Acculturation drives the evolution of intergroup conflict.

Conflict between groups of individuals is a prevalent feature in human societies. A common theoretical explanation for intergroup conflict is that it provides benefits to individuals within groups in the form of reproduction-enhancing resources, such as food, territory, or mates. However, it is not always the case that conflict results from resource scarcity. Here, we show that intergroup conflict can evolve, despite not providing any benefits to individuals or their groups. The mechanism underlying this process is acculturation: the adoption, through coercion or imitation, of the victor's cultural traits. Acculturation acts as a cultural driver (in analogy to meiotic drivers) favoring the transmission of conflict, despite a potential cost to both the host group as a whole and to individuals in that group. We illustrate this process with a two-level model incorporating state-dependent event rates and evolving traits for both individuals and groups. Individuals can become "warriors" who specialize in intergroup conflicts, but are costly otherwise. Additionally, groups are characterized by cultural traits, such as their tendency to engage in conflict with other groups and their tendency for acculturation. We show that, if groups engage in conflicts, group selection will favor the production of warriors. Then, we show that group engagement can evolve if it is associated with acculturation. Finally, we study the coevolution of engagement and acculturation. Our model shows that horizontal transmission of culture between interacting groups can act as a cultural driver and lead to the maintenance of costly behaviors by both individuals and groups.

Hamilton's rule.

This paper reviews and addresses a variety of issues relating to inclusive fitness. The main question is: are there limits to the generality of inclusive fitness, and if so, what are the perimeters of the domain within which inclusive fitness works? This question is addressed using two well-known tools from evolutionary theory: the replicator dynamics, and adaptive dynamics. Both are combined with population structure. How generally Hamilton's rule applies depends on how costs and benefits are defined. We therefore consider costs and benefits following from Karlin and Matessi's (1983) "counterfactual method", and costs and benefits as defined by the "regression method" (Gardner et al., 2011). With the latter definition of costs and benefits, Hamilton's rule always indicates the direction of selection correctly, and with the former it does not. How these two definitions can meaningfully be interpreted is also discussed. We also consider cases where the qualitative claim that relatedness fosters cooperation holds, even if Hamilton's rule as a quantitative prediction does not. We furthermore find out what the relation is between Hamilton's rule and Fisher's Fundamental Theorem of Natural Selection. We also consider cancellation effects - which is the most important deepening of our understanding of when altruism is selected for. Finally we also explore the remarkable (im)possibilities for empirical testing with either definition of costs and benefits in Hamilton's rule.

Group-level events are catalysts in the evolution of cooperation.

Group-level events, like fission and extinction, catalyze the evolution of cooperation in group-structured populations by creating new paths from uncooperative population states to more cooperative states. Group-level events allow cooperation to thrive under unfavorable conditions such as low intra-group assortment and moderate rates of migration, and can greatly speed up the evolution of cooperation when conditions are more favorable. The time-dependent effects of fission and extinction events are studied and illustrated here using a PDE model of a group-structured population based loosely on populations of hunter-gatherer tribes. By solving the PDE numerically we can compare models with and without group-level events, and explicitly calculate quantities associated with dynamics, like how long it takes a small population of cooperators to become a majority, as well as equilibrium population densities.

Continuous-time models of group selection, and the dynamical insufficiency of kin selection models.

Traditionally, the process of group selection has been described mathematically by discrete-time models, and analyzed using tools like the Price equation. This approach makes implicit assumptions about the process that are not valid in general, like the central role of synchronized mass-dispersion and group re-formation events. In many important examples (like hunter-gatherer tribes) there are no mass-dispersion events, and the group-level events that do occur, like fission, fusion, and extinction, occur asynchronously. Examples like these can be fully analyzed by the equations of two-level population dynamics (described here) so their models are dynamically sufficient. However, it will be shown that examples like these cannot be fully analyzed by kin selection (inclusive fitness) methods because kin selection versions of group selection models are not dynamically sufficient. This is a critical mathematical difference between group selection and kin selection models, which implies that the two theories are not mathematically equivalent.

A simple model of group selection that cannot be analyzed with inclusive fitness.

A widespread claim in evolutionary theory is that every group selection model can be recast in terms of inclusive fitness. Although there are interesting classes of group selection models for which this is possible, we show that it is not true in general. With a simple set of group selection models, we show two distinct limitations that prevent recasting in terms of inclusive fitness. The first is a limitation across models. We show that if inclusive fitness is to always give the correct prediction, the definition of relatedness needs to change, continuously, along with changes in the parameters of the model. This results in infinitely many different definitions of relatedness - one for every parameter value - which strips relatedness of its meaning. The second limitation is across time. We show that one can find the trajectory for the group selection model by solving a partial differential equation, and that it is mathematically impossible to do this using inclusive fitness.

Studying the emergence of complicated group-level cultural traits requires a mathematical framework.

Understanding the cultural evolution of complicated group-level traits requires the mathematical formulation of a dynamical system with birth and death events at multiple levels, that is, at the level of individual humans and at the level of groups of humans. Both levels are characterized by cultural traits that have complicated transmission, innovation, and inheritance mechanisms and that can undergo a form of Lamarckian evolution.

Fission as a source of variation for group selection.

Without heritable variation, natural selection cannot effect evolutionary change. In the case of group selection, there must be variation in the population of groups. Where does this variation come from? One source of variation is from the stochastic birth-death processes that occur within groups. This is where variation between groups comes from in most mathematical models of group selection. Here we argue that another important source of variation between groups is fission, the (generally random) group-level reproduction where parent groups split into two or more offspring groups. We construct a simple model of the fissioning process with a parameter that controls how much variation is produced among the offspring groups. We then illustrate the effect of that parameter with some examples. In most models of group selection in the literature, no variation is produced during group reproduction events, i.e., groups "clone" themselves when they reproduce. Fission is often a more biologically realistic method of group reproduction, and it can significantly increase the efficacy of group selection.

Hamilton's rule in multi-level selection models.

Hamilton's rule is regarded as a useful tool in the understanding of social evolution, but it relies on restrictive, overly simple assumptions. Here we model more realistic situations, in which the traditional Hamilton's rule generally fails to predict the direction of selection. We offer modifications that allow accurate predictions, but also show that these Hamilton's rule type inequalities do not predict long-term outcomes. To illustrate these issues we propose a two-level selection model for the evolution of cooperation. The model describes the dynamics of a population of groups of cooperators and defectors of various sizes and compositions and contains birth-death processes at both the individual level and the group level. We derive Hamilton-like inequalities that accurately predict short-term evolutionary change, but do not reliably predict long-term evolutionary dynamics. Over evolutionary time, cooperators and defectors can repeatedly change roles as the favored type, because the amount of assortment between cooperators changes in complicated ways due to both individual-level and group-level processes. The equation that governs the dynamics of cooperator/defector assortment is a certain partial differential equation, which can be solved numerically, but whose behaviour cannot be predicted by Hamilton's rules, because Hamilton's rules only contain first-derivative information. In addition, Hamilton's rules are sensitive to demographic fitness effects such as local crowding, and hence models that assume constant group sizes are not equivalent to models like ours that relax that assumption. In the long-run, the group distribution typically reaches an equilibrium, in which case Hamilton's rules necessarily become equalities.

A stochastic model of evolutionary dynamics with deterministic large-population asymptotics.

An evolutionary birth-death process is proposed as a model of evolutionary dynamics. Agents residing in a continuous spatial environment X, play a game G, with a continuous strategy set S, against other agents in the environment. The agents' positions and strategies continuously change in response to other agents and to random effects. Agents spawn asexually at rates that depend on their current fitness, and agents die at rates that depend on their local population density. Agents' individual evolutionary trajectories in X and S are governed by a system of stochastic ODEs. When the number of agents is large and distributed in a smooth density on (X,S), the collective dynamics of the entire population is governed by a certain (deterministic) PDE, which we call a fitness-diffusion equation.

Towards a general theory of group selection.

The longstanding debate about the importance of group (multilevel) selection suffers from a lack of formal models that describe explicit selection events at multiple levels. Here, we describe a general class of models for two-level evolutionary processes which include birth and death events at both levels. The models incorporate the state-dependent rates at which these events occur. The models come in two closely related forms: (1) a continuous-time Markov chain, and (2) a partial differential equation (PDE) derived from (1) by taking a limit. We argue that the mathematical structure of this PDE is the same for all models of two-level population processes, regardless of the kinds of events featured in the model. The mathematical structure of the PDE allows for a simple and unambiguous way to distinguish between individual- and group-level events in any two-level population model. This distinction, in turn, suggests a new and intuitively appealing way to define group selection in terms of the effects of group-level events. We illustrate our theory of group selection by applying it to models of the evolution of cooperation and the evolution of simple multicellular organisms, and then demonstrate that this kind of group selection is not mathematically equivalent to individual-level (kin) selection.