Long Swings in Homicide 1 1

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

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 Bureau of Justice Statistics, BJS “Homicide Trends in the United States, 1980-2008”, 11-16-2011 “Homicide Trends in the United States”, 7-1-2007 4 4

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

Report to the Nation ….p.15 6 6

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1980 8 8

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

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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

Cesare Marchetti 12 12

Erie Canal 13

10% 90% 1890 1859 1921 14 14

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

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Richard Easterlin 20 year swings 18

Cycles in Nature Canadian Lynx and Snowshoe Hare, data from the Hudson Bay Company, nearly a century of annual data, 1845-1935 The Lotka-Volterra Model (Sarah Jenson and Stacy Randolph, Berkeley ppt., Slides 4-9) 19

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

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

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

2008-2014: Hard Winter 24 24

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Cesare Marchetti “Fifty-Year Pulsation In Human Affairs” Futures 17(3):376-388 (1986) www.cesaremarchetti.org/archive/ scan/MARCHETTI-069.pdf Example: the construction of railroad miles is logistically distributed 26 26

Cesare Marchetti 27 27

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

Part Four: More About Ecological Cycles 29

Well Documented Cycles 30

Similar Data from North Canada 31

Weather: “The Butterfly Effect” 32

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

NetLogo Predator-Prey Model

Questions? How to Model?

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

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 (1860-1940) famous Italian mathematician Retired from pure mathematics in 1920 Son-in-law: D’Ancona

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

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

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

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

lnP(t) = lnP(1960) + at

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

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

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

[Hare(1865)-Hare(1863)]/Hare(1864) Vs. Lynx (1864) etc. 1863-1934 ∆ hare(t)/hare(t-1) = 0.77 – 0.025 Lynx a = 0.77, b = 0.025 (a = 0.63, b = 0.022)

[Lynx(1847)-Lynx(1845)]/Hare(1846) Vs. Lynx (1846) etc. 1846-1906 ∆ Lynx(t)/Lynx(t-1) = -0.24 + 0.005 Hare c = 0.24, d= 0.005 ( c = 0.27,d = 0.006)

Simulations: 1845-1935 Mathematica http://mathworld.wolfram.com/Lotka- VolterraEquations.html Predator-prey equations Predator-prey model

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

Part Six: A Lotka-Volterra Model For Homicide?