long swings in homicide

Long Swings in Homicide

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

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

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

Bureau of Justice Statistics

Peak to Peak: 50 years

Report to the Nation ….p.15

1900 1920 1940 1960 1980 2000

HOMICIDECA HOMICIDEUSA

California

Homicide and Non-negligent Manslaaaughter, Rates Per 100,000

55 60 65 70 75 80 85 90 95 00 05

HOMICIDE

California Homicide rate per 100,000: 1952-2007

Executions in the US 1930-2007

http://www.ojp.usdoj.gov/bjs

Peak to Peak: About 65 years

60 70 80 90 00 10 20 30 40

CAPRISONERS

California Prisoners: 1851-1945

Part Two: Evidence of Long Swings In Other Disciplines

Engineering50 year cycles in transportation technology

50 year cycles in energy technology

Economic DemographySimon Kuznets, “Long Swings in the Growth of Population and Related Economic Variables”

Richard Easterlin, Population, labor Force, and Long Swings in Economic Growth

EcologyHudson Bay Company

Cesare Marchetti

Erie Canal

-10 -5 0 5

RAILMILES

constructed

90%10%

Cesare Marchetti: Energy Technology: Coal, Oil, Gas,

Nuclear52 years 57 years 56 years

Richard Easterlin

20 year swings

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)

Cycles in Nature

What Causes These Cycles in Nature?

At least two kinds of cyclesHarmonics or sin and cosine waves

Deterministic but chaotic cycles

Part Three: Thinking About Long Waves In Economics

Kondratieff Wave

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

2008-2014:Hard Winter

252525

Cesare Marchetti“Fifty-Year Pulsation In

Human Affairs”Futures 17(3):376-388

(1986)www.cesaremarchetti.org/arc

hive/scan/MARCHETTI-069.pdf Example: the construction of railroad miles is

logistically distributed

Cesare Marchetti

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

Part Four: More About Ecological Cycles

Well Documented Cycles

Similar Data from North Canada

Weather: “The Butterfly Effect”

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 conceptsExponential 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 ModelLotka-Volterra Model

Vito Volterra Vito Volterra

(1860-1940)(1860-1940)

famous Italian famous Italian mathematicianmathematician

Retired from pure Retired from pure mathematics in 1920mathematics in 1920

Son-in-law: D’AnconaSon-in-law: D’Ancona

Alfred J. Lotka Alfred J. Lotka

(1880-1949)(1880-1949)

American mathematical American mathematical biologistbiologist

primary example: plant primary example: plant population/herbivorous population/herbivorous animal dependent on that animal dependent on that plant for foodplant for food

383838

Predator-Prey1926: 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

393939

Applications of Predator-Prey

Resource-consumer

Plant-herbivore

Parasite-host

Tumor cells or virus-immune system

Susceptible-infectious interactions

404040

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

California Population 1960-2007

5,000,000

10,000,000

15,000,000

20,000,000

25,000,000

30,000,000

35,000,000

40,000,000

......

……

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

lnP(t) = lnP(1960) + at

60 65 70 75 80 85 90 95 00 05 10 15

LNCAPOP

LnP(t) = ln[P(1960)e(at)] = lnP(0) + at

lnP(t)

CA Population: exponential rate of growth, 1995-2007

is 1.4%Natural Logarithm of California Population Vs Time, 1995-2007

y = 0.0141x + 17.269

R2 = 0.9967

0 2 4 6 8 10 12 14

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

1850 1860 1870 1880 1890 1900 1910 1920 1930

HARE LYNX

HudsonBay Company Data: Snowshoe Hare & Canadian Lynx, 1845-1935

[Hare(1865)-Hare(1863)]/Hare(1864)

Vs. Lynx (1864) etc. 1863-1934{Hare(t+1)-Hare(t-1)]/Hare(t) Vs. Lynx(t), 1863-1934

y = -0.0249x + 0.7677R2 = 0.2142

0 10 20 30 40 50 60 70 80 90

∆ 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+1) - Lynx(t-1)]/Lynx(t) Vs. Hare(t) 1846-1906

y = 0.005x - 0.2412R2 = 0.1341

0 20 40 60 80 100 120 140 160 180

∆ 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?

long swings in homicide

outlineevidence of long

long waves

documented cycles

chaotic cycles

netlogo predatorprey

growth of population

prey populations

similar data

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