User:Egm6341.s10.team3.heejun/Meeting 38 transcript

 '''MTG 38: Tue. 6 Apr 10'''


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Note:  Typeset of Transparencies, not lecture transcript  Heejun Chung 20:57, 11 August 2010 (UTC)

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Newton-Raphson(NR) Method: $$ \displaystyle F(\hat{Z}) = 0 $$



Similar triangles:


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 * $$ \displaystyle \frac{Z^{(k)}-Z^{(k+1)}}{1} = \frac{F(Z^{(k)})}{\frac{dF (Z^{(k)})}{dZ}} $$
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 * $$ \displaystyle Z^{(k+1)} = Z^{(k)} - (\frac{dF (Z^{(k)})}{dZ})^{-1} F(Z^{(k)}) $$
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Time-step algorithm (continued) (3) [[media: Egm6341.s10.mtg37.djvu | Page 37-2]]:  NR method


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 * $$ \displaystyle Z_{i+1}^{(k+1)} = Z^{(k)}_{i+1} - [\frac{dF(Z^{(k)}_{i+1})}{dZ}]^{-1} F(Z^{(k)}_{i+1}) $$
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 * Initial guess: $$ \displaystyle  Z^{(0)}_{i+1} = Z_{i} $$
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Convergence:


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 * $$ \displaystyle |Z_{i+1}^{(k+1)} - Z_{i+1}^{(k)}|\,\, \leq\,\, $$   AbsTol (Absolute Tolerance)
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or


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 * $$ \displaystyle \frac{|Z_{i+1}^{(k+1)} - Z_{i+1}^{(k)}|}{|Z_{i+1}^{(k)}|}\,\, \leq\,\, $$   RelTol (Relative Tolerance)
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Application: Mathematical Biology Population Dynamics Reference: Murray 2002 History: Leonardo of Pisa 1202, rabiit population

Continuous growth model

$$ \displaystyle x(t): $$   population at time $$ \displaystyle t $$ (number of members of a species)

$$ \displaystyle \frac{dx(t)}{dt} = $$   births - deaths + migration

Simplest model: (Malthus 1798)


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$$ \displaystyle births = bx(t) $$ $$ \displaystyle deaths = dx(t) $$
 * $$ \displaystyle migration = 0 $$
 * $$ \displaystyle migration = 0 $$
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$$ \displaystyle b$$   and $$ \displaystyle d $$ are constant


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 * $$ \displaystyle x(t) = x_{0}\,\, exp[(b-d)t] $$
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As $$ \displaystyle t \to \infty, \,\, x \to \infty $$ not possible population constrain by resources (land, food,...) fertility rate below replacement level 2.1

$$ \displaystyle \exists a $$   limit to population growth

Verhulst 1838, New model with growth limit


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 * $$ \displaystyle \underbrace{\frac{dx}{dt}}_{\color{blue}{\dot{x}}} = \underbrace{rx (1- \frac{x}{x_{max}})}_{\color{blue}{f(x)}} $$
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Equilibrium points:
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$$ \displaystyle \hat{x}=0\,\, \Rightarrow\,\, f(0)=0 $$ $$ \displaystyle \hat{x}=x_{max}\,\, \Rightarrow\,\, f(x_{max})=0 $$
 * $$ \displaystyle f(x)=0\,\, (rate\,\, \dot{x}=0) $$
 * $$ \displaystyle f(x)=0\,\, (rate\,\, \dot{x}=0) $$
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Linearization about equilibrium point $$ \displaystyle \hat{x} $$


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Where $$ \displaystyle \color{blue}{y=} $$ small perturbation about $$ \displaystyle \hat{x} $$ At $$ \displaystyle \hat{x}=0,\,\, f(x) = f(0+y) = ry(1- \frac{y}{x_{max}}) $$
 * $$ \displaystyle x= \hat{x} + y $$
 * $$ \displaystyle x= \hat{x} + y $$
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$$ \displaystyle \frac{dx}{dt} = \frac{d(0+y)}{dt} = \frac{dy}{dt} $$ $$ \displaystyle \Rightarrow \frac{dy}{dt} = ry \Rightarrow \underbrace{y=y_{0}e^{rt}}_{\color{blue}{unstable\,\, growth}} $$
 * $$ \displaystyle f(x) \simeq ry $$
 * $$ \displaystyle f(x) \simeq ry $$
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HW: Linearization about the equilibrium point $$ \displaystyle \hat{x} = x_{max} $$
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$$ \displaystyle y(t) \to 0\,\, as\,\, t \to \infty $$ $$ \displaystyle x(t) \to x_{max}\,\, as\,\, t \to \infty $$
 * $$ \displaystyle \frac{dy}{dt} = -ry \Rightarrow \underbrace{y=y_{0}e^{-rt}}_{\color{blue}{stable\,\, growth}} $$
 * $$ \displaystyle \frac{dy}{dt} = -ry \Rightarrow \underbrace{y=y_{0}e^{-rt}}_{\color{blue}{stable\,\, growth}} $$
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