Alexiou Konstantinos, Cooney Daniel B
Department of Mathematics, University of St. Andrews, St. Andrews, Scotland, UK.
Department of Mathematics and Carl R. Woese Institute for Genomic Biology, University of Illinois Urbana-Champaign, Urbana, IL, USA.
Bull Math Biol. 2025 Jun 19;87(7):97. doi: 10.1007/s11538-025-01476-4.
Evolutionary competition often occurs simultaneously at multiple levels of organization, in which traits or behaviors that are costly for an individual can provide collective benefits to groups to which the individual belongs. Building off of recent work that has used ideas from game theory to study evolutionary competition within and among groups, we study a PDE model for multilevel selection that considers group-level evolutionary dynamics through a pairwise conflict depending on the strategic composition of the competing groups. This model allows for incorporation of group-level frequency dependence, facilitating the exploration for how the form of probabilities for victory in a group-level conflict can impact the long-time support for cooperation via multilevel selection. We characterize well-posedness properties for measure-valued solutions of our PDE model and apply these properties to show that the population will converge to a delta-function at the all-defector equilibrium when between-group selection is sufficiently weak. We further provide necessary conditions for the existence of bounded steady state densities for the multilevel dynamics of Prisoners' Dilemma and Hawk-Dove scenarios, using a mix of analytical and numerical techniques to characterize the relative strength of between-group selection required to ensure the long-time survival of cooperation via multilevel selection. We also see that the average payoff at steady state appears to be limited by the average payoff of the all-cooperator group, even for games in which groups achieve maximal average payoff at intermediate levels of cooperation, generalizing behavior that has previously been observed in PDE models of multilevel selection with frequency-independent group-level competition.
进化竞争通常在多个组织层次上同时发生，在这种情况下，对个体而言成本高昂的特征或行为可以为该个体所属的群体带来集体利益。基于最近利用博弈论思想研究群体内部和群体之间进化竞争的工作，我们研究了一个用于多层次选择的偏微分方程模型，该模型通过取决于竞争群体战略组成的成对冲突来考虑群体层面的进化动态。这个模型允许纳入群体层面的频率依赖性，便于探索群体层面冲突中胜利概率的形式如何通过多层次选择影响对合作的长期支持。我们刻画了我们的偏微分方程模型的测度值解的适定性性质，并应用这些性质表明，当群体间选择足够弱时，种群将在全背叛者均衡处收敛到一个狄拉克函数。我们进一步提供了囚徒困境和鹰鸽博弈场景多层次动态存在有界稳态密度的必要条件，使用分析和数值技术的组合来刻画通过多层次选择确保合作长期生存所需的群体间选择的相对强度。我们还看到，即使对于群体在中间合作水平达到最大平均收益的博弈，稳态下的平均收益似乎也受到全合作者群体平均收益的限制，这推广了先前在具有频率独立群体层面竞争的多层次选择偏微分方程模型中观察到的行为。
