User:EGM6341.s11.team1.Chiu/S10 Mtg38

Mtg 38: Tue, 6 Apr 10

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$$ \underbrace{ \color{blue}Newton-Raphson}_{ \color{blue} NR} { \color{blue}method}: F(\hat{Z})=0 \ $$


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Similar triangles:


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$$ \frac{Z^{(k)}-Z^{(k+1)}}{1}= \frac{F(Z^{(k)})}{ \frac{dF(Z^{(k)})}{dZ}} \ $$


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$$ Z^{(k+1)}=Z^{(k)}-( \frac{dF(Z^{(k)})}{dZ})^{ \color{blue} -1} F(Z^{(k)}) \ $$


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End Newton-Raphson method

'''Time-step. algorithm (continued)'''

[[media: Egm6341.s10.mtg37.djvu| (3) page37-2:]] NR method


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$$ Z_{i+1}^{(k+1)}=Z_{i+1}^{(k)}-( \frac{dF(Z_{i+1}^{(k)})}{dZ})^{ \color{blue} -1} F(Z_{i+1}^{(k)}) \ $$


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Initial guess: $$ Z_{i+1}^{(0)}=Z_{i} \ $$


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Convergence: $$ \left | Z_{i+1}^{(k+1)}-Z_{i+1}^{(k)} \right | \leq \ \ $$  AbsTol, Absolute Tolerance


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or


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$$ \frac{ \left | Z_{i+1}^{(k+1)}-Z_{i+1}^{(k)} \right |}{ \left | Z_{i+1}^{(k)} \right |} \leq \ \ $$   RelTol, Relative Tolerance


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'''End Time-step. algorithm'''

Application: Mathematic Biology, Population Dynamics

Reference: Murray 2002

History: Leonardo of Pisa 1202, rabbit population

Continuous growth model


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


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


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End Continuous growth model

Malthus 1798, Simplest model:


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migration=0 births= $$ bx(t) \ $$ deaths= $$ dx(t) \ $$ $$ b,d \ $$ are constant.


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$$ x(t)=x_{0} \ exp \left [ (b-d)t \right ] \ $$


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End Simplest model

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As $$ t \to \infty, x \to \infty \ $$ not possible

Population construction by resources (land, food, ...)

fertility rate below replacement, level 2.1

$$ \color{blue} \exists \ $$ a limit to population growth.

Verhulst 1838, New model with growth limit


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$$ \underbrace{ \frac{dx}{dx \to { \color{blue} dt}}}_{ \color{blue} \dot{x}}= \underbrace{rx(1- \frac{x}{x_{max}})}_{ \color{blue} f(x)} \ $$

(1)
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Equilibrium points:  $$ \ f(x)=0 \ $$   (rate $$ \dot{x}=0 \ $$)


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$$ \hat{x}=0 \ \Rightarrow \ f(0)=0 \ $$

$$ \hat{x}=x_{max} \ \Rightarrow \ f(x_{max})=0 \ $$


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Linearization about equilibrium point $$ \hat{x} \ $$


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$$ x= \hat{x}+ \underset{ \color{blue} \underset{small \ perturbation \ about \ \hat{x}}{\uparrow}}y \ $$


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At $$ \hat{x}=0, f(x)=f(0+y)=ry(1- \frac{y}{x_{max}}) \ $$


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$$ f(x) \cong ry \ $$


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$$ \frac{dx}{dt}= \frac{d(0+y)}{dt}= \frac{dy}{dt} \ $$


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$$ \Rightarrow \frac{dy}{dt}=ry \ \Rightarrow \ y=y_{0}e^{rt} \ $$  unstable growth


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End New model with growth limit


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HW: Linearization about $$ \hat{x}=x_{max} \ $$


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$$ \frac{dy}{dt}=-ry \ \Rightarrow \ y=y_{0}e^{-rt} \ $$  stable growth


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$$ y(t) \to 0 \ $$ as $$ t \to \infty \ $$


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$$ x(t) \to x_{max} \ $$ as $$ t \to \infty \ $$


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