Download 2.2

2.2. Competition Phenomena
1.
2.
3.
4.
5.
Volterra-Lotka Competition Equations
Population Dynamics of Fox Rabies in Europe
Selection and Evolution of Biological Molecules
Laser Beam Competition Equations
Rapoport's Model for the Arms Race
2.2.1. Volterra-Lotka Competition
Equations
Predator-Prey Relationship
If
N B   B N B  g B N L N B
big (predator) fish
N L  L N L  gL N B N L
little (prey) fish
g B  g L  0 then
N B  t   N B  0 exp  B t 
N L  t   N L  0 exp L t 
Solutions to predator-prey problems are usually
cyclic with a difference in phase between NB and NL.
Rabbits-Foxes Equations
r  2 r   rf
f   f   rf
MF04.nb
MF04.mws
02-2.nb
Generalizations
Limited Resources:
Verhulst term
  NL
N L  L N L
 gL N B N L

 is the saturation number
Time-lag:
Problem 2-26
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
N  X Y  Z
Z  Y    b   N  Z
Symbol
Meaning
Value
a
average per capita birth rate
1 yr1
b
average per capita natural death rate
0.5 yr1

rabies transmission coefficient
79.67 km2 yr1

inverse of latent period (~28 to 30 days)
13 yr1

death rate rabid foxes
(average life expectancy ~ 5 days )
73 yr1

coefficient of limited food supply
0.1 to 5 km2 yr1
Population Dynamics For Fox Rabies
( X healthy, Y infected foxes )
  0.1
MF05.nb
MF05.mws
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
Xk(t)
Ak
Meaning
concentration of species k
total reproduction rate of species k,
including mutations
Dk
fraction of copies that are precise
(quality factor of species k)
decomposition (death) rate of species k
kl
mutation 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
Qk
Linear rate equation for producing species k
X k  t   Wk X k  t  
N

l   k 1
kl
X l t 
Since 1  Qk is the fraction of mutations:
 A 1  Q  X
k
k
k
k


k , l  k l 
kl X l
Conservation relation (Eigen and Schuster’s selection criteria)
X
k
 n  const
k

d

Xk    Xk  0


dt  k
 k
X
k
 Wk X k   Ak 1  Qk  X k
k
k
k
   Ak Qk  Dk  X k   Ak 1  Qk  X k
k
k
   Ak  Dk  X k
k
If Dk > Ak , the species will die out even without competition.
Let Dk < Ak , or Ek  Ak – Dk > 0 for all k, then
X
k

k
  Ek X k  0
k
“Dilution term” needed to satisfy constraints
X k  t   Wk X k  t  
N

l  k 1
kl
X l  t   X k
0    X k
Assume for simplicity:
k

0    Ak  Dk  X k   Ek X k
k
k
E X
E t  


X
k
k
k
k
k
X k  t   Wk  E  X k  t  
N

l  k 1
kl
X l t 
(Quasi-species model)
Case N  2 is a Riccati equation:
X  aX  f1  t  X  f 2  t   0
2
See Problem 2-31
To be solved in Chapter 5
2.2.4. Laser Beam Competition
Equations
Ruby Laser:
Gas cell:
  6943A.
liquid CCl4 colored with trace I2.
Stimulated Thermal Scattering
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.
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
dI L
  gI L I S   I L
dz
 0
dI S
  gI L I S   I S
dz
Laser beams travelling in same direction inside cell
dI L
  gI L I S   I L
dz
dI S
  gI L I S   I S
dz
Bernoulli equation
[see chapter 5]
dy
 f1  z  y  f 2 y n
dz
2.2.5. Rapoport's Model for the
Arms Race
L.F. Richardson:
Defense budgets of European nations for 1909-13.
X  a1Y
Y  a2 X
aj > 0
For a1  a2  k
X  t   Y  t    X  0   Y  0   e
kt
Rapoport
Budget growth rates:
accelerated in times of crisis
decelerated during peace times
X  m1 X  a1Y  b1Y
2
Y  m2Y  a2 X  b2 X
2
MF06.nb
MF06.mws.