Grant: Discovery Grants Program - Individual

The overarching goal of my research program is to develop realistic models, underlying mathematical theory and analytical tools to help us understand and predict the long-term behavior of populations of aquatic organisms in river ecosystems. Mathematically, I aim at expanding and contributing to the qualitative theory of reaction-diffusion-advection and integro-differential equations used in mathematical ecology. Ecologically, my research advances the understanding of how geometric and hydrological features of advective habitats affect community composition, biodiversity and species abundance, and attempts to explain the mechanisms behind species persistence and spatial distribution. My interdisciplinary program provides ample training opportunities for undergraduate and graduate students.Modeling spatial population dynamics in advective habitats brings us closer to understanding the various mechanisms and adaptations involved in species persistence. The practical importance of such models goes beyond biological curiosity. Increasing water and power demands by humans can change flow speed. Thus, it is crucial to be able to predict how these changes affect river ecosystems and their biodiversity. My past work on competition in river ecosystems showed that flow speed changes can alter the ecological balance by giving some species a competitive advantage, or even lead to extinctions or invasions by new species. The essence of my proposal is to develop spatial population dynamics models by starting with a very basic idealized setting, and expanding it in three different directions, each of which reflects one aspect of biological complexity: (1) the complexity of the nature of population growth (linear growth, logistic growth or modified logistic growth known as Allee effect) and/or presence of different life stages (e.g. larvae and adults) or compartments (e.g. mobile and immobile) of the same species; (2) the number of interacting (competing) species present in the habitat; (3) the geometric complexity of the habitat (single river reach or a complex river network). During the lifetime of this award and beyond I aim to work with various combinations of these levels of complexity so that a general theory of population dynamics in rivers can emerge. I focus on nonlinear models as more realistic representations of population dynamics. I study their long-term behavior which usually comes in the form of stationary (steady state) solutions. Specific mathematical tools I use include geometric methods for analyzing steady states solutions, and reduction of spatial models to spatially implicit versions. This proposal consists of three parts: (1) dynamics of single species in a river reach; (2) single species in a river network; (3) competition in both a river patch and a river network. Each component includes projects appropriate for training at undergraduate and graduate levels.