Transcript Ageclasses

Subjects
see chapters


Basic about models
Discrete processes
 Deterministic
models
 Stochastic models
 Many equations
 Linear
algebra
 Matrix, eigenvalues eigenvectors

Continuous processes
 Deterministic

models
(Stochastic models)
Stages, States and Classes
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Can we always treat a population as a single entity?
Do we need to divide it into different stages or classes?
 Age-classes
 Size-classes
 Subdivided in space
 Morphological classes
The subpopulations (stages-classes) differ from each other
in aspects important for the purpose and dynamics of the
modell. For example in fecundity, survival, dispersal, or risk
of predation, or environmental variation, or….. Specific
example. Young individuals give birth to fewer than mean
aged individuals
Stages, States and
Classes
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We can use linear algebra, matrix calculations to:
Determine equlibriums
(eigenvectors)
Time to equilibrium.
(eigenvalues)
Run simulations
(matrix multiplication)
Calculate velocity constants (eigenvalues)
Distribution of the
population
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A population can be treated as one unity
if only number of individuals define its
property, for example if 50 individuals
give birth to twice as many as 25 do.
If the population has a constant
distribution of individuals in its relevant
classes/stages, it can be treated as one
unity.

For example if it’s always 30% newborns,
20% young, 20% newly reproductive, 20%
highly reproductive and 10%
postreproductive.
Distribution of the
population
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n If a population of 50 consist of 10
adults/reproductively mature/ the population will
reproduce less then if it consists of 20 adults. If the
population varies in proportion of adults it will
reproduce differently per capita over time.
If the distribution (proportion in stages/classes) of the
population varies over time the population either have
to include stages/subpopulation or one have to show
that it is reasonable to approximate with a simpler
non-stage model.
Distribution of the
population
The dichotomy
Stable proportions
of classes/
subpopulations
Stable per capita
growth rates and
dispersal rates etc
Non-structured model
Variation in proportion
of individuals in stages/
subpopulations
Variation in per capita
growth rates and
dispersal rates etc
Structured model
Ageclasses
method: structured population
Three ageclasses, n1, n2 och n3.

Next timestep is calculated as
b1
1
n1(t+1)= b1 n1(t)+ b2 n2(t)+ b3n3(t)
n2(t+1)= s12 n1(t)
n3(t+1)= s23 n2(t)
Note, one time step
correspondence to
size/span of an ageclass.
b2
b3
s12 2
s23
3
bi = how many newborns from
ageclass i during one timestep
(span of an ageclass)
sij = probability for an individual
in age-class i to survive into the
next age-class, j
Ageclasses
method: structured population
Three ageclasses, n1, n2 och n3.

Next timestep is calculated as
n1(t+1)= b1 n1(t)+ b2 n2(t)+ b3n3(t)
n2(t+1)= s12 n1(t)
n3(t+1)= s23 n2(t)
 n (t  1) 
 b1

 
 n2 (t  1)    s12
 n (t  1)   0
 3
 
1
this is a
linear system of equations,
one can use linear algebra.
Matrix multiplication.
b2 b3   n1 (t ) 
 

0 0    n2 (t ) 
s23 0   n3 (t ) 
Ageclasses
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Next timestep is calculated as
n1(t+1)= b1 n1(t)+ b2 n2(t)+ b3n3(t)
n2(t+1)= s12 n1(t)
n3(t+1)= s23 n2(t)
 n1 (t  1)   b1

 
 n2 (t  1)    s12
 n (t  1)   0
 3
 
b2 b3   n1 (t ) 
 

0 0    n2 (t ) 
s23 0   n3 (t ) 
Ageclasses
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Next timestep is calculated as
n1(t+1)= b1 n1(t)+ b2 n2(t)+ b3n3(t)
n2(t+1)= s12 n1(t)
n3(t+1)= s23 n2(t)
 n1 (t  1)   b1

 
 n2 (t  1)    s12
 n (t  1)   0
 3
 
b2 b3   n1 (t ) 
 

0 0    n2 (t ) 
s23 0   n3 (t ) 
Ageclasses, an example
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Ageclass 1 do not reproduce
Ageclass 2 give birth to 2
Ageclass 3 give birth to 8
40% of individuals in ageclass 1
survives to ageclass 2 80% of
individuals in ageclass 2 survives to
ageclass 3 100% of the individuals
in agecass 3 dies. Start population
conisist of 10 young, 8 subadults
and 6 adults..
2 8  10 
 64   0
  
  
 4    0.4 0 0    8 
 6.4   0 0.8 0   6 
  
  
Ageclasses, matrix multiplication
– run a simulation
2 8   64 
 59.2   0

 
  
 25.6    0.4 0 0    4 
 3.2   0 0.8 0   6.4 

 
  
 59.2   0.67 
The right hand side distributions

 

will be the same for all
 25.6    0.29 
 3.2   0.04 
following timesteps

 

One can calculate this for ever
after a while a constant
distribution will evolve
Note that the number of
individuals may change
(density) but the distribution
over classes becomes stable
 64   0.86 
  

 4    0.05 
 6.4   0.09 
  

densities proportions
Ageclasses, eigenvalues and
eigenvectors
If the distribution becomes stable then the per capita
growth rate also stabilize and becomes a constant value
2 8
 0


 0. 4 0 0 
 0 0 .8 0 


 0.72 


1  1.56, v1  0.18 
 0.1 


If the per capita growth rate becomes
stable/constant over time, one can use that instead
of the matrix
Ageclasses, eigenvalues and
eigenvectors
2 8
 0


 0. 4 0 0 
 0 0 .8 0 


 0.72 


1  1.56, v1   0.18 
 0.1 


The other two eigenvalues are complex values and generates the
oscillations that occurs prior the stabilisation.
 - 0.7  0.6i 
 - 0.7  0.6i 


2  0.78  i, v2   0.3  0.05i  3  0.78  i, v3   0.3  0.05i 
 - 0.08 - 0.2i 
 - 0.08  0.2i 




From the beginning again:
Solution space and eigenvectors.
Assume nay population distribution (not an eigenvector)
2 8   64 
 59.2   0

 
  
 25.6    0.4 0 0    4 
 3.2   0 0.8 0   6.4 

 
  
The lefthand side, vector (59 26 3), exist in a solution space
spanned by the three eigenvectors. This means that you can reach
the point (59 26 3) in the 3D space by moving along the directions
of the three vectors
Mathematically this is expressed by:
n (1)  c1v1  c2v2  c3v3
Solution to n(t)=Atn(0)
t
2 8   64 
 0

  
n (t )   0.4 0 0    4 
 0 0.8 0   6.4 

  
We know that matris*eigenvector equals eigenvalue*eigenvector
Av1=λ1v1.
And that: n (1)  c1v1  c2v2  c3v3
Combine these two and x(t)=Atn(0) can be written as
n (t  1)  c1t1 v1  c2 t2 v2  c3t3 v3
What happends at
large t (long
time???
Stage models
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A stage model have classes
of different time span, not
equals the time step. Hence
some of the individuals may
stay in the original stage
after a timestep. A
proportion gi may stay.
Note one have to consider
survíval, during one time
step, in both p and g
parameters.
b2
g1
1
p12
b3
2
p23
3
g3
g2
 g1

 p12
 0

b2
g2
p23
b3 

0
g3 
More general
model
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Simple Markov chains
Handles probabilities for an organism
to change state, for example running to
sleeping or standing or.., healthy to
sick to recovered to..
Can also deal with dispersal. A specific
place/habitat is then a state
All numbers are then between 0 and 1
since probability to change from one
state to another.
Closed systems, hence no losses or
addition.
Simple Markov chains
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Handles probabilities for
an organism to ‘move’
between different states
All numbers along the
arrows have to be
between 0 and 1
All numbers out from a
state have to sum up to 1.
(otherwise a loss or
addition)
0.9
0.5
0.3
1
3
0.2
0.2
0.5
2
0.3
 0.5 0.5 0.9 


 0.2 0.3 0 
 0.3 0.2 0.1


0.1
Simple Markov chains
=1
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A row is the input to a
state.
A column is the output
of the state.
The row can sum to
[0,>1]
The columns always
sum to 1
=1
=1
 0.5 0.5 0.9 


 0.2 0.3 0 
 0.3 0.2 0.1


0.9
0.5
1
0.3
3
0.2
0.2
0.5
2
0.3
0.1
Simple Markov chains,
absorbing states
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A state is absorbing if
the probability is 1 to
stay in the state.
With time the
probability, where the
individuals are will
move, to this absorbing
state.
 0.5 0.5 0 


 0. 2 0 . 3 0 
 0. 3 0 . 2 1 


0.5
1
0.3
3
0.2
0.2
0.5
2
0.3
1
Simple Markov chains,
equilibriums
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What happends over time?
x(t)=Atx(0)?
Is there any equilibrium, x’=Ax’?
If At after a time t only consist of
positive elements (>0), the a
equlibrium exists. This equilibrium
is the eigenvector with eigenvalue 1
of matrix A.
This equilibrium is also a column in
At, for large t. At is then the steady
state matrix
This equilibrium is a kind of ultimate
probability between the states. For
example that that there is a 60%
probability that an individual is in
state 1,…..
 0.5 0.5 0 


 0. 2 0 . 3 0 
 0. 3 0 . 2 1 


0.5
1
0.3
3
0.2
0.2
0.5
2
0.3
1
Simple Markov chains,
eigenvalues eigenvectors
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An equilibrium exists if all states
are connected (direct or
 0.5 0.5
indirect). No state is completed

isolated. No groups of stes ae
 0. 2 0 . 3
isolated from the other.
 0. 3 0 . 2

Calculate eigenvectors and
eigenvalues by matlab code,
[x,y]=eig(A)
0.5
Several equilibriums may exists
0.3
if there are several absorbing
1
states
0.2
0

0
1 
3
0.2
0.5
2
0.3
1
Absorbing state,
equilibrium
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It is possible to
calculate the probability
that a system reaches
the different
equilibriums
In the example the
question is what the
probability is to end up
in state 2 or three?
More on page 126 and
127, yet this you can
read briefly.
 0.5 0 0 


 0.2 1 0 
 0.3 0 1 


0.5
1
0.3
3
0.2
1
2
1
Summarizing
classes/stages/state-matrices
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Ageclass/stages
Population growth –
eigenvalue.
Population distribution
eigenvector
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The state of individuals and
populations-Markov chains
Probability for the state of the
individual
Equlibrium-eigenvector
3.1, 3.2, 3.3, (3.4), 3.5, 3.6, 3.9, 3.10.
note that 3.4 is too small, you have to do an additional one