BSC 417/517 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_rate*growth_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, it’s 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 = 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

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-lagged_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 Behavior:Parameter Space Diagram + 3 Sustained + + + Overdamped Lag time (yr) 2 Damped Sustained + 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 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

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 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

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