BSC 417/517 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 and influence 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 growth rate

intrinsic growth 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

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0.2

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0 0.2 0.4 0.6 0.8 1

Fraction occupied

Gro

wth

rat

e m

ult

ipli

er

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 negative one, then we have recovered precisely the logistic growth equation

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 = N0ert/(1 + N0(ert –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 such a system using the graphical function approach

“Growth Rate Multiplier Approach” is More Flexible Than the Classical Logistic Equation

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Fraction occupied

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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/yr and 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 rate and rainfall using a nonlinear graphical function

Relationship Between Intrinsic Growth Rate and Rainfall

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0 5 10 15 20 25 30 35 40

Rainfall (inches/year)

Intr

insi

ce g

row

th r

ate

(1/y

r)

Flower Model With Variable Rainfall

area of flowersgrowth decay

decay rateactual growth rate

~

intrinsic growth rate

fraction occupied

~

growth rate multiplier suitable area

rainfall

Flower Model With Variable Rainfall

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60.

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1.D

2.D

1: rainfall 2: intrinsic growth rate 3: actual growth rate

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2 2 2 2

33

3

3

Period = 5 yrPeriod = 2.5 yr

Flower Model With Variable Rainfall

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1: area of flowers 2: decay 3: growth

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Flower Model With Variable Rainfall

• Sinusoidal changes in rainfall causes large swings 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 (compared to 5 years for rainfall)

• Equilibrium flower area is lower than that obtained with model employing constant optimal 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 variations can 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 the impact 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 translates into a change in growth rate• lagged_value_of_fraction = smth1(fraction_occupied,lag_time)

Structure of First-Order Exponential Smoothing Process

lagged value of

fraction 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 Exponential Smoothing Process

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Time

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1: lagged value of fraction occupied

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Flower Model With Lagged Effect of Area Coverage

area of flowersgrowth decay

decay rate

actual growth rate

intrinsic growth rate

fraction occupied

~

growth rate multiplier

area available

lagged value of fraction

lag time

Flower Model With First Order Lagged Effect of Area Coverage

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1: area of flowers 2: decay 3: growth

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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, which eventually results in an increase in growth

• Variations in growth and decay eventually fade away as the system approaches dynamic equilibrium = “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 Exponential Smoothing Process

lagged value of

fraction occupied 1

change in fraction occupied 1

fraction occupied

lag time

lagged value of

fraction occupied 2

change in fraction occupied 2

lagged value of

fraction occupied 3

change in fraction occupied 3

Structure of Third Order Exponential Smoothing Process

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Time

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Flower Model With First vs. Third Order Lagged Effect of Area Coverage

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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 through large oscillations before dynamic equilibrium is achieved

• Increased volatility arises because of the longer lag implicit in the third order smoothing

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

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

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Effects of Volatility Illustrated• Plot growth and decay together with flower area for simulation wi

th 3 year time lag• Flower area and growth rate increase in parallel even after

carrying capacity is reached; flowers do not “feel” the effect of space limitation due to the time lag

• Once effect of space limitation kicks in, growth rate drops rapidly 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”

Effects of Volatility Illustrated

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Effects of Volatility Illustrated• Key reason for sustained volatility of the model with

long 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, oscillations dampen over time

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

Influence of Intrinsic Growth Rate on Volatility

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

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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 and intrinsic 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

Criticaldampeningcurve

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

and found that oscillations occurred only when the product of the intrinsic growth rate and time lag (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 his findings as a working hypothesis for position of the critical dampening curve

• Define FMVI = “Flower Model Volatility Index” as the product of the time lag and the intrinsic growth rate 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 to Hastings’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 previously support 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