Temporal and Spatial Dynamics in Mathematical Ecology

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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.