We present a new modelling framework combining replicator dynamics, the standard model of frequency dependent selection, with an age-structured population model. The new framework allows for the modelling of populations consisting of competing strategies carried by individuals who change across their life cycle. Firstly the discretization of the McKendrick von Foerster model is derived. We show that the Euler–Lotka equation is satisfied when the new model reaches a steady state (i.e. stable frequencies between the age classes). This discretization consists of unit age classes where the timescale is chosen so that only a fraction of individuals play a single game round. This implies a linear dynamics and individuals not killed during the round are moved to the next age class; linearity means that the system is equivalent to a large Bernadelli–Lewis–Leslie matrix. Then we use the methodology of multipopulation games to derive two, mutually equivalent systems of equations. The first contains equations describing the evolution of the strategy frequencies in the whole population, completed by subsystems of equations describing the evolution of the age structure for each strategy. The second contains equations describing the changes of the general population’s age structure, completed with subsystems of equations describing the selection of the strategies within each age class. We then present the obtained system of replicator dynamics in the form of the mixed ODE-PDE system which is independent of the chosen timescale, and much simpler. The obtained results are illustrated by the example of the sex ratio model which shows that when different mortalities of the sexes are assumed, the sex ratio of 0.5 is obtained but that Fisher’s mechanism, driven by the reproductive value of the different sexes, is not in equilibrium.

我们提出了一个新的建模框架，将复制器动力学（频率相关选择的标准模型）与年龄结构的人口模型相结合。新框架允许对由在整个生命周期中发生变化的个体所采用的竞争策略组成的群体进行建模。首先推导出 McKendrick von Foerster 模型的离散化。我们表明，当新模型达到稳定状态（即年龄等级之间的稳定频率）时，满足 Euler-Lotka 方程。这种离散化由单位年龄类组成，其中选择了时间尺度，以便只有一小部分人玩一个游戏回合。这意味着一个线性动态，在回合中没有被杀死的个体被转移到下一个年龄段；线性意味着系统等效于一个大的 Bernadelli-Lewis-Leslie 矩阵。然后我们使用多种群博弈的方法推导出两个相互等价的方程组。第一个包含描述整个人口中策略频率演变的方程，由描述每个策略年龄结构演变的方程子系统完成。第二个包含描述一般人口年龄结构变化的方程，完成了描述每个年龄段内策略选择的方程子系统。然后，我们以混合 ODE-PDE 系统的形式呈现所获得的复制器动力学系统，该系统独立于所选的时间尺度，并且更简单。