We develop new equations for the eco-evolutionary dynamics of populations and their traits. These equations resolve the change in the phenotypic differentiation within a population, which better estimates how the variance of the trait distribution changes. We note that traits may be bounded, assume they may be described by beta distributions with small variances, and develop a coupled ordinary differential equation system to describe the dynamics of the total population, the mean trait value, and a measure of phenotype differentiation. The variance of the trait in the population is calculated from its mean and the population’s phenotype differentiation.

We consider an example of two competing plant populations to demonstrate the efficacy of the new approach. Each population may trade-off its growth rate against its susceptibility to direct competition from the other population. We create two models of this system: a population model based on our new eco-evolutionary equations; and a phenotype model, in which the growth or demise of each fraction of each population with a defined phenotype is simulated as it interacts with a shared limiting resource and its competing phenotypes and populations.

Comparison of four simulation scenarios reveals excellent agreement between the predicted quantities from both models: total populations, the average trait values, the trait variances, and the degree of phenotypic differentiation within each population. In each of the four scenarios simulated, three of which are initially subject to competitive exclusion in the absence of evolution, the populations adapt to coexist. One population maximises growth and dominates, while the other minimises competitive losses. These simulations suggest that our new eco-evolutionary equations may provide an excellent approximation to phenotype changes in populations.

我们为种群及其特征的生态进化动力学开发了新方程。这些方程解决了群体内表型分化的变化，从而更好地估计性状分布的方差如何变化。我们注意到性状可能是有界的，假设它们可以用小方差的 beta 分布来描述，并开发一个耦合的常微分方程系统来描述总人口的动态、平均性状值和表型分化的度量。群体中性状的方差是根据其均值和群体的表型分化来计算的。

我们考虑了两个相互竞争的植物种群的例子来证明新方法的有效性。每个种群都可以在其增长率与其对来自其他种群的直接竞争的敏感性之间进行权衡。我们为这个系统创建了两个模型：一个基于我们新的生态进化方程的人口模型；和一个表型模型，其中每个具有定义表型的种群的每个部分的生长或消亡都被模拟，因为它与共享的限制资源及其竞争表型和种群相互作用。

四种模拟场景的比较揭示了两种模型的预测数量之间的极好一致性：总种群、平均性状值、性状方差和每个种群内的表型分化程度。在模拟的四个场景中的每一个中，其中三个在没有进化的情况下最初受到竞争排斥，种群适应共存。一个群体最大化增长并占据主导地位，而另一个群体最小化竞争损失。这些模拟表明，我们新的生态进化方程可以很好地近似种群的表型变化。