1. Long Swings in Homicide1 1

2. Outline Evidence of Long Swings in Homicide
Evidence of Long Swings in Other Disciplines
Long Swing Cycle Concepts: Kondratieff Waves
More about ecological cycles
Models 2 2

3. Part I. Evidence of Long Swings in Homicide
US Bureau of Justice Statistics
Report to the Nation On Crime and Justice, second edition
California Department of Justice, Homicide in California 3 3

4. 4 Bureau of Justice Statistics, BJS
“Homicide Trends in the United States, ”,
“Homicide Trends in the United States”, 4 4

5. Bureau of Justice Statistics
Peak to Peak: 50 years 5 5

6. Report to the Nation ….p.15 6 6

7. 7

8. 1980 8 8

9. Executions in the US 1930-2007 9 http://www.ojp.usdoj.gov/bjs
Peak to Peak: About 65 years 9 9

10. 10

11. Part Two: Evidence of Long Swings In Other Disciplines
Engineering
50 year cycles in transportation technology
50 year cycles in energy technology
Economic Demography
Simon Kuznets, “Long Swings in the Growth of Population and Related Economic Variables”
Richard Easterlin, Population, labor Force, and Long Swings in Economic Growth
Ecology
Hudson Bay Company 11

12. Cesare Marchetti 12 12

13. Erie Canal 13

14. 10%
90%
1890
1859
1921 14 14

15. Cesare Marchetti: Energy Technology: Coal, Oil, Gas, Nuclear
52 years
57 years
56 years 15 15

16. 16

17. 17

18. Richard Easterlin
20 year swings 18

19. Cycles in Nature
Canadian Lynx and Snowshoe Hare, data from the Hudson Bay Company, nearly a century of annual data,
The Lotka-Volterra Model (Sarah Jenson and Stacy Randolph, Berkeley ppt., Slides 4-9) 19

20. 20

21. What Causes These Cycles in Nature?
At least two kinds of cycles
Harmonics or sin and cosine waves
Deterministic but chaotic cycles 21

22. Part Three: Thinking About Long Waves In Economics
Kondratieff Wave 22 22

23. 23 Nikolai Kondratieff (1892-1938) Brought to attention in
Joseph Schumpeter’s Business
Cycles (1939) 23 23

24. :
Hard Winter 24 24

25. 25 25

26. Cesare Marchetti “Fifty-Year Pulsation In Human Affairs” Futures 17(3): (1986) scan/MARCHETTI-069.pdf
Example: the construction of railroad miles is logistically distributed 26 26

27. Cesare Marchetti 27 27

28. Theodore Modis
Figure 4. The data points represent the percentage deviation of energy consumption in the US from the natural growth-trend indicated by a fitted S-curve. The gray band is an 8% interval around a sine wave with period 56 years. The black dots and black triangles show what happened after the graph was first put together in 1988.[7] Presently we are entering a “spring” season. WWI occurred in late “summer” whereas WWII in late “winter”. 28 28

29. Part Four: More About Ecological Cycles29

30. Well Documented Cycles30

31. Similar Data from North Canada31

32. Weather: “The Butterfly Effect”32

33. The Predator-Prey Relationship
Predator-prey relationships have always occupied a special place in ecology
Ideal topic for systems dynamics
Examine interaction between deer and predators on Kaibab Plateau
Learn about possible behavior of predator and prey populations if predators had not been removed in the early 1900s

34. NetLogo Predator-Prey Model

35. Questions? How to Model?

36. Part Five: The Lotka-Volterra Model
Built on economic concepts
Exponential population growth
Exponential decay
Adds in the interaction effect
We can estimate the model parameters using regression
We can use simulation to study cyclical behavior

37. Lotka-Volterra Model Alfred J. Lotka (1880-1949)
American mathematical biologist
primary example: plant population/herbivorous animal dependent on that plant for food
Vito Volterra
( )
famous Italian mathematician
Retired from pure mathematics in 1920
Son-in-law: D’Ancona

38. Predator-Prey
1926: Vito Volterra, model of prey fish and predator fish in the Adriatic during WWI
1925: Alfred Lotka, model of chemical Rx. Where chemical concentrations oscillate 38 38 38

39. Applications of Predator-Prey
Resource-consumer
Plant-herbivore
Parasite-host
Tumor cells or virus-immune system
Susceptible-infectious interactions 39 39 39

40. Non-Linear Differential Equations
dx/dt = x(α – βy), where x is the # of some prey (Hare)
dy/dt = -y(γ – δx), where y is the # of some predator (Lynx)
α, β, γ, and δ are parameters describing the interaction of the two species
d/dt ln x = (dx/dt)/x =(α – βy), without predator, y, exponential growth at rate α
d/dt ln y = (dy/dt)/y = - (γ – δx), without prey, x, exponential decay like an isotope at rate  40 40 40

41. Population Growth: P(t) = P(0)eat

42. lnP(t) = lnP(1960) + at

43. CA Population: exponential rate of growth, 1995-2007 is 1.4%

44. Prey (Hare Equation)
Hare(t) = Hare(t=0) ea*t , where a is the exponential growth rate
Ln Hare(t) = ln Hare(t=0) + a*t, where a is slope of ln Hare(t) vs. t
∆ ln hare(t) = a, where a is the fractional rate of growth of hares
So ∆ ln hare(t) = ∆ hare(t)/hare(t-1)=[hare(t) – hare(t-1)]/hare(t-1)
Add in interaction effect of predators; ∆ ln Hare(t) = a – b*Lynx
So the lynx eating the hares keep the hares from growing so fast
To estimate parameters a and b, regress ∆ hare(t)/hare(t-1) against Lynx

45. Hudson Bay Co. Data: Snowshoe Hare & Canadian Lynx, 1845-1935

46. [Hare(1865)-Hare(1863)]/Hare(1864) Vs. Lynx (1864) etc. 1863-1934
∆ hare(t)/hare(t-1) = 0.77 – Lynx
a = 0.77, b = (a = 0.63, b = 0.022)

47. [Lynx(1847)-Lynx(1845)]/Hare(1846) Vs. Lynx (1846) etc. 1846-1906
∆ Lynx(t)/Lynx(t-1) = Hare
c = 0.24, d= ( c = 0.27,d = 0.006)

48. Simulations:
Mathematica VolterraEquations.html
Predator-prey equations
Predator-prey model

51. Simulating the Model: 1900-1920
Mathematica
a = 0.5, b = 0.02
c = 0.03, d= 0.9

54. Part Six: A Lotka-Volterra Model For Homicide?