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Harvesting in diffusive population models (Cont'd)

$ f(x,u)=a(x)-b(x)u$, $ a(x)$ denotes the population growth rate, $ b(x)> 0$ for $ x\in\overline{\Omega}$ describes the efforts of crowding, $ c(x)h(x)$ represents a constant yield harvesting rate, and $ u_0(x)$ is the initial population. Here, $ c(x)$ reflects the harvesting quota (a humanly controllable factor) and $ h(x)$ reflects the intrinsic properties of the region. We assume that $ 0 \leq h(x) \leq 1$ for all $ x\in\overline{\Omega}$. The Dirichlet condition $ u(x,t)=0$ on the boundary $ \partial \Omega $ refers to a hostile boundary, and the Neumann condition $ \partial u(t, x)/ \partial n = 0$ on $ \partial \Omega $ corresponds to the case of no migration through the boundary. The functions $ a(x), b(x),$ and $ h(x)$ are assumed continuous on $ \overline{\Omega}$, and $ c(x)$ is assumed to be piecewise continuous on $ \overline{\Omega}$.

We certainly do not expect the REU students to analyze the above problems as stated. Their involvement in mathematical research will mostly be in the one-dimensional case. However, some of their computational analysis will include the two-dimensional case. The students will learn the importance of sensitivity analysis as well as the stability and asymptotic properties of the solutions to these models and their relevance in population dynamics. Such an experience will surely inspire them to more advanced research, at a later stage, combining both mathematical analysis and computational simulations.

The 2003-2005 NSF REU students at MSU achieved great success on this research. We are confident that this research topic will be highly successful again with the 2009-2011 NSF REU students.

Reference Material:

Bluefin Tuna
2005 REU Paper (pdf file)
2004 REU Paper (pdf file)
2003 REU Paper (pdf file)
REU 2003-05
REU 2003-05 Reports

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