Modeling Natural Selection at Multiple Levels of Organization

By Daniel B. Cooney, Simon A. Levin, Yoichiro Mori, and Joshua B. Plotkin 

Natural selection in complex biological and social systems can simultaneously operate across multiple levels of organization, ranging from genes and cells to animal groups and complex human societies. A common aspect of these scenarios is an evolutionary tug-of-war in which a trait or behavior that benefits a single individual may detrimentally affect the group to which the individual belongs (and vice versa). Such conflicts arise in the supply of common goods—like the production of diffusible metabolic enzymes in yeast populations [15] and collective hunting in animal groups [2]—as well as the evolution of virulence, during which pathogens that spread rapidly within a host’s cells may decrease the opportunity for onward transmission to subsequent hosts [10, 12]. One can also fundamentally think of cancer as a problem of multilevel selection; tumor cells benefit in the short term from rapid replication at the detriment of the organism’s long-term health [1]. This tension between the interests of individuals and groups features heavily in the study of major evolutionary transitions, during which new levels of selection arise through innovations in biological complexity (such as the emergence of multicellular life and the evolution of language) [19].

To investigate the evolutionary competition between the interests of individuals and groups, we use game theory to formulate the incentives of cheating and cooperative behaviors. We consider a game in which individuals pair up and play one of two possible strategies: they can cooperate and pay a cost to confer a benefit to fellow group members or defect, pay no cost, and confer no benefit. This choice gives rise to a social dilemma; defectors receive a higher individual payoff than cooperators, but groups with many cooperators have higher average payoffs than groups with many defectors.

Models of Multilevel Selection

We generate a model of multilevel selection to explore the tension between the individual incentive to defect and the collective incentive to cooperate. By generalizing the work of Shishi Luo and Jonathan C. Mattingly, we consider a nested birth-death process in which reproductive competition occurs among both individuals and groups [13, 14]. Individuals produce copies of themselves at a rate that depends on their personal payoff from social interactions, and the offspring individual replaces a randomly chosen group member. Groups also produce copies of themselves at a rate that is proportional to the average payoff of their members, and the offspring group replaces a randomly chosen group in the population. One may interpret this group-level reproduction as a literal reproduction event in the context of cell division [8], a collective viral transmission [16], or the occurrence of one group imitating the strategy composition of another in the context of social or cultural evolution [11]. Figure 1 depicts individual- and group-level reproduction events. The fraction of defectors within groups tends to increase in frequency due to the individual-level advantage of defection, while the fraction of groups that feature many cooperators may also increase in the population due to the collective advantage conferred by cooperators under group-level reproduction or imitation events.

Figure 1. Schematic depictions of sample birth-death events that are taking place at the level of individuals (1a) and groups (1b). The populations of three groups (large black circles), are composed of three individuals who are either cooperators (small blue circles) or defectors (small red circles). The urns represent the possible numbers of cooperators per group (between zero and three), and each ball in the urns describes the composition of a single group. 1a depicts the birth of a defector and death of a cooperator in the leftmost group and 1b depicts the reproduction of a two-cooperator group and the death of an all-defector group. Figure adapted from [5].

For the limit wherein both the size and number of groups become infinite, we can describe the composition of strategies across the population through the probability density \(f(t,x)\) of groups that feature fractions that comprise \(x\) cooperators and \(1-x\) defectors at time \(t\). Starting from the stochastic process in Figure 1, we are able to show that the density \(f(t,x)\) in this large population limit evolves according to the following partial differential equation (PDE):

\[\frac{\partial f(t,x)}{\partial t} = - \overbrace{  \frac{\partial}{\partial x} \left( x(1-x) ( \pi_C(x) - \pi_D(x) )  f(t,x) \right)}^{\text{Within-Group Competition}}  + \lambda \underbrace{ f(t,x) \left[ G(x)  - \int_0^1 G(y) f(t,y) dy  \right]}_{\text{Between-Group Competition}},\tag1\]

Here, \(\pi_C(x)\) and \(\pi_D(x)\) illustrate the individual-level birth rates of cooperators and defectors in a group with \(x\) cooperators, \(G(x)\) is the collective reproduction rate of an \(x\)-cooperator group, and \(\lambda\) represents the relative strength of selection at the two levels [3, 4, 6]. The first term describes the way in which individual-level competition favors an increase of defectors within groups (as \(\pi_D(x) > \pi_C(x)\)), and the second term promotes group compositions \(x\) for which \(G(x)\) exceeds the collective reproduction rate of the population \(\int_0^1 G(y) f(t,y) dy\). 

One of our main goals is to analyze the long-time behavior of \((1)\) in response to the strength of selection on group-level versus individual reproduction, as quantified by \(\lambda\). Analyzing this problem allows us to explore the  between-group competition’s impact on the ability to promote cooperation within groups. The long-time dynamics of \((1)\) have many possible solutions that depend upon the initial strategy distribution, whose key property is a feature of its tail near \(x=1\) called the Hölder exponent.

We define the Hölder exponent \(\theta\) as the infimum of \(\Theta \geq 0\) for which 

\[\lim_{x \to 0} x^{-\Theta} \int_{1-x}^1 f(t,y) dy > 0;\]

the dynamics of \((1)\) preserve this quantity. The family of densities \(f_{\theta}(x) = \theta (1-x)^{\theta - 1}\) provides an example of densities with Hölder exponent \(\theta\) (including the uniform density \(f_{1} = 1\) for \(\theta =1\)) and illustrates how one may think of the Hölder exponent as an inverse of the concentration of groups near the all-cooperator group.

Given an initial density with Hölder exponent \(\theta\), a critical threshold exists for the relative strength \(\lambda\) of between-group selection that permits the long-run survival of cooperation [6]. This threshold is given by

\[\lambda^* := \frac{( \overbrace{\pi_D(1) - \pi_C(1)}^{\substack{\text{Individual Incentive} \\ \text{to Defect}}}) \theta}{\underbrace{G(1) - G(0)}_{\substack{\text{Group Incentive} \\ \text{to Cooperate} }}}\tag2\]

and highlights the tug-of-war between the collective advantage of all-cooperator groups and the individual advantage of defection in a group with many cooperators. If group-level selection is too weak, \(\lambda < \lambda^*\) and defectors take over the whole population as the density \(f(t,x)\) converges to a delta function at the all-defector group composition. But when \(\lambda > \lambda^*\), \(f(t,x)\) converges to a steady-state density that features positive levels of cooperation (assuming that the limit that defines the Hölder exponent exists). In the case of initial densities for which this limit does not exist, we can still show that cooperation survives in the long-run population in the sense of weak persistence [9] for sufficiently strong between-group competition.

A Shadow of Lower-Level Selection?

Having established that cooperation can survive at steady state for sufficiently strong between-group competition, we now address a follow-up question: What happens in the limit of strong competition between groups? We might hope that cooperation gets restored to the socially-optimal level in each group. But Figure 2 demonstrates that this is not always the case. We plot steady-state densities for multilevel dynamics of scenarios in which the collective reproduction rate \(G(x)\) is maximized by the all-cooperator group (when "many hands make light work," as shown in Figure 2a) or by a group that features 75 percent cooperation (when "too many cooks spoil the broth," as shown in Figure 2b). Interestingly, the former case achieves high levels of steady-state cooperation for relatively weak between-group competition. In the latter case, however, the steady-state density seems to concentrate at suboptimal 50 percent cooperation — even for arbitrarily strong between-group competition.

Figure 2. Steady-state densities that are achieved by the dynamics of \((1)\) with an initial Hölder exponent \(\theta = 2\). Replication rates are from a game-theoretic example [3] with average payoff \(G(x)\) that is maximized by 100 percent cooperation (2a) or by 75 percent cooperation (2b). In 2b, the dashed vertical line on the left describes composition \(\overline{x} = 0.5\) where the steady state density concentrates, while the right dashed vertical line corresponds to optimal composition \(x^* = 0.75\). Figure adapted from [6].

Figure 2b highlights a discrepancy between the group composition that maximizes group-level replication and the composition of the most abundant group type in steady state. For an initial density with Hölder exponent \(\theta\), the long-time collective fitness of the population satisfies

\[\lim_{t \to \infty} \int_0^1 G(y) f(t,y) dy = \left\{ \begin{array}{cr} G(0) & : \lambda < \lambda^* \\ \left(\frac{\lambda^*}{\lambda}\right) G(0)+   \left( 1 - \frac{\lambda^*}{\lambda} \right) G(1) &: \lambda > \lambda^*. \end{array} \right.\tag3\]

This means that the long-time collective outcome cannot exceed the collective fitness of an all-cooperator group. Therefore, if groups are best-off with a mix of cooperators and defectors, no level of between-group competition can achieve this optimal outcome. Even in the limit of infinitely strong between-group competition, the multilevel dynamics cannot necessarily erase the shadow that is cast by the individual-level incentive to defect.

Applications and Future Work

Our PDE formulation and analysis \((1)\) provide a basis for understanding evolutionary dynamics when natural selection operates on both individuals and groups. In ongoing research, we are applying this framework to study the evolutionary transition to cellular life [8] and show that linking genes into chromosomes [7, 18] can help overcome the shadow of lower-level selection. These models are also useful for the study of cooperative dilemmas in human social systems, where group-level imitation serves as a mechanism to maintain cooperative social norms [17] or promote sustainable management of common-pool resources like fisheries [20]. From a mathematical perspective, our work on the deterministic limit of \((1)\) motivates exploration of the finite-population stochastic model in Figure 1 and other models that incorporate richer biological detail and feature different games or different norms of social interaction. We hope that this work provides a jumping-off point for further study of mathematical and biological questions in the context of cross-scale features of evolutionary dynamics.

Acknowledgments: This work was supported by the Simons Foundation through the Math+X grant (DBC, YM); the National Science Foundation through grants DMS-1514606 (DBC), DMS-1907583 (YM), and DMS 2042144 (YM); and the Army Research Office through grant W911NF-18-1-0325 (DBC, SAL). 

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Daniel B. Cooney is a Simons Postdoctoral Fellow in Mathematical Biology at the University of Pennsylvania. His research uses dynamical systems and partial differential equations to study biological evolution and collective behavior in social systems. Simon A. Levin is the James S. McDonnell Distinguished University Professor in Ecology and Evolutionary Biology at Princeton University. His research interests are in theoretical and applied ecology, especially the interface with socio-economic systems. Levin is a SIAM Fellow and a 2014 National Medal of Science recipient. Yoichiro Mori is the Calabi-Simons Professor in Mathematics and Biology at the University of Pennsylvania. His research interests are in mathematical biology and applied/numerical analysis. Joshua B. Plotkin is the Walter H. and Leonore C. Annenberg Professor of Natural Sciences at the University of Pennsylvania. His work uses mathematics and computation to understand broad patterns of biological, cultural, and social evolution.