Temporal and Spatial Dynamics in Mathematical Ecology

该计划侧重于数学生态学中的时间和空间动态。主要目标是对生态模型进行机制的制定和严格的分析。新的数学方法包括生态化学计量学和生态毒理学中的非光滑微分方程，混合偏微分方程和离散时间模型的持续理论和传播速度，以及用于描述基于记忆的动物运动的延迟扩散算子的延迟微分方程分岔理论。丰富的动态和新的定性理论有待探索。化学计量理论为理解宏观现象提供了一种基本的微观方法。它包括多种生物尺度，并允许通过化学和物理定律制定强大的机械，预测和实验可测试的模型。应用包括最优觅食策略，水生和陆地生态系统的多尺度比较，全球二氧化碳变化的影响，和蓝藻华。生态毒理学是保护和恢复工业污染下健康生态系统的重要研究领域。通过持久性和分岔分析，我将对有毒物质对生物体及其相互作用进行风险评估。空间异质性和自然延迟对于生物学的机制建模是重要的。例子包括具有空间记忆的复杂动物运动和景观的空间异质性。大多数拟议的研究将与经验生物学家或政府研究人员合作，用于数据验证和校准新模型以及决策。

This program focuses on temporal and spatial dynamics in mathematical ecology. The main objective is mechanistic formulation and rigorous analysis of ecological models. Novel mathematical approaches include nonsmooth differential equations in ecological stoichiometry and ecotoxicology, persistence theory and spreading speeds of hybrid partial differential equation and discrete-time models, and bifurcation theory of delay differential equations with applications to delayed diffusion operators for describing memory-based animal movement. Rich dynamics and new qualitative theory are waiting to be explored. Stoichiometric theory provides a fundamental microscopic approach to understand macroscopic phenomena. It includes multiple biological scales and allows the formulation of robust mechanistic, predictive, and experimentally testable models via chemical and physical laws. Applications include optimal foraging strategies, multi-scale comparison of aquatic and terrestrial ecosystems, impact of global carbon dioxide change, and cyanobacterial blooms. Ecotoxicology is a significant research area for the preservation and restoration of healthy ecosystems from industrial pollution. Via persistence and bifurcation analysis, I will perform risk assessment of toxicants on living organisms and their interactions. Spatial heterogeneity and natural delays are important for mechanistic modelling in biology. Examples include complex animal movement with spatial memory and spatial heterogeneity due to landscapes. Most of the proposed research will be in collaboration with empirical biologists or government researchers for data validation and calibration of new models and for decision making.