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Unformatted text preview: MATH314, Modeling Realizable Phenomena Gidon Eshel, Physics Department, Bard College AnnandaleonHudson, NY 125045000, x7232, [email protected] . Lecture Set 1 1. Most Fundamental Steps in Model Building Typically but not universally: (1) Observe a phenomenon, plot it, view it, prod it; get curious about its working (2) Devise governing equations (3) Modify them as needed to be solved (e.g., make continuous spatial dependence discrete; parameterize irresolvable physics) (4) Write computer code to represent the modified equations in the chosen spacetime do main (5) Visualize and analyze the results, ascertain the model skill (6) Derive mechanistic insights and understanding into the physical phenomenon. 2. The Principle of Parsimony or Occam’s Razor The cardinal, agreed upon but not necessarily persuasive rule: if the simulation (model output) resembles reality well, and if the model cannot be simplified it wins! 3. Forward vs. Inverse Models 3.1. Forward. ICs BCs specified physics (model equations) simulated outcome 1 2 !"#$%&’(!)# +$)$%’, #(.,’/$012#34# #35.,’/$0 !"#$%&’(!)# %$#30.’,# 653#7/8 !"#"!"$%&’# ) % *% ! + ,+ * 3.2. Inverse. 4. Deterministic vs. Probabilistic (Stochastic) Models, Mixed Models 5. Nondimensionalizing 5.1. Posing the Problem. As an example, let’s consider an ecosystem comprising one preda tor and one prey species. Let N ( t ) be the number of individuals of the prey species, and P ( t ) denote the number of individuals of the predator species. Suppose that in isolation , the prey species is unaffected by the finiteness of resources, thus growing exponentially according to (1) dN dt = rN. Conversely, having no food in the absence of prey, the predator decreases exponentially accord ing to (2) dP dt = mP. Each of these is a separable equation readily solved analytically. For example, for N , this means (3) dN dt = rN = ⇒ dN N = rdt. Integrating both sides, (4) Z N ( t ) N (0) dN N = Z N ( t ) N (0) d ln N = r Z t dt where the second step follows from d ln x/dx = 1 /x . The solution is (5) ln N ( t ) ln N (0) ≡ ln N t ln N o = ln N t N o = rt. 3 Exponentiating both sides, (6) N t N o = e rt = ⇒ N t = e rt N o . Substituting this into the equation yields (7) dN dt = rN = ⇒ r e rt N o = r e rt N o which, by all accounts, holds, substantiating our solution. This solution lends itself to addressing stability of the individual species’ populations. If r > 0, e rt > 1 (because t ≥ 0 by definition), and N increases exponentially with time. This unbounded solution is said to be unstable . If, on the other hand, r < 0, e rt < 1, and the bounded solution relaxes back to zero, thus garnering the title stable . An additional richness is introduced by a complex r , with which the solution both grows or decays exponentially (corresponding to Re( r ) > 0 and Re( r ) M < 0, respectively) and oscillates with a period governed by Im( r )....
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 Fall '10
 MBELK
 Math, Exponential Function, Trigraph, Nondimensionalization, Gidon Eshel

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