Steady-State and Dynamical Behavior of a PDE Model of Multilevel Selection with Pairwise Group-Level Competition

Area of Science:

• Evolutionary Biology
• Game Theory
• Mathematical Modeling

Background:

• Evolutionary competition occurs at multiple levels, with individual costs potentially yielding group benefits.
• Game theory has been applied to understand within- and between-group evolutionary dynamics.
• Previous models often simplified group-level competition, particularly frequency dependence.

Purpose of the Study:

• To develop and analyze a partial differential equation (PDE) model for multilevel selection incorporating group-level frequency dependence.
• To investigate how the probability of group victory influences the long-term persistence of cooperation.
• To determine conditions under which cooperation can be sustained via multilevel selection in evolutionary games.

Main Methods:

• Developed a PDE model for multilevel selection with pairwise group conflicts.
• Analyzed measure-valued solutions to establish well-posedness properties.
• Employed analytical and numerical techniques to study evolutionary game dynamics (Prisoners' Dilemma, Hawk-Dove).

Main Results:

• The population converges to an all-defector state when between-group selection is weak.
• Identified conditions for the existence of bounded steady-state densities in multilevel dynamics.
• The average payoff at steady state is capped by the all-cooperator group's payoff, even with intermediate cooperation optima.

Conclusions:

• The strength of between-group selection is critical for maintaining cooperation in multilevel selection scenarios.
• The study generalizes previous findings on PDE models of multilevel selection, particularly regarding frequency-dependent competition.
• Cooperation's long-term survival is contingent on sufficiently strong selection pressures acting at the group level.