Evolution of Conditional Dispersal and Population Dynamics

本项目研究了条件扩散对单个和多个相互作用物种的动力学的影响。在种群动力学的理论研究中，一个常见的基本假设是扩散速度在空间上是一致的。然而，这种简化可能会产生误导性的结果，因为周围的环境可能会在空间和时间上发生变化。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，它们的行动往往是随机和定向的组合。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的随机和定向运动以及种群动力学。本项目将研究三种条件扩散策略。第一个是种群沿环境梯度或依赖密度的增长率梯度的直接移动，目标是确定这种有偏见的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，其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。第三种是理想自由分布理论的动力学模型，该模型假设物种选择栖息地的方式是每个个体都试图最大化其繁殖适宜性。其目的是更好地了解这种扩散策略与种群动态之间的相互作用。为了解决这些生物学问题，首席研究者将使用数学方法，包括椭圆和抛物型算子的正则性理论、特征值问题的分析、最大值原理、分歧分析、单调系统理论、持久性理论和扰动分析。本项目的目的是增加我们对种群如何随着空间变化的环境而分散的理解，确定哪些扩散策略可以提供一些选择或生态优势，并对栖息地碎片化和新物种入侵等生物多样性问题提供见解。初步研究表明，栖息地的几何形状在物种的扩散演化中起着重要作用，物种的强烈定向运动也可以诱导与其竞争对手共存。这个项目的材料将被修改，并作为数学生物科学学院数学和生物专业教师和研究生暑期项目的团队项目。

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.