Investigating evolution and complexity with stochastic models

这个项目的动机是目前缺乏共识的机制背后的明显发展越来越复杂的生物实体随着时间的推移。在动物的语境中，复杂性并不是一个定义明确的数量：没有一个单一的数字可以比较鹪鹩和章鱼的复杂性。然而，有一些复杂性的替代品可以在物种之间进行测量和比较。一个例子是重复的结构，如脊柱或分割的身体部位，如千足虫。测量数量，如重复单位之间的差异，单位的数量和计算指数从这些数字可以比较不同的物种和潜在的跟踪数量沿着血统。这是一个很好的起点，研究机制驱动这种演变。这个项目的目的是使用数学模型，涉及随机过程，以了解如何系统演变，如果他们有特点，如约束（例如肢体的最小或最大长度），驱动力（如环境变化或机制导致全球某些特征的方向变化）和被动探索所有可能变化的空间。对这些系统的理解可以用来确定导致物种发生变化的过程，并通过创建可以与数据进行比较的模型，可以确定每个驱动变化的机制有多大的影响。数学模型可以基于对具有上述特征的随机过程的模拟，以及描述具有这些特征的系统的方程的解析解（如果可以找到的话），使用基于数据（例如，关于哺乳动物脊柱的大量可用数据）的复杂性指数将允许对此类模型的结果进行比较。生产模型，有效地描述措施的演变沿着血统将有望允许更明智的结论，我们究竟如何之间的单细胞和生态多样性，我们今天享受。一个足够好的模型还可以预测未来的适应，这些适应将根据当今自然世界面临的变化进行，并可以根据栖息地发生的变化告诉我们不同物种的相对风险。例如，了解环境因素限制种群发展的方式可以提供一种方法，使用边界条件来调查环境中随时间变化对不同物种的影响。要做到这一点，需要对复杂性和适应性之间的关系进行更深入的调查。了解不同类型的环境变化如何通过边界条件来描述，也将提供一种理论来预测不同环境变化可能影响种群的方式。因此，我们可以设想哪些类型的物种最有可能面临种群下降的风险，以及哪些环境变化可能对物种造成最大的问题。

The motivation for this project is the current lack of consensus about the mechanisms behind the apparent development of ever more complex biological entities over time. Complexity is not a well-defined quantity in the context of animals: there is no single number that can compare the complexity of a wren to an octopus. There are however proxies for complexity which can be measured and compared between species. An example is repeating structures such as vertebral columns or segmented body parts as in a millipede. Measuring quantities such as the differences between repeating units, the number of units and calculating indices derived from these numbers allows comparison of different species and potentially the tracking of the quantities along lineages. This is an excellent starting point for studying the mechanism driving this evolution.The aim of this project is to use mathematical models involving stochastic processes to understand how systems evolve if they have features such as constraints (such as a minimum or maximum length of a limb), driving forces (such as environmental changes or mechanisms causing global directional change in certain features) and passive exploration of the space of all possible changes. The resulting understanding of these systems can be used to identify the processes causing changes to occur to species and by creating models which can be compared to data, it may be possible to determine how much of an effect each of the mechanisms driving change has. The mathematical models can be based on simulations of stochastic processes with the features described above, as well as analytic solutions to the equations describing systems with these features if they can be found. Using indices of complexity based on data such as the vast data available on the vertebral columns of mammals will allow for comparison of the results of such models. Producing models which effectively describe the evolution of measures along lineages will hopefully allow for more informed conclusions to be made about exactly how we have moved between single cells and the ecological diversity we enjoy today. A good enough model may also allow for the prediction of future adaptions which will be made in light of the changes facing the natural world today and could inform us about the relative risks to different species based on changes occurring in their habitat. For example, understanding the way that environmental factors constrain development in populations could provide a way to use boundary conditions to investigate the impact of a time dependent change in environment on different species. To do this, a greater investigation of the relationship between complexity and adaptability would need to be carried out. Understanding how different types of environmental change can be described by boundary conditions would also provide a theory to predict the way different environmental changes could affect populations. Thus, we may be able to envisage which types of species are most at risk of population decline, as well which environmental changes are likely to cause the most problems for species.