In this paper, we study the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains. The LANS-α equations are able to accurately reproduce the large-scale motion (at scales larger than α > 0) of the Navier-Stokes equations while filtering or averaging over the motion of the fluid at scales smaller than α, an a priori fixed spatial scale.We prove the global well-posedness of weak H 1 solutions for the case of no-slip boundary conditions in three dimensions, generalizing the periodic-box results of [8]. We make use of the new formulation of the LANS-α equations on bounded domains given in [20] and [14], which reveals the additional boundary conditions necessary to obtain well-posedness. The uniform estimates yield global attractors; the bound for the dimension of the global attractor in 3D exactly follows the periodic box case of [8]. In 2D, our bound is α-independent and is similar to the bound for the global attractor for the 2D Navier-Stokes equations.1991 Mathematics Subject Classification. 35Q35. Key words and phrases. Lagrangian averaged Navier-Stokes, Turbulence, Model. 1 Some authors had previously referred to the LANS-α equations as the Viscous Camassa-Holm equations.
Swimmer's itch is an emerging disease caused by flatworm parasites that often use water birds as definitive hosts. When parasite larvae penetrate human skin they initiate localized inflammation that leads to intense itching. Concerns about this issue have been growing recently due to an apparent increase in the global occurrence of swimmer's itch and its subsequent impacts on recreational activities and associated revenues. Past study has identified the common merganser as a key definitive host for these worms in the United States; a number of snail species serve as intermediate hosts. Although previous attempts at controlling swimmer's itch have targeted snails, a handful of efforts have concentrated on treating water birds with the anthelmintic drug, praziquantel. We construct a mathematical model of swimmer's itch and its treatment within the infected merganser population. Our goal is to identify merganser treatment regimes that minimize the number of infected snails thereby reducing the risk of human infections. Optimal control of bird hosts is defined analytically and we include numerical simulations assuming different resource‐allocation strategies. Results from the study may help identify treatment protocols that lower merganser infection rates and ultimately reduce the occurrence of swimmer's itch in freshwater systems throughout the Midwest. Recommendations for Resource Managers Regardless of the time and monetary resources available, praziquantel treatment frequency should increase as mergansers arrive on the lake with continued treatments (albeit at reduced levels) until the end of the residency period. Allocating plenty of resources towards the treatment of mergansers predicted a sharp drop in infected birds, which then remained close to zero throughout the remainder of the residency period. This approach reduced schistosome infection in mergansers and kept snail infections within the idealized range during times of peak recreational activity. Consequently, human cases of swimmer's itch would be expected to be low to nonexistent. Furthermore, our treatment‐longevity computation suggested that subsequent praziquantel dosing would not be required for a number of years. Under more limited resources, the number of birds treated per day was much smaller throughout the residency period; however, even under these circumstances (which equated to treating approximately one bird every 5 days), simulated infected merganser densities were reduced to the point where snail infections remained below epidemic levels through to the end of the recreational period. Treatment longevity was shorter compared with the high‐resource option, but still extended 122 days into Season 2 (posttreatment). We also used our model to investigate situations where lake managers and/or federal agencies might be taxed in terms of the time available to continuously treat mergansers on a given lake. An individual scientist may only have a single day (or two) to dose birds, rather than continuously administering praziquantel throug...
Mathematical and computational methods are vital to many areas of contemporary biological research, such as genomics, molecular modeling, structural biology, ecology, evolutionary biology, neurobiology, and systems biology. As such, the contemporary life science student needs to be exposed to, if not well-versed in, many areas of mathematics to keep pace. However, traditional ways of teaching mathematics may not be able to provide life science majors the skills and experiences necessary to effectively use mathematics in their careers as practitioners and/or researchers, as these skills and experiences (for example, mathematical modeling and interdisciplinary collaboration) are difficult to teach using lecture-style approaches. In this paper the authors describe the implementation and assessment of a flipped-classroom approach to teaching a sophomore-level mathematical biology course for life science majors.
ABSTRACT:Bithynia tentaculata is an aquatic invasive snail first detected in the upper Mississippi River (UMR) in 2002. The snail harbors a number of parasitic trematode species, including Sphaeridiotrema pseudoglobulus, that have been implicated in waterfowl mortality in the region. We assessed the capacity of S. pseudoglobulus cercariae to infect B. tentaculata and native snails found in the UMR. Four snail species (one invasive and three native) were individually exposed to S. pseudoglobulus larvae and all were successfully infected. A subsequent experiment examining infection patterns in invasive and native hosts exposed singly or in mixed treatments revealed no difference in parasite establishment among snail species. Our results add to our understanding of S. pseudoglobulus transmission and provide insight into processes underlying waterfowl disease in the UMR.
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