slide1
Download
Skip this Video
Download Presentation
The number of B ’s born ~ the number of A-B pairs: ~ a ( x,t ) b ( x,t ) l d t

Loading in 2 Seconds...

play fullscreen
1 / 44

The number of B ’s born ~ the number of A-B pairs: ~ a ( x,t ) b ( x,t ) l d t - PowerPoint PPT Presentation


  • 271 Views
  • Uploaded on

a ( x,t ) = the number of A’ s ; b ( x,t )= the number of B ‘s The number of B ’s dying ~ proportional to their number ~ - m b ( x,t ) d t The number of B ’s born ~ the number of A-B pairs: ~ a ( x,t ) b ( x,t ) l d t In total:

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'The number of B ’s born ~ the number of A-B pairs: ~ a ( x,t ) b ( x,t ) l d t' - andrew


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide15
a ( x,t ) = the number of A’s ; b ( x,t )= the number of B ‘s

The number of B’s dying~ proportional to their number ~ - m b ( x,t )d t

The number of B’s born~the number of A-B pairs: ~ a ( x,t ) b ( x,t ) l d t

In total:

d b ( x,t ) ~ ( a ( x,t ) l - m )b ( x,t ) d t+diffusion

Continuum d t  0: b. = ( a l- m)b + Db D b

a. = Da D a

slide18
Since the classical works of Malthus and Verhulst it has been recognized that most of the processes in living systems are autocatalytic and thus are characterized by exponential growth. In fact, the appearance of an autocatalytic molecule may be considered as the origin of life. Autocatalytic system are shown to admit self organization in the presence of fluctuating environment. The exponential amplification of ``good'' fluctuations in the catalysis parameters prevails, in the situations discussed above, the globally hostile environment and is robust against the random motion of both the reactants and the catalysts. Our result may be interpreted as an indication that ``life'' (in the above sense) is resilient and is able to adapt itself to the changing environment. The applicability of this model ranges from biological evolution to the role of enzymes in chemical reactions and even in social or financial settings.
slide19
Since Malthus and Verhulst:
  • autocatalytic and thus
  • exponential growth.
  • Autocatalytic system admit self organization in the presence of fluctuating environment. Exponential amplification of ``good'' fluctuations in the catalysis parameters
  • prevails, the globally hostile environment and
  • is robust against the random motion of both the reactants and the catalysts.
  • The applicability of this model ranges from biological evolution to the role of enzymes in chemical reactions and even in social or financial settings.
slide20
Since Malthus and Verhulst:
  • autocatalytic and thus
  • exponential growth.
  • Autocatalytic system admit self organization in the presence of fluctuating environment. Exponential amplification of ``good'' fluctuations in the catalysis parameters
  • prevails, the globally hostile environment and
  • is robust against the random motion of both the reactants and the catalysts.
  • The applicability of this model ranges from biological evolution to the role of enzymes in chemical reactions and even in social or financial settings.
slide21
Malthus exponential growth when >

b.(t)= a b(t) => exponential increase for a > 0

Here  Super-Malthus: always exponential growth:Natural death mb(x,t)does not lead to saturation. For very largeb(x,t)’s competition necessary (Verhuulst 1838):

Logistic equationb.(t)= a b(t)– g b(t)2

Montroll: “almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth''.

Lotka and Volterra: b(t)= the size of an animal/plant population, a = the aggregated effects of birth and natural death, -g b(t)2 = competition for limited resources.

slide22
Malthus exponential growth when >

b.(t)= a b(t) => exponential increase for a > 0

Here  Super-Malthus: always exponential growth:Natural death mb(x,t)does not lead to saturation. For very largeb(x,t)’s competition necessary (Verhuulst 1838):

Logistic equationb.(t)= a b(t)– g b(t)2

Montroll: “almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth''.

Lotka and Volterra: b(t)= the size of an animal/plant population, a = the aggregated effects of birth and natural death, -g b(t)2 = competition for limited resources.

slide23
Economics, Aoki - b(t) = the total product demand in a market. -a b(t) = emergence of new products ~ size of the market. -g b(t)2 the products have to compete with one another within a finite total potential market.

Finance, Solomon and Levy have suggested that - b(t) = the total capital within a financial system. -a b(t) = the average returns that the system offers, -g b(t)2 the effects of competition and

slide24
Economics, Aoki - b(t) = the total product demand in a market. -a b(t) = emergence of new products ~ size of the market. -g b(t)2 the products have to compete with one another within a finite total potential market.

Finance, Solomon and Levy have suggested that - b(t) = the total capital within a financial system. -a b(t) = the average returns that the system offers, -g b(t)2 the effects of competition and

slide25
Local Competition

Many of the mechanisms limiting the B proliferation can be described as competition. Competition for a resource means that whenever 2 B agents need the resource either to survive or to proliferate only one of them will get it and the other will die (or fail to proliferate). In ecology competition can be over food or water, while in the immune system competition can be over access to an antigen. Such a system is sometimes called a Lotka Voltera system [18, 19] and can be classically described using a local differential equation:

b.(x,t)= (la(x,t)- m)b(x,t)– g b(x,t)2 + Db D b(x,t)

If (la(x,t)– m) < 0

The adaptive islands appear here too and increase greatly the sustainability of the B population. However, the height of the B islands is finite and for discrete B’s and finite volume the population will eventually die by some cataclysm which brings it to the b=0 fixed point.

slide26
Local Competition

Lotka Voltera system (use PDE notation to symbolize particles interactions in spite of its inadequacy as dynamical predictor):

b.(x,t)= (la(x,t)- m)b(x,t)– g b(x,t)2 + Db D b(x,t)If (la(x,t)– m) < 0

Adaptive islands appear here too and increase greatly the sustainability of the B population. However, the height of the B islands is finite and for discrete B’s and finite volume the population will eventually die by some cataclysm which brings it to the b=0 fixed point.

slide27
Global (long range/“mean field !?”) Competition

The globally averaged version of AB system with B competition (logistic, Lotka-Volterra system):b.(t)= a b(t)– g b(t)2where b(t) = < b(x,t) >x

The linear term a b(t)can be obtained by averaging over the spatial locations x in: b.(x,t)= (l a(x,t)- m )b(x,t)– g b(x,t)2 + Db D b(x,t)and denotinga= (l a(t)- m )

the termb(t)2 = < b(x,t) >x2cannot be the result of averaging over local (nonlinear) terms:b2 (t) = < b(x,t) >x2 < b(x,t)2 >x

slide28
Nonlinear global terms =averages over infinite range interactions:b(t)2 = < b(x,t) >x2 = < b(x,t) x >x , mean field!

b.(x,t)=(la(x,t)-m )b(x,t)–g b(x,t) x+ Db D b(x,t)Stable state (neglect diffusion): b.(x,t)=0=[(la(x,t)-m )– gx] b(x,t)

Except (measure 0 set of x’s) where(l a(x,t) - m ) - g x =0

=> localization:for almost all xb(x,t) = 0

In fact for (l a(x,t) - m ) - gx> 0, b(x,t) = 0 is unstable (b.(x,t) >0 if b(x,0) =e >0) => b(x,t) evolves until: x = (l a(xmax,t) - m )The entire B population is an island aroundxmaxwith populationb(xmax,t) = (l a(xmax,t) - m ) x Volume

slide29
Rather they can appear as averages over infinite range interactions:b(t)2 = < b(x,t) >x2 = < b(x,t)b(t) >x ,

mean field => greatly enhancing the localization effects

b.(x,t)= (la(x,t)- m )b(x,t)– g b(x,t) + Db D b(x,t)Fixed point: b.(x,t) =0 => (neglect diffusion)

(l a(x,t)- m ) = g or b(x,t) = 0

I.e b vanishes in the entire space except the singular locations where a(x,t) takes a specific value (which turns out to be its maximum).

The entire B population is concentrated in one island.The continuum limit is NEVER correct.(even for (l a0- m > 0)

slide30
Globalization of Competition => Localization of Growth

The linear term a b(t)can be obtained by averaging over the spatial locations x in: b.(x,t)= (l a(x,t)- m )b(x,t)– g b(x,t)2 + Db D b(x,t)

the nonlinear termg b(t)2cannot bethe result of averaging overlocal (nonlinear) terms:b2 (t) = < b(x,t)>x2 < b(x,t)2 >x

Global Relevant: O2/O3, greenhouse, polution, inflation,capital

The globally averaged version of AB system with B competition (logistic, Lotka-Volterra, Verhuulst 1838/Malthus):b.(t)= a b(t)– g b(t)2where b(t) = < b(x,t) >x

slide31
Finite Competition Range R

b.(x,t)=(la(x,t)- m )b(x,t)– gb(x,t)R+ Db D b(x,t)

R= S|y-x-1|

slide32
The necessity for B to find a way to the new A pile emphasizes the difference between

- random (space time-uncorrelated) noise – where A’s would disappear from one site and would reappear in another randomly uncorrelated site.

  • and
  • diffusive (space-time correlated) noise – where A just jumps between neighboring sites and allows (descendents of) the B’s to follow them.

Continuityof the random hostile environment in time is a crucial feature for adaptability and life emergence.

slide34
Local Competition

One spatially extended generalization of the Lotka Voltera system:

b.(x,t)= (la(x,t)- m)b(x,t)– g b(x,t)2 + Db D b(x,t)

a(x,t) is not a continuum function:

it is a stochastic integer at each site x and it changes in time according to the jumping diffusion rules discussed before.

If (la(x,t)– m) < 0 => localized adaptive islands of finite hight. - Improved life expectation but b=0 fixed point for discrete B.

slide35
The Local Competition equation

b.(x,t)= (l a(x,t)- m )b(x,t)– g b(x,t)2 + Db D b(x,t)

is not the natural space extended version ofb.(t)= a b(t)– g b(t)2 .

Indeed averaging does notgive g b(t)2g< b(x,t) > 2

average overthe nonlocal (infinite range) term < g b(x,t) x >x = g< b(x,t) >x2does.

One is lead to the global competition equation:b.(x,t)= (l a(x,t)- m )b(x,t)-gb(x,t)x+DbDb(x,t)

slide36
b.(x,t)=(la(x,t)-m )b(x,t)–g b(x,t) x+ Db D b(x,t)Stable state (neglecting diffusion) is given by : b.(x,t)=0=[(l a(x,t)-m )– gx]b(x,t) (1)

Concentrate on the site xmax with the property:a(xmax,t) > a(x,t):  x  xmaxaccording (1) its population keeps increasing until[(l a(xmax,t) m )– gx] = 0 (2)

This fixes x . But this means that for  x  xmax[(l a(x,t) m )– gx]< 0

And according (1)b(x,t) will decrease until b(x,t) =0  x  xmax

The entireBpopulation is in an island aroundxmaxwith populationb(xmax,t) ~ (l a(xmax,t) - m ) x Volume

slide37
Behavioral Finance

B= financial traders,

Patches= "herding" behavior

(despite the fact that we do not introduce communicationbetween B’s)..

  • May explain some paradoxes in finance E.g
  • between theefficient market hypothesis (absence of systematic profit opportunitiesin equilibrium markets) and
  • the actual profits that investorsextract daily from themarket.
slide39
Malthus exponential growth when >

b.(t)= a b(t) => exponential increase for a > 0

Here  Super-Malthus: always exponential growth:Natural death mb(x,t)does not lead to saturation. For very largeb(x,t)’s competition necessary (Verhuulst 1838):

Logistic equationb.(t)= a b(t)– g b(t)2

Montroll: “almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth''.

Tasmania Sheep

Lotka and Volterra: b(t)= the size of an animal/plant population, a = the aggregated effects of birth and natural death, -g b(t)2 = competition for limited resources.

slide40
Economics, Aoki - b(t) = the total product demand in a market. -a b(t) = emergence of new products ~ size of the market. -g b(t)2 the products have to compete with one another within a finite total potential market.

Finance, Solomon and Levy have suggested that - b(t) = the total capital within a financial system. -a b(t) = the average returns that the system offers, -g b(t)2 the effects of competition and

slide41
Local Competition

One spatially extended generalization of the Lotka Voltera system:

b.(x,t)= (la(x,t)- m)b(x,t)– g b(x,t)2 + Db D b(x,t)

a(x,t) is not a continuum function:

it is a stochastic integer at each site x and it changes in time according to the jumping diffusion rules discussed before.

If (la(x,t)– m) < 0 => localized adaptive islands of finite hight. - Improved life expectation but b=0 fixed point for discrete B.

slide42
The Local Competition equation

b.(x,t)= (l a(x,t)- m )b(x,t)– g b(x,t)2 + Db D b(x,t)

is not the natural space extended version ofb.(t)= a b(t)– g b(t)2 .

Indeed averaging does notgive g b(t)2g< b(x,t) > 2

average overthe nonlocal (infinite range) term < g b(x,t) x >x = g< b(x,t) >x2does.

One is lead to the global competition equation:b.(x,t)= (l a(x,t)- m )b(x,t)-gb(x,t)x+DbDb(x,t)

slide43
b.(x,t)=(la(x,t)-m )b(x,t)–g b(x,t) x+ Db D b(x,t)Stable state (neglecting diffusion) is given by : b.(x,t)=0=[(l a(x,t)-m )– gx]b(x,t) (1)

Concentrate on the site xmax with the property:a(xmax,t) > a(x,t):  x  xmaxaccording (1) its population keeps increasing until[(l a(xmax,t) -m )– gx] = 0 (2)

This fixes x . But this means that for  x  xmax[(l a(x,t) -m )– gx]< 0

And according (1)b(x,t) will decrease until b(x,t) =0  x  xmax

The entireBpopulation is in an island aroundxmaxwith populationb(xmax,t) ~ (l a(xmax,t) - m ) x Volume

ad