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Cooperation dilemma in finite populations under fluctuating environments

We present a novel approach allowing the study of rare events like fixation under fluctuating environments, modeled as extrinsic noise, in evolutionary processes characterized by the dominance of one species. Our treatment consists of mapping the system onto an auxiliary model, exhibiting metastable species coexistence, that can be analyzed semiclassically. This approach enables us to study the interplay between extrinsic and demographic noise on the statistics of interest. We illustrate our theory by considering the paradigmatic prisoner's dilemma game whose evolution is described by the probability that cooperators fixate the population and replace all defectors. We analytically and numerically demonstrate that extrinsic noise may drastically enhance the cooperation fixation probability and even change its functional dependence on the population size. These results, which generalize earlier works in population genetics, indicate that extrinsic noise may help sustain and promote a much higher level of cooperation than static settings.

We present a novel approach allowing the accurate study of rare events in a broad class of evolutionary processes under fluctuating environments. Our approach consists of mapping these processes, not necessarily displaying metastability, onto auxiliary models characterized by metastability that can be analyzed semiclassically. This enables us to study the interplay between extrinsic (environmental) noise and intrinsic (demographic) fluctuations on the statistics of interest. We illustrate our theory for the paradigmatic prisoner's dilemma game whose evolution is described by the probability that cooperators take over and fixate the entire population. We demonstrate both analytically and by using Monte Carlo simulations that extrinsic noise may drastically enhance such a fixation probability. In particular, we show that under strong, short-correlated extrinsic noise, the cooperation fixation probability decays algebraically with the population size, rather than exponentially. Finally, we show how our approach generalizes earlier works, notably in population genetics. Understanding the origin of cooperative behavior and how it is influenced by the population's intrinsic properties and by environmental factors are major scientific puzzles [1] that are suitably investigated in the framework of evolutionary game theory (EGT) [2]. In EGT, successful species spread at the expense of others, and the optimization of the reproductive potential (fitness) at an individual level can result in the reduction of the population's overall fitness, a phenomenon metaphorically captured by the prisoner's dilemma (PD) game [1,2]. While in EGT the dynamics is traditionally studied in terms of differential equations, demographic fluctuations -intrinsic noise (IN) -are known to affect the evolution in finite populations. In this case, the dynamics is often described by a Markov chain and characterized by the fixation probability of a given trait, which is the probability that the trait invades the entire population [3]. For the classic PD (with IN only), the cooperation fixation probability (CFP) vanishes exponentially with the population size, see e.g. [4], and defection prevails leading to a cooperation dilemma. This prediction, at odds with many observations, has motivated the investigation of various mechanisms that can promote cooperation [5].
In addition to IN, there often is also environmental (extrinsic) noise (EN) which results in random fluctuations of the interaction parameters. In theoretical population genetics [3,[6][7][8] and ecology [9][10][11], it has been shown that the combined effect of IN and EN can significantly affect the extinction properties of a population. While many earlier works focused on systems exhibiting metastability and/or short-correlated EN [3,[6][7][8][9][10], here we present a novel approach allowing one to analyze the combined influence of IN and EN, with arbitrary correlation time and magnitude, on a broad class of systems not necessarily displaying metastability. This is done by mapping the system at hand onto one possessing a long-lived metastable state and treating the latter semiclassically. We illustrate our approach with the paradigmatic PD game, and find that EN drastically enhances the CFP and may even alter its functional dependence on the population size from an exponential to a power law.
The paradigm of social dilemma is provided by the classic PD, whose main features are captured by assuming that the pairwise interaction between cooperators and defectors is described in terms of the benefit b and cost c of cooperation, with b > c > 0 [2]. Here, mutual cooperation leads to a payoff b−c > 0 and mutual defection gives a payoff 0, while when one player defects and the other cooperates, the former gets a payoff b and the latter gets −c. The quantity r ≡ c/b is the cost-to-benefit ratio [2] and the dilemma arises from the fact that each individual is better off defecting, even though r < 1 and mutual cooperation enhances the population overall payoff.
Here, we consider a finite population comprising N individuals on a complete graph with n cooperators and N − n defectors. At mean field level (N → ∞), defection prevails and the fraction x ≡ n/N of cooperators in the population evolves to extinction according to the replicator rate equation Here, Π C = bx−c and Π D = bx are the cooperator and defector average payoffs, respectively [2], and we henceforth assume that b, c = O(1).
When the population size is finite, fluctuations alter the mean field predictions and the dynamics is usually where the selection intensity s > 0 has been intro-arXiv:1305.6580v1 [q-bio.PE] 28 May 2013 duced and 1 accounts for a baseline fitness contribution [2,13,14], while the population average fitness is In a finite population, fluctuations always drive the system to either the absorbing states n = 0 or n = N , and the noisy dynamics is described by the corresponding master equation [15]. For weak selection such that N s 2 1, the latter can be approximated [14,16] by a Fokker-Planck equation (FPE) for the probability P (x, t) of having cooperator density x at time t [3,17]: [2,17].
An important notion to characterize evolutionary dynamics is the fixation probability φ C (x 0 ) that cooperation prevails starting from an initial fraction x 0 of cooperators. In the absence of environmental fluctuations, φ C can be calculated exactly [4,15]. Yet, it is useful to proceed with an asymptotic calculation (N 1) of φ C (x 0 ) using an auxiliary problem. For this, we consider the modified model obtained by supplementing the original PD model with a reflecting boundary at n 0 = N x 0 by imposing T − n=n0 = 0. Hence, the only absorbing state of the modified model is the state n = N and as a result, sincė x < 0, a quasi-stationary distribution (QSD) peaked at x 0 (for any value of x 0 ) forms after a relaxation time of order O(s −1 ). This QSD slowly decays due to a slow leakage of probability into the absorbing state at x = 1, with a rate given by the inverse of the cooperation mean fixation time (MFT). Therefore, after a long time growing exponentially with N , cooperators eventually take over. Importantly, we now demonstrate that φ C (x 0 ) in the original PD problem coincides to leading order with the inverse of the MFT in the modified problem.
The MFT in the modified problem can be computed by employing the metastable ansatz P (x, t) π(x)e −t/τ , where π(x) is the QSD and τ is the MFT. Then, Substituting the WKB ansatz π(x) ∼ e −N S(x) in Eq. (2) yields a stationary Hamilton-Jacobi equation [18,19]. Fixation occurs along the zero-energy trajectory ("optimal path") , which yields, for N s 1 [18,20] Therefore, in the original PD model (no reflecting boundary), the CFP is given by the inverse of Eq. (1−x0) , which coincides with the exact result [4,17].
We now add EN into the dynamics in the form of a fluctuating selection intensity in (1), s → s(t) = s 0 + ξ(t), where ξ is an Ornstein-Uhlenbeck (OU) process with mean zero, variance ξ(t)ξ(t ) = σ 2 e −|t−t |/τc and correlation time τ c > 0 [15]. The EN magnitude is assumed to satisfy σ s 0 1 so that s(t) can become negative when σ = O(s 0 ). The OU's Langevin equation (LE) readṡ where η 1 (t) is a white Gaussian noise with η 1 (t) = 0 and η 1 (t)η 1 (t ) = δ(t − t ) [21]. To account for the joint effects of IN and EN, Eq. (4) is coupled with the LĖ where η 2 (t) is also white Gaussian noise, and A(x, ξ) = T + (x, ξ) − T − (x, ξ) and B(x, ξ) = T + (x, ξ) + T − (x, ξ) read for s 1: We now proceed as in the absence of EN and compute the MFT τ of the modified PD model supplemented with a reflecting boundary at x 0 . As with only IN, for N s 0 1, φ C (x 0 ) and τ −1 in the original and modified models exhibit the same asymptotic dependence also in the presence of EN. Indeed, see Fig. 1, numerical simulations [22] with various EN strengths indicate that φ C (x 0 ) = ατ −1 , where α varies slowly with the model parameters.
The description in terms of the coupled LEs (4,5) is equivalent to the following bivariate FPE for the probability P (x, ξ, t) to find cooperator density x and selection strength s = s 0 + ξ at time t: where we have defined V ≡ N σ 2 1. For N 1, we can use the semi-classical ansatz for the QSD π(x, ξ) ∼ e −N S(x,ξ) in Eq. (7), which yields the HJE H(x, ξ, p x , p ξ ) = 0 with the Hamiltonian H = p x A(x, ξ) − p ξ ξ/τ c + (p 2 x /2)B(x, ξ) + (V /τ c )p 2 ξ , (8) where we have defined the momenta p x = ∂ x S and p ξ = ∂ ξ S. The optimal path to fixation satisfies the HJE whose solutions obey the Hamilton equations (HE) where the last line has been obtained by combining the HE forξ andṗ ξ into a single equation and by keeping terms up to O(p x ) = O(s), see below. The solution to the HJE for generic EN (with arbitrary τ c ) is found by solving numerically (9), yielding the action function S(x, ξ) = p x (x, ξ)dx + p ξ (x, ξ)dξ [23, 24]. Analytical progress can be done in two important limits: short-correlated (white) and long-correlated (adiabatic) EN.
In the limit of short-correlated EN characterized by τ c s −1 0 ,ξ is negligible in the third of Eqs. (9) [25] yielding the effective noise strength ξ ξ eff −2cV τ c p x x(1− x) [10,26,27]. Below we find p x > 0, implying that ξ eff < 0. As a result, the EN is exploited to enhance the CFP by decreasing the selection strength.
We now assume the noise strength satisfies τ c V s 0 , in which case the EN markedly affects the dynamics, see below. Substituting ξ eff into the first of Eqs. (9) one findṡ The corresponding effective "white-noise" Hamiltonian is Thus, the CFP of the original PD model is given by the accumulated action along p x = cs 0 /(1 + c 2 τ c V x(1 − x)): where γ = 1 + 4/(c 2 τ c V ). In Fig. 2 we compare (10) with stochastic simulations [22] as a function of the relative EN strength σ/s 0 and find a very good agreement. The most striking effect of the fluctuating environment appears in the limit of strong EN where γ → 1 [28]. In such a limit, the dependence of φ C on N is no longer exponential but decays as a pure power-law: when τ c V 1 and x 0 > 0 in (10), one indeed finds . This result is confirmed by stochastic simulations, see Fig. 3. The behavior of (10) in the limit x 0 1 (small initial density of C's) is also particularly relevant and yields  In the strong EN limit (τ c V 1, γ 1), Eq. (11) leads to the power-law φ C N (σc) 2 τ c −2(s0/σ 2 )/(cτc) .
In the opposite long-correlated (adiabatic) EN limit, ξ(t) varies slowly and we can integrate over the fixation probability φ C (ξ 0 ) ∼ e −cN (s0+ξ0)(1−x0) with the Gaussian weight e −ξ 2 0 /(2σ 2 ) for the noise ξ 0 . Within a saddle-point approximation, we find an optimal noise magnitude at This result is valid when | ln φ C | 1 which requires V < s 0 , i.e. not too strong EN. Eq. (12) shows that adiabatic EN can also significantly increase φ C , but the dependence on N remains exponential. A different scenario arises when there is strong adiabatic EN. In this case the IN can be neglected and the fixation of C's is solely governed by EN and thus the OU process (4) [27].
For completeness, we now briefly illustrate how the results obtained for fluctuating s can be directly used to study the case of fluctuating cost-to-benefit ratio r = c/b due to EN, according to r → r(t) = r 0 +ξ(t). Here r 0 < 1 with r 0 ∼ O(1), and ξ is the same OU process as in (4) with ξ(t)ξ(t ) = σ 2 r e −|t−t |/τc , where V r ≡ N σ 2 r . In addition we assume σ r r 0 to guarantee 0 < r(t) < 1, and that b is fixed so that c(t) = br(t) fluctuates.
The dynamics is thus described by Eqs. (4) and (5) . As above, in the limit of weak selection N s 2 1 and shortcorrelated EN τ c s −1 , one finds the effective value for the EN from (9) . Substituting ξ ξ eff in the first HE (9) leads to an effective Hamiltonian whose accumulated action gives the CFP As before, we have assumed that the EN is sufficiently strong so that V r τ c s −1 [29]. Eq. (13) can be treated in the same manner as (10) and when EN is strong enough it also predicts a power-law dependence of φ C on N [24].
Our approach generalizes earlier works in population genetics where the combined role of IN and EN was investigated by considering a fluctuating selection intensity, see e.g. [3,[6][7][8]. In these studies the dynamics was implemented with the Wright-Fisher model (WFM) with discrete time and non-overlapping generations [3]. In such a setting, a diffusion theory was devised in the weak selection limit to account for IN and time-uncorrelated (white) EN by averaging separately on the two sources of noise [3,[6][7][8]. When N σ 2 N s 2 0 1, the results of this approach coincide with (10) where τ c = 1 and N → N/2 [30]. Yet, our approach [Eqs. (9)] is more general and not limited to white EN, since it allows to study EN with arbitrary correlation time [24].
We have studied the effect of environmental noise (EN) in a broad class of evolutionary processes not necessarily displaying metastability. Our approach relies on a suitable mapping onto a system exhibiting metastability and on its semiclassical treatment. Using the paradigmatic prisoner's dilemma as an example, we have demonstrated how EN can drastically enhance the cooperators fixation probability (CFP). In particular, we have shown that for strong short-correlated EN the CFP decays algebraically (instead of exponentially) with the population size, as confirmed by stochastic simulations. Our results may provide further insight for resolving the contradiction between the observed examples of cooperation and predictions of many theoretical models.
gle trajectory, the current number of cooperators n is stochastically updated using the birth/death transition propensities (rates) described in the text. EN is added by permitting the selection intensity parameter s to fluctuate. A pseudo-reaction fires at intervals much less than the EN correlation time and s is updated as if it had been following an OU process satisfying Eq. (4) using the method of [32]. At each update both the birth and death transition propensities are recalculated using the new value of s and the next reaction time is adjusted to account for the change. To calculate φ C , many trajectories starting at n0 are run to fixation and the fraction in which cooperation results is calculated directly. For the MFT calculation, a reflecting boundary is placed at n0 and the time until the population fixates in the cooperation state is tracked. The mean time for many (≥1000) trajectories to reach fixation is then calculated. [25] Using the expression of ξ eff together with Eqs. (6) and (9), one a posteriori finds thatξ = O(V τcs 3 0 ) V s0/τc which is indeed negligible in the third equation of (9).
[28] The expression (10) is well-defined also when V → 0 (γ → 1) and x0 → 0, as can be seen by rewriting [29] Since Vrs 2 and V scale in the same manner in s, in the case of fluctuating r the EN is considered strong when Vrτc s −1 and weak when Vrτc s −1 .
[30] As time is discrete in the WFM and the EN between two successive generations is uncorrelated, the correspondence requires to set τc = 1 into (10). Furthermore, a population of N individuals in the WFM maps onto a population of size N/2 with the MM [3,14].