AI 中文总结
研究N个种群的选择-突变积分-微分Lotka-Volterra系统长期动力学，在常见假设下其解的长期行为由广义Lotka-Volterra方程决定，还表明不同环境下突变率的选择情况，关键是确定总种群大小受该方程控制。
AI 中文摘要
我们研究了N个种群的选择-突变积分-微分Lotka-Volterra系统的长期动力学。在我们的模型中，适应度取决于连续的表型性状，但一个种群对另一个种群的影响与此性状无关。我们确定，在关于种群间相互作用的一些常见假设下，解的长期行为完全由经过充分研究的广义Lotka-Volterra（常微分）方程决定。我们还表明，在其他条件相同的情况下，在静态环境中选择最小突变率，而在变化的环境中，可能会选择中间突变率。我们证明的关键步骤是确定总种群大小渐近地由广义Lotka-Volterra方程控制，这进而允许使用关于渐近自治动力系统的一般结果。
英文摘要
We study the long time dynamics of a selection-mutation integro-differential Lotka-Volterra system of $N$ populations. In our model, fitness depends on a continuous phenotypic trait, but the effect of one population on another is independent of this trait. We establish that, under some usual assumptions on the interactions between populations, the long time behaviour of solutions is exactly determined by the well studied Generalised Lotka-Volterra (ordinary differential) Equation. We also show that, all else being equal, the minimal mutation rate is selected for in a static environment, whereas in an a changing environment, an intermediate mutation rate could be selected instead. The key step in our proofs is establishing that the total population sizes are asymptotically governed by a Generalised Lotka-Volterra Equation, which then allows the use of a general result on asymptotically autonomous dynamical systems.
