3'2 Unconstrained Growth

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3.2 Unconstrained Growth
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Malthusian Population Model

• The power of population is indefinitely greater
  than the power in the earth to produce
  subsistence for man
• T. Malthus
• Mathematically

Thomas Malthus (1766 1834)
or
Differential equation
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Malthusian Population Model
(instantaneous, continuous) growth rate constant
of proportionality
r 0.1 Initial condition P(0) 100
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Finite Difference Equation

• In a system-dynamics tool like Vensim (or in a
  computer program), we simulate continuous time
  via small, discrete steps.
• So instead of using dP/dt for growth, we have
  DP/Dt

• Then solve for population(t) .

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Finite Difference Equation
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Finite Difference Equation

• In other words
• new_population old_population
  change_in_population

• In general, a finite difference equation has the
  form
• new_value old_value change_in_value
• Such an equation is an approximation to a
  differential equation (equal in the limit as Dt
  approaches 0)

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Quick Review Question 1

• Consider the differential equation dQ/dt
  -0.0004Q, with Q0 200.
• Using delta notation, give a finite difference
  equation corresponding to the differential
  equation.
• At time t 9.0 sec, give the time at the
  previous time step, where Dt 0.5 sec.
• If Q(t-Dt) 199.32 and Q(t) 199.28, give DQ.

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Quick Review Question 2

• Evaluate to six decimal places population(.045),
  the population at the next time interval after
  the end of Table 3.2.1.

Table 3.2.1 Table of Estimated Populations,
Where the Initial Population is 100, the
Continuous Growth Rate is 10 per Hour, and the
Time Step is 0.005 hr ____________________________
_______________________________________________ t
population(t) population(t-Dt) (growth)
Dt 0.000 100.000000 0.005 100.050000 100.050
000 10.000000 0.005

• Lets do it in Excel.

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Quick Review Question 2
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Simulation Algorithms Background

• An algorithm is an explicit step-by-step
  procedure for solving a problem.
• Basic building blocks are
• Sequencing (one instruction after another)
• Conditionals (IF THEN ELSE)
• Looping (For 100 steps, do the following)
• Assignment statements use left arrows x ? x 1

Al-Khwarizmi (ca. 780-850)
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Algorithm 1 Unconstrained Growth
initialize simulationLength initialize
population initialize growthRate initialize
length of time step Dt numIterations ?
simulationLength / Dt for i going from 1 through
numIterations do growth ? growthRate
population population ? population growth
Dt t ? i Dt display t, growth, and population
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Removing Loop Invariants

• If we dont need to display growth, we can
  remove the implicit, loop-invariant product
  growthRate Dt used to compute population

for i going from 1 through numIterations
do growth ? growthRate population population
? population growth Dt t ? i Dt display t,
growth, and population
population ? population growthRate Dt
population
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Removing Loop Invariants

• Then we save time by computing growthRate Dt
  just once, before the loop

initialize simulationLength initialize
population initialize growthRate initialize
length of time step Dt numIterations ?
simulationLength / Dt growthRatePerStep ?
growthRate Dt for i going from 1 through
numIterations do population ? population
growthRatePerStep population t ? i Dt display
t and population
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Analytical Solutions

• Some problems can be solved analytically,
  without simulation
• For example, calculus tells us that the solution
  to dP/dt 0.10P with initial
  condition P0 100 is P
  100e0.10t
• If such solutions exist, we should use them.
  But the point of modeling a system is usually
  that no analytical solution exists.

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Analytical Solution via Indefinite Integrals

• Separation of variables move dependent variable
  (P(t)) and independent variable (t) to opposite
  sides of equal sign

• Then integrate both sides

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Analytical Solution via Indefinite Integrals

• Solve the integral, using e.g. a free online
  tool like http//integrals.wolfram.com

• Some tools use log for ln
• Dont forget to add constant C

• Solve for P using algebra fact that eln(x) x

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Completion of Analytical Solution

• Need constant k in
• We know P 100 at t 0, so

• So analytical solution is

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General Solution to Differential Equation for
Unconstrained Growth

• In general, the solution to

with initial population P0
is

• We can observe how e emerges from the limit as
  Dt approaches 0.

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Deriving e

• Consider (again) the compound interest example
  how much money from initial principle P0 after t
  years at yearly rate r ?
• Compounded yearly P P0(1 r)t
• Compounded monthly P P0(1 r/12)12t
• Compounded weekly P P0(1 r/52)52t
• So what is the limit as Dt approaches 0 of
  (1rDt)t/Dt ?
• We can try this out in Excel.

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Deriving e
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Quick Review Question 3

• Give the solution to the differential equation

where P0 57
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Unconstrained Decay

• For some systems r is negative
• E.g., radioactive decay of carbon-14

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Unconstrained Decay
half-life (time to decay to half original amount)
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Quick Review Question 4

• Radium-226 has a continuous decay rate of about
  0.0427869 per year. Determine its half-life in
  whole years.

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Quick Review Question 4

• Radium-226 has a continuous decay rate of about
  0.0427869 per year. Determine its half-life in
  whole years.

• Answer 1620 years