Interplay between geometry and randomness in fitness landscapes for expanding populations

进化的过程是一场持续不断的冲突，这种冲突是由选择优势的确定性力量与突变和繁殖的随机性之间的相互作用形成的。在这种情况下，一个核心问题是可预测性：考虑到健康状况的几何形状，我们能预测进化的轨迹吗？从生物学和数学的角度来看，人们都很好地理解了两种机制，但这两种机制本质上都是极限决定论；一旦我们确定了参数和适应环境，进化就是预先确定的。我们建议的重点是突变足够频繁，适应度格局中的局部最优数目足够大的情况，这样种群就不一定移动，因为一个和几个物种或准物种可能会同时存在。在这种情况下，尽管模拟已经显示了有趣的结果，但严重缺乏严格的数学工作。这项提议的目的是开发新的严格的数学工具，以洞察进化过程可能采取的随机路径。

The course of evolution is a constant conflict shaped by the interplay between the deterministic forces of selective advantage, characterised by the geometry of a fitness landscape, and the randomness of mutation and reproduction. A central question in this context is predictability: given the geometry of the fitness landscape, can we predict the trajectory of evolution? Two regimes are very well understood, from both biological and mathematical perspectives, but both these regimes are essentially deterministic in the limit; once we fix the parameters and the fitness landscape, evolution is pre-determined. The focus of our proposal is on cases where mutations are sufficiently frequent and the number of local optima in the fitness landscape is sufficiently large, so that the population does not necessarily move as one and several species or quasispecies may exist concurrently. There is a significant lack of rigorous mathematical work in this situation, although simulations have shown intriguing results. The aim of this proposal is to develop new rigorous mathematical tools to gain insight into the possible random pathways taken by evolutionary processes.