Models for Social, Ecological, and Biological Systems: Narrowing the Gap Between Theory and Applications

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This research is aimed at development of mathematical tools for understanding and predicting behavior of a variety of important and apparently dissimilar biological phenomena and sociological processes, such as the spread of diseases, ecological changes, or the distribution and prevention of crime. Some salient features of these diverse phenomena can be described by mathematical models based on reaction-advection-diffusion equations that have been used extensively to investigate fundamental and ubiquitous phenomena in several areas of biology, and more recently in the social sciences. While these models are simplified versions of reality, their mathematical analysis has contributed to the understanding of many important phenomena. At the same time, the underlying equations can be extremely interesting and challenging from the mathematical point of view as their solutions can exhibit rich behaviors, e.g., pattern formation and traveling wave structures. The behavior of solutions reflects critical features of the system that mathematics models and have clear and convincing parallels with the evolution of real-world systems. However, there is still a large gap between real world systems and their more tractable mathematical models. One of the goals of this project is bridging this gap through the use of accumulated concrete data and the corresponding calibration and modification of mathematical models. Of particular importance is the analysis of systems which include heterogeneous environments, the understanding of the effects that non-local dispersal has on the behavior of the solutions, and the validation of these models with real-world data. Efforts will be focused on four reaction-advection-diffusion systems where the heterogeneities are due to: climate change in an ecological context, non-local and asymmetrical spread of information in the context of riots, income heterogeneities in the context of social segregation, and environment heterogeneities due to specific concentrated inputs. The primary goal of the project is to understand how heterogeneous environments and non-local dispersal impact solution patterns in both a general class of nonlinear reaction-diffusion equations as well in specific ecological and social contexts. Throughout this project the investigators will maintain and foster contacts with social scientists in order to conduct the much needed discourse between the mathematical theory and the applications. This project will focus on the development of an extensive theory for reaction-advection-diffusion systems in heterogeneous environments with applications in ecology and sociology. In particular, the objective of this research is three-fold: to expand the current mathematical theory for local and non-local reaction-advection-diffusion systems in heterogeneous environments, to gain insight into various (ecological, sociological, and biological) complex systems by modeling them using these types of systems, and to take an initial step toward bridging the gap between basic mathematical models and the complex real-world systems they aim to describe by incorporating the use of data. From the qualitative perspective, this work will be mainly concerned with exploring the effects that various dispersal mechanism have on the propagation (and lack thereof) of a solution in a heterogeneous environment: for example, determining the spreading speed, the existence of traveling wave solutions, pulsating fronts, traveling pulses, or generalized fronts in heterogeneous systems. Additionally, the fundamental issues of the global well-posedness of solutions to these systems, intermediate and long-term asymptotics, and existence and uniqueness of non-trivial steady-state solutions will be rigorously analyzed.