Structured Population Dynamics in a Real World Context

Ana Carolina Loureiro Martins

Thesis to obtain the Master of Science Degree in

Mathematics and Applications

Supervisor: Professor Henrique Manuel dos Santos Silveira de Oliveira

Examination Committee

Chairperson: Professor Maria Cristina de Sales Viana Serodio SernadasMembers: Professor Joao Maria da Cruz Teixeira Pinto

Professor Luıs Humberto Viseu MeloSupervisor: Professor Henrique Manuel dos Santos Silveira de Oliveira

July 2017

ii

Acknowledgments

After these five tough years, it was a pleasure to make part of this academical community in which

I found what excellence and resilience truly are. I thank every given opportunity to meet new people,

faculty and colleagues, and to enrich myself with new ways of thinking.

I want to specially thank the invaluable and helpful support of my supervisor, Professor Henrique

Oliveira, which gave me the opportunity of working in my thesis while contributing and applying my newly

acquired knowledge in the project called Conhecer Arroios, promoted by Instituto Superior Tecnico de

Lisboa and Junta de Freguesia de Arroios.

Furthermore, I couldn’t have finished this degree without the constant support and hope of my par-

ents, Ana and Joao, and of someone very close to me, Marılia. I also want to express my gratitude to

Miguel Pereira, that read everything carefully to avoid any grammar and spelling mistakes and helped

me finish writing this thesis in the computer when I couldn’t, due to a shoulder surgery.

Last but not least, I want to also thank all of my friends, from LMAC and MMA, that went with me on

this journey.

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iv

Resumo

O estudo de como decorre a evolucao de uma populacao ao longo do tempo e de quao vulneravel

esta se encontra face a mudancas nos parametros que a descrevem e essencial para a compreensao

das necessidades dos habitantes e para a tomada de decisoes segundo um plano a longo prazo. Por-

tanto, a exploracao de ferramentas que permitem esta visao global dos habitantes de uma determinada

regiao e crucial para o desenvolvimento dos mesmos. Nesta tese, e feita uma pequena pesquisa so-

bre modelos populacionais, seguindo-se a apresentacao de novos modelos que contemplam imigracao

e emigracao. Alem disso, o conceito original de matrizes de Leslie e adaptado de forma a integrar

os fluxos migratorios. Uma deducao das caracterısticas da entropia evolucionaria e desenvolvida, e

a entropia da taxa de crescimento e analisada. Um algoritmo que comprime as matrizes de Leslie e

tambem explicado. Finalmente, este conhecimento e aplicado num contexto do mundo real, permitindo

a obtencao de projeccoes populacionais e medidas intrınsecas do ecossistema em estudo.

Palavras-chave: Modelos Populacionais, Matrizes de Leslie, Entropia Evolucionaria, Aplicacao

a um Contexto da Vida Real, Compressao de Matrizes de Leslie.

v

vi

Abstract

Understanding how a population evolves with time and how vulnerable to changes in certain param-

eters this ecosystem is is essential to truly comprehend the needs and make decisions according to

long term plans. Therefore, the exploration of tools that help this overview of the inhabitants of a certain

area is crucial to their development. In this thesis, a brief survey about population models is made,

being followed by the presentation of new models that include immigration and emigration. Also, the

original concept of Leslie matrix is adapted in order to integrate migration flows. A deduction of the char-

acteristics of the evolutionary entropy is developed, and the sensitivity of the growth rate is analysed.

An algorithm that compresses Leslie matrices is further explained. Finally, this knowledge is applied

in a real world context, allowing the obtention of population projections and intrinsic measures of the

ecosystem in study.

Keywords: Population Models, Leslie Matrices, Evolutionary Entropy, Application to Real World

Context, Compression of Leslie Matrices.

vii

viii

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Claim of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 3

2.1 Concepts and relevant theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.3 Population matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.4 Relation between Leslie Matrices and Graphs . . . . . . . . . . . . . . . . . . . . . 7

2.1.5 Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Model proposed by H. Leslie [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Model proposed by L.P. Lefkovitch [13] . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Model proposed by H. Caswell and N. Sanchez Gassen [23] . . . . . . . . . . . . 11

2.2.4 Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Proposed Models 13

3.1 Model without migratory flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Model with one migratory flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Model with immigration and correction flows . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Model with internal and external flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Model with Leslie matrix that includes immigration and emigration I . . . . . . . . . . . . . 15

3.6 Model with Leslie matrix that includes immigration and emigration II . . . . . . . . . . . . 16

ix

4 Partial Derivates of Evolutionary Entropy 19

4.1 Evolutionary Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Characteristics of the Evolutionary Entropy . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Compression of the Leslie matrices 25

5.1 Re-weighting the edges of the graph G2c . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Real Word Context 33

6.1 Construction of Arroios’ Leslie Matrices based on 2001 and 2011 data . . . . . . . . . . . 33

6.1.1 Fertility Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.1.2 Mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.1.3 Survival Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.1.4 Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.1.5 Emigration to Foreign Countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.1.6 Error Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.2 Application of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.2.1 Model without migratory flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.2.2 Model with one migratory flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2.3 Model with immigration and correction flows . . . . . . . . . . . . . . . . . . . . . . 44

6.2.4 Model with internal and external flows . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2.5 Model with Leslie matrix that includes immigration and emigration I . . . . . . . . . 49

6.2.6 Model with Leslie matrix that includes immigration and emigration II . . . . . . . . 52

6.3 Partial Derivatives of Evolutionary Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.3.1 Leslie matrix without migrations 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.3.2 Leslie matrix that includes immigration and emigration I 6.2.5 . . . . . . . . . . . . 56

6.3.3 Leslie matrix that includes immigration and emigration II 6.2.6 . . . . . . . . . . . . 56

6.4 Population Sample from 2017 and the Application of Models . . . . . . . . . . . . . . . . . 57

7 Conclusions 61

Bibliography 63

A Iterative Method 65

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List of Tables

5.1 Table with the number of children born, of women that survived from age-class Ci to Ci+1,

and of women by c age-classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Table with the number of children born, of women that survived from age-class Ci to Ci+1,

and of women by 2c age-classes, related with the women subdivided by c age-classes. . . 30

6.1 Fertility rates obtained from 2001’s data without considering migratory flows. . . . . . . . 38

6.2 Survival rates obtained from 2001’s data without considering migratory flows. . . . . . . . 39

6.3 Proportion of the flux relatively to the estimated female residents in 2011 of the model

with one migratory flow in Arroios obtained using 2001 and 2011 Census data. . . . . . . 42

6.4 Proportion of the immigration flow relatively to the estimated female residents in 2011 of

the model with immigration and correction flows in Arroios obtained using 2001 and 2011

Census data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.5 Proportion of a correction flow that includes emigration and errors in estimation of immi-

gration relatively to the estimated female residents in 2011 of the model with immigration

and correction flows in Arroios obtained using 2001 and 2011 Census data. . . . . . . . . 45

6.6 Proportion of the internal flow relatively to the total female residents of the model with

internal and external flows in Arroios obtained using 2001 and 2011 Census data. . . . . 47

6.7 Proportion of the external flow relatively to the total female residents of the model with

internal and external flows in Arroios obtained using 2001 and 2011 Census data. . . . . 47

6.8 Fertility rates using the model with Leslie matrix that includes immigration and emigration

II and an iterative method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.9 Survival rates using the model with Leslie matrix that includes immigration and emigration

II and an iterative method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.10 Fertility rates using the model with Leslie matrix that includes immigration and emigration

II and an iterative method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.11 Survival rates using the model with Leslie matrix that includes immigration and emigration

II and an iterative method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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List of Figures

4.1 S = 0, where S is a measure of the number of replicative cycles. . . . . . . . . . . . . . . 21

4.2 S > 0, where S is a measure of the number of replicative cycles. . . . . . . . . . . . . . . 21

6.1 Distribution of female newborns using the non-adjusted number and the adjusted number

of female newborns, using 2001 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.2 Distribution of female deaths using the non-adjusted number and the adjusted number of

female deaths, using 2001 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.3 Distribution of female inhabitants by age classes in Arroios using 2011’s Census data,

and using the model without migrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.4 Projected total of female inhabitants in Arroios using the model without migrations in 2016,

2021, 2026, 2031 and 2036.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.5 Projected distribution of female inhabitants by age classes in Arroios using the model

without migrations in 2016, 2021, 2026, 2031 and 2036. . . . . . . . . . . . . . . . . . . . 41

6.6 Distribution of female inhabitants by age classes in Arroios using 2011’s Census data,

and using the model with one migratory flow. . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.7 Projected total of female inhabitants in Arroios using the model with one migratory flow in

2016, 2021, 2026, 2031 and 2036. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.8 Projected distribution of female inhabitants by age classes in Arroios using the model with

one migratory flow in 2016, 2021, 2026, 2031 and 2036. . . . . . . . . . . . . . . . . . . . 43

6.9 Projected distribution of female inhabitants by age classes in Arroios using the model with

one migratory flow in 2016, 2021, 2026, 2031 and 2036. . . . . . . . . . . . . . . . . . . . 46

6.10 Estimated internal and external flows of female inhabitants by age classes in Arroios using

the model with internal and external flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.11 Distribution of female inhabitants by age classes in Arroios using 2011’s Census data,

and using the model with Leslie matrix that includes immigration and emigration I. . . . . 50

6.12 Projected total of female inhabitants in Arroios using the model with Leslie matrix that

includes immigration and emigration I in 2016, 2021, 2026, 2031 and 2036. . . . . . . . . 51

6.13 Projected distribution of female inhabitants by age classes in Arroios using the model with

Leslie matrix that includes immigration and emigration I in 2016, 2021, 2026, 2031 and

2036. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xiii

6.14 Distribution of female inhabitants by age classes in Arroios using 2011’s Census data,

and using the model with Leslie matrix that includes immigration and emigration II. . . . . 53

6.15 Projected total of female inhabitants in Arroios using the model with Leslie matrix that

includes immigration and emigration II in 2016, 2021, 2026, 2031 and 2036. . . . . . . . . 54

6.16 Projected distribution of female inhabitants by age classes in Arroios using the model with

Leslie matrix that includes immigration and emigration II in 2016, 2021, 2026, 2031 and

2036. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.17 Projected distribution of female inhabitants using models without migration and with one

migratory flow in 2016, and distribution of female inhabitants of the population sample

taken from 2017, by age classes in Arroios. . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.18 Projected distribution of female inhabitants using models with Leslie matrix that includes

immigration and emigration I and II in 2016, and distribution of female inhabitants of the

population sample taken from 2017, by age classes in Arroios. . . . . . . . . . . . . . . . 59

xiv

Chapter 1

Introduction

1.1 Motivation

The knowledge of how a population will evolve over the years is essential to decision makers, since

they can create and follow strategies accordingly.

It can be achieved by forecasting a population, i.e. by making predictions based on current and past

data. Then, one can be aware of certain intrinsic aspects of an ecosystem, such as: the potential growth,

the age distribution of its individuals, survivors and descendants, fertility and mortality rates. Nonethe-

less, the projection of these intrinsic parameters is in fact conditional to one or more assumptions which

can indeed change. Sensitivity analysis measures how these aspects will react to perturbations (migra-

tion for example).

Assuming that age of individuals is known, there are two main approaches to exploit the dynamics

of the population in question: using a continuous-time integral equation, first introduced by F. R. Sharpe

and A. J. Lotka [1]; or developing a matrix formulation with age classes, presented by H. Bernardelli [2],

E.G. Lewis [3] and by P.H. Leslie [4].

In order to properly study population dynamics, models using Leslie matrices must be applied, based

on current and past data found at the Census, or other databases.

This thesis is closely related to a project called Conhecer Arroios, which was a partnership between

Junta de Freguesia de Arroios and Instituto Superior Tecnico de Lisboa. The main goal of this project

was to know how Arroios, a parish situated in Lisbon with more than 30000 inhabitants, was evolving

with time and predict the population growth in the future, by analysing 2001 and 2011 Census’ data and

by retrieving intercensus data from a population sample in 2017. In fact, I was coauthor of this study [5],

which conveyed the analysis of the evolution of Arroios parish along time and its projections, based on

2001 and 2011 Census’ data, and a population sample retrieved in 2017.

Thus, the decision makers can invest on infrastructures and services with guidance about the future

needs of their citizens. Otherwise, the lack of knowledge about the forecasted residents in a parish,

or even in a country, could lead to unnecessary expenditure and to depreciation of the domains on

demand at a later time. Therefore, in order to forecast useful pointers about the population’s future

1

behaviour, useful and effective models must be found based on the current inhabitants. Also, there is

the need to analyse the response or sensitivity of evolutionary entropy to perturbations in the microscopic

parameters.

Another subject related to the predicted models is how to compress a Leslie matrix and, consequently

these models. That is how to, from a 5 year-age class matrix, achieve a 10 year-age class matrix without

any further input, by instance. Therefore, the applied model can be adapted according to the focus of

the study in question without needing to recalculate every parameter for every age-class, which can lead

to a reduction of the error produced by the iteration of the model.

1.2 Claim of Contributions

In what follows, we can point several contributions of our work.

• We propose several models that include migration flows;

• We analyse and applied several measures of evolutionary entropy-sensitivity to Arroios parish;

• We develop an algorithm that allows the compression of Leslie matrices;

• We implement the models and measures of evolutionary entropy in a real world context.

1.3 Thesis Outline

We conclude by presenting a brief overview of this dissertation.

In Chapter 2, the background, regarding the concepts and relevant theorems, and the state of the art

of population models applied before, are explored.

In Chapter 3, we develop and explain population models where the first two, model without migratory

flows and model with one migratory flow, can be already found in literature, and the remaining ones

contemplate the need for accounting both immigration and emigration.

In Chapter 4, we approach the partial derivatives of evolutionary entropy, deriving the characteristics

of the evolutionary entropy, and analysing the growth rate sensitivity.

In Chapter 5, the algorithm allowing the compression of the Leslie matrices is detailed.

In Chapter 6, we present and discuss the results given a real world context, obtained by applying the

models and measures, explained on previous Chapters, to a parish in Lisbon called Arroios.

2

Chapter 2

Background

In this chapter we introduce some fundamental concepts and theorems related to matrix population.

Also, we introduce the Leslie and Lefkovitch matrices. A number of applied models by several authors

is presented.

2.1 Concepts and relevant theorems

2.1.1 Matrices

Definition 2.1. A square matrix, called A, is reducible iff it can be placed into block upper-triangular

form by simultaneous row/column permutations.

A square irreducible matrix is a non-reducible matrix. [6] That is, an irreducible matrix is a square

nonnegative matrix [7] such that

∀i,j∃k>0 : Ak(i, j) > 0. (2.1)

Definition 2.2. A square primitive matrix Aij [7] is a nonnegative matrix where some power of A = (i, j)

is positive.

Theorem 2.3. A positive square matrix is primitive and a primitive matrix is irreducible.[8]

Definition 2.4. A right eigenvector XR [9] is a column vector verifying

AXR = λRXR (2.2)

Notice that the term eigenvector is used to refer to a right eigenvector.

Definition 2.5. A left eigenvector XL [9] is a row vector verifying

XLA = λLXL (2.3)

3

Theorem 2.6. (Perron-Frobenius Theorem) If all elements aij of an irreducible matrixA are nonnegative,

then

R = min{Mλ}. (2.4)

is an eigenvalue of A and all the eigenvalues of A lie on the disk

|z| ≤ R, (2.5)

where, if λ = (λ1, ..., λn) is a set of nonnegative numbers (which are not all zero),

Mλ = inf{µ : µλ >n∑j=1|aij |, 1 ≤ i ≤ n}. (2.6)

Furthermore, if A has exactly p eigenvalues (p ≤ n) on the circle |z| = R, then the set of all its

eigenvalues is invariant under rotations by 2πp about the origin. [9]

2.1.2 Graphs

A graph [10] is a collection of points, called vertices or nodes, that can be connected using lines,

called edges or arcs. There are three main types of graphs:

• simple graphs: there is at most one edge connecting two vertices;

• multigraphs: two vertices can have more than one edge connecting them, without any loops (self-

connected vertices);

• pseudographs: multigraphs that allow loops.

If a graph has labelled edges and vertices, it is called a labelled graph. Otherwise, it is an unlabelled

graph.

If the edges of a graph are undirected, then it is called an undirected graph. Otherwise it can be

a directed graph (the edges have arrows representing a unilateral or bilateral direction), or an oriented

graph (directed graph in which each edge has a unique direction).

Each edge can also have a weight associated. In that case, the graph is called weighted graph.

Otherwise, it is called unweighted graph.

A graph can be represented using an adjacency matrix, which is a matrix with rows and columns

labelled by graph vertices with 0 or a number different from 0 that represents the weight of an edge

connecting the vertices vi and vj . If the matrix has a 0 on position (vi, vj) then vi and vj are not

connected, that is do not have any edge between them. Otherwise, the value on (vi, vj) is the weight of

the edge connecting vi and vj .

A symmetric adjacency matrix represents an undirected graph and loops are represented with values

different from 0 on the matrix diagonal.

4

2.1.3 Population matrices

Next we will exploit two types of population matrices: Leslie and Lefkovitch matrices. Due to the

regularity of the age intervals in the accessed data, there is no need to use the Lefkovitch matrix.

Instead, in this thesis we will use the Leslie matrix.

Leslie Matrix

Leslie Matrix rises as a possible answer for how are the survivors’ and descendants’ ages of a

certain population distributed, considering successive intervals of time and supposing that the rates

of fertility and mortality are applied to all elements equally and constant over time. The details are

further developed by H. Leslie [11] and H. Caswell [12]. In this model, only the female population will be

studied, being expressed by m + 1 linear equations organised in age classes. From now on, we adopt

the convention used on H. Leslie’s article [4].

Let:

• nxtbe the number of females alive in the age group x to x+ 1 at time t;

• Px be the survival probability of a female from an age class between x and x + 1 at time t to the

next age class x+ 1 to x+ 2 at time t+ 1. It has values between 0 < Px < 1;

• Fx ≥ 0 be the number of daughters born between t and t + 1 that are still alive in the age class

from 0 to 1 at time t+ 1 and whose mothers lived aged x to x+ 1.

Hence, the age distribution at the end of one unit interval can be expressed as:

m∑x=0

Fxnx0 = n01 (2.7)

P0n00 = n11 (2.8)

P1n10 = n21 (2.9)

P2n20 = n31 (2.10)

. . . (2.11)

Pm−1nm−10 = nm1 , (2.12)

which can be rewritten as the following m + 1 square matrix L × n0 = n1, where n0 and n1 are column

vectors providing the age distribution at t = 0, 1:

5

L =

F0 F1 F2 . . . Fm−2 Fm−1 Fm

P0 0 0 . . . 0 0 0

0 P1 0 . . . 0 0 0

0 0 P2 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . Pk 0 0

0 0 0 . . . 0 Pm−1 0

. (2.13)

If Fm = 0, the determinant of the matrix L is 0. Hence, the L is singular. Moreover, the age distribution

at time t is given by

Lt ×

n00

n10

n20

...

nm0

, (2.14)

that is, the multiplication between the age distribution at t = 0 and the matrix Lt. Then, the number of

female alive elements in the population at time t can be calculated as:

nj−10 ×m∑i=1

Lti,j . (2.15)

We are now going to derive the elements Fx and Px of the matrix.

Notice that, if there are nx,0 women alive in the age class from x to x + 1 at time t = 0, then the

survivors of this class will be the females in x+ 1 to x+ 2 age class at time t = 1. Then,

Pxnx,0 = nx+1,1. (2.16)

For Px’s derivation (Px is the survival probability of a female from an age class between x and x+ 1

at time t), we have:

Px = Lx+1

Lx, (2.17)

where Lx and Lx+1 is the number of females alive in the age group x to x + 1 and x + 1 to x + 2,

respectively, in the data collected from several databases.

Consider now Fx, which is the number of daughters born between t and t+ 1 that are still alive in the

age class from 0 to 1 at time t+ 1 and whose mothers lived aged x to x+ 1.

Let:

• mx be the fecundity per-capita for females of age class x;

6

• lx = p0p1...px−1 be the survival from class 0 to x− 1.

Then we can calculate Fx:

Fx = Pxmx+1. (2.18)

Lefkovitch Matrix

A Lefkovitch matrix, explained and applied in several ecological studies on Lefkovitch’s article [13],

is a squared matrix describing populations with stage or size structure, where:

• Fx is the number of daughters that survived from age class 0− 1 and whose mothers are from age

class x to x+ 1;

• Px,x is the survival probability of females of the age class x in time t that still belong to class x in

time t+ 1;

• Px,x+1 is the survival probability of females from age class x in time t to next age class x + 1 in

time t+ 1

Thereby we obtain the following matrix:

L =

F0 + P0,0 F1 F2 . . . Fm−2 Fm−1 Fm

P0,1 P1,1 0 . . . 0 0 0

0 P1,2 P2,2 . . . 0 0 0

0 0 P2,3. . . 0 0 0

......

.... . .

......

...

0 0 0 . . . Pm−2,m−1 Pm−1,m−1 0

0 0 0 . . . 0 Pm−1,m Pm,m

. (2.19)

Notice that Px,x+Px,x+1 gives the total survival rate for the age class x and that, since newborns can-

not reproduce themselves, F0 + P0,0 = P0,0. Also, observe that the diagonal entries give the probability

of a female remaining in the same class from year t to t+ 1.

A pseudo-Leslie matrix is a matrix that can be decomposed in a row matrix (does not need to be a

non-negative and a subdiagonal positive matrix). Also, Lefkovitch matrices are similar to Leslie matrices

as pseudo-Leslie matrices. [14]

2.1.4 Relation between Leslie Matrices and Graphs

Notice that Lt in 2.13 can be seen as an adjacency matrix of a oriented and weighted graph. There-

fore, another way of representing a Leslie matrix Gc can be:

7

C0 C1 C2 Cm−2 Cm−1 Cm

F0

P0 P1

F1F2

Pm−2

Fm−2

Pm−1

Fm−1

Fm

. . .

2.1.5 Population Dynamics

Evolutionary Entropy

Definition 2.7. Evolutionary entropy H is a statistical parameter that describes the diversity of pathways

of energy flow between the elements that compose the microlevel (e.g. individuals in the population),

and characterises the rate at which macroscopic variables, after a random perturbation, return to their

steady-state condition, that is, the robustness or stability of the hierarchy.

Note that this statistical measure is positively correlated with the elements that compose the mi-

crolevel.

Furthermore, evolutionary entropy can be defined as the rate at which the stochastic process P

generates information.

Analytically,

H =d∑i=1

πiHi (2.20)

with

Hi = −n∑j=1

pij log pij . (2.21)

Hi is called the Shannon-entropy associated with state X of the Markov Chain.[15]

Moreover, H is the weighted average, taken over all the stationary states πi, of Hi. Thus,

H =d∑i=1

πiHi = −n∑j=1

πipij log pij (2.22)

Robustness

Definition 2.8. The robustness of the system R is the fluctuation decay rate of Pε(t) on a logarithmic

time scale and characterises the insensitivity of a system to changes in the microlevel parameter.

Note that the entropy H and the robustness R are positively correlated.

Theorem 2.9. (Entropy-Robustness Theorem) Let ∆H = H ∗−H and ∆R = R∗−R be the change on

entropy and robustness, respectively, resulting from a perturbation in the parameters that describe the

network. Then

8

∆H ×∆R > 0. (2.23)

Therefore, this theorem states that an increase in entropy leads to an increase in robustness. Hence,

the system is more insensitive to changes in the microlevel parameters that describe the network.[15]

Resilience

Definition 2.10. The resilience of the system is an aggregate property that depends on the response of

entropy to perturbations of the various linkages that compose the network.

Sensitivity

Definition 2.11. The sensitivity Sij relative to the changes of a system defined by a squared matrix Aij

is given by the derivatives of λ respective to the entries of A, where λ corresponds the eigenvalues of

A. That is,

Sij = ∂λ

∂aij= viuj , (2.24)

where vi and uj are the components of the left and right dominant eigenvectors, which are always

positive. [16]

Elasticity

Definition 2.12. The elasticity eij of a system defined by a squared matrix Aij [15] is given by

eij = aij∂r

∂aij= aij

λSij , (2.25)

where

r = log λ. (2.26)

2.2 State of the art

The idea of projecting a structured population size via a model with discrete time steps was intro-

duced by E.G. Lewis [3] and P.H. Leslie [4]. After that, some extensions of these basic models were

published by M.H. Williamson [17], M.B. Usher [18], L.P. Lefkovitch [13], C.A. Bosch [19], and recently

J.F. Alves and H. Oliveira [14].

In order to study populations in more detail, we need to decide upon the structure of the population

model. That is we need to choose the variables that confine all the previous data of an inhabitant that are

important to predict his future, as explained on Metz and Diekmann [20], and on Chapter 3 of H.Caswell’s

book [12].

9

As H. Caswell pointed out in his article [21], it has been shown that it is beneficial to opt for age and

stage as the variables, since these models allow to take more information about the population than the

ones that rely only on stage as a variable. Also, if the vital rates are dependent on age and stage, then

it becomes essential to explore its correlation and therefore, the model must rely on these two variables.

Notice that stage is a relevant criteria to that population, for example, size, physiological condition,

fertility, or even spatial location, as seen on A.Rogers’ article [22].

Population growth, age and stage structure, and reproductive value belong to population dynamics,

while cohort dynamics refer to survivorship, life expectancy, age at death and generation time. Both pop-

ulation and cohort dynamics are studied calling upon a matrix model. As the name indicates, population

dynamics depends on society as a whole, including deaths and born individuals, while cohort dynamics

depends on individuals by themselves.

Since we are dealing with Human populations, we will classify individuals into the discrete stages

age and size. Population dynamics, projecting the population from time t to t + 1, can be showed as a

square matrix At as follows:

nt+1 = At × nt (2.27)

where nt+1 and nt are vectors with entries nit+1 and nit representing the number of individuals in a

certain stage i at time t+1 and t, where t+1 is the next iteration of t. The individuals can be classified into

age classes, done in Spain’s projection by H. Caswell and N. Sanchez Gassen [23], or the reproduction

for example.

We can classify these models into two categories based on the nature of At:

1. aij(t) are constants: the matrix population model is linear and time-invariant.

2. aij(t) are not constants: the matrix population model is non-linear and density-dependent.

In this thesis, only linear and time-invariant models are approached.

These models may have a change on the entries aij(t) due to external environmental periodic or

stochastic parameters, or to internal density or frequency dependence, as explained by H. Leslie on

[24].

Notice that a population modelled by a matrix with these characteristics evolves according to a expo-

nencial growth rate which is given by the maximum eigenvalue of At, where At is supposed irreducible

and primitive.

If the individuals are only subdivided into age classes then we use a Leslie matrix to model the

population. This approached was followed by P.H. Leslie [4].

Otherwise, if we want other criteria, such as size or stage, to classify the individuals, we must use a

Lefkovitch matrix.

10

2.2.1 Model proposed by H. Leslie [11]

The growth rate of a population can be achieved by finding the eigenvalues and eigenvectors of the

matrix L, explained on 2.1.3, which correspond to population growth rate, stable life distribution and

reproductive value.

Let the population model be written as:

n(t+ 1) = L× n(t). (2.28)

Then, if we want to find the eigenvalues we solve:

(L− λ Id) n(t) = 0. (2.29)

Notice that the roots of the equation can be complex numbers.

The growth rate of a certain population λ is the eigenvalue that has the largest absolute value and it

will determine the expected behaviour of the population in projections. The other eigenvalues determine

the dynamics of the population. Also, the right eigenvector represents the stable age distribution, while

the left one describes the reproductive value.

2.2.2 Model proposed by L.P. Lefkovitch [13]

Let n(t) represent the column vector of a certain female population at time t subdivided by age

classes and the matrix L as presented on 2.1.3. Then

n(t+ 1) = L× n(t), (2.30)

that is, the distribution of the population at the next time step t + 1 is given by the product of the L and

n(t), the population at time t.

2.2.3 Model proposed by H. Caswell and N. Sanchez Gassen [23]

In [23] of H. Caswell and N. Sanchez Gassen, it was applied a model where the Leslie matrix used

has a particularity: it accounts for the emmigration when calculating the survival rate for each age-

class. That is, the survival rate is obtained by subtracting to 1 the mortality and the emigration to foreign

countries for each age-class. Also, a column vector was added to the model, representing immigrations

from foreign countries distributed by age-classes. Hence, the model can be written as:

n(t+ 1) = L× n(t) + Cimmig(t) (2.31)

with n(t = 0) = n0 and where n(t+ 1) is the vector of the female population subdivided by age-classes

at time t + 1, L is the Leslie Matrix with emigration calculated at time t, n(t) is the vector of the female

population subdivided by age-classes at time t + 1, and Cimmig(t) is the column vector containing the

immigrations divided by age-classes.

11

This model does not adapt to the population inside a parish because there is the need to account not

only for flows coming from and to foreign countries, but those from and to the same country and different

parishes.

2.2.4 Population Dynamics

A crucial analysis of the population dynamics must be made in order to know how a perturbation can

affect the intrinsic variables of the population in study.

Thus, some analysis on the evolution of biological aging, i.e. the increase in mortality after maturation

of an organism, can be made. In fact, W.D. Hamilton, in [25], reported that higher fertility will lead to

increases in biological aging unless the resulting extra mortality happens only in immature ages. This

conclusions lead to the age-classified life cycles.

L. Demetrius showed in [26] that, for populations characterized by a stable size or by small fluc-

tuations of it, an unidirectional increase in population entropy for a large period of time leads to an

evolutionary change caused by mutation and natural selection. Also, for populations who experienced

an exponential growth, the same effect can be achieved by an unidirectional increase in growth rate and

a decrease in entropy for periods of time.

The competition between individuals for limited resources can be seen in two ways: classically, where

the process is deterministic and where its prediction relies only on the growth rate of the several popu-

lations (Malthusian selection principle); and non-classically, where the process is considered stochastic

and deeply related to the populations size and their evolutionary entropy and also by the availability of

the resource in question. As explored in L. Demetrius and S. Legendre [27], and in L. Demetrius [28],

the non-classical view which they call entropic selection principle, encapsulates two main contrasting

relations, where variants will have a selective advantage and an increase in frequency if:

• the populations have higher entropy and the resources are limited, constant and diversified;

• the populations have lower entropy and the resources are singular and suffer from disparities of

availability.

12

Chapter 3

Proposed Models

This chapter is mainly focused on presenting the proposed models which convey closed systems

and systems with migration, we are designing models for real world application, namely to the parish of

Arroios in Lisbon where the field work was performed.

All models use a Leslie matrix instead of a Lefkovitch matrix since we are considering regular age

classes, as it is presented on Census data, and, by doing so, we simplify the model.

Moreover, given that only women can produce new human beings and their number is highly corre-

lated to the growth of a population, we shall use a single gender projection, namely the projection of the

female individuals, as employed in the literature [29].

3.1 Model without migratory flows

The model without migratory flows, considered on H. Leslie’s article [4], is written as:

n(t+ 1) = L× n(t), (3.1)

with n(t = 0) = n0 and where n(t+ 1) is the vector of the female population subdivided by age-classes

at time t + 1, L is the Leslie Matrix, in 2.1.3, calculated at time t and n(t) is the vector of the female

population subdivided by age-classes at time t+ 1.

This model is the simplest presented on this thesis. It can only be applied to closed systems, that is,

systems without any kind of flows, e.g. immigration and emigration.

Therefore, if we consider the Arroios parish, this model won’t be very well adjusted to its population,

since these ignored flows are very relevant and of high impact on the overall projection.

3.2 Model with one migratory flow

To cover the migration flows and based on 3.1, a Leslie matrix with the intrinsic population (consider-

ing a closed system) adds to a column vector with the proportions of the flows (subtraction of immigration

to emigration). That is:

13

n(t+ 1) = L(

Id + α diag fluximm−em)

n(t), (3.2)

with n(t = 0) = n0 and where n(t+ 1) is the vector of the female population subdivided by age-classes

at time t + 1, L is the Leslie Matrix calculated at time t, n(t) is the vector of the female population

subdivided by age-classes at time t + 1, fluximm−em is the proportion of the flux relatively to the total

female residents and α is the adjustment factor of the flows along the years.

The main disadvantage of this model is that we have no separate control over the immigration and

emigration, since the only column vector contains both of them.

3.3 Model with immigration and correction flows

This model allows us to take control over the immigration and correction flows, solving in some way

the issue found with 3.2.

It can be written as:

n(t+ 1) = L(

Id + α diag fluximm + β diag fluxcor)

n(t), (3.3)

with n(t = 0) = n0 and where n(t+1) is the vector of the female population subdivided by age-classes at

time t+1, L is the Leslie Matrix calculated at time t, n(t) is the vector of the female population subdivided

by age-classes at time t+ 1, fluximm is the proportion of the immigration flux relatively to the projected

total female residents and α is its adjustment factor along the years, whilst fluxcor is the proportion

of a correction vector that includes emigration and errors in estimation of immigration relatively to the

projected total female residents and β is its adjustment factor along the years.

However, internal immigration, between parishes in the same country, is less prone to fluctuations

comparing to external immigration, between two different countries. The same applies for internal and

external emigration. Also, the vector fluximm was obtained based on the multiplication of the proportion

of Portugal’s female external immigrants for each age class, by the total number of female inhabitants in

Arroios that didn’t live in this parish five years ago. Consequently, some age classes may be predicted

having more immigration than the one observed in reality, being compensated by the opposite scenario

(higher emigration than in reality), since the total number of immigrants in Arroios is available to us.

Therefore, each of them has different rates and different adjustment factors along the years, even

though they are considered in the same column vectors and with the same adjustment factors in this

model, fluximm and fluxcor.

3.4 Model with internal and external flows

We can consider two types of migration: internal, between parishes in the same country, and exter-

nal, coming from or going to another country. These flows experience different fluctuations with time.

The external flows depend on various factors, for example the economy and working conditions of both

14

countries. Nonetheless, the factors between parishes of the same country do not differ greatly through-

out the years and so, internal flows are more stable. In other words, external flows are more prone to

change when considering 10 years, by instance.

Hence, we can consider the following model:

n(t+ 1) = L(

Id + α diag fluxint + β diag fluxext)

n(t), (3.4)

with n(t = 0) = n0 and where n(t+ 1) is the vector of the female population subdivided by age-classes

at time t + 1, L is the Leslie Matrix calculated at time t, n(t) is the vector of the female population

subdivided by age-classes at time t+ 1, fluxint is the proportion of the internal flux relatively to the total

female residents and α is its adjustment factor along the years, whilst fluxext is the proportion of the

external flux relatively to the total female residents and β is its adjustment factor along the years.

This model allows us to adjust our projections according to the expected internal and external flows,

while giving us pointers to the impact of migrations to the intrinsic population in study. In this way, we

can also test what may happen to the inhabitants if there is an increase or decrease in migration.

3.5 Model with Leslie matrix that includes immigration and emi-

gration I

In order to study the sensitivity analysis of the inhabitants including the ones that are subject to the

migration flows, there is the need to place the column vectors mentioned on 3.4 inside the Leslie matrix.

That is, to account the internal and external flows in the survival probability of a female from one class

to the next one, and to account for the fertility rates of the total population, which includes the intrinsic

inhabitants and the ones coming from the migration flows.

On that account, we obtain the following Leslie matrix:

L′ =

F ′0 F ′1 F ′2 . . . F ′m−2 F ′m−1 F ′m

P ′0 0 0 . . . 0 0 0

0 P ′1 0 . . . 0 0 0

0 0 P ′2 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . P ′k 0 0

0 0 0 . . . 0 P ′m−1 0

, (3.5)

where F ′x ≥ 0 is the number of daughters born between t and t + 1 that are still alive in the age class

from 0 to 1 at time t+ 1 and whose mothers lived aged x to x+ 1 and were an inhabitant or a immigrant

at time t + 1, and P ′x is the survival probability plus the migration flows of a female from an age class

between x and x+ 1 at time t to the next age class x+ 1 to x+ 2 at time t+ 1.

The model is:

15

n(t+ 1) = L’× n(t). (3.6)

With this model, we can study the influence of changing the parameters in the population with migra-

tion and how much it alters from the population without flows.

Computationally, this method uses immigration and emigration flows resulting from the model with

internal and external flows, described on 3.4, and integrates them into the Leslie matrix by performing a

proportion that reduces to half the time span that these refer to. After, two time steps (that is 10 years),

the number of daughters resultant from this altered matrix is obtained and compared to the first entry

of the population. In this way, new proportions for the fertility rates containing all female inhabitants are

revealed.

3.6 Model with Leslie matrix that includes immigration and emi-

gration II

This model bears in mind 3.5 but follows a different computacional approach.

Therefore, the Leslie matrix is given by:

L′′ =

F ′′0 F ′′1 F ′′2 . . . F ′′m−2 F ′′m−1 F ′′m

P ′′0 0 0 . . . 0 0 0

0 P ′′1 0 . . . 0 0 0

0 0 P ′′2 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . P ′′k 0 0

0 0 0 . . . 0 P ′′m−1 0

, (3.7)

where F ′′x ≥ 0 is the number of daughters born between t and t + 1 that are still alive in the age class

from 0 to 1 at time t+ 1 and whose mothers lived aged x to x+ 1 and were an inhabitant or a immigrant

at time t + 1, and P ′′x is the survival probability plus the migration flows of a female from an age class

between x and x+ 1 at time t to the next age class x+ 1 to x+ 2 at time t+ 1.

We obtain the following model:

n(t+ 1) = L”× n(t). (3.8)

This method can be computationally described as follows. First, we add unknowns parameters to

the survival rates Pi, from the Leslie matrix without any migration explained in 3.1. Then we obtain the

resulting vector of the population after two time steps (i.e. 10 years), and, as in 3.5, we compare the

number of daughters resultant from this altered matrix and the first entry of the population. Afterwards,

we put an unknown in the first entry of the matrix F ′′0 in order to be able to calculate a system with all the

unknowns. Since no daughters with less than 4 years old can produce offsprings, the unknown placed

16

in F ′′0 is not correct when taken in practice. Therefore, the value obtained for this unknown is distributed

between all non-zero entries referring to fertility according to their previous proportion between each

other, and was placed a 0 in the first entry of the matrix.

17

18

Chapter 4

Partial Derivates of Evolutionary

Entropy

Evolutionary entropy describes the diversity of pathways of energy flows between the elements that

compose the microlevel. That is, evolutionary entropy is a function of the change caused by perturba-

tions on a system that can be a biological, a metabolic or even an economic system. It allows for the

specification of robustness, the rate at the macroscopic variables that constitute the system return to

their steady state after a random perturbation.

The next sections will lead to the analysis of the sensitivity of evolutionary entropy to perturbations

in the microscopic parameters that describe the system.

This is based on L.A. Demetrius [15], and on H. M. Oliveira and L.A. Demetrius [30].

4.1 Evolutionary Entropy

Consider the following irreducible incidence matrix, associated with a graph,

A = (aij) ≥ 0. (4.1)

By the Perron-Frobenius theorem in 2.6, A has a dominant eigenvalue λ, and corresponding right

and left eigenvectors u = (u1, . . . , ud) and v = (v1, . . . , vd), respectively, such that:

Au = λu (4.2)

vA = λv (4.3)

(u,v) = 1. (4.4)

Also, this theorem ensures the differentiability of λ respective to the entries of A.

In many models, the incidence matrix A is considered as the matrix that specifies the steady state

19

of the dynamic system, where the vector

u(t) = {ui(t)} (4.5)

represents its phase state. Therefore, the system will progress according to the equation

u(t+ 1) = Au(t). (4.6)

Now, consider the diagonal matrix U

U =

u0 0 . . . 0 0

0 u1 . . . 0 0...

......

. . ....

0 0 . . . ud−1 0

0 0 . . . 0 ud

, (4.7)

and the stochastic matrix P = (pij) = aijujλui

can be represented by

P = 1λ

U−1AU. (4.8)

Notice that P describes a Markov process with transition rate (pij)and stationary distribution Π =

(πi), where ΠP = Π and πi = viui.

Therefore, the evolutionary entropy H, weighted average of Hi, is the rate at which P generates

information, which analytically translates into

H =d∑i=1

πiHi = −d∑

i,j=1πipij log pij , (4.9)

where

Hi = −d∑j=1

pij log pij . (4.10)

As a consequence, Hi is the Shannon-entropy associated with the state Xi of the Markov chain.

4.1.1 Characteristics of the Evolutionary Entropy

4.1.1.1. Evolutionary entropy and generation time If S is the measure of the number of replicative

cycles in the network and T is the cycle time (the mean return time of the Markov process associated

with the matrix P), the evolutionary entropy H can be displayed as:

H = S

T. (4.11)

Let us show the characteristic above. From now on, fix the vertex a ∈ A where X = (1, . . . , d). Also,

20

let X be the set of all states such that

X = { [a, β1, . . . , βn−1, a] : a→ β1 → · · · → βn−1 → a where βi 6= a for all i, and n ≥ 1 } . (4.12)

A state a ∈ X is a path of the graph G, which starts and ends at a, and does not visit a inbetween.

The probability of a given cycle which starts at a is given by

pa = pβ1pβ1β2 . . . pβn−1βn. (4.13)

The number of replicative cycles associated with the network will then given by

S = −∑a∈X

pa

log pa. (4.14)

In the graphs described by a unique replicative cycle S = 0, as illustrated in the graph below.

1 2 d. . .

Figure 4.1: S = 0, where S is a measure of the number of replicative cycles.

S increases proportionally to the number of replicative cycles increases as represented below.

1 2 d. . .

Figure 4.2: S > 0, where S is a measure of the number of replicative cycles.

Let

T =∑a∈X

|a|pa. (4.15)

In this case, we have |a| = n, the length of the cycle, for n such that

a = [a, β1, . . . , βn−1, a]. (4.16)

The quantitiesH, measure of the rate at which the process is generating information, and S, measure

of the uncertainty in the length of a randomly chosen cycle, differ only in terms of the value T , as

demonstrated on L.A. Demetrius, V.M. Gundlach and M. Ziehe’s article [31]

H = S

T. (4.17)

21

4.1.1.2. Variational principles and evolutionary entropy The matrix P = (pij) when is derived from

the interaction matrix A = (aij) ≥ 0 represents a fundamental feature of its network. If r = log λ and Φ

is the reproductive potential, we obtain:

r = Φ + S

T. (4.18)

Consider the set MA of all stochastic matrices P = (pij) which satisfy the property

aij = 0⇔ pij = 0. (4.19)

The parameter r = log λ satisfies a variational principle, as demonstrated by L. Arnold, L.A. Demetrius

and V. M. Gundlach in [32]. Then:

r = log λ = supP∈MA

∑i,j

πipij(log pij − log aij). (4.20)

Furthermore, the supremum in 4.20 can be obtained by the unique stochastic matrix P = (pij),

defined by the relation

pij = aijujλui

. (4.21)

Using the two equations above, we have:

r = log λ = H + Φ, (4.22)

where H is the evolutionary entropy and Φ is the reproductive potential given by

Φ =∑i,j

πipij log aij . (4.23)

Using 4.18, we can get the last equation as

r = Φ + S

T. (4.24)

4.1.1.3. Evolutionary entropy and robustness An increase in entropy entails an increase in robust-

ness and, therefore, a greater insensitivity of an observable to perturbations in the microlevel parameters

that describe the network.

Let Pε(t) denote the probability that the sample mean deviates from its unperturbed value by more

than ε at time t.

As L.A. Demetrius, V.M. Gundlach and G. Ochs showed in [33], as t increases, Pε(t) converges to

zero.

Robustness can be quantified by analysing deviations of the observables of the system following an

22

instantaneous perturbation of the microlevel parameters.

More formally, robustness R characterizes the insensitivity of an observable to structural changes

in the microlevel parameter and it is defined as the fluctuation decay rate of Pε(t) on a logarithmic time

scale. It can be written as:

R = limt→∞

− logPε(t)t

. (4.25)

The entropy H and the robustness R are positively correlated, as was also shown in [33].

In this case, using the Entropy-Robustness theorem in 2.9, we have the values

∆H = H∗ −H and ∆R = R∗ −R (4.26)

which are the changes in the variables that provoke a change in the parameters that describe the net-

work.

By this theorem, an increase in entropy leads to an increase in robustness, and, consequently, to a

larger insensitivity to perturbations in the microlevel parameters that characterize the network behaviour.

4.2 Sensitivity Analysis

For example, if we consider demographic networks, then the growth rate experiences stronger effect

due to variations in the fertility rates in lower age-classes than in higher age-classes.

Recall equation 4.22:

r = log λ = H + Φ,

which can be written as

H = r − Φ. (4.27)

Next we will exploit the sensitivity of the growth rate r = log λ.

4.2.0.1. Growth rate sensitivity[30] The sensitivities Sij are given by

Sij = ∂λ

∂aij= viuj , (4.28)

where vi > 0 and uj > 0 are the components of the left and right dominant eigenvectors, respectively.

Consider the matrix A, complying with the conditions mentioned on 4.1 , and recall the concept of

entropy in 2.11: there are exactly d2 such derivatives

Sij = ∂λ

∂aij.

Given the differentials in Au = λu such that

23

dAu +Adu = dλu + λdu. (4.29)

Recall that vu = 1, where v is the left eigenvalue. Using the matrix notation for the usual Euclidean

inner product, we obtain

vdAu + vAdu = vdλu + vλdu↔ vdAu + λvdu = vdλu + λvdu (4.30)

that is,

vdAu = vdλu. (4.31)

Then, the intended result follows:

Sij = ∂λ

∂aij= viuj . (4.32)

We construct the sensitivity matrix S as:

S = (viuj)d×d = v⊗ u =( ∂λ

∂aij

)d×d

. (4.33)

Using r = log λ, we get:

∂r

∂aij= 1λviuj = 1

λSij , (4.34)

and the concept of elasticity eij

eij = aij∂r

∂aij= aij

λSij . (4.35)

One interesting remark is that the sensitivity of the elasticity is closely related to the sensitivity of the

growth rate.

The sensitivity of the elasticity measures the effective perturbation on the growth rate weighted by

the size of each structure matrix entry.

24

Chapter 5

Compression of the Leslie matrices

For the sake of simplification, we shall use Leslie graphs which have Lt as incidence matrix.

Imagine that we want to compress the graph represented on 2.2.3, with c as the interval of each

age-class considered into a graph with n×c as the interval of each age-class. Without loss of generality,

let’s use n = 2.

Therefore, we want to discover F ′0, F ′1, . . . , F ′m2 and P ′0, P ′1, . . . , F ′m2 , illustrated in the graph G2c below,

such that they correspond to the values directly obtained using the fertility and survival rates for a Leslie

matrix with 2c-age-classes.

C ′0 C ′1 C ′2 C ′m2 −2 C ′m

2 −1 C ′m2

F ′0

P ′0 P ′1

F ′1F ′2

P ′m2 −2

F ′m2 −2

P ′m2 −1

F ′m2 −1

F ′m2

. . .

Suppose that m is an even number. Notice that, to obtain F ′0, F ′1, . . . , F ′m2 and P ′0, P ′1, . . . , F ′m2 from a

Leslie matrix with c-age-classes, these weights must be written in order of the number of females alive in

each age group (nx0 , nx1 , . . . , nxm ), and in order of the coefficients of a Leslie matrix with c-age-classes

(F0, F1, . . . , Fm2

and P0, P1, . . . , Fm2

).

A fertility rate Fi can be obtained by:

Fi = #(daughters given by women in age-class i)#(female population in age-class i) , (5.1)

which means that, if we double the time-interval and also double the age-class c, we can obtain the

fertility rates for a Leslie matrix subdivided into 2c age-classes depicted above in the graph by:

25

F ′j = #(daughters given by women in age-class 2j) + #(daughters given by women in age-class 2j+1)#(female population in age-class 2j) + #(female population in age-class 2j+1)

if 0 ≤ j ≤ m

2 − 1 (5.2)

and

F ′m2

= #(daughters given by women in age-class m)#(female population in age-class m) . (5.3)

Therefore, relating to the graph representing the Leslie matrix subdivided into c age-classes:

F ′j = F2j + F2j+1

nx2j + nx2j+1

if 0 ≤ j ≤ m

2 − 1

F ′m2

= Fm

(5.4)

(5.5)

.

The same way of thinking can be applied to achieve the values of the survival rates (considering that

the system is closed for migrations), that are calculated as follows:

Pi = 1− #(women that passed away in age-class i)#(midyear female population in age-class i) . (5.6)

Consequently,

P ′j = P2j + P2j+1

nx2j + nx2j+1

if 0 ≤ j ≤ m

2 − 1 (5.7)

If m was an odd number, then we would obtain, from the c age-class Leslie graph Gc, the following

graph for the 2c age-class Leslie matrix G′2c:

C ′0 C ′1 C ′2 C ′m+12 −3 C ′m+1

2 −2 C ′m+12 −1

F ′0

P ′0 P ′1

F ′1F ′2

P ′m+12 −3

F ′m+12 −3

P ′m+12 −2

F ′m+12 −2

F ′m+12 −1

. . .

and the weights representing the fertility rates in the graph represented above are given as, for 0 ≤ j ≤m+1

2 − 1:

F ′j = F2j + F2j+1

nx2j+ nx2j+1

. (5.8)

Mutatis mutantis, the survival rates, for 0 ≤ j ≤ m+12 − 2, are calculated as:

26

P ′j = P2j + P2j+1

nx2j + nx2j+1

. (5.9)

5.1 Re-weighting the edges of the graph G2c

In order to perform the proposed compression of the Leslie graph Gc, we must find an algorithm

containing a function that allows the re-weighting of the edges with the least possible error. This function

must only depend on the coefficients presented on the matrix L. From a graph’s point of view, this

procedure can only depend on the current weight of the edge wi of Gc, and the number of female

individuals on the population subdivided into c age-classes.

The idea is to perform a mean of the two edges, i and i + 1, that are going to be joined with a

correction factor. To find this factor, g, and as consequence the weight of the corresponding edge w′j of

G2c, the following system must be solved in order of wi, wi+1, nxiand nxi+1 (notice that the weights of

the edges considered are always rates and ai is the numerator of these):

wi = ainxi

wi+1 = ai+1

nxi+1

.

w′j = wi + wi+1

g

(5.10)

(5.11)

(5.12)

By solving the system above, we achieved the following result for g:

g =(nxi

+ nxi+1)(wi + wi+1)nxiwi + nxi+1wi+1

. (5.13)

That is, the weight of the edge w′j of G2c, if m is even, is given by:

w′j = w2j + w2j+1

(nx2j+ nx2j+1)(w2j + w2j+1)

nx2jw2j + nx2j+1w2j+1

if 0 ≤ j ≤ m

2 − 1.

w′m2

= wm if such wm exists

(5.14)

(5.15)

If m is odd, the new weight is given by, for 0 ≤ j ≤ m+12 − 1:

w′j = w2j + w2j+1(nx2j + nx2j+1)(w2j + w2j+1)

nx2jw2j + nx2j+1w2j+1

if such wm exists. (5.16)

Notice that, even if the studied Leslie matrix included migrations, which wasn’t a closed system, the

same procedure could be applied to re-weight each edge. This happens because the migrations (migx)

are added to the sub-diagonal as follows

Pj = 1−mortj −migj , (5.17)

27

that is the probability of survival of some class j to the next class j + 1 is obtained by subtracting to 1

the mortality rate (mortx) and the migration rate, where the migration rate is the emigration rate (emx)

minus the immigration rate (immx):

migx = emj − immj . (5.18)

In more detail, the probability of survival of some class j to the next class j + 1, in an open system,

is not only dependent on the mortality rate, but also on the migration flow (the proportion of inhabitants

leaving the system and the proportion of individuals deciding to be a part of the system). But this

information can be compressed the same way as P ′j in a closed system 5.9:

P ′j = P2j + P2j+1

nx2j + nx2j+1

.

Therefore, we can apply 5.14 and 5.16 in open systems as well.

5.2 Proposed Algorithm

Suppose that the input Leslie matrix, lc, has dimensions n × n (a Leslie matrix is always a square

matrix). The complexity with regard to its input of the inner While of this algorithm is given by O(dn2 e),

while the outer While has complexity O(2). Therefore, the complexity of this algorithm is O(2dn2 e).

28

Algorithm 1 Compression of the Leslie Matrix

1: procedure LESLIECOMPRESSED(lc,p)

Input:lc, Leslie matrix subdivided into c age-classes in the considered initial year t, and p, thecorresponding column vector containing the number of females

Output:l2c, Leslie matrix subdivided into 2c age-classes

2: Solve[

(a2j−1+a2j)(nx2j−1 +nx2j

) == (w2j−1+w2j)g , w2j−1 == a2j−1

nx2j−1, w2j == a2j

nx2j, a2j−1, a2j , nx2j−1 , nx2j

, g]

3: g[nx2j−1 , nx2j, w2j−1, w2j ] = (nx2j−1 +nx2j

)(w2j−1+w2j)(nx2j−1w2j−1)+(nx2j

w2j) ;4: q = 1;5: r = {}6: l = {lc1,Table[lcj,j−1, j, 2,Dimensions[lc]1]7: while q ≤ Dimensions[l]1 do8: i = 19: m1 = {}

10: w = lq11: while i ≤ dDimensions[w]1

2 e do12: if i 6= dDimensions[w]1

2 e then13: if w2i−1 6= 0&&w2i 6= 0 then14: m1 = Append

[m1, w2i−1+w2i

g[p2i−1,1,p2i,1,w2i−1,w2i]

]15: else16: if w2i−1 == 0&&w2i 6= 0 then17: m1 = Append

[m1, w2i

]18: else19: if w2i−1 6= 0&&w2i == 0 then20: m1 = Append

[m1, w2i−1

]21: else22: m1 = Append

[m1, 0

]23: end if24: end if25: end if26: else27: if isEven

[Dimensions[w]1

]then

28: m1 = Append[m1, w2i−1+w2i

g[p2i−1,1,p2i,1,w2i−1,w2i]

]29: else30: m1 = Append

[m1, w2i−1

]31: end if32: end if33: i+ +34: end while35: r = Append[r,m1]36: q + +37: end while38: if isEven[Dimensions[w]1] then39: lc = Table[PadLeft[PadRight[{r2,u+1},

Ceiling[Dimensions[lc]]12 − u], Ceiling[Dimensions[lc]]1

2 ], {u, 0, Dimensions[r2]1 − 1}]

40: else41: lc = Table[PadLeft[PadRight[{r2,u+1},

Ceiling[Dimensions[lc]]12 − u], Ceiling[Dimensions[lc]]1

2 ], {u, 0, Dimensions[r2]1 − 2}]

42: end if43: lc = Prepend[lc, r1]44: end procedure

29

Theorem 5.1. The proposed algorithm for the compression of Leslie matrices has null error.

Proof:

Consider the following two tables concerning the data that we are going to use.

Age-classes C0 C1 C2 . . . Cm

Number of children born whose mothers belong to this class a0 a1 a2 . . . am

Number of women that survived from age-class Ci to Ci+1 s0 s1 s2 . . . sm

Number of women of this class n0 n1 n2 . . . nm

Table 5.1: Table with the number of children born, of women that survived from age-class Ci to Ci+1,and of women by c age-classes.

Age-classes C ′0 . . . C ′m2

Number of children born whose mothers belong to this class a′0 = a0 + a1 . . . a′m2

= am

Number of women that survived from age-class Ci to Ci+1 s′0 = s0 + s1 . . . s′m2

= sm

Number of women of this class n′0 = n0 + n1 . . . n′m2

= nm

Table 5.2: Table with the number of children born, of women that survived from age-class Ci to Ci+1,and of women by 2c age-classes, related with the women subdivided by c age-classes.

Therefore, the Leslie matrix subdivided into c age-class is given by:

Lc =

f0 = a0n0

f1 = a1n1

. . . fm−1 = am−1nm−1

fm = am

nm

p0 = s0n0

0. . . 0 0

0 p1 = s1n1

. . . 0 0

0 0. . . 0 0

0 0. . . pm−1 = sm−1

nm−10

. (5.19)

We have two cases to consider:

a. m is even

Let us suppose that m is even.

The Leslie matrix subdivided into 2c age-classes, using the c age-classes’ data:

L2c =

f ′0 = a0+a1n0+n1

f ′1 = a2+a3n2+n3

. . . f ′m/2−1 = am−2+am−1nm−2+nm−1

f ′m/2 = am

nm

p′0 = s0+s1n0+n1

0. . . 0 0

0 p′1 = s2+s3n2+n3

. . . 0 0

0 0. . . 0 0

0 0. . . p′m/2−1 = sm−2+sm−1

nm−2+nm−10

. (5.20)

30

Using as input Lc and nc =

n0

n1

n2...

nm

for the proposed algorithm, we obtain for the fertilities:

f ′′0 = (f0n0 + f1n1)(f0 + f1)(n0 + n1)(f0 + f1) =

( a0n0n0 + a1

n1n1)( a0

n0+ a1

n1)

(n0 + n1)( a0n0

+ a1n1

) = a0 + a1

n0 + n1= f ′0 (5.21)

f ′′1 = (f2n2 + f3n3)(f2 + f3)(n2 + n3)(f2 + f3) =

( a2n3n2 + a3

n3n3)( a2

n2+ a3

n3)

(n2 + n3)( a2n2

+ a3n3

) = a2 + a3

n2 + n3= f ′1 (5.22)

. . .

f ′′m2

= fm = f ′m2, (5.23)

and for the survival probabilities:

p′′0 = (p0n0 + p1n1)(p0 + p1)(n0 + n1)(p0 + p1) =

( s0n0n0 + s1

n1n1)( s0

n0+ s1

n1)

(n0 + n1)( s0n0

+ s1n1

) = s0 + s1

n0 + n1= p′0 (5.24)

p′′1 = (p1n1 + p2n2)(p1 + p2)(n1 + n2)(p1 + p2) =

( s1n1n1 + s2

n2n2)( s1

n1+ s2

n2)

(n1 + n2)( s1n1

+ s2n2

) = s1 + s2

n1 + n2= p′1 (5.25)

. . .

p′′m2 −1 =

(pm2 −1nm2 −1 + pm

2nm

2)(pm

2 −1 + pm2

)

(nm2 −1 + nm

2)(pm

2 −1 + pm2

) =

=(sm

2 −1

nm2nm

2 −1 +sm

2nm

2nm

2)(sm

2 −1

nm2 −1

+sm

2nm

2)

(nm2 −1 + nm

2)(sm

2 −1

nm2

+sm

2 −1

nm2

)=

=sm

2 −1 + sm2

nm2 −1 + nm

2

=

= p′m2 −1

(5.26)

which corresponds to the Leslie matrix L2c, as expected.

b. m is odd

Let m is odd and consider the following Leslie matrix subdivided into 2c age-classes, using the c

age-classes’ data.

31

L2c =

f ′0 = a0+a1n0+n1

f ′1 = a2+a3n2+n3

. . . f ′(m + 1)/2−2 = am−3+am−2nm−3+nm−2

f ′(m + 1)/2−1 = am−2+am−1nm−2+nm−1

p′0 = s0+s1n0+n1

0. . . 0 0

0 p′1 = s2+s3n2+n3

. . . 0 0

0 0. . . 0 0

0 0. . . p′(m + 1)/2−2 = sm−3+sm−2

nm−3+nm−20

. (5.27)

Following the procedure using as input Lc and nc =

n0

n1

n2

. . .

n4

, we obtain for the fertilities:

f ′′0 = (f0n0 + f1n1)(f0 + f1)(n0 + n1)(f0 + f1) =

( a0n0n0 + a1

n1n1)( a0

n0+ a1

n1)

(n0 + n1)( a0n0

+ a1n1

) = a0 + a1

n0 + n1= f ′0 (5.28)

f ′′1 = (f2n2 + f3n3)(f2 + f3)(n2 + n3)(f2 + f3) =

( a2n3n2 + a3

n3n3)( a2

n2+ a3

n3)

(n2 + n3)( a2n2

+ a3n3

) = a2 + a3

n2 + n3= f ′1 (5.29)

. . .

f ′′(m + 1)/2−2 =(f(m + 1)/2−1n(m + 1)/2−1 + f(m + 1)/2n(m + 1)/2)(f(m + 1)/2−1 + f(m + 1)/2)

(n(m + 1)/2−1 + n(m + 1)/2)(f(m + 1)/2−1 + f(m + 1)/2) =

=(a(m + 1)/2−1n(m + 1)/2

n(m + 1)/2−1 + a(m + 1)/2

n(m + 1)/2n(m + 1)/2)( a(m + 1)/2−1

n(m + 1)/2−1+ a(m + 1)/2

n(m + 1)/2)

(n(m + 1)/2−1 + n(m + 1)/2)( a(m + 1)/2−1n(m + 1)/2−1

+ a(m + 1)/2

n(m + 1)/2)

=

=a(m + 1)/2−1 + a(m + 1)/2

n(m + 1)/2−1 + n(m + 1)/2

= f ′(m + 1)/2−2

(5.30)

and for the survival probability:

p′′0 = (p0n0 + p1n1)(p0 + p1)(n0 + n1)(p0 + p1) =

( s0n0n0 + s1

n1n1)( s0

n0+ s1

n1)

(n0 + n1)( s0n0

+ s1n1

) = s0 + s1

n0 + n1= p′0 (5.31)

. . .

p′′(m + 1)/2−2 =(p(m + 1)/2−1n(m + 1)/2−1 + p(m + 1)/2n(m + 1)/2)(p(m + 1)/2−1 + p(m + 1)/2)

(n(m + 1)/2−1 + n(m + 1)/2)(p(m + 1)/2−1 + p(m + 1)/2) =

=( s(m + 1)/2−1n(m + 1)/2−1

n(m + 1)/2−1 + s(m + 1)/2

n(m + 1)/2n(m + 1)/2)( s(m + 1)/2−1

n(m + 1)/2−1+ s(m + 1)/2

n(m + 1)/2)

(n(m + 1)/2−1 + n(m + 1)/2)( s(m + 1)/2

n(m + 1)/2+ s(m + 1)/2−1

n(m + 1)/2−1)

=

=s(m + 1)/2−1 + s(m + 1)/2

n(m + 1)/2−1 + n(m + 1)/2

=

= p′(m + 1)/2−2

(5.32)

which corresponds to the Leslie matrix L2c, as expected.

32

Chapter 6

Real Word Context

This thesis was developed alongside with a project called Conhecer Arroios, where an intercensus

study, referring to the year 2017, of the population of Arroios was made. I was coauthor of this study [5],

which conveyed the analysis of the evolution of Arroios parish along time and its projections, based on

2001 and 2011 Census’ data, and a population sample retrieved in 2017. This was a great advantage

since it allowed us to apply the models presented on 3 to a real world context.

In the next sections, we will explain how we obtained the data to apply to our projections. After,

we will approach the results obtained by projecting each model on 3, exploring age distributions and, if

applicable, migration flows. Next, the effect of changes to the parameters that constitute the population

will be studied, using the obtained three Leslie matrices. Finally, a comparison between the data from a

population sample in 2017 and the real world application of our models will be made.

6.1 Construction of Arroios’ Leslie Matrices based on 2001 and

2011 data

In order to make several projections of the population in Arroios, we must obtain some possible

Leslie matrices based on 2001 and 2011 data. Our models take into account some indices derived from

Census 2001 and 2011, for instance fertility and mortality indices, immigration and emigration.

The data from the Census is very incomplete regarding the considered parish and the required

variables to build this model (e.g. emigration and immigration). Therefore, we made several estimations

which are detailed on the following subsections.

6.1.1 Fertility Indices

There are multiple indices to calculate fertility and we explore some beginning with less precision

and then go forward and reach to one that has enough precision for our study.

In our case, we are only interested on newborn females, since they are the ones renovating the

population.

33

We can obtain the Gross Reproduction Rate (GRR):

GRRi =∑j

#(daughters given by women in age-class j in year i)#(midyear female population in age-class j in year i) , (6.1)

which gives the number of girls born per woman. However, if we take into account mortality among

female births, then we calculate the Net Reproduction Rate or Net Replacement Rate:

NRRi =∑j

#(daughters alive given by women in age-class j in year i)#(midyear female population in age-class j in year i) (6.2)

Since the estimated number of stillborn children in Arroios during the years of 2001 and 2011 is almost

0 (in 2001 there was one stillborn daughter), we shall procede the calculations of fertility based on the

Gross Reproduction Rate.

Let us suppose that the age-specific fertility is different from 0 between 15 and 49 years old.

Since there was no available data for the number of daughters per their mothers’ age-class, for

Arroios, our calculations were based under the assumption that the distribution of the proportion of

daughters and mothers of this area is similar to Lisbon’s distribution. After obtaining this value, we

applied a correction to the extrapolation because the estimate of total number of daughters born was

not equal to the number of female babies born for parish considered. That is, we multiplied the number

of daughters per their mothers’ age-class by a factor

α = #(total estimated number of daughters born)#(total number of daughters born)

. (6.3)

The following plot gives a more clear idea of the distribution shift in order to match the known number

of daughters born.

34

Figure 6.1: Distribution of female newborns using the non-adjusted number and the adjusted number offemale newborns, using 2001 data.

6.1.2 Mortality

The procedure applied for finding the mortality rates for each age class was similar to the fertility

indices.

Let us consider the Gross Mortality Rate (GMR):

GMRi =∑j

#(women that passed away in age-class j in year i)#(midyear female population in age-class j in year i) . (6.4)

Therefore, GMRi gives us the proportion of women in age-class j that died comparing to the total

number of women in that age-class, considering the same year i.

However, since there is no available data to cover the number of deaths per age-class, we assumed

that the distribution of the proportion of mortalities is similar to Lisbon’s distribution, and then we applied

a correction of this value since the estimate of the total number of deaths was not equal to the number

of deaths for each parish. The correction applied may be expressed as:

α = #(total estimated of women that passed away)#(total of women that passed away)

. (6.5)

The corresponding shift in mortality is illustrated in the figure below.

35

Figure 6.2: Distribution of female deaths using the non-adjusted number and the adjusted number offemale deaths, using 2001 data.

6.1.3 Survival Indices

Remember the concept of Leslie matrix given on 2.1.3:

L =

F0 F1 F2 . . . Fm−2 Fm−1 Fm

P0 0 0 . . . 0 0 0

0 P1 0 . . . 0 0 0

0 0 P2 . . . 0 0 0...

......

. . ....

......

0 0 0 . . . Pk 0 0

0 0 0 . . . 0 Pm−1 0

. (6.6)

is a Leslie matrix where Pi,i+1 is the survival index and Fi is the fertility rate for each age-class.

When we consider a closed system (without any migrations), the survival index Pi of this matrix was

calculated by subtracting to 1 the mortality rate for each age-class.

Otherwise, when the migration flow was considered inside the matrix, the survival index Pi was

obtained by subtracting mortality and emigration rates and adding the immigration rate to 1.

36

6.1.4 Immigration

We first obtained the total female population that didn’t live in each parish 5 years ago, which was

accomplished by multiplying, for 2001 and 2011, the total female population by the given proportion of

female residents that didn’t live in each parish. Observe that this subpopulation contains portuguese

and foreign nationalities.

In order to separate the vectors containing immigration from foreign nationalities and immigration

from portuguese women, we multiplied the proportion of foreign women living in each parish by the total

female population that wasn’t a resident of that parish five years ago. It had to be assumed that the

immigration was uniformly proportional over time to the number of foreign women.

We fitted our data using a quadratic function because there is no data available for 2006, needed

for the construction of the model. Applying this expression, a value for the number of foreign immigrant

women was obtained.

Then, we subtracted the total female immigrant population by the previous value and, therefore, the

number of portuguese immigrants of this parish was calculated.

Due to the lack of data for the distribution of immigration by age-classes in the parishes, we inferred

using Portugal’s immigration data, which only considers women arriving from foreign countries. There-

fore, it is assumed that the internal immigration (women coming from other parishes in Portugal) follows

a similar distribution.

Consequently, a vector containing the immigration of women of foreign nationalities divided by age-

classes between 2001 and 2011, and a vector containing the immigration of portuguese women divided

by age-classes between 2001 and 2011, were calculated.

Afterwards, the immigration ratio was calculated based on the number of female inhabitants in 2001.

6.1.5 Emigration to Foreign Countries

Using a scale parameter which accounts for the population of each parish and Portugal, we multiplied

it by the Portuguese emigration of each year, assuming that the parish follows the distribution of Portugal

over the years. A similar assumption was made regarding the age-class distribution of emigration, where

the proportion of the cumulative distribution between 2008 and 2015 by age-class was used in order to

obtain the vector containing the emigration of women of foreign nationalities divided by age-classes,

between 2001 and 2011.

6.1.6 Error Estimate

We used a forecast error in order to evaluate the error of the predicted model, which is the difference

between the observed value and its forecast. That is:

error =

∑21i=1|nesti2011

− ni2011 |ni2011

21 (6.7)

37

where nesti2001, and ni2011 , is the estimated, and the observed, distribution of the female population

of the age-classes i, respectively.

6.2 Application of the models

In 3, several models were presented, conveying closed systems and systems with migration. It is

important to establish a connection between the theoretical point of view and its application to the real

world, in order to evaluate and compare models and attain better projections.

Having as goal to decrease the error, and since the Leslie matrix calculated for 2001 is considered

constant over time, we consider all iterations beginning from 2011, that is:

n2011+5i = L2011i × n2011. (6.8)

Also, notice that we suppose, in our calculations, the emigration and immigration flows follow an

uniform distribution with time.

Moreover, since our models only consider women, we can obtain the total population by multiplying

the number of female inhabitants by 1.86, which is the observed proportion between women and the

total population in 2011.

6.2.1 Model without migratory flows

Following the model without migratory flows explained on 3.1, we present the values obtained for

each parameter of these matrices, considering a Leslie matrix for closed systems. Notice that only

age-classes of 5 and 10 years will be considered in this research.

Therefore, the parameters for Leslie Matrices are displayed in the following tables.

F3 F4 F5 F6 F7 F8 F9

Leslie Matrix I 0.0101563 0.0286868 0.0443657 0.0401565 0.0126653 0.00325448 0.0000779663

Table 6.1: Fertility rates obtained from 2001’s data without considering migratory flows.

P0 P1 P2 P3 P4 P5 P6

Leslie Matrix I 0.998069 0.999676 0.999902 0.999466 0.999733 0.999054 0.998966

P7 P8 P9 P10 P11 P12 P13

Leslie Matrix I 0.998043 0.997554 0.995478 0.99625 0.995094 0.991306 0.989036

38

P14 P15 P16 P17 P18 P19

Leslie Matrix I 0.978306 0.960356 0.921155 0.859404 0.744179 0.573544

Table 6.2: Survival rates obtained from 2001’s data without considering migratory flows.

If we compare the distribution of female residents in Arroios found in 2011, using Census data, and

using this model without migrations, we can closely analyse the effect of migrations.

• The model without migrations underestimates the number of newborns and relatively young people

(until the age class of 45-49 years old), which can induce to the conclusion that the effects of

immigration are stronger than the ones felt from emigration in these age classes, where young

couples bring their children to live in Arroios;

• The model without migrations gives a close number to the reality found in 2011 between age

classes of 50 and 59 years old;

• The model without migrations overestimates the number of relatively old inhabitants (after 60 years

old), from which we can infer a stronger role of emigration comparing to immigration. This can be

due to lack of nursing homes in Arroios and exodus of the elderly to caregivers’ house.

The shift between the estimated female population using this model in 2011 and the actual female

population encountered in Census 2011 is pictured below, and the error estimate between these two

populations is 94.068%.

39

Figure 6.3: Distribution of female inhabitants by age classes in Arroios using 2011’s Census data, andusing the model without migrations.

Next, we present projections, using this model for closed systems, obtained for 2016, 2021, 2026,

2031 and 2036.

Figure 6.4: Projected total of female inhabitants in Arroios using the model without migrations in 2016,2021, 2026, 2031 and 2036..

40

The figure above shows that this model without migrations predicts a significant decrease in female

population from 2001 to 2036. Moreover, if we observe the projected distribution of female inhabitants by

age classes, there is a shift in the most expressive class, that is projected as being from 35 to 39 years

old in 2016 and, in 2036, is from 55 to 59 years old. Also, notice that this model projects an increase

in elderly people, and a decrease in younger people and in female births. This ageing is due to the

relatively low fertility rate, which translates into a smaller number of newborns relatively to the elderly.

Therefore, this effect is propagated along the years, leading to the ageing of female population.

Figure 6.5: Projected distribution of female inhabitants by age classes in Arroios using the model withoutmigrations in 2016, 2021, 2026, 2031 and 2036.

The model without migrations is clearly inadequate to a parish, since it lacks to explore migration

flows of its inhabitants, which, in a long term, results in large errors in projections.

6.2.2 Model with one migratory flow

Recall the model with one migratory flow detailed in 3.2, which considers an open system. The

values obtained for the Leslie matrix are the same as the ones presented in 6.2.1. The main difference

between this model and the one without migrations, is the added column vector multiplied by a factor,

representing the migration flow. This column vector is calculated based on 2001 and 2011 Census’ data,

and, therefore, in the following projections, it is assumed that its behaviour is constant along time. These

matrix’ values are given in the table below.

41

flux0 flux1 flux2 flux3 flux4 flux5 flux6

Leslie Matrix I 3.63668 2.63716 0.0517544 0.143053 0.711082 0.878175 0.136603

flux7 flux8 flux9 flux10 flux11 flux12 flux13

Leslie Matrix I 0.055245 0.113591 0.0690748 0.00489464 -0.0414223 -0.084362 -0.118182

flux14 flux15 flux16 flux17 flux18 flux19 flux20

Leslie Matrix I -0.1638 -0.224038 -0.314274 -0.501457 -0.651249 -0.837236 -0.88391

Table 6.3: Proportion of the flux relatively to the estimated female residents in 2011 of the model withone migratory flow in Arroios obtained using 2001 and 2011 Census data.

If we plot 2011’s female population data and the projected female inhabitants using this model with

one migratory flow, we obtain an error of 6.14342 × 10−17, which doesn’t translate into changes in the

female’s number by age class, as seen in the graphic below.

Figure 6.6: Distribution of female inhabitants by age classes in Arroios using 2011’s Census data, andusing the model with one migratory flow.

Let us now analyse the obtained projections for 2016, 2021, 2026, 2031 and 2036.

42

Figure 6.7: Projected total of female inhabitants in Arroios using the model with one migratory flow in2016, 2021, 2026, 2031 and 2036.

Figure 6.8: Projected distribution of female inhabitants by age classes in Arroios using the model withone migratory flow in 2016, 2021, 2026, 2031 and 2036.

43

According to this model, the female population increased in 2006, reaching a total of 19784 women,

which was the maximum number of inhabitants predicted. Afterwards, in 2011 the female population suf-

fered a decrease of 12.61%, approximately, being followed by another increase, reaching 19062 women

in 2016. From this year, the number of female inhabitants is predicted to decrease until it reaches 12603

women, according to this model and maintaining the migratory flow, fertility and mortality rates of the

population.

In 2016, the projected population has 5 to 10 years-old as most expressive age class, while in 2021

the most expressive age class is projected as 40 to 44 years-old. Therefore, this model predicts the

ageing of the female population between these years, along with a decrease in the total number of

female inhabitants, contradicting the predicted increase tendency verified between 2011 and 2016. In

2026, the most predominant age class is expected to be from 45 to 49 years old, which corresponds to

the natural ageing of the most expressive age class in 2021. The same effect happens in 2031 and in

2036, where the most predominant age class is from 50 to 54 years-old and from 55 to 59 years old,

respectively. However, in 2036, the age class of 25 to 29 years old suffers a slight increase over 5 years,

almost reaching the same expressivity in female inhabitants as the most predominant age class.

The assumption of uniform distributed and steady immigration and emigration flows between 2001

and 2011, can be misleading in these projections, since a socioeconomic crisis occurred from 2010

implying more emigration compared to immigration. Therefore, it is being applied and amplified with

time with the usage of this model.

The model with one migratory flow can be improved to adjust more to the changes in immigration

and emigration flows verified with time.

6.2.3 Model with immigration and correction flows

The model detailed in 3.3 assesses a system open to migrations, considering it through two vectors:

proportion of the immigration vector, fluximm, and proportion of a correction vector that includes em-

igration and errors in estimation of immigration (it was based on Portugal’s data which can differ from

Arroios parish), fluxcor.

Again, the Leslie matrix has parameters presented in 6.2.1. The proportion of immigration and

correction flows were calculated based on 2001 and 2011 Census’ data, being presented in the following

tables. In these projections, it is assumed that their behaviour is constant and uniform with time. The

error obtained for this model, comparing with the observed female population in 2011, was 1.3486 ×

10−16.

Notice that the immigration over 85 years old was assumed to be 0, since the last age group consid-

ered in Portugal’s Census is 85 or more years old.

fluximm0 fluximm1 fluximm2 fluximm3 fluximm4 fluximm5 fluximm6

Leslie Matrix I 1.50613 1.16503 0.405383 1.05065 2.45095 1.41552 0.468952

44

fluximm7 fluximm8 fluximm9 fluximm10 fluximm11 fluximm12 fluximm13

Leslie Matrix I 0.335111 0.214589 0.148746 0.108777 0.0991554 0.106146 0.0889245

fluximm14 fluximm15 fluximm16 fluximm17 fluximm18 fluximm19 fluximm20

Leslie Matrix I 0.0443614 0.0190279 0.00930098 0.00288786 0 0 0

Table 6.4: Proportion of the immigration flow relatively to the estimated female residents in 2011 of themodel with immigration and correction flows in Arroios obtained using 2001 and 2011 Census data.

fluxcor0 fluxcor1 fluxcor2 fluxcor3 fluxcor4 fluxcor5 fluxcor6

Leslie Matrix I 2.13055 1.47213 -0.353629 -0.907592 -1.73987 -0.537348 -0.332349

fluxcor7 fluxcor8 fluxcor9 fluxcor10 fluxcor11 fluxcor12 fluxcor13

Leslie Matrix I -0.279866 -0.100998 -0.079671 -0.103882 -0.140578 -0.190508 -0.207106

fluxcor14 fluxcor15 fluxcor16 fluxcor17 fluxcor18 fluxcor19 fluxcor20

Leslie Matrix I -0.208162 -0.243066 -0.323575 -0.504345 -0.651249 -0.837236 -0.88391

Table 6.5: Proportion of a correction flow that includes emigration and errors in estimation of immigrationrelatively to the estimated female residents in 2011 of the model with immigration and correction flowsin Arroios obtained using 2001 and 2011 Census data.

Notice that the total immigration of Arroios was obtained multiplying the proportion of Portugal’s

female external immigrants for each age class, by the number of female inhabitants in Arroios that didn’t

live in this parish five years ago. A more clear idea of how adjusted the estimation of the immigration of

Arroios between 2001 and 2011 can be given by the graphic below.

45

Figure 6.9: Projected distribution of female inhabitants by age classes in Arroios using the model withone migratory flow in 2016, 2021, 2026, 2031 and 2036.

For example, the number of female babies entering Arroios surpasses the estimated value for the

observed age class’ immigration, since the correction factor is non-negative and even greater than the

estimated proportion for immigration. By the absolute minimum achieved between the age class of

20 to 24 years old in the proportion of the correction factor, we have two possible scenarios: whether

the population experienced strong immigration and emigration fluxes, prevailing a positive flux, or the

predicted immigration wasn’t as high as the observed one, possibly meaning that the immigration flows

in Arroios were less than the ones registered for Portugal. Notice that, after the age-class of 50 to 54

years-old, the emigration flow prevails over the immigration flow, since the correction entry is negative

and its absolute value is greater, which means that the older female classes are leaving Arroios.

This model reduces itself to the previous one (the correction flux is obtained by subtracting the esti-

mated immigration of Arroios to the estimated migratory balance) if the growth of the flows is maintained

constant with time, projecting the population the same way as pictured in Figures 6.6, 6.7 and 6.8.

However, contrarily to the model with one migratory flow in 3.2, we can project the population using

different coefficients α and β, since the immigration occurs at a different rate than the one observed for

the applied correction, and vice-versa.

We can fine-tune the projection of the female population by, instead of considering vectors of pro-

portions of immigration and correction flows, considering vectors of proportions of internal and external

46

flows. This can be explained by the almost constant rate at which internal fluxes are detected and by

the fluctuations that the external flux is subject to, namely the socio-economic conditions of countries of

origin and destination.

6.2.4 Model with internal and external flows

Following the model with internal and external flows explained on 3.4, several projections were ob-

tained using a constant rate, an increase of 5% and a decrease of 5% in external rates, whilst the internal

rates were maintained constant with time.

The parameters of the Leslie matrix, closed for migrations, are the ones presented in 6.2.1. The

parameters for the internal and external flows are introduced below. The error obtained for this model

considering the observed female population in 2011 was 1.54418× 10−16.

fluxint0 fluxint1 fluxint2 fluxint3 fluxint4 fluxint5 fluxint6

Leslie Matrix I 3.51086 2.5375 0.0132182 0.0464248 0.493284 0.760772 0.0994145

fluxint7 fluxint8 fluxint9 fluxint10 fluxint11 fluxint12 fluxint13

Leslie Matrix I 0.0293956 0.101607 0.0628015 -0.0015631 -0.0494154 -0.0948082 -0.126845

fluxint14 fluxint15 fluxint16 fluxint17 fluxint18 fluxint19 fluxint20

Leslie Matrix I -0.168028 -0.225817 -0.315153 -0.501579 -0.65138 -0.837389 -0.884314

Table 6.6: Proportion of the internal flow relatively to the total female residents of the model with internaland external flows in Arroios obtained using 2001 and 2011 Census data.

fluxext0 fluxext1 fluxext2 fluxext3 fluxext4 fluxext5 fluxext6

Leslie Matrix I 0.125815 0.0996613 0.0385363 0.0966285 0.217798 0.117403 0.0371888

fluxext7 fluxext8 fluxext9 fluxext10 fluxext11 fluxext12 fluxext13

Leslie Matrix I 0.0258494 0.0119839 0.00627329 0.00645774 0.00799309 0.0104463 0.00866303

fluxext14 fluxext15 fluxext16 fluxext17 fluxext18 fluxext19 fluxext20

Leslie Matrix I 0.00422779 0.00177914 0.000878334 0.000122026 0.00013138 0.000153188 0.000403932

Table 6.7: Proportion of the external flow relatively to the total female residents of the model with internaland external flows in Arroios obtained using 2001 and 2011 Census data.

To achieve a better visualisation of the estimated internal and external flows verified between 2001

and 2011, the column vectors above were plotted as shown bellow.

47

Figure 6.10: Estimated internal and external flows of female inhabitants by age classes in Arroios usingthe model with internal and external flows.

The movements between parishes to or from Arroios (internal flow) had more impact on the number

of female inhabitants than those coming or going to the outside of Portugal (external flow). This was

expected since it is easier to adapt to a new place of residence inside the same country (there are

almost no cultural clashes) than to a new country with a new cultural identity, for example.

The external flow has the age class between 20 and 24 years old as maximum, indicating that female

immigrants are more expressive in the total inhabitants in this age class. After the age class of 45 years-

old, the external flow remains positive, even though it doesn’t suffer significant changes in its value,

meaning that there were no significant immigration from or emigration to foreign countries. Curiously,

there is a small increase in female inhabitants coming from foreign countries in age classes between 55

48

and 69 years old, which can be explained to the improved life and less expensive conditions Portugal

offers when compared with their country of origin.

The age class of 0 to 4 years old is more expressive in the total of female inhabitants in that age class

when we analyse the internal flow. Moreover, if we notice that the other local maximum occurred with 25

to 29 years old, we can assume that the mothers moved from other parishes in Portugal bringing small

children with them. From 55 years old, women started to move out of Arroios, probably to caregivers’

houses.

Notice that the internal flux suffers less fluctuations with time because the social and economic

conditions that affect the country are more or less similar for all parishes, and, usually, similar reasons

for leaving or arriving to a certain parish apply along time. However, this scenario of almost reaching

stability in flows does not occur in the case of external flows. Notice that there are a lot of social-

economic differences between countries with time, which makes the countries more or less attractive to

its possible inhabitants. Therefore, these flows are affected by crisis. For example, the social-economic

crisis felt in Portugal, that started in 2010, affected our results, since these flow calculations were based

in a period contemplating 2010 and 2011, where external immigration decreased and external emigration

increased.

Using a constant rate, the same projections of the model with one migratory flow, pictured in Figures

6.6, 6.7 and 6.8, are obtained.

If we project the female population with an increase of 5% and a decrease of 5% in the external flow,

maintaining the internal flow, the results do not differ much from the ones presented above, being the

reason why these are not displayed in this thesis.

6.2.5 Model with Leslie matrix that includes immigration and emigration I

If we apply the model with Leslie matrix that includes immigration and emigration I, detailed on 3.5,

we can obtain the Leslie matrix parameters below. To minimize the overall error between the calcu-

lated female population and the observed one in 2011, an iterative method was applied, and its output

corresponds to the parameters below.

F3 F4 F5 F6 F7 F8 F9

Leslie Matrix I 0.0648405 0.115457 0.177891 0.199013 0.0636024 0.0156161 0.000346013

Table 6.8: Fertility rates using the model with Leslie matrix that includes immigration and emigration IIand an iterative method.

P0 P1 P2 P3 P4 P5 P6

Leslie Matrix I 0.816699 1.18027 0.976328 1.75146 1.07149 1.05948 0.994028

P7 P8 P9 P10 P11 P12 P13

Leslie Matrix I 1.11693 0.952943 1.04718 0.907832 0.999886 0.869962 0.942388

49

P14 P15 P16 P17 P18 P19

Leslie Matrix I 0.796705 0.808649 0.545391 0.506219 0.205634 0.24096

Table 6.9: Survival rates using the model with Leslie matrix that includes immigration and emigration IIand an iterative method.

The error for this Leslie matrix that includes migration, without performing any tuning, is 0.0101619,

and after using the iterative method is 0.00976734. Once more, notice that we are supposing that the

migrations are steady and uniform with time.

Figure 6.11: Distribution of female inhabitants by age classes in Arroios using 2011’s Census data, andusing the model with Leslie matrix that includes immigration and emigration I.

The predicted distribution obtained using the model in question is close between the age classes

of 40 to 84 years old and of more than 90 years old. The number of female inhabitants was more

undervalued between the age class of 10 to 14 years old, and more overvalued between 85 to 89 years

old. These shifts may occur because, when we integrate the internal and external flow vectors, referring

to 10 years, we must ”convert” them to characterise 5 years only, which is accomplished by simply

supposing that the female population follows an uniform distribution and, in that way, we only need to

divide the proportions by two.

50

Next, we introduce the projections obtained for 2016, 2021, 2026, 2031 and 2036 using this Leslie

matrix.

Figure 6.12: Projected total of female inhabitants in Arroios using the model with Leslie matrix thatincludes immigration and emigration I in 2016, 2021, 2026, 2031 and 2036.

The figure above shows that this model with a Leslie matrix containing migrations predicts a signifi-

cant decrease in female population from 2001 to 2036, only having a slight increase between 2011 and

2016.

51

Figure 6.13: Projected distribution of female inhabitants by age classes in Arroios using the model withLeslie matrix that includes immigration and emigration I in 2016, 2021, 2026, 2031 and 2036.

Moreover, if we observe the projected distribution of female inhabitants by age classes, there is a

shift in the most expressive class, that is projected as being from 35 to 39 years old in 2016 and, in 2036,

is from 55 to 59 years old. This model projects that the most predominant classes will tend to be those

between 20 to 70 years old, that is, there is no increase in the number of female children and elderly.

This might be due to the strong impact of immigration and emigration that occur in these age classes.

6.2.6 Model with Leslie matrix that includes immigration and emigration II

The model with Leslie matrix that includes immigration and emigration II, detailed on 3.6 is now

going to be applied in order to obtain projections for the female inhabitants. The coefficients of the

Leslie matrix presented below were obtained through the referred model and by applying an iterative

method that diminishes the overall error between the calculated female population and the observed

one in 2011.

F3 F4 F5 F6 F7 F8 F9

Leslie Matrix I 0.0608461 0.109119 0.168138 0.187409 0.0598817 0.0147132 0.000326433

Table 6.10: Fertility rates using the model with Leslie matrix that includes immigration and emigration IIand an iterative method.

52

P0 P1 P2 P3 P4 P5 P6

Leslie Matrix I 0.868069 1.00968 1.03572 1.08096 1.36505 0.83779 1.07718

P7 P8 P9 P10 P11 P12 P13

Leslie Matrix I 1.03561 1.03418 0.969863 0.98868 0.924553 0.949489 0.870746

P14 P15 P16 P17 P18 P19

Leslie Matrix I 0.868008 0.751969 0.593521 0.4676 0.2264 0.225816

Table 6.11: Survival rates using the model with Leslie matrix that includes immigration and emigration IIand an iterative method.

The error for this Leslie matrix that includes migration, without performing any tuning, is 0.39237,

and after using the iterative method is 0.0526008. Once more, notice that we are supposing that the

migrations are steady and uniform with time.

Figure 6.14: Distribution of female inhabitants by age classes in Arroios using 2011’s Census data, andusing the model with Leslie matrix that includes immigration and emigration II.

The predicted distribution calculated using the model detailed above is very similar to the one ob-

served in 2011. Moreover, only the age classes of 0 to 4 years old, 10 to 14 years old and 15 to 19 years

53

old, suffer some shift between those two distributions.

The projections obtained for 2016, 2021, 2026, 2031 and 2036 using this Leslie matrix are introduced

next.

Figure 6.15: Projected total of female inhabitants in Arroios using the model with Leslie matrix thatincludes immigration and emigration II in 2016, 2021, 2026, 2031 and 2036.

Contrarily to the models previously shown, the model with a Leslie matrix containing migrations II

projects the total number of female inhabitants with a slight decrease between 2001 and 2026, and,

from 2026 to 2036, a small increase is verified.

54

Figure 6.16: Projected distribution of female inhabitants by age classes in Arroios using the model withLeslie matrix that includes immigration and emigration II in 2016, 2021, 2026, 2031 and 2036.

If we analyse the projected distribution of female inhabitants by age classes, we can observe that

the classes with more expression inside the female population are always between 20 to 69 years old,

corresponding to the age where ative inhabitants have. There is also a shift of the most expressive

classes between 2016 and 2036: in 2016, the most expressive class was 40 to 44 years old, while in

2036, it is expected to be 50 to 54 years old. This model projects a stable number of daughters and a

decrease on the number of elderly women. This model is very well adjusted to our case study, Arroios.

6.3 Partial Derivatives of Evolutionary Entropy

To achieve a better knowledge of the population and its models, some parameters of perturbation

to changes must be studied. In this section, we will explore, in the light of these parameters subject

to change, three Leslie matrices introduced before: without migrations 6.2.1, including immigration and

emigration I 6.2.5, and including immigration and emigration II 6.2.6.

6.3.1 Leslie matrix without migrations 6.2.1

The parameter r = log λ, the growth rate, satisfies a variational principle and, in this case, considering

a Leslie matrix without migrations,

55

r ≈ log 0.734435 ≈ −0.308654. (6.9)

Therefore, we can reach to the conclusion that, if the population follows this model, it is decreasing

with time, with no sufficient renovation of the inhabitants to compensate the deaths.

The value of the reproductive potential is:

Φ = −0.497105. (6.10)

The evolutionary entropy has as value:

H = 0.188451. (6.11)

6.3.2 Leslie matrix that includes immigration and emigration I 6.2.5

The growth rate r = log λ of Leslie matrix that includes immigration and emigration I is:

r ≈ log 0.942799 ≈ −0.0589022. (6.12)

Even though the decline of the projected population is less abrupt than in the Leslie model without

migrations, there is still a decrease in inhabitants of this parish.

The value of the reproductive potential is:

Φ = −0.282154, (6.13)

which confirms the insufficient renovation of population, leading to a decrease in the total number of

inhabitants.

The evolutionary entropy has as value:

H = 0.223252. (6.14)

6.3.3 Leslie matrix that includes immigration and emigration II 6.2.6

The Leslie matrix that includes immigration and emigration II has as growth rate r = log λ:

r ≈ log 1.01356 ≈ 0.0134648. (6.15)

The value of the growth rate is positive, which means that the population is growing according to this

projection. However, it is not a significant increase in the number of inhabitants of this parish but rather a

stabilisation of it, even though it contradicts the population decline predicted in the previous Leslie matrix

models.

Nevertheless, the reproductive potential is negative:

56

Φ = −0.210466. (6.16)

The evolutionary entropy has as value:

H = 0.22393. (6.17)

6.4 Population Sample from 2017 and the Application of Models

The relation of this thesis to a project called Conhecer Arroios, a partnership between Junta de

Freguesia de Arroios and Instituto Superior Tecnico de Lisboa, allowed us to obtain a population sample

of 2344 inhabitants. Notice that the total population in 2011 was 32262 inhabitants. The interviews

were made between 1st April 2017 and 13th May 2017. Therefore, these results will be compared to

the projections obtained for 2016. For the sake of the simplification, only female population will be

considered.

The sampling error, since we assume that the population is infinite, is given by:

E ≈ 1.29√2344

≈ 2.66%. (6.18)

The sampling method is further detailed on [5].

Comparing the projected distribution, for 2016, of the female population using models without migra-

tions and with one migratory flow and the observed female distribution, we reach the conclusion that

these are quite different.

The model without migrations predicts a population more mature and with less young people. On

the other hand, the model with one migratory flow overestimates the number of women between 10 to

14 years old, and underestimates the middle age class. However, neither of these two models correctly

project the most expressive age class, which was, according to the population sample retrieved in 2017,

from 40 to 44 years old.

57

Figure 6.17: Projected distribution of female inhabitants using models without migration and with onemigratory flow in 2016, and distribution of female inhabitants of the population sample taken from 2017,by age classes in Arroios.

The projections for 2016 obtained using the models with Leslie matrix that includes immigration and

emigration I and II describe better the observed female distribution than using the two models previously

exploited.

The model with Leslie matrix that includes immigration and emigration I underestimates the number

of female inhabitants until the age class of 55 to 59 years old. Afterwards, the predicted number of

women is higher than the reality found in this sample. The most expressive age class in the female

sample was correctly predicted by the model with Leslie matrix that includes immigration and emigration

II. However, this model underestimates the youngest age classes and overestimates the eldest age

classes.

58

Figure 6.18: Projected distribution of female inhabitants using models with Leslie matrix that includesimmigration and emigration I and II in 2016, and distribution of female inhabitants of the populationsample taken from 2017, by age classes in Arroios.

Since we assumed, in the construction of the models, that the migration flow was steady and it would

have the same behaviour of the one observed between 2001 and 2011, the comparison between our

projections and the real world results may lead us to the conclusion that more young couples, with and

without children, entered this parish than between the same period ten years before. Furthermore, the

emigration of people with more that 55 years old increased. Almost certainly, Arroios parish is even

more rejuvenated than our projections.

59

60

Chapter 7

Conclusions

In this work, we explored several models and their characteristics, which were previously used by

Demetrius, Lewis, and Leslie, for example. The first two were already seen in literature: the basic model

without migratory flows, previously explored by Leslie; the model with one migratory flow, proposed by

Caswell. However, these do not include emigration, which is relevant to our real world application on the

Arroios parish. Therefore, some new models were explored: the model with immigration and correction

flows; the model with internal and external flows; the models with Leslie matrix that includes immigration

and emigration I and II. These last two do not use a conventional Leslie matrix such as the one originally

proposed by Leslie. Instead, those matrices integrate immigration and emigration flows, which is also a

new way, as far as we know, of modelling the population. This innovation allows us to check the impact

of migration in terms of partial derivatives of evolutionary entropy.

After, we derived and developed some measures to assess the changes to perturbations in micro-

scopic parameters that describe the system, such as evolutionary entropy, H, and generation time,

variational principles (growth rate, r, and reproductive potential, Φ), and robustness, R. Furthermore,

we developed the theoretical analysis of growth rate sensitivity. We also implemented some of these

measures in Mathematica, given the Leslie matrix as input. In the future, we would have wanted to

extend this implementation to other population matrices, such as Lefkovitch matrices.

Another new and original work, as far as we know, was the compression of the Leslie matrices, in

which we designed a method based on graphs that reduced the number of nodes by half with null error.

In a future work, we pretend to implement the proposed algorithm in languages such as R.

In this thesis we had the chance to apply and test all previously mentioned theoretical models and

measures in a real world context, namely with data retrieved concerning a parish in Lisbon called Arroios.

The fact that we applied our theoretical models to a real population generated very interesting remarks

such as the emigration of the elderly, due to the lack of nursing homes in this region, and high immigration

of children, due to the high number of young couples with children that were immigrants.

Our projections, in a general way, predict a decrease in the number of female inhabitants, which

is corroborated by measures such as the growth rate of the Leslie matrices. In fact, the Leslie matrix

without migrations has the smallest growth rate of the three studied, suggesting that the decrease in

61

Arroios’ population is higher when only the inhabitants, without migrants, are considered. Furthermore,

this population without migrations has the lowest reproductive potential, which indicates a low capacity of

reproduction of the individuals and slow renovation of the intrinsic population with time. The evolutionary

entropy is also the lowest, suggesting that this population without migrations is more susceptible to

changes in microscopic parameters that describe the system, being more exposed to extinction. On

the other hand, the Leslie matrix that includes immigration and emigration II has the highest growth rate

and, even though it is close to zero, it is positive, leading to the conclusion that the population is slightly

growing but tending to stability. Another important measure of population growth is the reproductive

potential, which is the smallest but still negative, meaning that the renovation of individuals is not at a

rate that compensates the exits and deaths verified in this society with migrations. Also, the evolutionary

entropy value is the highest but yet corresponds to a population that is very vulnerable to perturbations.

Notice that the evolutionary entropy found for Arroios is low comparing to other endangered species. For

example, the Desert tortoise (Gopherus agassizii) has a higher evolutionary entropy value (H = 0.548)

[30].

Finally, we would like to apply our models and measures to other parishes and refine them with

more data concerning different years and, in that way, improve the robustness of our models and its

projections, and help the decision makers to know certain intrinsic aspects of a population.

62

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64

Appendix A

Iterative Method

Function that receives the Leslie Matrix, the female population in 2001 and in 2011, the maximum

difference allowed between decrease the overall error, and the rational number to be added or subtracted

to each parameter until the model projects 2011 with a minimum error. Returns the matrix optimized

taking into account the error obtained using the data of 2011.

65

iteradif = Function[{la, p2001, p2011, eps, step, indice}, Module[{dif, ma, pa,a, b, c, d, e, f, g, h, in, jn, l, m, n, o, p, q, r, s, t, u, el, ij, fi},

{a, b, c, d, e, f, g, h, in, jn, l, m, n, o, p, q, r, s, t, u} = PadRight[{}, 20];dif = Table[(MatrixPower[la, indice].p2001)[[i]] - p2011[[i]],

{i, 1, Dimensions[p2001][[1]]}];pa = {a, b, c, d, e, f, g, h, in, jn, l, m, n, o, p, q, r, s, t, u};ma =

Extract[Table[PadRight[PadLeft[{pa[[j]]}, j], Dimensions[p2001][[1]]], {i,1, Dimensions[p2001][[1]] - 1}, {j, 1, Dimensions[p2001][[1]] - 1}], {1}];

ma = Prepend[ma, PadRight[{}, Dimensions[p2001][[1]]]];ij = 2;While[ij ≤ Dimensions[p2001][[1]],If[Extract[dif[[ij]], {1}] != 0,el = Extract[dif[[ij]], {1}];While[Abs[el] > eps,If[Extract[dif[[ij]], {1}] > 0,

pa[[ij - 1]] = pa[[ij - 1]] + step;ma[[ij, ij - 1]] = pa[[ij - 1]];fi = la - ma;dif = Table[(MatrixPower[fi, indice].p2001)[[i]] - p2011[[i]],

{i, 1, Dimensions[p2001][[1]]}];el = Extract[dif[[ij]], {1}],

pa[[ij - 1]] = pa[[ij - 1]] - step;ma[[ij, ij - 1]] = pa[[ij - 1]];fi = la - ma;dif = Table[(MatrixPower[fi, indice].p2001)[[i]] - p2011[[i]],

{i, 1, Dimensions[p2001][[1]]}];el = dif[[ij]];

];];

];ij = ij + 1;

];Print[Extract[(Sum[Abs[((MatrixPower[fi, indice].p2001 - p2011) / p2011)[[i]]],

{i, 1, Dimensions[p2001][[1]]}]), {1}] / Dimensions[p2001][[1]]];Print[MatrixForm[fi]];fi

]

];

66