L32 - Hand Examples Fall 2003 Math 308/501502 3...

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Fall 2003 Math 308/501–502 3 Mathematical Models 3.2 Compartmental Analysis Mon, 15/Sep c 2003, Art Belmonte Summary Mixing Problems Mixing problems deal with the amount x ( t ) of a substance in solution in a tank. The solution may be a salt solution, sugar solution, blood, etc. Our starting point is the classical balance law, which says that net rate = rate in - rate out dx dt = rate in - rate out where the rates are given as flow rate × concentration. Malthusian Population Model A population p , initially of size p 0 , whose rate of change is given by p 0 = kp , experiences exponential growth: p ( t ) = p 0 e kt . The parameter k is a positive constant called the reproductive rate . This model assumes that there is no limit on food or space in the population’s environment. Logistic Population Model If there are limits on food and/or space, another model for a population p , initially of size p 0 , is to assume the rate of growth is given by p 0 = - Ap ( p - p 1 ) . Here the parameters A and p 1 (the carrying capactity ) are positive constants. Solving for p gives p ( t ) = p 0 p 1 p 0 + ( p 1 - p 0 ) e - Ap 1 t . The parameter A is determined by the population’s birth and death rates as well as by environmental factors. As t → ∞ , we see that p ( t ) p 1 . In other words, the carrying capacity is the limiting population. Evaluation of parameters Using experimental data (measuring the population at various times), we may determine numerical values of the parameters involved in the models. This may involve algebra, linear regression, and/or the method of least squares. Fortunately, MATLAB and/or calculators assist us in this regard. Hand Examples Example A A tank initially contains 100 gal of a salt-water solution containing 0 . 05 = 1 20 lb of salt for each gallon of water. At time zero, pure water [containing no salt] is poured into the tank at a flow rate of 2 gal per minute. Simultaneously, a drain is opened at the bottom of the tank that allows salt-water solution to leave the tank at a flow rate of 3 gal per minute. What will be the salt content in the tank when precisely 50 gal of salt solution remain?
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