A social dilemma structure in diffusible public goods

A public goods game (hereafter PGG) can be used not only in economics but also in several interdisciplinary fields [1]. Mathematically, the PGG is a dilemma game classified as a multi-player Prisoner's Dilemma (n-Prisoner's Dilemma or n-PD) game [2]. An agent can adopt one of two strategies: Cooperation (C) or Defection (D). A cooperator (or C agent) offers to invest a certain amount of their funds (called a cost) into a common pool. The cumulative investment dedicated by the cooperators is multiplied by r, and this amplified benefit is distributed to all participants equally, irrespective of their strategies. Therefore, adopting the D strategy and taking a free ride on the benefits must dominate even though the system will collapse if all do the same (the tragedy of commons [3]).

The dilemma situation associated with the PGG is observed in the world of microbes [4–7]. It is well known that the species Saccharomyces cerevisiae, a type of budding yeast, helps others by producing invertase enzymes that hydrolyze sucrose to fructose and glucose. This monosaccharide (glucose and fructose) provided by hydrolysis is a necessary resource for the survival of the other types of yeast and is distributed via diffusion through the nutrient medium. In essence, it can be said that S. cerevisiae produces public goods and the others free ride. The relationship between the two types of microbes resembles the roles of the cooperator (donor) and the defector (recipient) in a social dilemma game. Therefore, when they appropriately account for the diffusion of "public goods" or resources in a field, PGGs may fairly describe this particular social dilemma in the world of microbes. Several previous studies have reported that the dynamics of resource diffusion can be interpreted as a redistribution of public goods, which has led to the establishment of advanced PGG models including mathematical models and cell automaton models [8–16]. Note that this system is not entirely defined as the class of PD, snow-drift result is also reported [17–19].

Most notable is a study by Allen et al. [11]. Allen et al. proposed a mathematical model to describe the social dilemma observed between the two types of microbes mentioned above, S. cerevisiae and its follower (or free rider) as a generalized PGG in which the dilemma structure, amazingly, is consistent with the inequality associated with network reciprocity Specifically, cooperation is favored only when the benefit-to-cost ratio is larger than the network degree b/c > k, a condition that Nowak proved to be one of the five fundamental mechanisms of "social viscosity" [20]. One novel point Allen identified is that the resource diffusion can be modeled by a transformed relation between the vertices connected by an edge, i.e., a network relation.

Despite its novelty, Allen et al.'s study seems less appropriate when applied to a general social dilemma, perhaps one universally applicable in the sphere of microbes, because, in their model, a cooperator can maintain a constant resource production rate irrespective of its inner metabolic level. It would be more appropriate if a cooperative-minded agent could not dedicate resources when exposed to a meager environment, while being able to behave in a fully cooperative manner when faced with abundant primary resources in its environment. In other words, the Allen model potentially assumes a system with two unique resources, where the cooperator relies on a different primary resource than the resource that both cooperators and defectors consume to enhance their fitness.

This study addresses a mono-resource system as opposed to a multi-resource system, and therefore defines only a single resource. Any agent can use this resource for survival, but a cooperator uses some part of the resource it imports from its environment to amplify it by means of an anabolic mechanism, and subsequently releases the product, which then diffuses throughout the field. We establish a new mathematical model and examine how rich phases can result. We investigated whether the population, which shares the resource through diffusion, goes extinct like the tragedy of commons or builds a sustainable system.

This report is organized as follows: the second section provides the model framework, the third section shows the preliminary results, the fourth section gives the simulation results and discussion, with a particular focus on the contour map, and we conclude in the last section.

We aimed to establish a general model for the evolutionary process of a species of microbes living on a plate of agar gel. Each microbe puts down its roots at a certain location in the 2-dimensional (2D) agar gel. The microbes therefore do not move on the gel, but rather the resource diffuses through the gel according to the archetypal 2D diffusion equation. We can define the concentration of the resource at any place on the gel. The microbes can be divided into two categories. The first is the defective type (hereafter called defectors, D), which consumes a certain amount of the resource by absorbing it from the gel. The second is the cooperative type (hereafter called cooperators, C), which likewise consumes the resource but simultaneously replenishes the resource through a specific anabolic effect. The resource, produced in this way, is released into the gel. For simplicity, this study assumes that the only evolutionary process that takes place is in the turnover of single individuals, and that the evolutionary process of cell division that results in the growth of the microbe colony into the surrounding vacant space does not occur.

We describe the details of this model below.

Mathematical model of diffusible public goods

We assume a 2D gel space divided into N ($=51 \times 51$; see fig. 1) von Neumann cells. We initially place n₀ ($=221$; see fig. 1) microbes in the center of the gel region in a diamond formation as shown in fig. 1. Each microbe occupies a single cell of the gel. We express the address of a cell as an index i, and assume a concentration of the resource in that cell $\Psi_{i}$. The strategy (C or D) of an agent located in cell i is denoted as $s_{i}\in \{+1,-1\}$. A cooperator (C: $s_{i}= 1$) produces and releases the resource P into its own cell. In contrast, a defector (D: $s_{i}= -1$) cannot produce the resource. Any arbitrary agent consumes a quantity of the resource, M_i, to maintain its fundamental metabolism. Denoting the equivalent overall diffusion coefficient as Λ, the governing equation can be written as follows:

$$\begin{equation} \frac{\text{d}\Psi_{i}}{\text{d}t}=\Lambda\left(\sum_{j\epsilon\{N_i\}}\Psi_{j} -\Psi_{i}\right)+\text{Abs}[s_{i}]\cdot\left(\frac{s_i+1}{2}\cdot P-M_{i}\right), \end{equation} \tag{ 1 }$$

where $\{N_{i}\}$ indicates the set of four von Neumann cells neighboring cell i.

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An agent, irrespective of its strategy, imports the resource, which depends on the intake efficiency μ and the concentration difference between $\Psi_{\text{body}}$ and $\Psi_{i}$, where $\Psi_{\text{body}}$ represents the internal resource concentration of agent i. The resource intake must at least equal the basal metabolism $M_{\min}$. Thus, we can evaluate M_i as follows:

$$\begin{equation} M_{i} = \text{Max}\left[\mu\left(\Psi_{i}-\Psi_{\text{body}}\right), M_{\min}\right]. \end{equation} \tag{ 2 }$$

An agent dies and is removed from the cell whenever it cannot import more than $M_{\min}$ from the cell in which it lives.

A cooperator devotes a part of its intake resources to resource production. We presume that a cooperator invests the maximum possible amount in this process, $M_{i}-M_{\min}$, which must also be no less than the Cost, which is the minimum requisite resource amount required to activate the anabolic effect. Our normalization in this study assumes $\textit{Cost} = 1$. In short, the amount invested by the cooperator i is Min[Cost, M_i − M_min]. The quantity of the resource produced, P, is given as the product of the investment and the productivity parameter b, which corresponds to the dilemma weakness. Thus, we get

$$\begin{equation} P=b\cdot\text{Min}[Cost, M_i - M_{\min}]. \end{equation} \tag{ 3 }$$

The fitness of agent i, $\pi_{i}$, which stipulates its evolutionary strength, depends on whether it is cooperative or defective as follows:

$$\begin{equation} \pi_{i} = M_{i} - \frac{s_{i}+1}{2}\cdot \text{Min}[Cost, M_{i}-M_{\min}]. \end{equation} \tag{ 4 }$$

Death-birth process

Simulation setting

Before reporting the simulation results of competition for the two species, to ensure the inherent sustainability of the population, we first conducted simulations to determine whether a population consisting of only cooperators could survive for various initial population densities ρ. Specifically, starting from a certain ρ in which Cs are randomly assigned in the diamond in fig. 1, we examined how many realizations of the 300 trials allowed the population to survive. The result, presented as the average frequency of extinction for each dilemma strength, is shown in fig. 2. In both figs. 2(A1) and (B1), we can clearly see that the population can survive in most of the parameter range, except when μ is very close to 0. This implies that too small a μ does not allow even cooperators to survive due to the lack of sufficient nutrition. From the indicated values at the two cross-marks in the panels, we can see that fig. 2(A1), which assumed the stronger dilemma, has a slightly higher possibility of extinction. Examining the details in figs. 2(A2) and (B2), we see that the population is more likely to go extinct when assuming a smaller initial population density. Moreover, in fig. 2(A2), which assumes the stronger dilemma, depending on the diffusivity of the public goods, the extinction frequency increases even when a reasonably large initial population density is given. This implies that even without exploiting agents, the cooperators cannot maintain their population by diffusing the resource out into the surrounding space. This study does not take into account the increasing number of agents in an episode from the viewpoint of exploring the inherent system dynamics brought on by the public goods diffusion; however, if we were to consider this, the possibility of the population surviving would be slightly higher.

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In this section, we discuss the social dilemma structure in public goods games with diffusion, namely, whether or not the population survives as a result of competition between two species.

Under a particular parameter set for Λ, μ, and b, we can compile statistics, expressed by the frequency of Extinction (either true extinction or only defectors surviving), Polymorphic (co-existing cooperators and defectors), or Cooperation (cooperator dominating the system) results over 300 independent simulation trials while varying the initial cooperation fraction. In a typical illustration, fig. 3 shows the equilibrium frequency distribution. In observing fig. 3(A), we can see that extinction is prevalent and does not depend on the initial cooperation fraction (hereafter P_{c_init}). However, when most agents are C ($P_{c\_\text{init}}$ is close to 1), this is not true. The frequency of cooperation exceeds that of extinction, which is inevitable because the stochastic adaptation process of death-birth is adopted. We classify this case as Extinction. In fig. 3(B), even though we can confirm that the system reaches various equilibriums according to the $P_{c\_\text{init}}$, intermediate $P_{c\_\text{init}}$ values show a high probability of a coexistent equilibrium. In addition, frequent Cooperation (Extinction) is seen in the range of high (low) $P_{c\_\text{init}}$. This complex tendency results from the subtle power balance between C and D due to this particular parameter set and $P_{c\_\text{init}}$. If C is small, it is hard to maintain the population because either C or D is inevitably weeded out. Conversely, a sufficient number of cooperators results in a sustainable system. This phase is called Polymorphic. Figure 3(C), contrary to fig. 3(A), shows consistent cooperation; however, the effect of the stochastic process is still observed. In fact, assuming a small initial cooperator density leads to frequent extinction and a sustainable system is rarely realized.

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In the subsequent section, we will examine what phases can be observed when the three parameters, Λ, μ, and b are varied.

In fig. 4, we see the average frequency of each equilibrium state defined by the above-mentioned procedure for fig. 3, as well as a phase diagram represented by the most dominant equilibrium (e.g., see ref. [12]).

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Strong dilemma

Weak dilemma

To reveal the underlying dilemma structure in a public goods game (PGG) in which the "public goods" diffuse throughout the system, we developed a new PGG model to reproduce the struggle between cooperators, who replenish a resource, and defectors, who free ride on the productivity of the cooperators.

To ensure sustainability at a certain level, preliminary simulations assuming that the population is composed entirely of cooperators were conducted. We confirmed that the population could be sustained, except when there was a small initial population density, and variations were observed due to the effect of diffusivity.

Our numerical simulations resulted in phase diagrams where the diffusivity, the intake rate of the resource, and the dilemma strength were all varied. Despite the payoff function based on the PGG, the effect of the spatial structure (the diffusion and concentration distribution) resulted in very different outcomes from those estimated from the payoff function, i.e., Cooperation, Polymorphic, and Extinction phases. We found that a stronger social dilemma requires a small diffusivity to maintain the population because the produced resources are very small. That is, fast diffusion encourages the survival of defectors, and a sustainable population cannot be achieved. An intermediate diffusivity establishes a symbiotic relationship, namely, clustering cooperators produce large quantities of resources and a few defectors get a free ride. A weaker social dilemma not only favors cooperation but also discourages "cloudy" equilibria, such as the Polymorphic phase, in favor of "clear" equilibria, either C-dominated or D-dominated.

As mentioned in the text, the presented model assumes that the only evolutionary process taking place is in the turnover of individual agents and disregards another important evolutionary process, cell division, which results in the growth of the microbe colony into the surrounding vacant space. In addition, it is expected that the number of individuals will express fluctuations, such as those seen in Lotka-Volterra systems. In the beginning of each episode in such a system, defectors overtake cooperators and population size diminishes due to die outs. In the next stage, cooperators in clusters generate new individuals with ample resources and the population size grows. However, defectors again exploit the cooperators and increase in number; and the decrease in the population size is repeated. In this way, a certain sustainability, as seen in biological systems, can be produced. This type of system will be examined further in a subsequent report.

This study was partially supported by a Grant-in-Aid for Scientific Research from JSPS, Japan, awarded to JT (Grant No. JP15K14077). In addition, we would like to express our gratitude to Dr Eriko Fukuda for several interactive discussions.