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Unformatted text preview: MATH314, Modeling Realizable Phenomena Practice Midterm Solutions Gidon Eshel, Physics Department, Bard College Annandale-on-Hudson, NY 12504-5000, x-7232, firstname.lastname@example.org . 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 = x 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 = x = dx x = dt. = Z x ( t ) x (0) dx x = Z x ( t ) x (0) d ln x = Z t dt ln x ( t )- ln x (0) ln x t- ln x o = ln x t x o = t = x t x o = e t = x t = e t x o . So x o is the piece of information needed. (c) define, and characterize the condition(s) for, instability. If > (which means that e t > 1 for t > ), the model is unstable. (3) For the model df dt = f- 10 fg dg dt = 10 fg- g, please (a) Write down the state vector. (1) x = f g ! . 1 2 (b) Identify the coefficients and describe briefly their biological roles. is f s intrinsic growth rate. 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,- fg/ 10 , and in g s evolution equa- tion, it is the 1st term, 10 fg/ 10 , because both contain mutually multiplied state variables....
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This note was uploaded on 10/14/2010 for the course MATH 314 taught by Professor Mbelk during the Spring '10 term at Bard College.
- Spring '10