Understanding how Nonlocal Diffusion Shapes Patterns in Biological Systems

University Of Houston, Houston TX

Investigators

Abstract

Many biological and physical systems are known for exhibiting interesting periodic structures, like striped and spot patterns. In many instances, like in the case of chemical reactions, these structures emerge from the interaction of two competing processes that lead to an instability. Roughly, the first mechanism relates to the intrinsic behavior of the system, while the second one corresponds to a transport, or diffusion process. While in most cases diffusion is the correct mechanism to consider, in many biological applications it does not capture the observed behavior. This is particularly true when modeling dispersion of populations or plant seeds, or when looking at hunting or foraging behavior of certain animals. In this project, the investigator studies how these nonlocal forms of dispersal affect pattern formation in physical and biological applications. The main motivation comes from arid ecosystems, which are known to exhibit patterns of banded vegetation. While most mathematical models use diffusion to represent dispersal of plant seeds, numerical experiments suggest that these equations, although more tractable, ignore critical information. For example, the analysis of these 'local' models shows that changes in the spacing between bands of vegetation due to reduced rainfall often signal a possible irreversible transition to a fully desert state. However, when nonlocal seed dispersal is considered, simulations indicate that the resilience of these ecosystems with respect to such changes is increased. The discrepancy between these two models highlights the need for more in-depth studies of long-range dispersal effects. To study these effects, a nonlocal Gray-Scott model will be used as a test case. This specific set of equations also provides a basis for interesting projects for graduate students and for summer research experiences for undergraduates. Because nonlocal processes are often modeled using convolution operators, the main obstacle for studying these systems comes from the limited set of mathematical and numerical tools available to analyze these maps. Indeed, most studies represent these operators using convolution kernels that have properties which facilitate their analysis. At the same time, numerical simulations often ignore the fact that convolution maps require information about the unknown outside the computational domain, and thus incorrectly implement boundary constraints. One of the project's goals is to provide methods for proving Fredholm properties for these maps by choosing as their domain a specific class of functions with algebraic decay. This work allows investigators to use the implicit function theorem to prove the existence of patterns. Since the class of functions being considered also provides information about the level of decay of solutions at infinity, this information can then be used to construct numerical schemes with appropriate boundary constraints. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.