The Price
equation and contextual analysis are not models, but methods of analyzing and
interpreting population change – data must come from experiments or
mathematical models.
The approach here:
- Using dynamical models of two-level population processes, we define group selection in terms of the evolutionary outcome of the process.
- If the outcome is due to group-level events (i.e., the outcome would be different without the group-level events), then group selection played an important role.
- The purpose is to create a general mathematical framework for studying two- level evolutionary dynamics, and to shed light on the mathematical distinction between individual- and group-level events.
Outline:
- Describe the math as a Markov process and PDE
- The PDE has a piece corresponding to individual-level selection and a piece corresponding to group-level selection (like the Price eq)
- This structure is used to characterize a general model
- The new theory is used to study two example systems
- Evolution of cooperation
- Evolution of simple multicellular organisms
- Show that the two-level population processes cannot be properly analyzed in terms of individual-level (i.e., inclusive) fitness measures alone, and therefore group selection, defined in terms of long-term evolutionary outcomes, is not mathematically equivalent to inclusive fitness approaches.
The model framework (The model is mathematized, but the
construct is as follows):
- Individuals live in groups.
- Each individual has some trait or type, which can take on values {1, …, k}
- Individuals have finite lifespans, can reproduce (asexually), and occasionally migrate to other groups.
- Group-level events include fissioning (splitting into two or more groups), fusion (combining two or more groups into one), and extinction.
- Individual birth, death, and migration:
- Birth and death rates typically depend on interactions taking place within groups (e.g., according to an evolutionary game such as a public goods game), but these rates may also depend on the state of the whole population.
- Offspring are usually the same type as the parent, but mutations can occur.
- Offspring are born into the same group as parent, but can migrate by choosing another group at random. An attempted migration is successful depending on the individual’s type and the state of the chosen group.
- Group-level events: Group fission, fusion, extinction, dispersal, group games
- Fissioning and extinction occur at rates that depend on the state of the group as well as the state of the whole population.
- When a group fissions, it breaks into two or more autonomous groups. The event is statistically specified by a density function that describes the number and types of group that can result from a fissioning event.
- Fusion: two given groups fuse at a rate determined by their respective types.
- Group dispersal can also occur – this is a mass migration of all the individuals in the group.
- Group-level interactions: groups could play games against each other with outcome determined by their compositions. For example, they could play a hawk-dove game, and the outcome could effect group-level events such as extinction or dispersal, or individual-level events such as birth rates.
Evolutionary definitions
- A trait evolves by group selection in a model of two-level population dynamics if it establishes itself when group- level events are present in the model, and does not establish itself in the same model when they are absent.
- A trait is assisted by group selection in a model of two-level population dynamics if it establishes itself in the model more quickly and/or more completely when group-level events are present in the model then when they are absent.
- Reminder: group-level events are fission, fusion, and extinction.
Examples
- They then apply their framework to two explicit models in which individual- and group-level effects are seen:
- The evolution of cooperation in hunter-gatherer tribes
- The evolution of “sticky” cells that form a crude multicellular organism.
This framework
is strikingly convergent with some of my own modeling work, particular
Smaldino, Newson, Schank, & Richerson and Makowsy & Smaldino. I agree
that this type of framework is very valuable for studying group-level effects.
However, things become difficult when groups become ephemeral, or when
individuals can belong to multiple groups.
I fully agree
that a mathematical modeling framework is vital for the characterization of
cultural evolution (as it is for other forms of evolution). The framework
presented by Simon and colleagues is quite a promising one. Time will tell how
successful it is at capturing GLTs and other nuances of cultural evolution, and
if it is flexible enough to capture those phenomena.
