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
Dec 20, 2015
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.
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
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)
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