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mockMidtermF10solutions - MATH314 Modeling Realizable...

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MATH314, Modeling Realizable Phenomena Practice Midterm Solutions Gidon Eshel, Physics Department, Bard College Annandale-on-Hudson, NY 12504-5000, x-7232, [email protected] . Remember: succinct and to the point and very clearly printed, no cursive ever. (1) Explain clearly in a sentence the Principle of Parsimony. If the simulation (model output) resembles reality well, and if the model cannot be simplified – it wins! (2) For the scalar linear model dx dt = αx please (a) point out the state vector, It is the 1-vector (a scalar) x . (b) obtain the general solution and state what information you need for it to be com- plete, dx dt = αx = dx x = αdt. = x ( t ) x (0) dx x = x ( t ) x (0) d ln x = α t 0 dt ln x ( t ) - ln x (0) ln x t - ln x o = ln x t x o = αt = x t x o = e αt = x t = e αt x o . So x o is the piece of information needed. (c) define, and characterize the condition(s) for, instability. If α > 0 (which means that e αt > 1 for t > 0 ), the model is unstable. (3) For the model df dt = αf - α 10 fg dg dt = 10 βfg - βg, please (a) Write down the state vector. (1) x = f g . 1
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2 (b) Identify the coefficients and describe briefly their biological roles. α is f ’s intrinsic growth rate. β is g ’s intrinsic death rate. The interaction terms’ coefficients are given in terms of the above two. (c) Point out sources of nonlinearities. In f ’s evolution equation, it is the 2nd term, - αfg/ 10 , and in g ’s evolution equa- tion, it is the 1st term, 10 βfg/ 10 , because both contain mutually multiplied state variables.
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