111 Long Swings in Homicide 1

222 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

333 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

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

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

666 Report to the Nation ….p.15 6

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10 Executions in the US Peak to Peak: About 65 years 10

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12 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

13 Cesare Marchetti 13

14 Erie Canal

15 90%10%

16 Cesare Marchetti: Energy Technology: Coal, Oil, Gas, Nuclear 52 years57 years56 years 16

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19 Richard Easterlin 20 year swings

20 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) Cycles in Nature 20

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22 What Causes These Cycles in Nature? At least two kinds of cycles Harmonics or sin and cosine waves Deterministic but chaotic cycles 22

23 Part Three: Thinking About Long Waves In Economics Kondratieff Wave 23

24 Nikolai Kondratieff ( ) Brought to attention in Joseph Schumpeter’s Business Cycles (1939) 24

: Hard Winter 25

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27 Cesare Marchetti “Fifty-Year Pulsation In Human Affairs” Futures 17(3): (1986) scan/MARCHETTI-069.pdf scan/MARCHETTI-069.pdf Example: the construction of railroad miles is logistically distributed 27

28 Cesare Marchetti 28

29 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”. 29

30 Part Four: More About Ecological Cycles 30

31 Well Documented Cycles 31

32 Similar Data from North Canada 32

33 Weather: “The Butterfly Effect”

34 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

35 NetLogo Predator-Prey Model

36 36 Crime Generation Crime Control Offense Rate Per Capita Expected Cost of Punishment Schematic of the Criminal Justice System: Simultaneity Causes ? (detention, Deterrence, Rehabilitation, And revenge) Expenditures Weak Link OF = f(CR, SV, CY, SE, MC) CR = g(OF, L)

37 Source: Report to the Nation on Crime and Justice Expect Get 37

38 Questions? How to Model?

39 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

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

41 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 41

42 Applications of Predator-Prey Resource-consumer Plant-herbivore Parasite-host Tumor cells or virus-immune system Susceptible-infectious interactions 42

43 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  43

44 Population Growth: P(t) = P(0)e at

45 lnP(t) = lnP(1960) + at

46 CA Population: exponential rate of growth, is 1.4%

47 Prey (Hare Equation) Hare(t) = Hare(t=0) e a*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

48 Hudson Bay Co. Data: Snowshoe Hare & Canadian Lynx,

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

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

51 Simulations: Mathematica VolterraEquations.htmlhttp://mathworld.wolfram.com/Lotka- VolterraEquations.html Predator-prey equations Predator-prey model

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54 Simulating the Model: Mathematica a = 0.5, b = 0.02 c = 0.03, d= 0.9

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57 Part Six: A Lotka-Volterra Model For Homicide? Do other violent crimes move with homicide?

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59 Distribution of Ratio of Rape to Homicide; Median = 4.2

60 Ratio of Rapes to Homicides

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62 Part Six: A Lotka-Volterra Model For Homicide? Do other violent crimes move with homicide? We have a measure of the rabbits: homicides. How about a measure for the foxes (coyotes)?

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66 Fractional Change in California Prisoners Trough to trough 16 years, a half a cycle

67 Fractional Change in California Prisoners Trough to trough 18 years, a half a cycle

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