Subjects see chapters

Subjectssee 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

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 andClasses

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 thepopulation 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 thepopulation

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 thepopulationThe dichotomy

Stable proportionsof classes/subpopulations

Stable per capitagrowth rates anddispersal rates etc

Non-structured model

Variation in proportion of individuals in stages/subpopulations

Variation in per capitagrowth rates anddispersal rates etc

Structured model

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)

Note, one time step correspondence to size/span of an ageclass.

1 2 3

b1

s23

sij = probability for an individual in age-class i to survive into the next age-class, j

s12

b2 b3

bi = how many newborns fromageclass i during one timestep(span of an ageclass)

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)

this is a linear system of equations,

one can use linear algebra. Matrix multiplication.

)()()(

0000

)1()1()1(

3

2

1

23

12

321

3

2

1

tntntn

ss

bbb

tntntn

Ageclasses 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)

)()()(

0000

)1()1()1(

3

2

1

23

12

321

3

2

1

tntntn

ss

bbb

tntntn

Ageclasses 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)

)()()(

0000

)1()1()1(

3

2

1

23

12

321

3

2

1

tntntn

ss

bbb

tntntn

Ageclasses, an example 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..

68

10

08.00004.0820

4.64

64

Ageclasses, matrix multiplication – run a simulation

One can calculate this for everafter a while a constantdistribution will evolve

The right hand side distributions will be the same for all following timesteps

Note that the number of individuals may change (density) but the distribution over classes becomes stable

4.64

64

08.00004.0820

2.36.252.59

09.005.086.0

4.64

64

04.029.067.0

2.36.252.59

densities proportions

Ageclasses, eigenvalues andeigenvectors

1.018.072.0

,56.1 11 v

08.00004.0820

If the distribution becomes stable then the per capita growth rate also stabilize and becomes a constant value

If the per capita growth rate becomesstable/constant over time, one can use that insteadof the matrix

Ageclasses, eigenvalues andeigenvectors

1.018.072.0

,56.1 11 v

08.00004.0820

0.2i - 0.08-

0.05i 0.3 0.6i 0.7-

,78.0 22 vi

0.2i 0.08-

0.05i 0.3 0.6i 0.7-

,78.0 33 vi

The other two eigenvalues are complex values and generates theoscillations that occurs prior the stabilisation.

From the beginning again:Solution space and eigenvectors.

The lefthand side, vector (59 26 3), exist in a solution spacespanned by the three eigenvectors. This means that you can reachthe point (59 26 3) in the 3D space by moving along the directionsof the three vectorsMathematically this is expressed by:

4.64

64

08.00004.0820

2.36.252.59

332211)1( vcvcvcn

Assume nay population distribution (not an eigenvector)

Solution to n(t)=Atn(0)

We know that matris*eigenvector equals eigenvalue*eigenvectorAv1=λ1v1.

And that:

4.64

64

08.00004.0820

)(

t

tn

332211)1( vcvcvcn

Combine these two and x(t)=Atn(0) can be written as

3322211 31)1( vcvcvctn ttt

What happends at large t (long time???

Stage models 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.

1 2 3

g1

p23p12

b2 b3

g2

g3

323

212

321

00gp

gpbbg

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.

More general model

Simple Markov chains 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)

1

2

3

0.5

0.20.2

0.3

0.9

0.1

0.3

0.5

1.02.03.003.02.09.05.05.0

Simple Markov chains

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

2

3

0.5

0.20.2

0.3

0.9

0.1

0.3

0.5

1.02.03.003.02.09.05.05.0

=1 =1 =1

Simple Markov chains,absorbing states

1

2

3

0.5

0.20.2

0.3

1

0.3

0.5

12.03.003.02.005.05.0

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.

Simple Markov chains,equilibriums

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,…..

1

2

3

0.5

0.20.2

0.3

1

0.3

0.5

12.03.003.02.005.05.0

Simple Markov chains,eigenvalues eigenvectors An equilibrium exists if all states

are connected (direct or indirect). No state is completed isolated. No groups of stes ae isolated from the other.

Calculate eigenvectors and eigenvalues by matlab code, [x,y]=eig(A)

Several equilibriums may exists if there are several absorbing states

1

2

3

0.5

0.20.2

0.3

1

0.3

0.5

12.03.003.02.005.05.0

Absorbing state,equilibrium

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.

1

2

3

0.5

0.20.3

1

1

103.0012.0005.0

Summarizingclasses/stages/state-matrices

Ageclass/stages Population growth –

eigenvalue. Population distribution

eigenvector

The state of individuals and populations-Markov chains

Probability for the state of the individual

Equlibrium-eigenvector