BSC 417517 Environmental Modeling

1
BSC 417/517 Environmental Modeling

• Introduction to Oscillations

2
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

3
First Example Influence of Variable Rainfall on
Flower Growth

• Flower growth model of S-shaped growth from
  Chapter 6

actual_growth_rate intrinsic_growth_rategrowth_
rate_multiplier
growth_rate_multiplier GRAPH(fraction_occupied)
4
Growth Rate Multiplier for Modeling S-Shaped
Growth
5
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

6
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

7
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, its easy to numerically simulate
  such a system using the graphical function
  approach

8
Growth Rate Multiplier Approach is More
Flexible Than the Classical Logistic Equation
9
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 10 SINWAVE(15,5)
• Rainfall 10 15SIN(2PI/5TIME)
• Assume optimal rainfall for flower growth is 20
  inches per year
• Define relationship between intrinsic growth rate
  and rainfall using a nonlinear graphical function

10
Relationship Between Intrinsic Growth Rate and
Rainfall
11
Flower Model With Variable Rainfall
12
Flower Model With Variable Rainfall
Period 5 yr
Period 2.5 yr
13
Flower Model With Variable Rainfall
14
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

15
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

16
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)

17
Structure of First-Order Exponential Smoothing
Process
0.0
2.0
1.0
change_in_fraction_occupied (fraction_occupied-l
agged_value_of_fraction_occupied)/lag_time
18
Structure of First-Order Exponential Smoothing
Process
19
Flower Model With Lagged Effect of Area Coverage
20
Flower Model With First Order Lagged Effect of
Area Coverage
21
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

22
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

23
Structure of Third Order Exponential Smoothing
Process
24
Structure of Third Order Exponential Smoothing
Process
25
Flower Model With First vs. Third Order Lagged
Effect of Area Coverage
26
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

27
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

28
Flower Model With Third Order Lagged Effect of
Area Coverage and Variable Lag Time
29
Effects of Volatility Illustrated

• Plot growth and decay together with flower area
  for simulation with 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

30
Effects of Volatility Illustrated
31
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)

32
Influence of Intrinsic Growth Rate on Volatility
r 1.5/yr
r 1.0/yr
r 0.5/yr
33
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

34
Oscillatory BehaviorParameter Space Diagram

3
Sustained



Overdamped
Lag time (yr)
2
Sustained
Damped

Critical dampening curve
Overdamped
1
0
0.5
1.0
1.5
Intrinsic growth rate (yr-1)
35
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 Hastingss
  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

36
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
  Hastingss 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

37
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