Evolution of Conditional Dispersal and Population Dynamics

本计画探讨条件扩散对单一及多重交互物种动态的影响。 在种群动力学的理论研究中，一个常见的基本假设是扩散率在空间上是均匀的。 然而，这种简化可能会产生误导性的结果，因为周围的环境可以在空间上和时间上变化。 事实上，动物倾向于通过定向分散来感知和响应当地的环境线索，它们的运动通常是随机和定向运动的组合。 反应-扩散-平流方程是理解扩散等时空过程的主要方法之一，在本研究中用于模拟物种和种群动态的随机和定向运动。 本计画将研究三种条件散布策略。 第一种是种群沿着环境梯度或沿着密度依赖的生长速率梯度的直接移动，目的是确定这种有偏移动对单个种群和多个竞争物种的影响。 第二种扩散策略涉及捕食者的区域限制搜索，目的是了解捕食者的有偏见的觅食行为如何诱导捕食者聚集。 第三个是理想自由分布理论的动力学模型，它假设物种选择栖息地的方式，每个人都试图最大限度地提高其繁殖适合度。 其目的是更好地了解这种扩散策略和种群动态之间的相互作用。为了解决这些生物学问题，首席研究员将使用数学方法，包括椭圆和抛物算子的正则性理论，特征值问题的分析，最大值原理，分叉分析，单调系统理论，持久性理论和扰动分析。这个项目的目的是增加我们对种群如何响应空间变化环境而分散的理解，以确定哪些模式的扩散战略可以赋予一些选择或生态优势，并提供有关生物多样性问题，如栖息地破碎化和新物种的入侵的见解。 初步研究表明，生境的几何形状可以在扩散的演变中发挥重要作用，而且一个物种的强烈定向运动可以诱导与竞争对手的共存。 从这个项目的材料将被修改，并作为团队项目的数学生物科学研究所暑期课程的大学教师和研究生在数学和生物学。

This project investigates the effects of conditional dispersal on dynamics of single and multiple interacting species. A common underlying assumption in theoretical studies of population dynamics is that dispersal rates are uniform across space. However, such simplification can yield misleading results because the surrounding environment can vary both spatially and temporally. Indeed, animals tend to sense and respond to local environmental cues by dispersing directionally, and their movements are often combinations of both random and directed ones. Reaction-diffusion-advection equations serve as one of the major approaches to understanding spatial-temporal processes such as dispersal, and are used in this research to model both random and directed movements of species and population dynamics. Three kinds of conditional dispersal strategies will be studied in this project. The first one is the direct movement of populations along environmental gradients or along density-dependent growth rate gradients, and the goal is to determine the effects of such biased movement on both single population and multiple competing species. The second dispersal strategy concerns area-restricted search of predators, and the purpose is to understand how biased foraging behaviors of predators can induce the aggregation of predators. The third is a dynamical model for ideal free distribution theory, which assumes that species choose habitats in such a way that each individual tries to maximize its reproduction fitness. The aim is to obtain a better understanding of interactions between such dispersal strategy and population dynamics. To address these biological questions, the principal investigator will use mathematical methods which include regularity theory for elliptic and parabolic operators, analysis of eigenvalue problems, maximum principles, bifurcation analysis, monotone system theory, permanence theory, and perturbation analysis.The purpose of this project is to increase our understanding of how populations disperse in response to spatially varying environments, to determine which patterns of dispersal strategy can confer some selective or ecological advantage, and to provide insights on biodiversity issues such as habitat fragmentation and invasions of new species. Preliminary investigations show that the geometry of habitat can play important roles in the evolution of dispersal, and also that strong directed movement of a species can induce coexistence with its competitors. Materials from this project will be modified and used as team projects for a Mathematical Biosciences Institute Summer Program for college teachers and graduate students in mathematics and biology.