Interplay between geometry and randomness in fitness landscapes for expanding populations

进化的过程是由选择优势的确定性力量之间的相互作用形成的持续冲突，以适应性景观的几何形状为特征，以及突变和繁殖的随机性。在这种情况下，一个核心问题是可预测性：给定适合度景观的几何形状，我们能预测进化的轨迹吗？从生物学和数学的角度来看，我们很好地理解了两种机制，但这两种机制在极限下本质上都是确定性的；一旦我们确定了参数和适合度，进化是预先决定的。我们的建议的重点是在突变足够频繁的情况下，在适应度景观中，局部最优的数量足够大，因此种群不一定是一个，几个物种或准物种可能同时存在。尽管模拟已经显示出有趣的结果，但在这种情况下，严格的数学工作明显缺乏。这项提议的目的是开发新的严格的数学工具，以深入了解进化过程中可能采取的随机途径。

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.