Residence and First Passage Time Functionals in Heterogeneous Ecological Dispersion Multiscale problems continue to motivate important mathematical modeling and research. This proposal aims to develop and analyze models relevant to several examples from biology, ecology, oceanography and epidemiology, that involve interfacial effects defined by discontinuities in values of coefficients in the models. These phenomena occur on highly heterogeneous domains in which sharp or abrupt discontinuities in certain physical, chemical, or biological properties of the landscape occur in the coefficients of the basic equations. The Pis will analyze functionals of the associated processes, both for fragmented or patchy domains and for discrete graphical structures, to quantify the effects that smaller scale interfacial discontinuities have on macro scale variables, such as resident and occupation time functionals. In the first part of the proposal, the PIs will develop stochastic approaches to the advection-dispersion-reaction equations with discontinuous coefficients that model different biological processes. Unlike more classical physical models where the micro-scale interface conditions can be determined by macro-scale conservation laws, data on biological responses to interfacial boundaries can be quite different. The determination of the appropriate models requires the development of new micro-scale methods of analysis involving local time and the Ito-Tanaka stochastic calculus to uncover the appropriate macro-scale equations governing population densities and characteristic functionals of dispersion. In the second part of the proposal the Pis will develop numerical methods, Monte-Carlo stochastic particle schemes, and new methods of statistical parameter estimation for advection-dispersion equations involving discontinuous coefficients with special interface geometries relevant to key biological field data.
Natural physical processes, as well as certain anthropogenic activities, result in fragmented habitats to which species (animal, plants and bacteria) adapt or modify their behavior. Changes in the habitat configuration and/or its conditions, present new challenges and pose important broad new questions to scientists, policy makers and resource managers concerned with natural resources. Several contemporary problems in the biological and environmental sciences and engineering where such effects are reported to occur include: Bio-remediation of contaminated sediments in heterogeneous landscapes; Spread of infectious disease over fragmented habitats causing shifts in community structures possibly leading to invasion by exotic species; Species dispersal and sustainability in a heterogeneous environment affecting persistence of endangered species; Spatial localization of oceanic chlorophyll blooms impacting the fisheries industry. The specific mathematical issues common to these examples involve appropriate modeling of interfacial processes, i.e., mathematical discontinuities in the coefficients of the model equations, that affect the large scale behavior of species movement. The mathematical framework to be developed in this research is particularly aimed at assessing and quantifying interfacial effects on the large scale caused by these abrupt small -scale changes. This research will provide a mathematical framework and tools to support field and laboratory efforts to quantify and resolve fundamental questions about species dispersal through a combination of numerical and statistical algorithms, together with a theoretical mathematical analysis involving tools from deterministic and stochastic calculus.
The research conducted under this grant was focussed on improved understanding of sharp changes (discontinuities) in the rates at which dispersion occurs in biological and physical populations. The spatial locations of such discontinuities are referred to as interfaces, and are often observed as points or curves. Examples irange from the patch boundaries of Lupin flower patches, the natural habitat of the endangered Fenders Blue butterfly populations, to sea shelf breaks in the offshore slope of a coastal waterway responsible for upwellings of cold water and nutrients that can support high concentrations of fish, to the dispersion of substances injected into a saturated porous media. Typical quanties of interest to scientissts and resource managers.involve the time it takes to arrive at a location a specified distrance from its initial location, referred to as a breakthrough time, or breakthrough curve when viewed as a function of the distances. Other such ``functionals'' of interest include residence times spent in a given reegion, and the so-called local time spent at or near an interface. The main mathematical resutls that were achieved in this research involve the mathematical characterization of the special role played by the local time functional in the various interfacial conditions that can occur, and to explicitly mathematically demnostrate how these conditions influence the other functionals of interest to scientists. Although there is much more to achieve in this general area, mathematical results of this type can be of great value to scientists involved in the observation and measurement of interfacial discontinuities in dispersion rates. This is both from the point of view of assessing the consequence on functionals of primary interest, as wall as in informing statistical estimates of the interfacial rates.