Sobre la Complejidad de la evolucion de mapas reducionistas desordenados

1. Instituto Balseiro, Universidad Nacional de Cuyo, Centro Atomico Bariloche, CNEA, CONICET

San Carlos de Bariloche, Argentina.

Manuel O. Cáceres (1,2)

2. Senior Associated to the ICTP, Trieste, Italy.

Consider an ensemble of positive random matrices M, with a well defined probability measure.

?)(lim

nfX nn

Given the random linear map:

how can we characterize the long time growth rate of the vector

nn XMX 1

Dynamics form linear positive maps

mjnC j 1 ; )( X ;0,1,2n ;XMX nn1n

Let be a state vector of dimension , characterizing a population at the time step . The linear dynamics is given by a recurrence relation:

)( Xn nC j mn

mjM j ,...,3,2 , , 1111

Because M is a positive matrix we can apply Perron Frobenius theorem then: If M is irreducible there exist a positive single eigenvalue associated to a positive eigenvector such that:

Then the Liapunov exponent can be defined as: 1logr

Then (if ) it is simple to prove the asymptotic behavior:

11lim nnn X

ljlj ,

A more robust prove can also be done using the Tauberian theorem, for divergent series,

and using the Green function techniques (Z-transform)

0 ;)exp()(0

n

nn anyayU

The Green function of a linear positive map

)(M)(1

z1

00

01

1n zGMXzXzGz

Xz n

nn

nn

Defining the z-transform vector:

Consider summing in the map , then:

So we arrive to the solution:

Then we can define the Green function (Matrix) as

0

nz)(n

nXzG

011)( XzMzG

11)(G zMz

0

10 n

nn

nn

n XMzXz

Asymptotic dynamics of a linear positive map

From the Perron Frobenius theorem we know that there exist a pole such that the Green function behaves like

Then:

This is the expected results (non-random case)

jljl Ctezzzz 111)(G

111

/1 ,1

lim zz

Xn

nn

1z

Consider now the random case

? )(lim nfX nn

Given the random linear map:

we like to characterize the long time behavior of the ensemble mean value vector:

nn XMX 1

Calculating the mean-value Green function

a) The mean-value Green matrix can be calculated from its own evolution equation, splitting M in order + disorder

and using the projection operators

in the evolution, we get two coupled equations. So we have to solve the problem:

1,G(z)B)z(H-1G(z)zM-1

1

)(G)(G zz

),(B)(H1)(

),(B)(H1)(

zGzzG

zGzzG

Calculating the mean-value Green matrix

10 H

1)(G

z

z

a) After some algebra we get an exact and close expressionb) Each term y the series represent a cumulant contribution c) This is given in term of the ordered Green function

d) The smallest pole of the averaged Green function is the key element in order to study the long time behavior of the averaged linear map.

,BBGH-1)(G

1

0

0

k

kzz

1) How can an age-structured population be described when the vital parameters have uncertainties?2) How can we make inferences about the global growth rate in a population? (i.e., to characterize the Malthusian rate)3) Can a constructive approach be made to get the time evolution of the mean-value age structured population vector?

Biological Motivations (Marine mammal)

12p 3p 1mp

1mp

2f3f

1mf

mf

1f

2 m1m31p

Vital Parameters: General properties for the elementsin a discrete age-structured population dynamics

((reproductive eventsreproductive events: ): )

((aging processesaging processes: ): )

mf

mp

1) The age-specific classification is stable {j=1,…,m}.

2) The reproduction is by birth-pulse: age-specific fecundity

3) The density does not affect the vital parameters (density independent vital parameters { , })

4) The evolution is performed on discrete times corresponding to the age structure

Assumptions in a discrete age-specific model

Why randomness in a Leslie matrix?

1) Very often vital-parameters are limited for long-lived species.

2) The estimation of all age-specific parameters are in general impossible to attain.

3) There are large sampling variances because of the small sample sizes.

4) The repeated multiplication of imprecise estimations lead to large error propagation in the vital parameters.

jf

jpjf

1) The Leslie matrix is an array of positive numbers.

2) are survival probabilities and are fecundities.

3) Example of a 4x4 Leslie matrix model (irreducible):

4) Then we can apply Perron-Frobenius theorem (primitive):

5) The real value is an estimate of the growth rate.

(the corresponding continuous growth rate is: )

6) is the relative population contribution made to the stationary state by each individual age-group.

Example of a non-random Leslie matrix

10 ;0 ;

000

000

000

3

2

1

4321

nn pf

p

p

p

ffff

M

0 2,3,...j ; 11 j

1

111 M

1

mp mf

1logr

Exact solution of the Leslie dynamics

12p 3p 1mp

1mp

2f3f

1mf

mf

1f

2 m1m31p

mjnC j 1 ; )( X ;XMX nn1n

Let be the state vector of dimension , characterizing the population at the time step . Each component represents the number of individual in each age-category . Then, Leslie’s dynamics is given by a recurrence relation:

)( Xn nC j mn

)(nC j

j

0

nX)(n

nzzG

1M -1)(G zz

0X)(M)( zGzzG

Time asymptotic behavior (non-random case)

By using the Tauberian theorem we can study the asymptotic behavior of each population group:

Then,

where is the smallest (positive) pole of the Green function Matrix; i.e., the Perron-Frobenius eigenvalue!

1z

0 11

1

)(1

dzzz

Limn

n

11

n

1X

n

n zLim

)(nCLim jn

1M -1)(G zz

Note that: if the elements in a Leslie matrix were random, and if we were able to find the distribution of this dominant eigenvalue, it would be naive to analyze the behavior of the population growth rate by calculating:

1) may have a different statistics compared with the one from the fecundity elements

2) Elements in each statistics and/or can be correlated!

3) The size of the uncertainty may be quite different in the survival probabilities than in the fecundities elements

What about a random Leslie matrix ?

np

nfnp

nf

Remarks

1) Is there and effective growth rate?

2) Can we calculate the mean value population vector state?

3) Does Perron-Frobenius’ theorem exist for the random case?

Questions

),( 1 mff )( 1 mpp

Asymptotic analysis in the random case

1n

~1X

n

en z

Lim

a) Where is the smallest positive pole of the mean-value Green function.b) Then, we can define , as the effective growth rate for a random Leslie model (effective Malthusian rate).

ez

By using the mean-value Green function we can study the asymptotic temporal behavior of the mean-value population by applying the Tauberian theorem. Then we can proof that:

)(G z

eez 1)(

)(G zDominant pole

How to calculate the effective growth rate

a) Having proved that the asymptotic time-evolution of the mv population goes like:

0 1det )3(

0 1det )2(

0 1det )1(

000

0

TT

T

BBGBGBBGBHzO

BBGBHzO

BHzO

nen zLim /1Xn

b) The dominant pole of can be obtained from the secular polynomial:

ez

0

0 BBGH-1detk

kz

We can proceed to calculate order-by-order the value: ,for example:

1)( ee z

)(G z

Extinction analysis

Resources

FertilityNaive

Extinction if <1 !

ensfluctuatio

nsfluctuatio

Resources

Fertility

MAXNaivemin e

G )(1

M 1

N

i

iN

e

The usual approximation to studythe growth rate and its variance...Sensitivity analysis, etc....

Our perturbative approach allows usto define an effective value and to project the extintion...

1) A constructive approach, in terms of the general properties of linear random positive maps has been presented2) The dynamics of an age-structured population in the presence of randomness, in the vital parameters, have been described.3) We got an effective growth rate for the asymptotic mean-value population vector state, so this value can be called the effective Lyapunov exponent.Unresolved issues (in the context of Linear positive maps):Dispersion analysis. Mean-value invariant vector. Extended Leslie matrices (associated with spatial structure)All of this facts can help in the study of indices in Eco-toxicology.

Our motivations have been solved…

12p 3p 1mp

1mp

2f3f

1mfmf

1f

2 m1m31p

?1e

Predicting the population-level effects of a toxic substance is challenging because the individuals-level effects are diverse and stage specific.

Theorem: Let M be a mxm positive matrix

a) M is irreducible

b) Let c the minimum cycle in the digraph of M

Then, M is primitive

In general almost all Leslie matrices are primitive:

Perron-Frobenius theorem:

a) Let M be a primitive positive matrix, then

b) Let M be irreducible but non-primitive with indices d

Then,

Remarks on irreducible Leslie matrices

0)M1( 1 m

0M)( 2)-c(mm

,m][jijinfn 1, with ,,for least at ,0

1,...,2,1 mjj

dkk ,...,2,1

Some Refs: Arnold et al. Ann. Appl. Prob. 4, 859, (1994).Caceres M.O., Elementos de Est. de no-equil., Reverte (2003).

From: H. Caswell, Lewis Publishers, NY. 1996