Logistic Eq, Pareto Law,GLV

1
Logistic Eq, Pareto Law,GLV

• dw /dt A w - B w 2

Montroll 1978 "almost all the social
phenomena,... obey the logistic growth".
Pareto 1897 sic! P(w) w - 1-a w i (t t )
r i (t) w i (t) - c(w. ,t) w i (t) a w(t)
2
w i (t t) - w i (t) ei (t)
c(w.,t) w i (t) ai w (t) w (t t) - w
(t) c(w.,t) w(t) a w (t) x i (t) w i
(t) /w(t) d x i(t) dw i / w( t) - w i(t) /w(t)
d w (t) / w(t) ei (t)c (w.,t) xi (t)ai - xi
(t) c(w.,t) a ei (t) - a x i ai More
general ai u ai Sjbjwj (t) x i (t) w i (t)
/u(t) gt same eq
3
x(t t ) - x(t) e (t) g (x(t)) f (x(t)) Dlt
e (t) 2 gt y(t) y (x(t)) dy (dy /d x) dx
1/2 d2y/dx2 (dx) 2 etc. . dy /dx e
(t) g (x(t)) f (x(t)
1/2 d2y /dx2 D g2 etc.dy/dx 1/ g gt
d2y /dx2 - dg/dx / g2dy e (t) f
(x(y))/g(x(y)) - D/2 dg/dxd y(t) e (t) j
(y(t)) P(y) dy exp 2/D S j (y) dy dy P(x)
dx P(y) dy exp 2/D S (f (x)/g (x) -
D/2(dg/dx )/g dx 1/g(x) dx exp 2/D S f
(x)/g2 (x) d x - ln g 1/g(x) dx
exp 2/D S f (x)/g2 (x) d x 1/g(x)2 dx
4
P(x) dx exp 2/D S f (x)/g2 (x) dx 1/g(x)2
dx exp 2/D S (a-x)/x2 dx 1/x2 dx exp
2/D S (a-x)/x2 dx 1/x2 dx x-1-a exp
-2a/(xD)