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。主要目标是生态模型的机制制定和严格的分析。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久性理论和传播速度，以及延迟微分方程的分支理论及其在描述基于记忆的动物运动的延迟扩散算子中的应用。丰富的动力学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基本的微观途径。它包括多个生物尺度，并允许通过化学和物理定律制定强大的机制，预测和实验可测试的模型。应用包括最佳觅食策略，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，以及蓝藻水华。生态毒理学是一个重要的研究领域，为保护和恢复健康的生态系统从工业污染。通过持久性和分歧分析，我将进行风险评估的有毒物质对生物体和它们的相互作用。空间异质性和自然延迟对于生物学中的机械建模是重要的。例子包括复杂的动物运动与空间记忆和空间异质性，由于景观。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，进行数据验证和新模型的校准以及决策制定。

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.