, the inverse first moment of the group size distribution. For r>r_{c}(g_{m}), the equation that determines the dynamical mixed states f_{c}(r) is given, with exact solutions existing for g_{m}≤5. The exact treatment allows the study of the phase boundary between the AllC state and the mixed states. The analytic results are checked against simulation results and exact agreements are demonstrated. The analytic form of the critical r_{c}(g_{m}) illustrates the necessity of having groups of a sole player in the evolutionary process. This result is supported by simulations with group sizes excluding the sole groups for which no AllC state emerges. A physically transparent picture of the importance of the sole players in inducing an AllC state is further presented based on the last surviving pattern before the AllC state is attained. The exact expression r_{c}(g_{m}) turns out to remain valid for nonuniform group-size distributions. Our analytical tractable generalization, therefore, sheds light on how a competing environment with variable group sizes could enhance cooperation and induce an AllC state.

Selfish risk-seeking can provide an evolutionary advantage in a conditional public goods game

While cooperation and risk aversion are considered to be evolutionarily advantageous in many circumstances, and selfish or risky behaviour can bring negative consequences for individuals and the community at large, selfish and risk-seeking behaviour is still often observed in human societies. In this paper we consider whether there are environmental and social conditions that favour selfish risk-seeking individuals within a community and whether tolerating such individuals may provide benefits to the community itself in some circumstances. We built an agent-based model including two types of agent-selfish risk-seeking and generous risk-averse-that harvest resources from the environment and share them (or not) with their community. We found that selfish risk-seekers can outperform generous risk-averse agents in conditions where their survival is moderately challenged, supporting the theory that selfish and risk-seeking traits combined are not dysfunctional but rather can be evolutionarily advantageous for agents. The benefit for communities is less clear, but when generous agents are unconditionally cooperative communities with a greater proportion of selfish risk-seeking agents grow to a larger population size suggesting some advantage to the community overall.

Expansion, Exploitation and Extinction: Niche Construction in Ephemeral Landscapes

We aim to understand general consequences of niche construction on metapopulation dynamics in ephemeral landscapes. To this effect, a contact process-like stochastic spatial model is introduced where local populations colonize and go extinct on a dynamic landscape of habitable and destroyed patches. In contrast to previous models, where the extinction threshold is a consequence of available niche rendered by global rates of patch destruction/renewal, here we investigate how the metapopulation persists when they are the sole generators of their own niche. Niche construction is full-filled by localized populations through the transformation of destroyed patches in their neighborhood to viable habitat for future colonization. With this theoretical framework we are able to address the dual nature of niche construction by investigating the ephemerality of the landscape (destruction rate) and the continuum of population level strategies, where construction comes at a cost to colonization. Using mean field theory and Monte Carlo simulations of the model, we are able to quantify optimal population level strategies in a wide range of ephemeral landscapes. Interestingly, we observe qualitative differences at the extinction threshold between analytic and numeric results. Investigating this discrepancy further, we find that increasing niche construction neighborhood in the spatial model leads to two interrelated effects i) an increased rate in range expansion ii) a loss in resiliency and return of the discontinuous transition at the extinction threshold. Furthermore, in the discontinuous regime of the model, spatial clustering prior to a critical transition disappears. This is a significant finding as spatial clustering has been considered to be an early warning signal before ecosystems reach their 'tipping point'. In addition to maintaining stability, we find local niche construction strategies have an advantage when in scramble competition with an exploiter strategy because of their ability to monopolize the constructed niche due to spatial adjacency. As the niche construction neighborhood expands this advantage disappears and the exploiter strategy out-competes the niche constructor. In some cases the exploiter pushes the niche constructor to extinction, thus a tragedy of the commons ensues leading to 'ecological suicide' and a collapse of the niche.

Turbulent coherent structures and early life below the Kolmogorov scale

Major evolutionary transitions, including the emergence of life, likely occurred in aqueous environments. While the role of water’s chemistry in early life is well studied, the effects of water’s ability to manipulate population structure are less clear. Population structure is known to be critical, as effective replicators must be insulated from parasites. Here, we propose that turbulent coherent structures, long-lasting flow patterns which trap particles, may serve many of the properties associated with compartments — collocalization, division, and merging — which are commonly thought to play a key role in the origins of life and other evolutionary transitions. We substantiate this idea by simulating multiple proposed metabolisms for early life in a simple model of a turbulent flow, and find that balancing the turnover times of biological particles and coherent structures can indeed enhance the likelihood of these metabolisms overcoming extinction either via parasitism or via a lack of metabolic support. Our results suggest that group selection models may be applicable with fewer physical and chemical constraints than previously thought, and apply much more widely in aqueous environments.

Models of the origin of life generally require a mechanism to structure emerging populations. Here, Krieger et al. develop spatial models showing that coherent structures arising in turbulent flows in aquatic environments could have provided compartments that facilitated the origin of life.

Neighborhood size-effects shape growing population dynamics in evolutionary public goods games

An evolutionary game emerges when a subset of individuals incur costs to provide benefits to all individuals. Public goods games (PGG) cover the essence of such dilemmas in which cooperators are prone to exploitation by defectors. We model the population dynamics of a non-linear PGG and consider density-dependence on the global level, while the game occurs within local neighborhoods. At low cooperation, increases in the public good provide increasing returns. At high cooperation, increases provide diminishing returns. This mechanism leads to diverse evolutionarily stable strategies, including monomorphic and polymorphic populations, and neighborhood-size-driven state changes, resulting in hysteresis between equilibria. Stochastic or strategy-dependent variations in neighborhood sizes favor coexistence by destabilizing monomorphic states. We integrate our model with experiments of cancer cell growth and confirm that our framework describes PGG dynamics observed in cellular populations. Our findings advance the understanding of how neighborhood-size effects in PGG shape the dynamics of growing populations.

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.

Disentangling eco-evolutionary effects on trait fixation

In population genetics, fixation of traits in a demographically changing population under frequency-independent selection has been extensively analysed. In evolutionary game theory, models of fixation have typically focused on fixed population sizes and frequency-dependent selection. A combination of demographic fluctuations with frequency-dependent interactions such as Lotka-Volterra dynamics has received comparatively little attention. We consider a stochastic, competitive Lotka-Volterra model with higher order interactions between two traits. The emerging individual-based model allows for stochastic fluctuations in the frequencies of the two traits and the total population size. We calculate the fixation probability of a trait under differing competition coefficients. This fixation probability resembles, qualitatively, the deterministic evolutionary dynamics. Furthermore, we partially disentangle the selection effects into their ecological and evolutionary components. We find that changing the evolutionary selection strength also changes the population dynamics and vice versa. Thus, a clean separation of the ecological and evolutionary effects is not possible. Instead, our results imply a nested interaction of the evolutionary and ecological effects. The entangled eco-evolutionary processes thus cannot be ignored when determining fixation properties in a co-evolutionary system.

Stationary frequencies and mixing times for neutral drift processes with spatial structure

We study a general setting of neutral evolution in which the population is of finite, constant size and can have spatial structure. Mutation leads to different genetic types (traits), which can be discrete or continuous. Under minimal assumptions, we show that the marginal trait distributions of the evolutionary process, which specify the probability that any given individual has a certain trait, all converge to the stationary distribution of the mutation process. In particular, the stationary frequencies of traits in the population are independent of its size, spatial structure and evolutionary update rule, and these frequencies can be calculated by evaluating a simple stochastic process describing a population of size one (i.e. the mutation process itself). We conclude by analysing mixing times, which characterize rates of convergence of the mutation process along the lineages, in terms of demographic variables of the evolutionary process.

Fixation probabilities in populations under demographic fluctuations

We study the fixation probability of a mutant type when introduced into a resident population. We implement a stochastic competitive Lotka-Volterra model with two types and intra- and interspecific competition. The model further allows for stochastically varying population sizes. The competition coefficients are interpreted in terms of inverse payoffs emerging from an evolutionary game. Since our study focuses on the impact of the competition values, we assume the same net growth rate for both types. In this general framework, we derive a formula for the fixation probability [Formula: see text] of the mutant type under weak selection. We find that the most important parameter deciding over the invasion success of the mutant is its death rate due to competition with the resident. Furthermore, we compare our approximation to results obtained by implementing population size changes deterministically in order to explore the parameter regime of validity of our method. Finally, we put our formula in the context of classical evolutionary game theory and observe similarities and differences to the results obtained in that constant population size setting.