UNDERGRADUATE RESEARCH IN APPLIED MATHEMATICS
SUMMER REU AT MISSISSIPPI STATE UNIVERSITY June 9August 7, 2004 

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Harvesting 
This project involves modeling of population dynamics in the presence of harvesting. This research topic is closely related to the important and fast developing areas of natural resource management and bioeconomics. Among many other applications in these areas is the fishery management and an understanding the fascinating migratory abilities of the Atlantic bluefin tunas (Thunnus thynnus). These fish can grow as big as 700 kg and longer than 3 meters, and they are preferred over other fish for sushi and sashimi; for example, one 200kg bluefin was recently sold at an auction in Japan for a record $390/lb. Despite a history of harvesting and exploitation spanning hundreds of years, little is known about the spatial dynamics of bluefin tuna movements. The status of current knowledge about these fish is outlined below:
The International Commission for the Conservation of Atlantic Tunas (ICCAT) manages this species of fish in two separate units: the eastern and western Atlantic, separated by the 45°W meridian. Due to a dramatic decline in the western bluefin population since the 1970s, the ICCAT set an allowable catch limit in the western Atlantic at approximately half the total catch of the 1970s. However, in the ICCAT stock assessment, the mixing of the western and eastern populations is assumed to be low, and the allowable catch limits in the western and eastern Atlantic are different. The data collected through satellites from the electronic tags inserted in samples of bluefin suggests that there is a greater mixing of the eastern and western bluefin populations than previously thought; thus, these two populations are interdependent. One conclusion is that, in order to rebuild the western Atlantic bluefin stock, in addition to limiting the catch in the western Atlantic, it is also necessary to curb the overfishing in the eastern Atlantic; this was also recently recommended by the ICCAT. There are two major breeding grounds for the Atlantic bluefin; namely, the Gulf of Mexico and the Mediterranean Sea. There is an urgent need to protect both these major spawning regions due to their direct influence on the western fishery.
The spatial dynamics and the mixing of the Atlantic bluefin tunas can be described by the reactiondiffusion partial differential equation
where we assume, for simplicity, that the parameters change only in one direction, i.e. we make the simplifying assumption that the fish move only either eastward or westward. Here u(t,x) denotes the density of bluefin at time t at location x in the bounded interval [0,1] representing the habitat, the positive parameter m is the rate of mixing, a(x) denotes the population growth rate, the positive function b(x) describes the efforts of crowding, c(x)h(x) represents a constant yield harvesting rate. In particular, c(x) reflects the harvesting quota (a humanly controllable factor) and h(x) reflects the intrinsic properties of the region. We assume that 0<h(x)<1. Let u_{0}(x) denote the initial population, i.e. u(0,x)=u_{0}(x). The Dirichlet boundary conditions u(t,0)=u(t,1)=0 refer to the hostile boundaries x=0 and x=1. The Neumann boundary conditions ¶u(t,0)/¶x=¶u(t,1)/¶x=0 correspond to the case of no migration through the boundary points x=0 and x=1.
Some mathematical problems can be summarized as follows:
Suppose that c(x)=c_{1} when xÎ[0,1/2] and c(x)=c_{2} when xÎ[1/2,1], where c_{1} and c_{2} are two constants. Study the existence of stable steadystate solution of the reactiondiffusion equation, and also analyze the spatial pattern of the steady state when c_{1} is much larger than c_{2}. The outcome of this study is expected to test the recommendation that overfishing in the eastern Atlantic needs to be ended. Another way to rebuild the fishery stock is to set up some nofishing zones (a recent experimentation has been done in the Florida Keys). The positive impact of nofishing zones on the bluefin population has been speculated, but not yet scientifically tested. This impact can be analyzed as follows: Suppose that there is a subregion, say the interval [1/3,2/3], on which c(x)=0. How does the nonfishing zone [1/3,2/3] influence the spatial pattern of the steadystate solution? Another important problem is related to the ideal location of the nofishing zone: should it be in the favorite breeding bed or in some other subregion? In the breeding region, the growth rate a(x) is much larger than that in other subregions. The location of larger concentration of bluefin can be investigated when some parameters are small or large. This helps to understand the behavior of these fish under extreme conditions. An important question is to understand the dependence of the bluefin concentration on the harvesting quota c(x) and on the initial population u_{0}(x). This helps us to understand what initial populations and what harvesting quotas can lead to the best use of the resources. Investigate the above problems corresponding to various movement patterns of bluefin tuna across the interface x=1/2 separating the two regions.
Some sample research problems for the students are outlined below:
Develop a numerical scheme (e.g. finite difference) to approximate solutions to the onedimensional harvesting problem on [0,1] for a specified harvesting pattern and a variety of initial population distributions. Carry out the simulations over different time periods and ask and answer questions such as "what happens as time increases,'' "how does the initial condition effect the longterm behavior,'' and "how does the solution of the problem with ¶u/¶t=0 relate to what is being observed as time gets large.'' Find the steadystate solutions to the problem. This can be done both analytically and numerically. Numerical approaches can include shooting methods, finitedifference methods, and collocation methods. For example, one can first explore what happens when c(x)=0 and then analyze the solutions with c(x)h(x)=c[xx^{2}] as the constant parameter c is increased. Explore the bluefin problem with h(x)=xx^{2} and c(x)=c_{1} when xÎ[0,1/2] and c(x)=c_{2} when xÎ[1/2,1], where c_{1} and c_{2} are two constants. If we interpret c_{1} and c_{2} as the regulated fishing quotas in two neighboring nations, would the case c_{1}¹c_{2} mean that one nation is actually subsidizing the aquaculture of the other? How can the two nations cooperate to make the best use of their fishing resources? Suppose that harvesting is prohibited in a subinterval of [0,1]. Can this result in a higher overall fish population? For some specified h(x), in what subinterval should one choose c(x)=0 so that the fish population would flourish?
For further information you may also contact Dr. R. Shivaji (email: shivaji@math.msstate.edu).
