I really enjoyed the discussions that led to this model, and it was really fun to think about how these kinds of multilevel selection models can describe systems ranging from replication competition between genes in cells to the evolution of cooperative social norms in complex animal societies.

By the way, in case you are still interested in models of ecology and evolutionary biology, I am co-organizing a conference session at Boston College in March 2026 with Alex McAvoy and Thomas Zdyrski.

www.ams.org/meetings/sec...

Thanks for sharing, @blakestacey.myatproto.social. I am particularly looking forward to the chapter on invasion fitness with a pair-approximation approach, as well as the discussion on different interpretations of MLS.

Wow, that takes me back to an earlier research life...

arxiv.org/abs/1509.02958

Donald Dawson also explored the behavior of the Fleming-Viot limit, using a dual process to study the extinction or survival of cooperation due to selection, mutation, and drift at both the individual and group levels.

link.springer.com/article/10.1...

Analysis of this two-level Moran model has led to a lot of other interesting work in probability and PDEs. In one interesting paper, Aurélien Velleret studied quasi-stationary distributions in a diffusive PDE model that arises in the limit of weak selection.

www.aimsciences.org/article/doi/...

We saw that the shadow/ghost(👻) prevented coexistence of the fast and slow genes if protocell success most favored an even mix of the genes.

However, introducing a third replicator (a dimer linking together the two genes), we saw that coexistence could occur in three-type multilevel dynamics.

The three of us ended up writing a paper with Simon Levin on applying these PDE models to protocell evolution, considering competition between "fast genes" and "slow genes" that were complementary for protocell-level replication.

link.springer.com/article/10.1...

Later on in grad school, I had many fun discussions with @fernpizza.bsky.social and Dylan Morris about applying multilevel selection models for coexistence to topics like the origins of chromosomes (such as in the linked paper by Maynard Smith and Szathmary).

www.sciencedirect.com/science/arti...

Basically, no level of group-level competition can fully erase the shadow of the individual incentive to defect.

Overall, it seemed like it was very hard to sustain coexistence of strategies that are complementary at the group level in the presence of an individual-level selective advantage.

When groups are best of with an intermediate mix of cooperators and defectors, no level of group-level selection could allow the population to achieve this optimal composition, and the population could not achieve an average payoff greater than the payoff achieved by an all-cooperator group.

Here is a blog post that I wrote with Yoichiro Mori, @jplotkin.bsky.social, and Simon Levin on the dynamical behaviors of this cross-scale model. In particular, we highlight a behavior that I like to call the shadow/ghost(👻) of lower-level selection.

www.siam.org/publications...

The PDE model was really fun to explore, because it was basically a two-level replicator equation: individual-level selection depends on personal payoff within groups and group-level selection favors groups whose achieve higher average payoffs.

I made an extendsion the Luo-Mattingly model to incorporate individual and group replication rates based on the payoff of two-strategy, two-player games.

Such games can feature dominance of defectors, coexistence of both strategies, or bistability within groups.

link.springer.com/article/10.1...

Early in grad school, Carl Veller pointed me to the Luo-Mattingly model and related cross-scale models by Burt Simon. Carl thought that evolutionary game theory could be helpful to describing different tugs-of-war between individual and group incentives.

www.sciencedirect.com/science/arti...

Luo and @jonathancm.bsky.social derived and analyzed PDE and Fleming-Viot scaling limits in the limit of infinitely many groups and infintie group size, characterizing the strength of group-level selection required for survival of cooperation in the PDE model.

iopscience.iop.org/article/10.1...

Mathematically, Luo's model is particularly tractable due to the choice of a nested birth-death process of individuals and groups.

Biologically, the plasmid system of @fernpizza.bsky.social et al fit this approach so nicely because the plasmids replicate independently but impact cellular fitness.

Luo considered two strategies: cheaters with an individual-level replication advantage and cooperators that conferring a collective replication advantage to its group. Luo formulated a stochastic model for individual and group dynamics, deriving ODE and PDE models in the large population limit.

Here is one of my favorite papers, "A unifying framework reveals key properties of multilevel selection" by Shishi Luo. Luo highlights examples of multilevel selection and introduces a nested birth-death process to model evolutionary competition across scales.

www.sciencedirect.com/science/arti...