Spatial social dilemmas promote diversity

Significance

Evolutionary branching and diversification in interactions with continuous behavioral traits is important for understanding the emergence of distinct, discrete, and coexisting strategies. In social dilemmas, this suggests an evolutionary pathway for the origin of cooperators and defectors. Here we study evolutionary diversification in structured populations and identify mechanisms driving spontaneous and persistent diversification. Through analytical and numerical techniques, we demonstrate that spatial structure admits new modes of diversification that complement classical evolutionary branching. In particular, when selection is strong, diversification tends to occur more readily than in unstructured populations.

Abstract

Cooperative investments in social dilemmas can spontaneously diversify into stably coexisting high and low contributors in well-mixed populations. Here we extend the analysis to emerging diversity in (spatially) structured populations. Using pair approximation, we derive analytical expressions for the invasion fitness of rare mutants in structured populations, which then yields a spatial adaptive dynamics framework. This allows us to predict changes arising from population structures in terms of existence and location of singular strategies, as well as their convergence and evolutionary stability as compared to well-mixed populations. Based on spatial adaptive dynamics and extensive individual-based simulations, we find that spatial structure has significant and varied impacts on evolutionary diversification in continuous social dilemmas. More specifically, spatial adaptive dynamics suggests that spontaneous diversification through evolutionary branching is suppressed, but simulations show that spatial dimensions offer new modes of diversification that are driven by an interplay of finite-size mutations and population structures. Even though spatial adaptive dynamics is unable to capture these new modes, they can still be understood based on an invasion analysis. In particular, population structures alter invasion fitness and can open up new regions in trait space where mutants can invade, but that may not be accessible to small mutational steps. Instead, stochastically appearing larger mutations or sequences of smaller mutations in a particular direction are required to bridge regions of unfavorable traits. The net effect is that spatial structure tends to promote diversification, especially when selection is strong.

Social dilemmas (1, 2) are important mathematical metaphors for studying the problem of cooperation. A famous example is the tragedy of the commons (3), which results in overexploitation of communal resources. The best-studied models of social dilemmas are the prisoner’s dilemma and the snowdrift game (4). Traditionally, such models are often restricted to the two distinct strategies of cooperate, $C$, and defect, $D$. However, these models can easily be extended to describe a continuous range of cooperative investments into a public good. In such continuous games, the aim is to study the evolutionary dynamics of the level of investment.

For example, in the donation game (5), which is the most prominent version of the prisoner’s dilemma, cooperators confer a benefit $b$ to their interaction partners at a cost $c$ to themselves. Defection neither entails costs nor confers benefits. With $b>c>0$, both individuals prefer mutual cooperation, which pays $b-c$, over a zero payoff for mutual defection. However, the temptation to shirk costs and free ride on benefits provided by the partner undermines cooperation to the detriment of all. In fact, defection is the dominant strategy because it results in higher payoffs regardless of the partner’s strategy. This conflict of interest between the individual and the group represents the hallmark of social dilemmas.

A natural translation of the donation game to continuous traits is based on cost and benefit functions, $C\left(x\right)$ and $B\left(x\right)$, where the trait $x\in \left[0,x_{max}\right]$ represents a level of cooperative investment that can vary continuously. In the spirit of the donation game, an individual with strategy $x$ interacting with a $y$ strategist obtains a payoff $P\left(x,y\right)=B\left(y\right)-C\left(x\right)$. Assuming that 1) zero investments incur no costs and provide no benefits, $B\left(0\right)=C\left(0\right)=0$, and 2) benefits exceed costs, $B\left(x\right)>C\left(x\right)\ge 0$, and 3) are increasing functions, $B^{\prime }\left(x\right),C^{\prime }\left(x\right)\ge 0$, recovers the social dilemma of the donation game for continuous investment levels: The level of investment invariably evolves to zero, which corresponds to pure defection, despite the fact that both players would be better off at nonzero investment levels. The reason is that an actor can only influence the cost $C\left(x\right)$ of an interaction, but not the received benefit $B\left(y\right)$, and hence selection can only act to minimize costs.

In a weaker form of a social dilemma, the snowdrift game (6, 7), cooperators also provide benefits, $b$, at a cost, $c$. However, benefits are now accessible to both individuals and accumulate in a discounted manner. For example, yeast secretes enzymes for extracellular digestion of sucrose (8). While access to nutrients is crucial, the marginal value of additional resources diminishes and may exceed the intake capacity (2). As a result, the social dilemma remains in effect, because the most favorable outcome of mutual cooperation remains prone to cheating, but the dilemma is relaxed because it pays to cooperate against defectors. Again, it is straightforward to formulate a continuous version of the snowdrift game. In contrast to the donation game, in the continuous snowdrift game, the benefits depend on the strategies of both interacting individuals. For example, for simplicity, one can assume that benefits depend on the aggregate investment levels of both players, so that the benefit is a function $B\left(x+y\right)$. Assuming a cost function $C\left(x\right)$, an $x$ strategist interacting with strategy $y$ obtains a payoff $P\left(x,y\right)=B\left(x+y\right)-C\left(x\right)$, again with $B\left(0\right)=C\left(0\right)=0$, $B\left(x\right)>C\left(x\right)\ge 0$ and $B^{\prime }\left(x\right),C^{\prime }\left(x\right)\ge 0$, as before, but with the additional constraint $C\left(x\right)>B\left(x+y\right)-B\left(y\right)$ for sufficiently large $y$ so that the increased return from investments of an individual do not outweigh its costs when interacting with high investors. Typically, this is achieved by saturating benefit functions, $B^{\prime \prime }\left(x\right)<0$.

The gradual evolution of continuous traits can be described using the framework of adaptive dynamics (9–12). The central quantity is the invasion fitness $f\left(x,y\right)$, which is the growth rate of a rare mutant $y$ in a monomorphic resident population with trait $x$. To derive the adaptive dynamics of the trait $x$, one considers the selection gradient $D\left(x\right)={\partial }_{y}f\left(x,y\right){|}_{y=x}$. If $D\left(x\right)>0$, nearby mutants with trait values $y>x$ can invade, whereas, for $D\left(x\right)<0$, nearby traits with $y<x$ invade. For example, it is easy to see that, in the continuous snowdrift game, $D\left(0\right)>0$ if $B^{\prime }\left(x\right)>C^{\prime }\left(x\right)\ge 0$, so that, in contrast to the continuous donation game, the pure defector strategy $x=0$ is not evolutionarily stable (13). Traits $x^{*}$ with $D\left(x^{*}\right)=0$ are called singular traits and are convergence stable if the Jacobian of the selection gradient, $C\hspace{0.17em}S\left(x^{*}\right)=dD\left(x\right)/dx{|}_{x=x^{*}}$, is negative, whereas evolutionary stability requires the Hessian of the fitness, $E\hspace{0.17em}S\left(x^{*}\right)={\partial }^{2}f\left(x^{*},y\right)/\partial y^2{|}_{y=x^{*}}$, to be negative, that is, that the invasion fitness $f\left(x,y\right)$ has a (local) maximum at $x^{*}$. Since the conditions for convergence stability and for evolutionary stability are different, in general, a singular trait $x^{*}$ may be convergence stable but not evolutionarily stable. This results in the intriguing dynamical feature of evolutionary branching, where an initially monomorphic population evolutionarily splits into two (or more) coexisting traits (14).

For social dilemmas with a discrete set of strategies, there exists extensive literature on the effects of spatial structures on the dynamics of the social dilemma (15–19). In particular, it has been shown, in many incarnations, that spatial structure can promote cooperative strategies, and hence that spatial structure can have a profound effect on game dynamics. However, far less is known about the effects of spatial structure on continuous games (20, 21), and on evolutionary diversification in particular. In well-mixed populations, evolutionary branching has been shown to occur generically in the continuous snowdrift game (13), in which cooperative investments can diversify into stably coexisting high and low contributors. Similar observations have been made for the “tragedy of the commune,” a game that is “dual” to the tragedy of the commons in the sense that the trait $x$ represents the amount of consumption of a public good. In this case, diversification results in a coexisting coalition of unequal consumers of a public good (22).

Here we study evolutionary diversification in continuous spatial games. To derive the corresponding spatial adaptive dynamics, we first use the technique of pair approximation (17, 23, 24) to describe the frequency dynamics of pairs of resident and mutant traits in order to obtain an analytical expression for the invasion fitness of a rare mutant in a structured resident population. We then use the invasion fitness function to determine singular trait values and their convergence and evolutionary stability. Analytical approximations and predictions are tested against individual-based simulations of the corresponding spatial games. It turns out that, in structured populations, evolutionary branching is suppressed but easily compensated for by other modes of diversification, especially for stronger selection, such that evolutionary diversification represents a robust feature of continuous spatial games.

Spatial Invasion Fitness and Adaptive Dynamics

The dynamics of spatial games is generally complex (25, 26) and depends on the details of the updating process (16, 18). In particular, the sequence of birth and death events can be of crucial importance in spatially structured populations. For example, in the classical donation game with selection on birth rates in spatially structured populations, cooperators invariably disappear for birth−death updating but are able to thrive for death−birth updating [provided that the benefits exceed the $k$-fold costs, $b>ck$ (16)]. Here we analyze the spatial evolutionary dynamics for the more interesting case of death−birth updating for social dilemmas with continuous traits, as exemplified by the continuous prisoner’s dilemma and the continuous snowdrift game. For birth−death updating, the effects of population structures are less pronounced, and the corresponding results are relegated to SI Appendix, section S4.

In the following, we assume that the total population size is constant, and that spatially structured populations are represented by lattices in which each site is occupied by one individual. Each individual interacts with a limited number of local neighbors, and we assume this number, $k$, to be the same for all individuals. We first consider a case where there are two types of players in the structured population: a mutant type with trait value $y$ and a resident type with trait value $x$ (where $x$ and $y$ denote investment strategies in a continuous game). If the mutant has $j$ other mutants among its $k$ neighbors, the mutant payoff is ${\pi }_m\left(j\right)=\left[\left(k-j\right)P\left(y,x\right)+jP\left(y,y\right)\right]/k$. Similarly, the payoff of a resident with $j$ mutant neighbors is given by ${\pi }_r\left(j\right)=\left[jP\left(x,y\right)+\left(k-j\right)P\left(x,x\right)\right]/k$. The payoffs, ${\pi }_m\left(j\right),{\pi }_r\left(j\right)$, of mutants and residents from interactions with their $k$ neighbors determine the birth rates as $b_m\left(j\right)=\exp \left(w{\pi }_m\left(j\right)\right)$ and $b_r\left(j\right)=\exp \left(w{\pi }_r\left(j\right)\right)$, where $w>0$ denotes the strength of selection. The birth rate is proportional to the probability of taking over an empty site for which a given mutant or resident individual competes. For $w\ll 1$, selection is weak, and differences in payoffs result in minor differences in birth rates, and hence in small differences in probabilities of winning competition for an empty site. With strong selection, $w\gg 1$, payoff differences are amplified in the corresponding birth rates. This exponential payoff-to-birth rate map has several convenient features: 1) ensures positive birth rates, 2) admits easy conversion to probabilities for reproduction, 3) selection can be arbitrarily strong, and 4) for weak selection, the more traditional form of birth rates $b_i\left(j\right)\approx 1+w{\pi }_i\left(j\right)$, $i=m,r$, is recovered. Note that, in the limit $y\to x$, differences in birth rates vanish. However, this does not imply weak selection. Instead, selection strength is determined by the magnitude of the invasion fitness gradient at $x$, which is proportional to $w$.

In well-mixed populations, the current state is simply given by the frequency of mutants and residents. In contrast, in structured populations, the state space is immense because it involves all possible configurations. Pair approximation offers a convenient framework to account for corrections arising from spatial arrangements (17, 23, 24). Instead of simply tracking the frequencies of mutants and residents, pair approximation considers the frequencies of neighboring strategy pairs. We denote the frequencies of mutant−mutant, mutant−resident, resident−mutant, and resident−resident pairs by $p_{mm}$, $p_{mr}$, $p_{rm}$, and $p_{rr}$, respectively. This reduces to two dynamical equations because $p_{mr}=p_{rm}$ and $p_{mm}+p_{mr}+p_{rm}+p_{rr}=1$ must hold. The most informative quantities are the global mutant frequency, $p_m=p_{mm}+p_{mr}$, and the local mutant density $q_{m|m}=p_{mm}/p_m$, that is, the conditional probability that a neighbor of a mutant is also a mutant. Note that, for rare mutants, $p_m\ll 1$, their local densities need not be small. The derivation of the corresponding dynamical equations depends on the details of the microscopic updating.

The death−birth process with selection on birth in structured populations results in local competition: An individual is selected to die uniformly randomly from the whole population, then its $k$ neighbors compete to repopulate the newly vacated site. They succeed with a probability proportional to their birth rates. Note that payoffs, and hence birth rates, are based on interactions with all neighbors, including the neighbor that may subsequently be chosen to die (uniformly at random) and its vacant site subject to recolonization by the offspring of one of its neighbors. To determine the dynamics of $p_m$ and $q_{m|m}$, we first note that configurations only change when a resident is replaced by a mutant, or when a mutant is replaced by a resident. The dynamical equations for the mutant dynamics in structured populations based on pair approximation with death−birth updating are derived in SI Appendix, section S1.

In order to obtain the invasion fitness, we note that the dynamics of spatial invasion unfolds in two stages: Mutants quickly establish a local (pseudo) equilibrium and then gradually increase (or decrease) in frequency (20, 27, 28). Formally, this is reflected in a separation of time scales in the limit of rare mutants, $p_m\ll 1$, between the slow global frequency dynamics and the fast local pair density (SI Appendix, section S1 and Eqs. S3 and S5). As a consequence, to calculate invasion fitness, we assume that the local pair density of the mutant is at its equilibrium $q_{m|m}^{*}$. As shown in SI Appendix, section S1, the invasion fitness of mutants, $f\left(x,y\right)$, defined as their per capita growth rate, ${\dot{p}}_m/p_m$, in the limit $p_m\to 0$, then becomes

$$\begin{array}{ll}\hfill f\left(x,y\right)=& \hspace{0.17em}\frac{k\left(1-q_{m|m}^{*}\right)b_m\left(\left(k-1\right)q_{m|m}^{*}\right)}{b_m\left(\left(k-1\right)q_{m|m}^{*}\right)+\left(k-1\right)b_r\left(0\right)}-\hfill \\ \hfill & \hspace{1em}\sum _{j=0}^k\left(\genfrac{}{}{0.0pt}{}{k}{j}\right){\left(q_{m|m}^{*}\right)}^j{\left(1-q_{m|m}^{*}\right)}^{k-j}\times \hfill \\ \hfill & \hspace{1em}\frac{\left(k-j\right)b_r\left(1\right)}{j{b}_m\left(1+\left(k-1\right)q_{m|m}^{*}\right)+\left(k-j\right)b_r\left(1\right)},\hfill \end{array}$$ [1]

where $b_m\left(v\right)$ and $b_r\left(v\right)$ denote the birth rates of mutants and residents, respectively, with an average number of $v$ mutants in their neighborhood. Even though the solution to ${\dot{q}}_{m|m}=0$ is analytically inaccessible, in general, the equilibrium $q_{m|m}^{*}$ can be approximated using a Taylor expansion if $|y-x|\ll 1$ (SI Appendix, section S1),

$$\begin{array}{ll}\hfill q_{m|m}^{*}=& \hspace{0.17em}\frac{1}{k-1}+w\left(y-x\right)\frac{k^2-4}{2{\left(k-1\right)}^2{k}^2}\times \hfill \\ \hfill & \left({\partial }_{z}P\left(x,z\right)+k{\partial }_{z}P\left(z,x\right)\right){|}_{z=x}+O\left({\left(y-x\right)}^2\right).\hfill \end{array}$$ [2]

It follows that, in the limit $y\to x$, mutants with at least one resident neighbor have, on average, one mutant neighbor among their $k-1$ other neighbors. Note that mutants with no resident neighbors are uninteresting because they are unable to initiate a change in the population configuration. Interestingly, this limit of the local pair configuration is fairly robust with respect to changes in the updating process (compare SI Appendix, Eq. S25 for birth−death updating). Moreover, in this limit, a rare mutation with positive invasion fitness is guaranteed to eventually take over (see ref. 10 and SI Appendix, section S1).

Using Eqs. 1 and 2, the selection gradient, $D_{\text{db}}\left(x\right)=\partial f\left(x,y\right)/\partial y{|}_{y=x}$, as well as its Jacobian, ${CS}_{\text{db}}\left(x^{*}\right)=\frac{d{D}_{\text{db}}\left(x\right)}{dx}{|}_{x=x^{*}}$, and the Hessian of fitness, ${ES}_{\text{db}}\left(x^{*}\right)=\frac{{\partial }^{2}f\left(x^{*},y\right)}{\partial y^2}{|}_{y=x^{*}}$, at a singular point $x^{*}$ can be calculated as

$$D_{\text{db}}\left(x\right)\hspace{0.17em}=\hspace{0.17em}w\frac{k-2}{k\left(k-1\right)}\left(k{\partial }_{y}P\left(y,x\right)+{\partial }_{y}P\left(x,y\right)\right){|}_{y=x}$$ [3]

$$\begin{array}{ll}\hfill {CS}_{\text{db}}\left(x^{*}\right)\hspace{0.17em}=& \hspace{0.17em}w\frac{k-2}{k\left(k-1\right)}\left(k{\partial }_y^{2}P\left(y,x^{*}\right)+{\partial }_y^{2}P\left(x^{*},y\right)+\right.\hfill \\ \hfill & \left.\left(k+1\right){\partial }_{y,z}P\left(y,z\right)\right){|}_{z=y=x^{*}}\hfill \end{array}$$ [4]

$${ES}_{\text{db}}\left(x^{*}\right)\hspace{0.17em}=\hspace{0.17em}C\hspace{0.17em}{S}_{\text{db}}\left(x^{*}\right)-w\frac{{\left(k-2\right)}^2\left(k+1\right)}{k^2\left(k-1\right)}{\partial }_{y,z}P\left(y,z\right){|}_{z=y=x^{*}}.$$ [5]

Continuous Social Dilemmas in Space

We now apply the approach described above to specific continuous spatial games. To get a sense of the predictive power of the analytical approximations, we compare the analytical approximations to results from corresponding individual-based models.

Continuous Prisoner’s Dilemma.

In the continuous prisoner’s dilemma, the payoff to an individual with strategy $y$ interacting with an $x$ strategist is $P\left(y,x\right)=B\left(x\right)-C\left(y\right)$. This implies that the selection gradient in well-mixed populations is proportional to $-C^{\prime }\left(x\right)$, and hence, assuming monotonously increasing costs, no singular strategy exists apart from the pure defection state $x=0$, and the population always evolves to that state, regardless of how large the benefits of cooperation are. In contrast, for death−birth updating in structured populations, the selection gradient, Eq. 3, becomes

$$D_{\text{db}}\left(x\right)=w\frac{k-2}{k\left(k-1\right)}\left(B^{\prime }\left(x\right)-k{C}^{\prime }\left(x\right)\right).$$ [6]

Thus, in structured populations, a singular strategy, $x^{*}$, may exist as a solution to $D_{\text{db}}\left(x^{*}\right)=0$. If $x^{*}$ exists and is convergence stable, then it is also evolutionarily stable, because the two stability conditions are identical (the mixed derivatives on the right-hand side of Eqs. 4 and 5 are zero). In particular, cooperation can be maintained in spatially structured populations, a result that is, of course, in line with classical theory (15). More specifically, cooperative investments can increase if the marginal benefits, $B^{\prime }\left(x\right)$, exceed the $k$-fold marginal costs, $C^{\prime }\left(x\right)$, which is reminiscent of the $b>ck$ rule in the traditional donation game (16).

Linear Costs and Benefits.

The evolutionary analysis becomes particularly simple for linear benefit and cost functions $B\left(x\right)=x$ and $C\left(x\right)=rx$, where $r$ denotes the (marginal) cost-to-benefit ratio. Again, population structures are capable of supporting cooperation for sufficiently small $r$. More precisely, the selection gradient $D_{db}\left(x\right)=w\left(k-2\right)\left(1-kr\right)/\left(k\left(k-1\right)\right)$ is reduced to a constant (compare Eq. 6), and hence no singular strategy exists. However, the gradient changes sign for $r^{*}=1/k$. Thus, for favorable cost-to-benefit ratios, $r<r^{*}$, investments increase to their maximum, whereas, for harsher conditions, $r>r^{*}$ cooperative investments cannot be sustained; see Fig. 1.

Fig. 1.

Equilibrium investment levels (mean $\pm$ SD) in individual-based simulations of the linear continuous prisoner’s dilemma ($B\left(x\right)=x,C\left(x\right)=rx$) as a function of the cost-to-benefit ratio, $r$, for weak, $w=1$ (red), moderate, $w=10$ (black), and strong, $w=100$ (blue) selection [see ref. 29 for interactive online simulations (30)]. (A) In well-mixed populations with $N=10^4$ individuals, cooperative investments are close to zero regardless of $r$ as predicted by adaptive dynamics (dashed line). The small variance further decreases for stronger selection, emphasizing the disadvantage of mutants with higher investments. (B) For populations with $k=4$ neighbors, spatial adaptive dynamics predicts a threshold $r*=1/k$ (dash-dotted line) below which investments reach the maximum but above which they disappear. Simulations on a $100\times 100$ lattice confirm the trend but reveal an interesting susceptibility to noise: For weak selection (red), the maximum investment is not reached; intermediate selection (black) essentially follows pair approximation, while strong selection (blue) maintains nonzero investment levels beyond the predicted threshold. The large variation suggests coexistence of different traits and is confirmed through sample snapshots of the spatial configuration in Fig. 2I.

Interestingly, these analytical predictions are not always borne out in individual-based models when the selection strength, $w$, is large. The reasons for the differences can be appreciated by looking at pairwise invasibility plots (PIP), which show the regions in $\left(x,y\right)$ space in which a mutant $y$ can invade a resident $x$, that is, regions for which the invasion fitness $f\left(x,y\right)$ is positive. To construct these plots, we first solve the dynamical equation ${\dot{q}}_{m|m}=0$ (SI Appendix, section S1 and Eq. S5) to find the equilibrium $q_{m|m}^{*}$ for given resident and mutant trait values $x,y$ and then determine the sign of the corresponding invasion fitness (Eq. 1). An example is shown in Fig. 2G with $r>1/k$ so that adaptive dynamics predicts evolution to pure defection.

Fig. 2.

Linear continuous prisoner’s dilemma in $100\times 100$ lattice populations with $k=4$ neighbors [see ref. 29 for interactive online simulations (30)]. Evolutionary dynamics is illustrated for (A−C) favorable cost-to-benefit ratios, $r=0.1<1/k$ and (D−F) for harsher conditions, $r=0.3>1/k$, with moderate selection, $w=10$ as well as (G−I) strong selection, $w=100$. Left shows the PIP, which indicate whether mutant traits are capable of invading a particular resident population (white regions) or not (black regions). Middle shows the evolutionary trajectory of the distribution of investments over time in individual-based simulations (darker shades indicate higher trait densities in the population, with the highest densities in yellow). Right depicts snapshots of the population configuration at the end of the simulation runs. The color hue indicates the investment levels ranging from low (red) to intermediate (green) and high (blue). For $r<1/k$, higher investors can always invade, and, eventually, the maximum investment is reached (A–C), regardless of selection strength. The situation is reversed for $r>1/k$ and weak to moderate selection where only lower investors can invade and investments dwindle to zero (D–F). Interestingly, for strong selection in lattice populations, not only lower investors can invade for $r>1/k$ but also those that invest significantly more than the resident (G–I). Note that the width of the region of unfavorable mutants decreases with increasing selection strength $w$; that is, the gap becomes easier to bridge (but does not depend on the resident trait $x$; gray area for $w=50$ and black area for $w=100$). Nevertheless, adaptive dynamics predicts that investments evolve to zero because of the assumption that mutations are small, which restricts the dynamics to the diagonal of the PIP. However, individual-based simulations show that rare larger mutations or a sequence of smaller ones can give rise to the coexistence of high and low investors. The outcome does not depend on the initial configuration of the population.

For mutants $y$ sufficiently close to the resident $x$, $f\left(x,y\right)>0$ only holds for $y<x$, so that evolution by (infinitesimally) small mutations, as envisaged in adaptive dynamics, indeed leads to ever smaller investment. However, the PIP also shows positive invasion fitness for mutants $y>x$ if $y$ is sufficiently larger than $x$ (white area above the diagonal in Fig. 2G). In individual-based models, mutations are normally distributed around the parental trait and of finite size such that stochastically appearing larger mutations or sequences of smaller mutations in the right direction can give rise to the coexistence of high and low investors (Fig. 2I). Thus, evolutionary diversification is possible in individual-based models despite the absence of evolutionary branching points in the adaptive dynamics, and even if the analytical prediction is evolution of pure defection. Note that this mode of evolutionary diversification is a feature of spatial structure that does not occur in the corresponding well-mixed models.

Saturating Benefits.

The continuous prisoner’s dilemma was first introduced in ref. 31 with saturating benefits $B\left(x\right)=b_0\left(1-\exp \left(-b_{1}x\right)\right)$ and linear costs $C\left(x\right)=c_{0}x$ ($b_0,b_1,c_0>0$). Using individual-based simulations, these authors showed that structured populations are capable of supporting cooperation, whereas well-mixed populations are not. Spatial adaptive dynamics provides an analytical perspective on these results. For the given cost and benefit functions, we have

$$D_{\text{db}}\left(x\right)=w\frac{k-2}{k\left(k-1\right)}\left(b_0{b}_1{e}^{-b_{1}x}-c_{0}k\right)$$ [7]

$${CS}_{\text{db}}\left(x^{*}\right)={ES}_{\text{db}}\left(x^{*}\right)=-w\frac{k-2}{k-1}{b}_1{c}_0<0.$$ [8]

There is a singular point at

$$x^{*}=\frac{1}{b_1}\log \left(\frac{b_0{b}_1}{c_{0}k}\right).$$ [9]

The singular point is always convergence stable as well as evolutionarily stable (Fig. 3B).

Fig. 3.

Continuous prisoner’s dilemma with saturating benefits $B\left(x\right)=b_0\left(1-\exp \left(-b_{1}x\right)\right)$ and linear costs $C\left(x\right)=c_{0}x$ for $b_0=8,b_1=1,c_0=0.7$ in $100\times 100$ lattice populations for (A−C) weak selection, $w=1$ and (D−F) strong selection, $w=100$ [see ref. 29 for interactive online simulations (30)]. The PIP (A and D) show that higher-investing mutants can invade for low resident investments, and lower−investing mutants can invaded high-investing residents. However, near the singular investment level, $x*=\log \left(b_0{b}_1/\left(k{c}_0\right)\right)/b_1\approx 1.050$, selection strength gives rise to interesting differences in the dynamics. A shows that, for weaker selection, $w=1$, $x*$ is evolutionarily stable. This is confirmed through individual-based simulations. B depicts the investment distribution over time (darker shades indicate higher trait densities in the population with the highest densities in yellow). C shows a snapshot of the spatial configuration at the end of the simulation. The color hue indicates the investment level ranging from low (red) to intermediate (green) to high (blue). In contrast, D shows that, for strong selection, $w=100$, $x*$ can be invaded, but only by higher-investing mutants. As a consequence, a degenerate form of evolutionary branching may occur. Individual-based simulations confirm branching already for $w=10$ (E and F).

Interestingly, the adaptive dynamics analysis again misses subtle but intriguing effects arising from strong selection. A PIP for strong selection is shown in Fig. 3D and reveals that $x^{*}$ is susceptible to invasion by mutants with slightly higher investments. In individual-based models, this effectively turns $x^{*}$ into a (degenerate) branching point, that is, a starting point for evolutionary diversification. The diversification into coexisting high and low investors has already been observed in ref. 31, but the underlying mechanism had not been addressed. The earlier results were based on a different, deterministic update rule, according to which a focal individual imitated the strategy of the best-performing neighbor, including itself, but this update rule essentially corresponds to death−birth updating with very strong selection (the only difference being that the focal individual is removed). Hence the diversification reported in ref. 31 is of the same type as the one seen here for strong selection.

Continuous Snowdrift Game.

In the continuous snowdrift game, the payoff to an individual with strategy $y$ interacting with an $x$ strategist is $P\left(y,x\right)=B\left(x+y\right)-C\left(y\right)$. Note that this generalization to continuous strategies does not imply that every interaction between an individual with strategy $x$ and another with strategy $y$ invariably results in a snowdrift game. In fact, snowdrift games in this narrower sense appear only if the two traits $x,y$ straddle the singular strategy $x^{*}$. Depending on the cost and benefit functions, as well as on the strategies $x$ and $y$, any kind of $2\times 2$ game can arise, including a prisoner’s dilemma, in which the lower investing strategy dominates (13, 32). In this sense, the framework of the continuous snowdrift game actually encompasses the gist of the continuous prisoner’s dilemma for sufficiently high costs—just as the classical snowdrift game turns into a prisoner’s dilemma (7).

The selection gradient for the continuous snowdrift game in structured populations, Eq. 3, is given by

$$D_{\text{db}}\left(x\right)=w\frac{k-2}{k\left(k-1\right)}\left(\left(k+1\right)B^{\prime }\left(2x\right)-k{C}^{\prime }\left(x\right)\right).$$ [10]

Thus, singular strategies $x^{*}$ may exist, and, if $x^{*}$ exists, it is convergence stable if

$$\begin{array}{ll}\hfill {CS}_{\text{db}}\left(x^{*}\right)& =w\frac{k-2}{k\left(k-1\right)}\times \hfill \\ \hfill & \hspace{1em}\left(2\left(k+1\right)B^{\prime \prime }\left(2x^{*}\right)-k{C}^{\prime \prime }\left(x^{*}\right)\right)<0,\hfill \end{array}$$ [11]

and evolutionarily stable if

$$\begin{array}{ll}\hfill {ES}_{\text{db}}\left(x^{*}\right)& =w\frac{k-2}{k^2\left(k-1\right)}\times \hfill \\ \hfill & \left(\left(k+2\right)\left(k+1\right)B^{\prime \prime }\left(2x^{*}\right)-k^2{C}^{\prime \prime }\left(x^{*}\right)\right)<0.\hfill \end{array}$$ [12]

In particular, the conditions for convergence and evolutionary stability are different, which indicates the potential for evolutionary branching and hence the evolutionary emergence and coexistence of high-investing cooperators and low-investing defectors. The above conditions are the spatial analogs of the analysis for well-mixed continuous snowdrift games reported in ref. 13. Also note that, for linear cost and benefit functions, it is not possible to satisfy all constraints for the continuous snowdrift game, and hence the simplest case is given by quadratic costs and benefits.

Quadratic Costs and Benefits.

For suitable cost and benefit functions, the (spatial) adaptive dynamics is analytically accessible. Here we focus on the quadratic cost and benefit functions used in ref. 13 for the well-mixed case: $B\left(x\right)=b_{1}x+b_2{x}^2$ and $C\left(x\right)=c_{1}x+c_2{x}^2$. We assume that the evolving trait is confined to the interval $\left[\mathrm{0,1}\right]$. The benefit and cost functions satisfy the criteria $B\left(0\right)=C\left(0\right)=0$, $B\left(x\right),C\left(x\right)\ge 0$, and $B^{\prime \prime }\left(x\right)<0$ for $x\in \left[\mathrm{0,1}\right]$ provided that $b_1,c_1>0$, $b_2,c_2<0$, $b_1<-4b_2$, and $c_1<-2c_2$ hold. The first two conditions ensure that costs and benefits are increasing yet saturating, while the latter two ensure that $B\left(2x\right)$ and $C\left(x\right)$ are increasing over the entire trait interval; that is, their respective maximum occurs for $x\ge 1$.

For small resident values $x$, the selection gradient Eq. 10 becomes $D_{db}\left(x\right)\approx w\left(k-2\right)\left(\left(k+1\right)b_1-k{c}_1\right)/\left(k\left(k-1\right)\right)$, and hence cooperative investments evolve away from zero provided that $b_1>k{c}_1/\left(k+1\right)$, which is slightly less restrictive than the corresponding condition $b_1>c_1$ in the well-mixed case (13).

For the above quadratic costs and benefits, the singular strategy is given by

$$x_{\text{db}}^{*}=\frac{b_1\left(k+1\right)-c_{1}k}{2c_{2}k-4b_2\left(k+1\right)}.$$ [13]

The numerator of Eq. 13 is positive if and only if the condition for evolution away from zero, $b_1>k{c}_1/\left(k+1\right)$, is satisfied. Similarly, the denominator of Eq. 13 is positive if and only if $2b_2\left(k+1\right)<c_{2}k$ (recall $b_2,c_2<0$). It is clear from Eq. 11 that this is also the condition for convergence stability. Thus, if a singular point, $x^{*}$, exists, cooperative traits evolve away from zero and $x^{*}$ is convergence stable, or the trait cannot evolve away from zero and $x^{*}$ is a repellor.

We note that, if the singular point exists, $x^{*}$, it is shifted toward smaller investments for a given set of parameters, as compared to well-mixed populations with the same parameters: $x^{*}=\left(b_1-c_1\right)/\left(2c_2-4b_2\right)$ (13). Furthermore, the condition for convergence stability, $2b_2\left(k+1\right)<c_{2}k$, is less restrictive than the corresponding condition $2b_2<c_2$ in the well-mixed case. Finally, from Eq. 12, we see that the condition for evolutionary stability is $b_2\left(k+2\right)\left(k+1\right)<c_2{k}^2$, which is again less restrictive than the corresponding condition $b_2<c_2$ in the well-mixed case. Combining the two stability conditions shows that evolutionary branching occurs for $2b_2\left(k+1\right)/k<c_2<b_2\left(k+1\right)\left(k+2\right)/k^2$. Thus, the analytical approach based on pair approximation suggests that population structures tend to inhibit evolutionary diversification by decreasing the range of parameters for which the singular point is an evolutionary branching point (see Fig. 5 A and C).

Fig. 5.

Spatial modes of diversification in the continuous snowdrift game with quadratic benefit and cost functions [see ref. 29 for interactive online simulations (30)]. A–D depict PIPs for the four scenarios illustrating increased spatial diversification due to strong selection (each panel refers to a parameter combination marked a–d in Fig. 4). In all cases, the width of the region of disadvantageous mutants decreases with selection strength (gray for $w=1$; black for $w=100$). (A) The PIP suggests gradual evolution toward minimal investments, except for smaller resident traits, where not only lower-investing mutants can invade but also those making markedly higher investments. (B) Higher-investing mutants can always invade, but so can those investing markedly less. (C) Selection strength distorts the PIP in the vicinity of the convergence and evolutionarily stable $x*$, resulting in a degenerate form of branching (compare Fig. 3). (D) The PIP indicates that $x*$ is a repellor such that residents with $x<x*$ are invaded by lower investors, while those with $x>x*$ are invaded by higher investors. However, as a consequence of strong selection, mutants with markedly higher (lower) investments can also invade. E–H depict corresponding plots of regions of mutual invasibility (PIP², white regions). Regions where mutants or residents are unable to invade (gray) are marked with $\left(+,-\right)$ and $\left(-,+\right)$, respectively. The vector field shows the divergence (see SI Appendix, section S2 for details) and indicates the direction of selection for two coexisting residents based on analytical approximations of the spatial invasion dynamics. In all cases, divergence drives the traits away from the diagonal and hence preserves diversity. Parameters are as follows: $b_2=-1/4,c_1=2$; (A) $b_1=1.55$, $c_2=-0.6$; (B) $b_1=1.65$, $c_2=-0.625$; (C) $b_1=1.9$, $c_2=-0.3$; (D) $b_1=1.5$, $c_2=-0.72$.

In order to explore the evolutionary dynamics more specifically and for a range of parameters, it is convenient to fix two parameters, say $b_2$ and $c_1$, which then leaves a range for the other two parameters (subject to the constraints listed above, i.e., $0<b_1<-4b_2$ and $0>c_2>-c_1/2$). Strictly speaking, $b_1<c_1$ violates one of the assumptions of the continuous snowdrift game at least for small $x$, namely, that $B\left(x\right)>C\left(x\right)$. It means that, for small $x$, lower investors dominate and investments are expected to dwindle and disappear, which actually recovers the continuous prisoner’s dilemma scenario, and cooperative investments cannot take off because $D\left(x\right)<0$. In this sense, the parameter space depicted in Fig. 4 encompasses the gist of the continuous analogs of both the prisoner’s dilemma and the snowdrift game.

Fig. 4.

Continuous snowdrift game with quadratic benefit and cost functions, $B\left(x\right)=b_{1}x+b_2{x}^2$ and $C\left(x\right)=c_{1}x+c_2{x}^2$ [see ref. 29 for interactive online simulations (30)]. Evolutionary outcomes are shown as a function of the benefit parameter $b_1$ and cost parameter $c_2$ with $b_2=-1/4$ and $c_1=2$. Note that $b_1<2$ violates the assumption $B\left(x\right)>C\left(x\right)$, at least for small $x$, and hence effectively mimics the characteristics of the prisoner’s dilemma. (A) Analytical predictions based on adaptive dynamics in well-mixed populations and (B) results from individual-based simulations for populations with $N=10^4$ individuals. (C) Analytical predictions based on spatial adaptive dynamics and complementing individual-based simulations on $100\times 100$ lattices for (D) moderate selection, $w=10$, and (E) strong selection, $w=100$. In lattice populations, the parameter region admitting singular strategies is shifted to smaller values both of $b_1$ and of $c_2$, and the size of the region admitting evolutionary branching is markedly smaller than in well-mixed populations (A and C). Interestingly, spatial adaptive dynamics predicts branching only for parameters where defection dominates in well-mixed populations, $b_1<c_1$, mimicking the continuous prisoner’s dilemma. For weak to moderate selection, predictions by adaptive dynamics (A and C) are in good agreement with results from individual-based simulations (B and D), where equilibrium investment levels range from minimum (black) to intermediate (gray) and maximum (white) augmented by convergence stability (red) and evolutionary instability (blue), with the overlapping region indicating evolutionary branching (maroon) in adaptive dynamics and diversification in simulations. For strong selection (E), striking differences arise, with a much increased region of diversification. The points labeled “a” through “d” indicate the parameter settings for the invasion analysis in Fig. 5. Note that the automated classification of investment distributions becomes more difficult whenever the singular investment $x*$ is close to zero or one (for details, see SI Appendix, section S2).

This results in five salient regimes for the evolutionary dynamics (13): If $x^{*}$ does not exist, investments evolve either to their 1) maximum or 2) minimum, depending on the sign of the selection gradient. The latter case recovers the essence of the continuous prisoner’s dilemma and arises if $b_1<c_1$ (at least for small $x$). If $x^{*}$ exists but is not convergence stable, it is 3) a repellor, and the evolutionary outcome depends on the initial investment, $x_0$, such that, for $x_0>x^{*}$, investments evolve to the maximum, and, for $x_0<x^{*}$, investments evolve to zero. Conversely, if $x^{*}$ is both convergence stable and evolutionarily stable, it is 4) an attractor of stable intermediate investments, which represent the evolutionary end state. Finally, if $x^{*}$ is convergence stable but not evolutionarily stable, it is 5) an evolutionary branching point and hence a potential starting point for evolutionary diversification into coexisting high and low investors.

Fig. 4 compares the different dynamical regimes for well-mixed and structured populations in terms of the parameters $b_1$ and $c_2$ based on analytical predictions derived from adaptive dynamics in well-mixed populations and for structured populations based on pair approximation as well as individual-based simulations (see SI Appendix, section S2 for simulation details). Interestingly, due to the shift of the singular point $x^{*}$ toward smaller investments as compared to well-mixed populations, branching now only occurs for parameters that result in negative selection gradients in well-mixed populations (i.e., drive investments to zero, at least for small $x$) and hence qualitatively mimic the prisoner’s dilemma.

For weaker selection strengths, predictions based on (spatial) adaptive dynamics and individual-based simulations turn out to be in surprisingly good agreement given that the effects of space are reduced to mere pair correlations (Fig. 4 B and D), while significant differences arise for stronger selection (Fig. 4E). In particular, the range of parameters leading to diversification is increased in individual-based models compared to the predictions based on pair approximation. This effect is particularly pronounced when selection is strong (Fig. 4E). With strong selection, diversification expands into parameter regions that result in monomorphic traits at minimal, maximal, or stable intermediate investments, as well as into regions of bistability (i.e., with $x^{*}$ as a repellor) for weaker selection.

Qualitatively, these discrepancies may not be that surprising, given that pair approximation cannot account for larger-scale spatial structures, such as large clusters of cooperators, or frozen, filamentous regions of defectors (Fig. 2I). Again, the main reason for the discrepancies is finite-size mutations and nonmonomorphic populations in individual-based models, which differ from the assumptions for adaptive dynamics. With finite mutations and finite population variances, unfavorable regions in the invasion fitness landscape can more easily be crossed by mutational steps to reach regions in which coexistence is feasible. This can be seen by using PIPs to depict regions of mutual invasibility between pairs of trait values. In essence, this means that population structures open new avenues for evolutionary diversification where divergent selection is not solely based on evolutionary branching.

Fig. 5 shows PIPs for four different regimes where spatial adaptive dynamics does not predict branching but diversification emerges in individual-based simulations on lattices (compare Fig. 4E with each parameter set marked by the letters a, b, c, and d, respectively).

In Fig. 5A, the selection gradient is negative everywhere, and, in Fig. 5B, it is positive everywhere (indicated by the white regions below and above the diagonal, respectively). In both cases, there is no singular point. However, for small values of the resident in Fig. 5A (large values in Fig. 5B), mutants with sufficiently larger (smaller) investment levels than the resident can invade and coexist with the resident. Regions of coexistence are characterized by the fact that mutants can invade residents and vice versa. Graphically, this can be captured by mutual invasibility plots, PIP²; see Fig. 5 E and F. Moreover, for persistent coexistence, evolution must drive divergence of the coexisting branches (as indicated by the vector field in Fig. 5 E and F). For strong selection, the impermissible regions shrink (gray and black areas in Fig. 5). In individual-based models with sufficiently strong selection, coexisting trait pairs eventually occur due to finite-size mutations. Divergence then results in persistent evolutionary diversification into distinct trait groups of high and low investors.

In Fig. 5 C and D, singular points exist and are evolutionarily stable or a repellor, respectively. In both cases, many mutant−resident pairings in the vicinity of the singular point can coexist. As a result, evolutionary stability is marginal in the former case, and, in the latter case, the region of impermissible mutants around the diagonal is narrow. The effect is again more pronounced for strong selection. In either case, finite-size mutations in individual-based models can easily lead to coexistence of diverging phenotypic branches (Fig. 5 G and H), and hence result in evolutionary diversification.

Discussion

In all scenarios considered here, we find that population structures can promote and facilitate spontaneous diversification in social dilemmas into high and low investors, especially when selection is strong. However, at the same time, classical evolutionary branching tends to be inhibited, but compensated for by other modes of diversification. We derive an extension of adaptive dynamics for continuous games in (spatially) structured populations based on pair approximation, which tracks the frequencies of mutant−resident pairs during invasion. It turns out that predictions derived from this spatial adaptive dynamics framework are independent of selection strength. More precisely, selection strength only scales the magnitude of the selection gradient as well as that of convergence and evolutionary stability but neither affects the location of singular strategies nor their stability. Nevertheless, from the invasion analysis of mutant traits $y$ into resident populations $x$, as well as from individual-based simulations, it is clear that selection strength has a significant impact on the dynamics in general, and on diversification in particular. During the invasion process, the local configuration probability of mutant pairs $q_{m|m}$ is of crucial importance and changes with selection strength; see Eq. 2. However, in the adaptive dynamics limit $y\to x$, the pair configuration probability reduces to a constant $q_{m|m}\to 1/\left(k-1\right)$. Even though this is the same value as obtained in the limit of weak selection, $w\to 0$, it is important to note that the limit $y\to x$ does not necessarily imply weak selection. Thus, the different modes of diversification cannot be understood based on spatial adaptive dynamics alone but require a more comprehensive analysis of invasion dynamics.

Structured populations offer new modes of diversification that are driven by an interplay of finite-size mutations and population structures. Trait variation is more easily maintained in structured populations, due to the slower spreading of advantageous traits as compared to well-mixed populations. Spatial adaptive dynamics is unable to capture these new modes of diversification because of the underlying assumption that the resident population is composed of discrete traits (monomorphic before branching and dimorphic or polymorphic after branching). Nevertheless, invasion analysis and PIPs, in particular, provide an intuitive interpretation for additional pathways to diversification that are introduced by spatial structures and further promoted for increasing selection strength. Even in the vicinity of evolutionarily stable investments, trait combinations exist that admit mutual invasion and hence can coexist. However, such states cannot evolve through adaptive dynamics but are nevertheless accessible to trait distributions around the evolutionarily stable trait and can drive a degenerate form of branching. Moreover, spatial invasion fitness can open up new regions in trait space for mutant invasion. However, those regions need not be accessible by small mutational steps, and instead require stochastically appearing larger mutations or sequences of smaller mutations that allow bridging of regions of unfavorable traits.

Previous attempts at amalgamating adaptive dynamics and spatial structure have not observed spontaneous diversification or evolutionary branching. In particular, Allen et al. (21) augment adaptive dynamics by structure coefficients (33), which restrict the analysis to weak selection. Moreover, their framework is fundamentally different from ours because it is based on fixation probabilities rather than invasion fitness. More specifically, their analysis is based on the fixation probabilities $p_x$ and $p_y$ of two types $x$ and $y$ in a population consisting of $x$ and $y$: $x$ is favored over $y$ if $p_x>p_y$ (see equation 4 and formulas following equation 7 in ref. 21). Clearly, it is then impossible to have both types be favored in an $\left(x,y\right)$ population, but this is a fundamental requirement for mutual coexistence. In fact, Allen et al. acknowledge that fixation probabilities are “less appropriate” for studying evolutionary branching and that, instead, “establishment probabilities” should be considered for studying diversification. As a consequence, their adaptive dynamics derivation precludes its most intriguing feature, that of evolutionary branching. However, in the limit of weak selection, the two approaches are compatible in some aspects. For example, the selection gradient and the condition for convergence stability reported for the continuous prisoner’s dilemma are identical (compare our Eqs. 10 and 11 and Allen et al.’s equations 11 and 13). In our framework, the use of invasion fitness allows determination of evolutionary stability, and the corresponding PIP reveal the potential for degenerate forms of branching—but only for stronger selection (Fig. 3). Another attempt at spatial adaptive dynamics considers spatial structure in the form of demes (34). However, quite naturally, such deme structures suppress branching, because, in the long run, it is impossible to maintain multiple coexisting traits in small demes, which quickly become homogeneous. Instead, branching is only possible if demes are large enough to support branching in every single one.

Interestingly, in well-mixed populations, evolutionary branching is only observed for the continuous snowdrift game, where two distinct traits of high and low investors can coexist and essentially engage in a classical (discrete) snowdrift game. In contrast, in structured populations with death−birth updating, evolutionary branching is only observed for prisoner’s dilemma−type interactions where lower investments invariably dominate higher ones, which applies both in the continuous prisoner’s dilemma and the continuous snowdrift game with sufficiently high costs. The reason for this surprising difference can be understood intuitively by considering the preferred spatial configurations in the two classical (discrete) games: In the prisoner’s dilemma, cooperators form compact clusters to reduce exploitation by defectors (minimize surface), while, in the snowdrift game, filament-like clusters form because it is advantageous to adopt a strategy that is different from that of the interaction partners (maximize surface). In the continuous variants of those games, it is naturally much harder to maintain and spread distinct traits in fragmented filament-like structures because they are more prone to effects of noise than compact clusters. Effectively, this fragmentation inhibits evolutionary branching, because diverging traits tend to trigger further fragmentation and, as a consequence, do not survive long enough to get established and form their own branch. In contrast, the compact clusters promoted by the prisoner’s dilemma provide structural protection for higher investors and thus help drive diversification.

Because of global competition, the spatial dynamics for birth−death updating is (unsurprisingly) much closer to results for well-mixed populations. For example, evolutionary branching was again only observed for continuous snowdrift game. Also, because of global competition, structured populations are updated in a nonuniform manner. In particular, regions of high payoffs experience a much higher turnover than regions of low payoffs. For strong selection, this can result in almost frozen parts of the population. As a consequence, unsuccessful traits are able to stay around for long times, and, in some cases, those traits turn out to be advantageous again at later times when the surroundings have sufficiently changed, so that the stragglers then contribute to diversification. This mode of diversification, however, introduces historical contingencies where the evolutionary end state can sensitively depend on the initial configuration.

Overall, we find that evolutionary diversification is a robust feature of continuous spatial games, and that spatial structure can sometimes hinder, but generally promote, diversification through modes of diversification that complement traditional evolutionary branching.

Data Availability

The source code for the individual-based simulations is available at GitHub, https://github.com/evoludolab/IBS-DSCG.