ESPM-EEP%202010%20Lecture%208

ESPM-EEP%202010%20Lecture%208 - ESPM 104/EEP 115 Fall...

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Unformatted text preview: ESPM 104/EEP 115 Fall 2010: Lecture 8 1. Relationship between discrete and continuous time models: Discrete model: b and d are the per ­capita birth and death rates respectively ⎧ x (t ) + births − deaths + net migration ⎪ x (t + 1) = ⎨ x (t ) + b − d x (t ) + net migration ⎪ ⎩ ( ) No migration but arbitrary Δt: x (t + Δt ) − x (t ) = ( b − d ) x (t ) Δt In the limit: ⎛ x (t + Δt ) − x (t ) ⎞ dx = lim ⎜ ⎟ = ( b − d ) x (t ) dt Δt → ∞ ⎝ Δt ⎠ Therefore defining r=b-d and x(0)=x0: dx = rx ⇔ x (t ) = x0 ert dt dx 2. Solve = rx , x (0) = x0 , by integration. dt t ′ 1 dx t′ dt = ∫ rdt . Integrate both sides: ∫ 0 x dt 0 LHS: t ′ 1 dx x(t′ ) 1 x (t ′ ) dt = ∫ dx = ln x (t ′ ) − ln x0 = ln provide x0 ≠ 0. ∫0 x dt x(0) x x0 RHS: ( ) ∫ t′ 0 rdt = r (t ′ − 0) = rt ′ ⎛ x (t ′ ) ⎞ rt ′ Therefore: x0 > 0 : ln ⎜ ⎟ = rt ′ ⇔ x (t ′ ) = x0 e . ⎝ x0 ⎠ dx = Rx ⇔ x (t ) = x0 e Rt : see SSG 6.1, Ex. 1 & 2. dt 4. Carlson’s yeast experiment. SSG 6.1 pp. 736 ­4; Ex. 3. Estimate R; Ex. 4. Doubling Time. 5. Principle of Parsimony: Occam’s Razor (http://en.wikipedia.org/wiki/Occam%27s_razor); KISS Principle (keep it simple stupid: http://en.wikipedia.org/wiki/KISS_principle) 6. Mixing Models: SSG Section 6.3 3. Behavior of 7. Lake Pollution Model: SSG 6.3, Ex. 4. Read for yourself. 8. Von Bertalanffy growth equation, SSG 6.3, Ex 6. (http://en.wikipedia.org/wiki/Ludwig_von_Bertalanffy). ⎛ ⎛ L − L0 ⎞ − kt ⎞ dL = k ( L∞ − L) ⇔ L(t ) = L∞ ⎜ 1 − ⎜ ∞ e ⎟ dt L∞ ⎟ ⎠ ⎝⎝ ⎠ ...
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This note was uploaded on 02/09/2011 for the course EEP 115 taught by Professor Waynem.getz during the Fall '10 term at Berkeley.

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