Evolutionary game theory is a powerful method for modelling animal conflicts. The original evolutionary game models were used to explain specific biological features of interest, such as the existence of ritualised contests, and were necessarily simple models that ignored many properties of real populations, including the duration of events and spatial and related structural effects. Both of these areas have subsequently received much attention. Spatial and structural effects have been considered in evolutionary graph theory, and a significant body of literature has been built up to deal with situations where the population is not homogeneous. More recently a theory of time constraints has been developed to take account of the fact that different events can take different times, and that interaction times can explicitly depend upon selected strategies, which can, in turn, influence the distribution of different opponent types within the population. Here, for the first time, we build a model of time constraint games which explicitly considers a spatial population, by considering a population evolving on an underlying graph, using two graph dynamics, birth–death and death-birth. We consider one short time scale along which frequencies of pairs and singles change as individuals interact with their neighbours, and another, evolutionary time scale, along which frequencies of strategies change in the population. We show that for graphs with large degree, both dynamics reproduce recent results from well-mixed time constraint models, including two ESSs being common in Hawk-Dove and Prisoner’s Dilemma games, but for low degree there can be marked differences. For birth–death processes the effect of the graph degree is small, whereas for death-birth dynamics there is a large effect. The general prediction for both Hawk-Dove and Prisoner’s dilemma games is that as the graph degree decreases, i.e., as the number of neighbours decreases, mixed ESS do appear. In particular, for the Prisoner’s dilemma game this means that cooperation is easier to establish in situations where individuals have low number of neighbours. We thus see that solutions depend non-trivially on the combination of graph degree, dynamics and game.

进化博弈论是模拟动物冲突的有力方法。原始的进化博弈模型用于解释特定的生物学特征，例如仪式性竞赛的存在，并且必然是忽略了真实种群的许多属性（包括事件的持续时间以及空间和相关结构效应）的简单模型。这两个领域随后都引起了很多关注。在进化图论中已经考虑了空间和结构的影响，并且已经建立了大量的文献来处理人口不均一的情况。最近，人们开发了一种时间约束理论，以考虑到不同事件可能需要不同时间的事实，而且互动时间可以明确地取决于所选策略，进而可以影响人口中不同对手类型的分布。在这里，我们首次建立了一个时间约束博弈模型，该模型通过考虑出生图和死亡图和出生图图两种动力学，通过考虑在基础图上演化的种群来明确考虑空间种群。我们考虑一个短期的时间尺度，随着个体与邻居的互动，配对和单身的频率会发生变化，而另一个进化的时间尺度，即人口中的策略发生频率会发生变化。我们表明，对于具有较大程度的图，这两种动力学都可以从混合时间约束模型（包括在Hawk-Dove和Prisoner's Dilemma游戏中很常见的两种ESS）再现最近的结果，但程度较低，可能会有明显的差异。对于出生-死亡过程，图形程度的影响很小，而对于死亡-出生动力学，则影响很大。对Hawk-Dove和Prisoner困境游戏的一般预测是，随着图度的降低，即随着邻居数的减少，确实会出现混合ESS。特别是，对于囚徒困境游戏，这意味着在个人邻居少的情况下更容易建立合作关系。因此，我们看到解决方案非常重要地取决于图度，动力学和博弈的组合。对Hawk-Dove和Prisoner困境游戏的一般预测是，随着图度的降低，即随着邻居数的减少，确实会出现混合ESS。特别是，对于囚徒困境游戏，这意味着在个人邻居少的情况下更容易建立合作关系。因此，我们看到解决方案非常重要地取决于图度，动力学和博弈的组合。对Hawk-Dove和Prisoner困境游戏的一般预测是，随着图度的降低（即，随着邻居数量的减少），确实会出现混合ESS。特别是，对于囚徒困境游戏，这意味着在个人邻居少的情况下更容易建立合作关系。因此，我们看到解决方案非常重要地取决于图度，动力学和博弈的组合。