Evolution of contribution timing in public goods games

Abstract

Life-history strategies are a crucial aspect of life, which are complicated in group-living species, where pay-offs additionally depend on others’ behaviours. Previous theoretical models of public goods games have generally focused on the amounts individuals contribute to the public good. Yet a much less-studied strategic aspect of public goods games, the timing of contributions, can also have dramatic consequences for individual and collective performance. Here, we develop two stage game theoretical models to explore how the timing of contributions evolves. In the first stage, individuals contribute to a threshold public good based on a performance schedule. The second stage begins once the threshold is met, and the individuals then compete as a function of their performance. We show how contributing rapidly is not necessarily optimal, because delayers can act as ‘cheats,’ avoiding contributing while reaping the benefits of the public good. However, delaying too long can put the delayers at a disadvantage as they may be ill-equipped to compete. These effects lead to bistability in a single group, and spatial diversity among multiple interacting groups.

1. Introduction

In all biological systems, timing plays a fundamental role, from molecular-level circadian rhythms controlling gene expression, to large-scale migratory movement and breeding timing. The right timing allows proteins to be ready by the time they are needed (e.g. digestive enzymes), and animals to arrive at foraging or breeding sites at an advantageous time, or to exhibit a specific phenotype at the right time.

Exogenous stimuli, such as temperature [1], are often involved in coordinating behaviours. Moreover, where large numbers of individuals coordinate behaviours, timing can also create issues, such as crowding and competition for limited resources [2]. In fact, modelling approaches suggest that density-dependence can lead to a timing diversification to avoid competition [3].

In group living animals, timing issues can lead to specific social dilemmas, where the interests of the group are in conflict with the interests of the individual. An example is when individuals contribute to a public good which benefits the group as a whole. Sometimes these benefits are only achieved if sufficient amounts of time or resources have been contributed to the public good. Examples for such interactions come from a variety of systems, from meerkat (Suricata suricatta) sentinel behaviour [4,5], to mitigating climate change [6] and herd immunity [7]. If sufficient time or resource is contributed to the public good, every group member can receive a share of the benefit. However, contributing generally comes at a personal cost, such as resources used in provisioning and time spent guarding or putting oneself in danger. Thus, an individual that does not contribute might have a fitness advantage relative to others that do once the public good is produced.

In such a public goods setting, timing of contributions can be a crucial part of the social strategy. For example, in pharaoh ant colonies, Monomorium pharaonis, many potentially unrelated queens live together and cooperate [8]. Queens continuously produce queen, male reproductives and worker eggs. However, workers curate these eggs and will only rear worker eggs until the colony grows and the egg to worker ratio falls below a certain threshold [9]. Here, workers represent a public good to the queens as they provide food and protection, and otherwise maintain the colony. However, the benefit to queens from the public good in terms of reproductive offspring, i.e. new queens, is only realized when the colony reaches a size threshold. Queens that delay expending energy into worker production will have more reproductive offspring (queens and reproductive males) and therefore higher fitness compared with those that contribute early on. Strikingly, the timing of reproductive effort can be influenced by Wolbachia endosymbionts, which expedite the egg laying of infected queens relative to uninfected queens [10]. Similarly, in the greater ani, several pairs of this communally breeding bird will protect a shared nest, incubate eggs, and provide food to chicks. While generally cooperative, females will roll eggs out of the nest until they have started to lay their own eggs and so reduce competition for their offspring [11]. The public good here is the initiation of the nest after which the birds can move to the next stage in the breeding cycle where all of the females care for the eggs laid. The nest typically is initiated by some female who will pay a cost in lost eggs, while females that lay their eggs later will avoid having their eggs ejected [12]. Further, delaying egg laying too long has been associated with increased chick starvation rates [13]. Thus, there is a high chance to lose offspring when laying eggs early (risk of ejection), but also when laying eggs late (risk of starvation), creating a coordination problem. Cooperation and public goods games are also important in the biology of microorganisms [14,15]. In particular, the behaviour of altruistic sacrifice where some cells forgo reproduction to the benefit of others such as in the formation of fruiting bodies under food deprivation in the slime mold Dictyostelium discoideum [16]. In groups of cooperative cells, fast production of a stalk would be the social optimum. Yet, delaying from movement to stalk forming may permit a cell to be a member of the spore-producing head of the stalk [17]. Here again, timing is crucial: waiting too long can cause a cell to miss joining the spore forming head in time, but switching too early means it would contribute to the stalk and not to the head.

In these examples, even if everyone could contribute the same amount, there is a conflict over the timing of contributions. If delaying contributions means more resources available for private benefits or competition once a public good is produced, individuals will have an incentive to delay their contributions. On the other hand, delayed contributions means that the group as a whole will have to wait longer until the public good is produced, and so the benefits from the public good are realized more slowly. In some cases, delaying contributions can also run afoul of time constraints, such as seasonal changes or predation risk, where fitness will be zero if contributions have been delayed for too long.

This conflict over the timing of contributions to public goods can have additional consequences for life-history decisions. Life-history strategies commonly describe the trade-off between investing resources, such as energy and time, into growth or reproduction. Whatever is not invested into growth is what is left for reproduction and vice versa. However, where fitness depends on the timely production of a public good, fitness may also depend on the timing of contributions relative to others. Therefore, in social organisms when components of fitness depend on the production of public goods, life-history schedules might be affected by the conflict over timing of contributions.

It is therefore important to understand how conflicts over the timing of contributions to public goods play out in evolutionary games. Yet most of the work on threshold public goods assumes that participants in a public goods game differ only in the magnitude of their contribution at a fixed point in time or integrates over the entire interaction [18,19]. The timing of contributions and the production of the public good has in contrast received relatively little attention (with some notable exceptions, [20–22]).

To study the effect of the timing of contributions, we develop a threshold public goods model where members of a group expend effort (or contribute resources) towards a common good. Once a specific total contribution level is reached, the group enters a different non-cooperative state, where individuals expend their effort into competition with one another over the common resource. We define an individual’s behavioural strategy as the timing of its contribution. That is, all individuals can contribute the same amount in principle, but vary in their schedule of when to make the contributions. This may mean that depending on others’ schedules an individual might end up contributing more or less than others until the public good is produced.

We use this model to explore under which conditions we would expect contributions to be delayed, and to investigate whether delayers can co-exist with non-delayers. For a single group, we find bistablity between the two strategies in the conflict over timing. We then extend this model to a spatial, multi-group setting, to test whether groups of delayers and non-delayers can co-exist. We find that the bistability of single group dynamics lead to spatial patterns of stable monomorphic clusters of delayers and non-delayers. Our results provide, to our knowledge, the first evolutionary game theoretic account for how conflict over timing of contributions to a public goods game plays out in well-mixed and spatially structured populations.

2. Material and methods

We begin with a model of a single group of players and a single public good. The model is a non-overlapping generations model where a group plays a game with two stages within each generation. The within-generation interactions happen in continuous time, whereas benefits are realized at discrete events. In the first stage, individuals contribute to a public good, according to their performance schedules (the evolving trait). The second stage is reached when a specific amount of total contribution is made to the public good, at which time individuals engage in scramble competition, where their competitive effort is again determined by their contribution schedule. The pay-offs from the scramble stage then determine the frequency of types in the following generation. We develop a model that tracks these generational changes. Later, we extend the single group model to many groups that occupy two-dimensional space, and use reaction–diffusion equations to study how between-group competition and spatial structure affect the evolution of timing.

(a). Single group and public good

Consider a group of players contributing to a public good, such as group vigilance or cooperative breeding. We assume that each player has the same amount of resources to contribute over their lifetime and that contributions follows a hump-shaped ‘performance schedule’ over time, f(t). That is, f(t) starts out small, peaks partway through, and then declines as the individual senesces. Mathematically stated, there exists a unique peak time $t^{†}$ such that f ′(t^†) = 0 and $f{}^{″}(t^{†})<0$. Further, f ′(t) > 0 for $t<t^{†}$ and f ′(t) < 0 for t > t^†. This reflects the biologically realistic assumption that the performance of behaviours or phenotypes typically will need to increase from a low or zero baseline, and will decline later in life. The actual shape of the performance schedule does not qualitatively affect our results, so long as the curves are unimodal. As we are interested in the evolution of contribution timing, we assume that individuals play one of two strategies: a ‘delayed schedule’ or ‘non-delayed schedule’. The delayed schedule is shifted along the time axis by an amount δ. Thus, f(t − δ) and f(t) are the performance schedules for delayers and non-delayers, respectively.

In our model, all efforts flow towards the public good until the threshold amount of resources is met at which point the group enters the second stage of the game, which is a non-cooperate scramble for resources (e.g. opportunities to reproduce). We assume that the scramble happens relatively quickly, so that it depends only on the performance at the time the threshold is reached.

To model the evolution of this process, we employ a discrete dynamical system that tracks the proportion of delayers at generation n, p_n, and the time at which the condition is met for generation n, θ_n. The condition is met when a total contribution level C is reached by the population:

$$C={\int }_0^{{\theta }_n}{p}_{n}Nf(t-\delta )+(1-p_n)Nf(t)\hspace{0.17em}\text{d}t,$$ 2.1

where N is the number of individuals in the group. Here we are concerned with the dynamics of the proportion of each type where N is a continuous variable. Thus, the condition is met and the public good realized when the per capita total contribution reaches a threshold, which we denote by α = C/N, i.e.

$$\alpha ={\int }_0^{{\theta }_n}{p}_{n}f(t-\delta )+(1-p_n)f(t)\hspace{0.17em}\text{d}t.$$ 2.2

Because pay-offs are proportional to f(θ_n) and f(θ_n − δ) for non-delayers and delayers, respectively, we calculate the proportion of the delaying players at the next generation p_n+1 as

$$p_{n+1}=\frac{\hspace{0.17em}{p}_{n}f({\theta }_n-\delta )}{p_{n}f({\theta }_n-\delta )+(1-p_n)f({\theta }_n)}.$$ 2.3

In the electronic supplementary material, A, we also consider a generalization of the model above that has competition over some duration of time after the threshold has been reached. Equation (2.3) arises in the limit of this model as the duration goes to zero. We also considered a fixed time threshold model, where θ is a fixed parameter. See the electronic supplementary material, B for an analysis of this model.

(b). Multiple groups and public goods

The single-group model does not consider the potential group-level effects of timing, which can favour non-delayers as groups of non-delayers would reach the public good threshold faster and thus have shorter generation times. The effects of such between-group effects will also depend on the spatial structure of the population, which is known to affect life-history traits in many interactions, such as in host-pathogen systems [23]. To explore the impact of delaying, we extend the model above by letting multiple groups play their own dynamic public goods game in a spatial setting. We assume that these groups are spread out in a square two-dimensional space with periodic boundaries. Each point in space has a carrying capacity, and players may randomly migrate to adjacent areas.

To model the growth dynamics as a result of social interactions together with migration, we use a reaction diffusion model that extends the model in §2a with the addition of growth rates and a carrying capacity. Reaction–diffusion models are well-studied models of spatial ecology [24]. For simplicity, we assume carrying capacity to be constant across space. We track the number of delayers, n₁, and non-delayers, n₂, at each point in space. Additionally, we take θ into account and treat it as an evolving variable. Here, θ is the length of time until pay-offs are earned rather than the time until a new generation. This definition of time is different from the time t in which the dynamical system evolves. If a group has a lower θ relative to another group, it will reproduce sooner and so grow more quickly.

Let us first consider the continuous time dynamics at a single location before we extend the model to a spatial version. We use a two-type Lotka–Volterra system that takes both the growth rates and carrying capacity, K, into account. This approximates the single group model at a single point, which we discussed above. The change of n₁ and n₂ over time are given by:

$${\dot{n}}_1=\frac{n_1}{\theta }(\hspace{0.17em}f(\theta -\delta )-\frac{\hspace{0.17em}f(\theta -\delta )n_1+f(\theta )n_2}{K})$$ 2.4

and

$${\dot{n}}_2=\frac{n_2}{\theta }(\hspace{0.17em}f(\theta )-\frac{\hspace{0.17em}f(\theta -\delta )n_1+f(\theta )n_2}{K}).$$ 2.5

Note that dividing by θ takes the generation time into account, i.e. the time until pay-offs are earned, thus inducing a cost to delaying contribution into the public good in one group relative to another group. The smaller θ, the faster the population will grow. θ is constrained by equation (2.2), making this a differential algebraic equation.

In the next step, we extend this continuous time approximation to multiple locations. Each location has an x, y coordinate. Thus, the number of delayers and number of non-delayers at (x, y) and time t are given by n₁(x, y, t) and n₂(x, y, t) respectively. To simulate movement of players between patches, we add a diffusion term to equations (2.4) and (2.5), which yields the following reaction–diffusion equations:

$$\frac{\mathrm{\partial }{n}_1}{\mathrm{\partial }t}=D{\mathrm{\nabla }}^2{n}_1+\frac{n_1}{\theta }(f(\theta -\delta )-\frac{\hspace{0.17em}f(\theta -\delta )n_1+f(\theta )n_2}{K})$$ 2.6

and

$$\frac{\mathrm{\partial }{n}_2}{\mathrm{\partial }t}=D{\mathrm{\nabla }}^2{n}_2+\frac{n_2}{\theta }(f(\theta )-\frac{\hspace{0.17em}f(\theta -\delta )n_1+f(\theta )n_2}{K}),$$ 2.7

where D is the diffusion constant, $\mathrm{\nabla }$ is the spatial gradient, and ${\mathrm{\nabla }}^2$ is the Laplace operator. Finally, θ is determined by local patch conditions via equation (2.2), yielding a partial differential-algebraic equation. We solved this system numerically. See the electronic supplementary material, C for details.

3. Results

(a). The single-group model

For the single-group model, we find that the groups will either be all delaying or all non-delaying at equilibrium (p*, θ*). The exception is where f(θ* − δ) = f(θ*). This is an edge-case where the dynamics devolve into p_n+1 = p_n, where any mix of both types is an equilibrium. Figure 1 portrays the stability of the monomorphic equilibria for low, intermediate, and high values of α. When α is low, the times θ at which the thresholds are met, for delayers and non-delayers (figure 1a,b, respectively), are short. As such, the all non-delaying equilibrium is stable, while the all delaying one is not. For intermediate levels of α, however, we observe bistability as both equilibria may be stable (figure 1c,d). For high α, only a monomorphic population of delayers is stable (figure 1e,f). See the electronic supplementary material, D.1 for a mathematical analysis of these equilbria.

Figure 1.

A graphical representation of the stability of the monomorphic equilibria for varying α (rows) and the delaying and non-delaying strategies (columns). Vertical bars indicate the relative contribution levels of delayers (blue) and non-delayers (red) when the contribution per capita threshold α is reached. This time point is marked with θ*, which differs depending on the required contribution, α, and the contribution schedule (delayed/non-delayed). All delaying is stable and all non-delaying is unstable where α is high (e,f), whereas the all non-delaying state is stable and the all delaying state is unstable when α is low (a,b). For intermediate values of α, we have bistability as both strategies are stable at this point (c,d). The invading mutant strategies are shown for each panel as dashed curves with the respective colour. (Online version in colour.)

Although the stability of the monomorphic equilibria depends on α, where a polymorphic equilibrium exists in a single group, it is always unstable (see the electronic supplementary material, D.1). The dependence of the two strategies on α is shown in figure 2 as a bifurcation diagram. For low α, there is only one equilibrium, monomorphic non-delaying. As α increases, the monomorphic delaying equilibrium becomes stable and an unstable polymorphic equilibrium emerges. Further increasing α destabilizes the monomophic non-delaying equilibrium. We lose the polymorphic equilibrium and are left with the stable monomorpic delaying equilibrium. The range of α where bistability is obtained is larger when δ is larger.

Figure 2.

The bifurcation diagram shows that when α is low, non-delayers (pink) dominate, while for high α, delayers (blue) dominate, whereas for intermediate values of α the system is bistable. Shown are results where the performance function, f(t), is the probability distribution function of the normal distribution $\mathcal{N}(\mu =10,\sigma =2$), and δ = 0.25 μ (25% delay). Solid lines indicate stable states, dashed lines indicate unstable states. (Online version in colour.)

We analysed the adaptive dynamics of this system, and found that for a low contribution per capita threshold α where f(θ*) is less than the peak value, expediting the schedule will evolve. If, however, α is high, f(θ*) is greater than the peak value, and so further contribution delaying will evolve. In these cases, evolution will only halt once some boundary effect is reached, i.e. delaying or expediting cannot be done indefinitely (see the electronic supplementary material, D.2 for an analysis).

(b). The multi-group reaction–diffusion model

The Lotka–Volterra model for a single group (equations (2.4) and (2.5)) behaves similarly to the single group model, i.e. bistability is contingent on α (see the electronic supplementary material, E). To analyse the spatial extension, we numerically solved the fixed-contribution model for a variety of initial conditions and parameter values. The higher functions f(t) and f(t − δ) are normal probability distribution functions (PDF). For non-delayers, the mean is μ = 10 and the standard deviation is σ = 2. We consider delayers with several different delays, $\delta =10\text{\%}$, 30%, and 50% of μ (i.e. f(t − δ) is a normal PDF with mean μ(1 + δ)). We also consider α = 0.4, 0.5 and 0.6 and diffusion constant D = 0.01. For greater or lower α, the population fixes to one or the other type. We fixed the carrying capacity K = 1000. Initial conditions were N = 5, and the proportion of delayers drawn from an L × L Gaussian field (see the electronic supplementary material, C for details). We explored the effects of spatially uncorrelated initial conditions using a multivariate beta distribution with parameters (0.5, 0.5), i.e. a U-shaped distribution on [0, 1] (see the electronic supplementary material, F.2).

In figure 3, we show heatmaps of the proportion of delayers across space at time t = 1000 for different values of α and δ. Note the coexistence and patterns that occur. For α outside of the bistable region, we do not observe such patterns as the populations evolve to a spatial monomorphism (results not shown here). The delay parameter δ has an impact on how sharp the boundaries between clusters of the two types are. When the delay is low, the boundaries are fuzzier than when it is higher. This is because increasing the delay increases local bistability resulting in patches of (nearly) all delayers or all non-delayers. This effect amplifies the formation of clusters, which can then be maintained. As observed in figure 3, the frequency of non-delayer clusters is higher in panel i (δ = 0.5) relative to panel f (δ = 0.3).

Figure 3.

Delayers and non-delayers co-exist for a range of parameters. Shown are the spatial polymorphisms of delayers (yellow) and non-delayers (black) for the multi-group, multi public goods game model for low, intermediate, and high α (columns), where delayers delay contributions with $\delta \in \{10\text{\%},30\text{\%},50\text{\%}\}$, and for diffusion constant D = 0.01. These snapshots are taken at time t = 1000. (Online version in colour.)

Figure 4 depicts the time series for various parameter combinations. The time to reach equilibrium is faster as α is high or low. Convergence is slowest in the middle of the bistable region at α = 0.5. In the single group model, we are concerned with discrete stages, and thus θ only has one role, determining pay-offs when the threshold is met. In the multi-group model, θ also impacts the growth rate. Because a smaller θ increases the group’s growth rate, non-delayers increase the group’s growth rate. Thus, groups with more non-delayers will grow faster than groups with fewer, and thereby spread more quickly throughout space. However, as the carrying capacity is reached, this effect abates and the local dynamics dominate (see the electronic supplementary material, F.1 for a time series of the spatial mean of N). This process leads to the fluctuations in the time series most evident for longer delays. Early on, growth dominates and non-delayers spread. However, as the carrying capacity is reached, the local dynamics can force the groups into more delayers.

Figure 4.

Fifty time series of the spatial average of p for low, intermediate, and high α (columns), and where delayers delay contributions with $\delta \in \{10\text{\%},30\text{\%},50\text{\%}\}$. D = 0.01. The curves are colour coded based on their final value (red for less than 50% delayers, and blue otherwise). The initial proportion of delayers is a normalized Gaussian random field. (Online version in colour.)

For intermediate α (figure 4), we observe that the system can evolve to a large range of relative abundance of delayer clusters, the results dependent on the initial frequency of each type. For initial conditions close to equal proportions of each type, we observe the most equal coexistence.

4. Discussion

Timing is important in biology, especially in social settings where not just what individuals do but also when they do it can have important pay-off consequences. Here we investigate these consequences in a two stage evolutionary game where individuals contribute to a threshold public good, and then compete with one another once the threshold is met. Our model sheds light on how the tension between individual- and group-level incentives for early versus delayed contribution plays out at the group and population level. Contributing early is optimal for growth on the group level as it allows the public good to be realized earlier, which helps the non-delaying strategy to spread. However, a player that is delaying contributions can free-ride on others’ contributions. While this sounds like a classic social dilemma, it comes with a twist. Within the bistability regime, which can be intuited as a public good of an intermediate size, it is best to be synchronized with the majority of players as they will primarily determine the groups’ dynamics. This happens because individuals’ have a fitness advantage if their performance is relatively high when the threshold public good is realized. Further, the time at which it is realized is determined by the aggregate behaviour of the group. Thus, when a majority of group members contribute early, it is best to also contribute early, and conversely if a majority of group members delay contributions, it is best to also delay. When we consider the continuous evolution of contribution timing we find a similar result: either contributing as early as possible or delaying as much as possible is selected for, depending on where the contribution threshold lies relative to the peak of the performance schedule. This positive frequency dependence and bistability are in contrast to the instantaneous threshold public goods games where the incentives typically give rise to negative frequency dependence. When groups exist in a spatially distributed population, this local bistability can result in a pattern of patchy coexistence of delayers and non-delayers.

A few previous studies [20–22] looked at the role of timing of contributions in collective-risk games such as the Volunteer’s Dilemma [25]. In these models, players observe each others’ behaviour over several turns before benefits are accrued and contributions are contingent on the behaviour of others [20,22]. However, because the pay-offs are a function of the total amount contributed plus the amount that is withheld, not contributing always increases an individual’s pay-off. In the models of Abou Chakra & Traulsen [20] and Hilbe et al. [22], the pay-offs do not directly depend on the timing of contributions, as pay-offs are realized at the end of the interaction and depend only on the total contributions. But, timing might affect the responses of individuals to each other. They find that strategies which delay contributions are selected for. By contrast, Abou Chakra et al. [21] consider a model inspired by abatement actions to avert climate change, where in each round there is a risk of incurring loss that players can abate by contributing to the public good. They show that such ongoing risks can incentivise early contributions, because current contributions mean higher expected endowments in later rounds. Our model highlights a different strategic issue that can arise with timing when contributions have to follow a performance schedule that needs to be ramped up first. We find that similar to other public goods games, delayers can act as cheaters, avoiding contributing to a public good while reaping its benefit. However, if the public good is realized early, then delayers can ‘miss out’ on capitalizing on the public good if their performance is too low (they have yet to mature), and thereby lose to non-delayers who contributed more.

In our basic model fitness is determined in a very short competitive phase once the public good is realized. In the electronic supplementary material, A, we consider a generalization where the competitive phase lasts for some finite period after the public good is realized. This analysis shows that our results will still hold as long as this period is not too long. Increasing the length of time during which pay-offs are determined will benefit delayers. The basin of attraction to the monomorphic delayer equilibrium will increase at the expense of the basin of attraction for the monomorphic non-delayer equilibrium. Conversely, the shorter this competition duration is, the stronger the positive frequency dependence we uncover in our basic model will be. Further increasing the duration of the competitive phase will give us the classic public goods dilemma in which it is always beneficial to cheat, i.e. delay. Interestingly, this limit is mathematically equivalent to the truncation selection dynamics [26,27] where social interactions result in a hump-shaped distribution of pay-offs, and only those above a certain pay-off get to reproduce (see the electronic supplementary material, A for details on this correspondence).

Our spatial model allows us to study the dynamics of between group competition, where the outcomes depend on the initial distribution of delayers and non-delayers. Notably, we observe two distinct dynamical regimes in our simulations, which we start below the carrying capacity. In the first regime, groups grow to saturate the landscape, where the non-delayers have an advantage because of their faster growth rate, as indicated by the initial dip (or slow increase) in the frequency of delayers in figure 4. However, as populations reach carrying capacity, migration starts to be more important, and what happens at this stage depends on the clumpiness of the distribution of delayers and non-delayers: large clusters of both delayers and non-delayers can at least temporarily resist invasion by the other type, which results in a persistent polymorphism across space. These results have implications for maintenance of life-history variation: recent work on steelhead trout showed that variation in life-history speed (spawning timing and parity) was maintained on a single river system by negative frequency dependence [28]. Our results suggest that when life history is a function of social interactions and threshold public goods, variation in speed can be maintained across space through positive frequency dependence.

An interesting application of our model is to pharaoh ants infected with the endosymbiotic bacteria, Wolbachia. Pharaoh ant queens, regardless of infection status, will contribute workers to the colony as a public good, forgoing reproduction until a threshold of workers to queens is reached. Wolbachia can accelerate this life cycle of the queens [10]. One would expect then that Wolbachia prevalence should increase. However, in the wild, both monomorphic infected and monomorphic non-infected colonies coexist in proximity to one another [29–31], unlike non-social insects such as mosquitoes, where spatial coexistence is not predicted from models [32]. With respect to our model, Wolbachia expedites the performance schedule relative to uninfected queens. Equivalently, f(t) and f(t − δ) are the performance schedules for infected and uninfected, respectively. The bistability and spatial patterns of our model can explain this observation.

Furthermore, timing of reproduction is frequently a crucial component of reproductive isolation and potential for speciation [33,34]. Our bistability result suggests a potential new mechanism in which timing differences in social behaviours can lead to reproductive isolation. Two sub-populations that evolved different timing strategies (e.g. in response to different thresholds in different environments) might not be able to interbreed even after they come in contact and experience the same environment.

The models that we have explored here can be expanded in a variety of ways such as unrealized public goods, and observability of contributions. In our model, individuals cannot change their performance schedule nor can they adapt to others’ behaviours. An extension to this model could incorporate feedback wherein players may observe and then adjust their life-history strategies. Further, we can consider the case where the contributions into the public good dissipate over time, i.e. the common pool into which players contribute is leaky. In this case, delaying imposes a further cost to the group and erodes past contributions, and thereby the threshold might never be reached. Relaxing our other assumptions may have interesting results. For example, we assumed that the benefit from the public good is relative to the growth rates at the threshold in the single group model and the growth rates and time in the multi-group model. Further, we assumed that the carrying capacity was fixed, i.e. it was neither a function of the total amount nor the timing of the contributions. Relaxing these assumptions would mean that group size and the contribution amount would be important. Different strategies could be favoured at different population sizes, and it would be important to explore the evolution of the threshold contribution level, C. Such a model could be further expanded by incorporating a mortality risk over time. There is much further work to explore the role of timing in evolutionary and ecological games with implications for biodiversity, speciation and cooperation.

In conclusion, our work illustrates the interesting interactions between timing, life histories, and public goods games. We have shown that the impact of delaying life history is non-trivial. High contribution games foster delaying, while low ones foster advancing life histories. We have opened up new dimensions for both public goods games, and for life-history strategies. More broadly, it is important to study the constraints that tie individuals together, and their impact on evolution and ecology.

Data accessibility

Our code is available at https://github.com/erolakcay/contributiontiming.

Authors' contributions

All authors contributed to the conception of the study. B.M. did the mathematical analysis and led the development of the simulation code. B.M. and M.S. analysed the data and wrote the first draft, and all authors contributed to the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Funding

This research was supported by the Army Research Office(grant no. W911NF1710017) and the University of Pennsylvania.