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ESPM-EEP%202010%20Lecture%208

ESPM-EEP%202010%20Lecture%208 - ESPM 104/EEP 115 Fall 2010...

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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 + 1) = x ( t ) + births deaths + net migration x ( t ) + b d ( ) x ( t ) + net migration No migration but arbitrary Δ t : x ( t + Δ t ) x ( t ) = ( b d ) x ( t ) ( ) Δ t In the limit: dx dt = Δ t →∞ lim x ( t + Δ t ) x ( t ) Δ t = ( b d ) x ( t ) Therefore defining r = b - d and x (0)= x 0 : dx dt = rx x ( t ) = x 0 e rt 2. Solve dx dt = rx , x (0) = x 0 , by integration. Integrate both sides: 1 x 0 t dx dt dt = r 0 t dt . LHS: 1 x 0 t dx dt dt = 1 x x (0) x ( t ) dx = ln x ( t ) ln x 0 = ln x ( t ) x 0 provide x 0 0. RHS: r 0 t dt = r ( t 0) = r t
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Therefore: x 0 > 0 : ln x ( t ) x 0 = r t x ( t ) = x 0 e r t . 3. Behavior of dx dt = Rx x ( t ) = x 0 e Rt : see SSG 6.1, Ex. 1 & 2. 4. Carlson’s yeast experiment. SSG 6.1 pp. 736-­૒4; Ex. 3.
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