PDE Models of Multilevel Selection: The Evolution of Cooperation and the Shadow of Selection

Abstract

In this thesis, we explore the fundamental evolutionary conflict between the individual incentive

to cheat and the collective incentive to be part of a cooperative group. Our focus is on multilevel selection,

in which individuals compete against peer group members and simultaneously compete together with their

group members against other groups. Here we formulate these two levels of competition by considering

group-structured populations in which each individual plays a two-strategy game with every other group

member. Then the within-group competition depends on individual payoff, while between-group

competition depends on the average payoff of group members. With this approach, we are able to explore a

variety of individual-level and group-level incentives and the impact of relative incentives at the two level

in establishing long-time defection or cooperation.

In Chapter 2, we first formulate a finite population model for multilevel selection and then derive a PDE for

the time evolution of the density of group compositions. The equation has an advection term corresponding

to within-group competition and a non-local term corresponding to between-group competition. We then

consider special cases of the Prisoners' Dilemma and Hawk-Dove game with solvable characteristic curves

and find a threshold relative strength of between-group competition separating a regime in which defection

dominates and one in which cooperation can survive in steady state. An interesting finding is the shadow of

lower-level selection: when groups are best off with a mix of cooperators and defectors: no level of

between-group selection strength can produce the optimal possible collective payoff.

In Chapter 3, we study a broader class of payoff matrices, and use comparison principles and a preserved

property of population distribution to show that the shadow of lower-level selection holds more generally.

In Chapter 4, we apply the methods of chapter 3 to study how assortment, reciprocity, and network structure

can work synergistically with multilevel selection. Chapter 5 presents alternate mathematical approaches

and numerical methods for our model. In Chapter 6, we consider a model of protocell evolution, and then

show how linking genes in a chromosome can help to overcome a cellular analogue to the shadow of lower-

level selection.