2.2. Competition Phenomena

1. Volterra-Lotka Competition Equations

2. Population Dynamics of Fox Rabies in Europe

3. Selection and Evolution of Biological Molecules

4. Laser Beam Competition Equations

5. Rapoport's Model for the Arms Race

2.2.1. Volterra-Lotka Competition Equations

Predator-Prey Relationship

B B B B L BN N g N N

L L L L B LN N g N N

big (predator) fish

little (prey) fish

0B Lg g

0 expB B BN t N t

0 expL L LN t N t

If then

Solutions to predator-prey problems are usually cyclic with a difference in phase between NB and NL.

Rabbits-Foxes Equations

2r r rf

f f rf

MF04.nb 02-2.nb

MF04.mws

Generalizations

LL L L L B L

NN N g N N

Verhulst term

is the saturation number

Time-lag: Problem 2-26

Limited Resources:

2.2.2. Population Dynamics of Fox Rabies in Europe

Rabies epidemic in central Europe.

Originated in Poland in 1939

Transmitted primarily by the fox population

Model

3 categories of fox population:1. Susceptibles: population density X.

Currently health but susceptible to infection.

2. Infected: population density Y Infected, cannot infect the susceptibles.

3. Infectuous: population density Z, Infected, can infect the susceptibles.

No recovered category -- high mortality rate

X aX b N X XZ

Y XZ b N Y

Z Y b N Z

N X Y Z

Symbol Meaning Value

a average per capita birth rate 1 yr1

b average per capita natural death rate 0.5 yr1

rabies transmission coefficient 79.67 km2 yr1

inverse of latent period (~28 to 30 days) 13 yr1

death rate rabid foxes (average life expectancy ~ 5 days )

73 yr1

coefficient of limited food supply 0.1 to 5 km2 yr1

Population Dynamics For Fox Rabies ( X healthy, Y infected foxes )

MF05.nb

MF05.mws

0.1

2.2.3. Selection and Evolution of Biological Molecules

Eigen and Schuster :• 1st carriers of genetic information:

self-replicating strands of RNA

• Mutations:slight errors in the duplication of the nucleotide sequences

• Food:energy-rich monomers

• Selection- evolution:Darwinian survival-of- the-fittest

Symbol Meaning

Xk(t) concentration of species k

Aktotal reproduction rate of species k,

including mutations

Qkfraction of copies that are precise

(quality factor of species k)

Dk decomposition (death) rate of species k

klmutation coefficient for producing species k

due to errors in the replication of species l

Wk Ak Qk Dk

net intrinsic rate of producing exact copies

Linear rate equation for producing species k

1

N

k k k kl ll k

X t W X t X t

Conservation relation (Eigen and Schuster’s selection criteria)

,

1k k k kl lk k l k l

A Q X X

Since 1 Qk is the fraction of mutations:

kk

X n const

0k kk k

dX X

dt

1k k k k k kk k k

X W X A Q X

1k k k k k k kk k

A Q D X A Q X

k k kk

A D X

If Dk > Ak , the species will die out even without competition.

Let Dk < Ak , or Ek Ak – Dk > 0 for all k, then

0k k kk k

X E X

“Dilution term” needed to satisfy constraints

1

N

k k k kl l kl k

X t W X t X t X

Assume for simplicity: 0 kk

X

0 k k kk

A D X k kk

E X

k k

k

kk

E XE t

X

1

N

k k k kl ll k

X t W E X t X t

(Quasi-species model)

Case N 2 is a Riccati equation:

21 2 0X aX f t X f t

See Problem 2-31To be solved in Chapter 5

2.2.4. Laser Beam Competition Equations

Ruby Laser: 6943A.Gas cell: liquid CCl4 colored with trace I2.

I2 molecules excited by absorbing photons of energy

L S

De-excitations via collisions with molecules of host liquid.

Thermal fluctuations modulation of refractive index of liquid scattering between the laser beams.

Stimulated Thermal Scattering

LL S L

dIgI I I

dz

SL S S

dIgI I I

dz

0

To establish steady state, duration of light pulses must be long compared to the lifetime of the thermal fluctuations.

Laser beams travelling in opposite direction inside cell

LL S L

dIgI I I

dz

SL S S

dIgI I I

dz

Laser beams travelling in same direction inside cell

1 2ndy

f z y f ydz

Bernoulli equation [see chapter 5]

2.2.5. Rapoport's Model for the Arms Race

L.F. Richardson: Defense budgets of European nations for 1909-13.

1X a Y2Y a X

aj > 0

1 2a a k For

0 0 k tX t Y t X Y e

Rapoport Budget growth rates:

accelerated in times of crisis decelerated during peace times

21 1 1X m X a Y bY

22 2 2Y m Y a X b X

MF06.nb MF06.mws.