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. Given the random linear map: how can we characterize the long time growth rate of the vector

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

Then (if ) it is simple to prove the asymptotic behavior: A more robust prove can also be done using the Tauberian theorem, for divergent series, and using the Green function techniques (Z-transform)

The Green function of a linear positive map 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

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)

Consider now the random case Given the random linear map: we like to characterize the long time behavior of the ensemble mean value vector:

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:

Calculating the mean-value Green matrix a)After some algebra we get an exact and close expression b)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.

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) Vital Parameters: General properties for the elements in a discrete age-structured population dynamics (reproductive events: ) (aging processes: )

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.

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

Exact solution of the Leslie dynamics 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:

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! 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 ? 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

Asymptotic analysis in the random case 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). 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: Dominant pole

How to calculate the effective growth rate a) Having proved that the asymptotic time-evolution of the mv population goes like: b) The dominant pole of can be obtained from the secular polynomial: We can proceed to calculate order-by-order the value:, for example:

Extinction analysis Extinction if <1 ! The usual approximation to study the growth rate and its variance... Sensitivity analysis, etc.... Our perturbative approach allows us to 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 presented 2) 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… 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 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