Introduction to Oscillation

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Environmental Modeling

Introduction to Oscillations



Oscillations are Common

• Oscillatory behavior is common in all types

of natural (physical, chemical, biological)

and human (engineering, industry,economic) systems

• Systems dynamics modeling is a powerful

tool to help understand the basis for andinfluence of oscillations on environmental

systems



First Example: Influence of

Variable Rainfall on Flower Growth• Flower growth model of “S-shaped” growth

from Chapter 6:area of flowers

growth decay

decay rateactual g rowth rate

intr insic g rowth rate

fraction occupied

~

growth rate multiplier suitable area

actual_growth_rate = intrinsic_growth_rate*growth_rate_multiplier

growth_rate_multiplier = GRAPH(fraction_occupied)



Growth Rate Multiplier for

Modeling S-Shaped Growth

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Fraction occupied

G

r o w t h r a t e m u l t i p l i e r



Analogy Between Logistic Growth Equation

and “Growth Rate Multiplier Approach”

• Logistic equation:

• dN/dt = r × N × f(N)

• f(N) = (1 – N/K)• K = carrying capacity

• Growth rate multiplier approach

• dN/dt = r × N × GRAPH(fraction_occupied)

• fraction_occupied = area_of_flowers/suitable_area

• If GRAPH(fraction_occupied) is linear with slope of negativeone, then we have recovered precisely the logistic growthequation



Analogy Between Logistic Growth Equation

and “Growth Rate Multiplier Approach”

• Growth rate multiplier approach

• dN/dt = r × N × (1 – area_of_flowers/suitable_area)

• Logistic equation:

• dN/dt = r × N × (1 – N/K)

• The two equations are identical because• N/K = area_of_flowers/suitable_area



“Growth Rate Multiplier Approach” is More

Flexible Than the Classical Logistic Equation

• Logistic equation has an analytical solution:

Nt = N0e

rt

/(1 + N0(e

rt

– 1))/K• However, no simple analytical solution exists if

growth rate multiplier is a nonlinear function of N

• In contrast, it’s easy to numerically simulate sucha system using the graphical function approach



“Growth Rate Multiplier Approach” is More

Flexible Than the Classical Logistic Equation

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Fraction occupied

G

r o w t h r a t e m u l t i p l i e r



First Example: Influence of Variable

Rainfall on Flower Growth• Assume rainfall varies sinusoidally around a mean

of 20 inches/yr with an amplitude of 15 inches/yrand a periodicity of 5 years:

• Rainfall = 20 + SINWAVE(15,5)

• Rainfall = 20 + 15*SIN(2*PI/5*TIME)

• Assume optimal rainfall for flower growth is 20

inches per year• Define relationship between intrinsic growth rateand rainfall using a nonlinear graphical function



Relationship Between Intrinsic

Growth Rate and Rainfall

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

Rainfall (inches/year)

I n t r i n s i c e g r o w t h r a t e ( 1 / y r )



Flower Model With Variable Rainfall

area of flowersgrowth decay

decay rate

actual growth rate

~

intrinsic growth rate

fraction occupied~

growth rate multiplier suitable area

rainfall




10:05 PM Sun, Oct 27, 2002

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0.00 5.00 10.00 15.00 20.00

Years

1:

1:

1:

2:

2:

2:

3:

3:

3:

0

30.

60.

0

1.D

2.D

1: rainfall 2: intrinsi c growth rate 3: actual growth rate

1 1 1 1

2 2 2 2

33

3

3

Period = 5 yrPeriod = 2.5 yr




10:10 PM Sun, Oct 27, 2002

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Years

1:

1:

1:

2:

2:

2:

3:

3:

3:

0

350

700

0

100

200

0

150

300

1: area of flowers 2: decay 3: growth

1

1

1 1

2

2

2 2

3

3 3 3




• Sinusoidal changes in rainfall causes largeswings in growth rate but only minor

swings in area and decay• General pattern of growth is S-shaped, with

a superimposed cycle of 2.5 year (comparedto 5 years for rainfall)

• Equilibrium flower area is lower than thatobtained with model employing constantoptimal intrinsic growth rate



General Conclusions

• Cycles imposed from outside the system

can be transformed as their affects “pass

through” the system • Periodicity can be modified as a result of

system dynamics

• Quantitative effect of external variationscan be moderated at the stocks in the system



Oscillations From Inside the System

• Consider oscillations that arise from structure

within the system

• New version of flower model in which in theimpact of the spreading area on growth is

lagged in time, i.e. there is a time lag (2 years)

before a change in fraction occupied translatesinto a change in growth rate• lagged_value_of_fraction = smth1(fraction_occupied,lag_time)



Structure of First-Order

Exponential Smoothing Process

lagged value offraction occupied

change in fraction occupied

fraction occupied lag time

change_in_fraction_occupied =

(fraction_occupied-lagged_value_of_fraction_occupied)/lag_time

1.0 2.0

0.0



Structure of First-Order


9:26 PM Mon, Oct 28, 2002

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0.00 2.50 5.00 7.50 10.00

Time

1:

1:

1:

0.00

0.50

1.00

1: lagged value of fraction occupied

1

1

11



Flower Model With Lagged

Effect of Area Coveragearea of flowers

growth decay

decay rate

actual growth rate

intrinsic growth rate

fraction occupied

~

growth rate multiplier

area available

lagged value of fraction

lag t ime



Flower Model With First Order

Lagged Effect of Area Coverage

9:37 PM Mon, Oct 28, 2002

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0.00 10.00 20.00 30.00 40.00

Years

1:

1:

1:

2:

2:

2:

3:

3:

3:

0

650

1300

1: area of flowers 2: decay 3: growth

1

11 1

2

2 2 2

3

3 3 3



Flower Model With First Order

Lagged Effect of Area Coverage• Area of flowers overshoots maximum

available area, which causes a major decline

in growth so that decay exceeds growth by 8th

year of simulation

• Area declines, which frees up space, whicheventually results in an increase in growth

• Variations in growth and decay eventuallyfade away as the system approaches dynamicequilibrium = “damped oscillation”



Higher Order Lags are Possible

• STELLA has built-in function for 1st, 3rd,

and nth order smoothing, which can be used

to produced any desired order of lag• The higher the order of the lag, the longer

the delay in impact

• Example = third order lag



Structure of Third Order


lagged value of

fraction occupied 1

change in fraction occupied 1

fraction occupied

lag t ime

lagged value of

fraction occupied 2

change in frac tion occupied 2

lagged value of

fraction occupied 3

change in frac tion occupied 3



Structure of Third Order


9:50 PM Mon, Oct 28, 2002

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0.00 2.50 5.00 7.50 10.00

Time

1:

1:

1:

2:

2:

2:

3:

3:

3:

0.00

0.50

1.00

0.00

0.45

0.90

1: lagg ed value of fraction occup… 2: lagg ed value of fraction occup… 3: lagg ed value of fraction occup…

1

1

11

2

2

2

2

3

3

3

3



Flower Model With First vs. Third Order

Lagged Effect of Area Coverage

9:54 PM Mon, Oct 28, 2002

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Years

1:

1:

1:

2:

2:

2:

0

750

1500

1: area of flowers fir rst order l ag 2: area of flowers third order lag

1

11 1

2

2 2

2



Flower Model With First vs. Third Order

Lagged Effect of Area Coverage• Third order lag shows more volatility

• Flower area shoots farther past the carrying

capacity of 1000 acres and goes throughlarge oscillations before dynamicequilibrium is achieved

• Increased volatility arises because of thelonger lag implicit in the third ordersmoothing



Further Examination of Lag Time Effect

• Compare simulations with third order

smoothing and lag times of 1, 2, or 3 years

• Longer lags lead to greater volatility

• Flower area in simulation with 3 year lag

time shoots up to greater than 2X the

carrying capacity



Flower Model With Third Order Lagged Effect

of Area Coverage and Variable Lag Time

10:01 PM Mon, Oct 28, 2002

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0.00 10.00 20.00 30.00 40.00

Years

1:

1:

1:

2:

2:

2:

3:

3:

3:

0

1250

2500

1: area of flowers 1 year third or… 2: area of flowers 2 year third or… 3: area of flowers 3 year third or…

1

1 1 1

2

2 22

3

3

33



Effects of Volatility Illustrated

• Plot growth and decay together with flower area for simulationwith 3 year time lag

• Flower area and growth rate increase in parallel even aftercarrying capacity is reached; flowers do not “feel” the effect ofspace limitation due to the time lag

• Once effect of space limitation kicks in, growth rate dropsrapidly to zero

• Active growth does not resume until ca. year 15, meanwhile

decay continues on• New growth spurt occurs at around year 20, utilizing space

freed-up during previous period of decline

• Magnitude of oscillations does not decline over time =“sustained oscillation”




10:16 PM Mon, Oct 28, 2002

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Years

1:

1:

1:

2:

2:

2:

3:

3:

3:

0

1250

2500

1: area of flowers 3 year third or… 2: growth 3 3: decay 3

1

1

1

1

22

2 2

33

3 3




• Key reason for sustained volatility of the model withlong time lag is the high intrinsic growth rate

• To illustrate, repeat simulation with different values of

the intrinsic growth rate and a 2 year lag time • Sustained oscillation (volatility) occurs with intrinsic

growth rate of 1.5/yr

• With intrinsic growth rate of 1.0/yr, oscillationsdampen over time

• With intrinsic growth rate of 0.5/yr, no oscillationsoccur (system is “overdamped”)



Influence of Intrinsic Growth

Rate on Volatility

10:28 PM Mon, Oct 28, 2002

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0.00 10.00 20.00 30.00 40.00

Years

1:

1:

1:

0

1500

3000

area of flowers 2 year third order lag 2: 1 - 2 - 3 -

1

1

1 1

2

2 2 2

3

3

3

3

r = 1.5/yr

r = 1.0/yr

r = 0.5/yr



Summary of Oscillatory Tendencies

• Simple flower model gives rise to three basic patterns of oscillatory behavior:

• Overdamped• Damped

• Sustained

depending on the values for lag time andintrinsic growth rate

• Can summarize the observed effects with a parameter space diagram



Oscillatory Behavior:

Parameter Space Diagram

+

+

+ +

+

0 0.5 1.0 1.5

Intrinsic growth rate (yr -1)

Lag time (yr)

1

2

3

Overdamped

Overdamped

Sustained

SustainedDamped

Critical

dampeningcurve



Critical Dampening Curve• Hastings (1997) analyzed a logistic growth model

with lags, and found that oscillations occurred onlywhen the product of the intrinsic growth rate and timelag (a dimensionless parameter ) was greater than 1.57

• Flower model is not identical to Hastings’s model, but there is sufficient similarity to warrant using hisfindings as a working hypothesis for position of thecritical dampening curve

• Define FMVI = “Flower Model Volatility Index” asthe product of the time lag and the intrinsic growthrate in the flower model

• FMVI = intrinsic growth rate x lag time



Curve For Critical Dampening

• Curve in our parameter space diagram was drawn

so that FMVI is 1.5 everywhere along the curve

• Assuming that the FMVI of 1.5 is analogous toHastings’s value of 1.57, hypothesize that

oscillations will appear only whenever the

parameter values land above the curve

• Results of the six simulations discussed previouslysupport this hypothesis



The Volatility Index

• The dimensionless parameter FMVI is a plausible

index of volatility because it reflects the tendency of

the system to overshoot its limit• Can be interpreted as the fractional growth of the

flowers during the time interval required for

information to feed back into the simulation

FMVI = growth rate (1/year) x lag time (year)

• The higher the index, the greater the tendency to

overshoot

Documents

Introduction to Oscillation