Jovan Zamac

Daniel Hallberg

Thomas Lindh
email: thomas.lindh@framtidssstudier.se

Low fertility and long-run growth in an economy with a large public sector

Keywords: Low fertility trap, Social norms, Relative income, Economic growth

Recently it has been suggested that low fertility countries may be caught in a trap that is hard to get out of. One important mechanism in such a trap would be social interaction and its effect on the ideal family size. Such social interaction mechanisms are hard to capture in formal models, therefore we use an agent-based simulation model to investigate the issue. In our experimental setup a stable growth and population pa th is calibrated to Swedish data and using the Swedish social policy setup. The model is provoked into a fertility trap by increasing relative child costs linked to positive growth. Even rather large increases in child benefits are then insufficient to get out of the trap. However, the small number of children temporarily enables the economy to grow faster for several decades. Removing the adaptation of social norms turns out to disarm the trap.

Basse fécondité et croissance à long terme dans une économie à secteur public très développé Il a été suggéré récemment que les pays à basse fécondité pourraient être victimes d'un piège dont ils auraient du mal à se dégager. Un mécanisme essentiel dans ce piège serait l'interaction sociale et son effet sur la taille idéale de famille. Des mécanismes de ce type sont difficiles à représenter dans un modèle formel, et c'est pourquoi nous avons eu recours à un modèle de simulation multi-agents pour explorer le processus. Dans notre dispositif expérimental, un modèle de croissance et de population stable est calibré aux données suédoises, en utilisant la configuration suédoise de politique sociale. Le modèle est entraîné dans un piège de fécondité en élevant les coûts relatifs de l'enfant en lien avec la croissance positive. Dans ce cas, même des augmentations importantes des prestations familiales sont insuffisantes pour sortir du piège. Toutefois, le petit nombre d'enfants permet temporairement à l'économie de croître plus rapidement pendant quelques décennies. L'arrêt de l'adaptation aux normes sociales conduit à une neutralisation du piège.

Mots-clés : Piège de basse fécondité, Normes sociales, Revenu relatif, Croissance économique

Introduction

There is by now plenty of evidence that gross domestic product (GDP) growth across countries has a negative relation to fertility rates and dependency ratios. [START_REF] Kelley | Evolution of Recent Economic-Demographic Modeling: A Synthesis[END_REF] summarize much of the evidence concerning the importance of demographic factors for growth in both developing and developed countries. [START_REF] Lindh | Demographically based global income forecasts up to the year 2050[END_REF] estimate a demographically based forecasting model which takes account of the changing economic impact of age distribution as longevity increases. The negative impact of large cohorts of children is a robust feature. In Barro-type crosscountry growth regressions, high fertility has also been established as a substantial negative factor. This is not really new, but provides confirmation, for example, of the shift-share analysis of [START_REF] Krueger | Factor endowments and per capita income differences among countries[END_REF] who makes the point that a very large part of the difference in production levels between developing and developed countries can be explained by demography and education. 1It is intuitively rather obvious that an economy where almost half of the population is below 15, as is the case in many African countries, has to carry a large burden of supporting and educating the young, a burden of which only a tiny fraction will appear as value-added in the national accounts. As emphasized by [START_REF] Bloom | The Demographic Dividend: A New Perspective on the Economic Consequences of Population Change[END_REF], the demographic dividend from falling fertility appears as the working population starts to grow faster than the dependent population. [START_REF] Mason | Transfers, Capital and Consumption over the Demographic Transition[END_REF] stress the possibility of a second demographic dividend as middle-aged people start saving for retirement and thereby contribute to increasing capital resources.

Because growth can be accelerated by lowering fertility it becomes theoretically possible to compensate for rising elderly dependency ratios by decreasing fertility. Indeed we observe that several developed countries both in Europe and Asia are experiencing very low fertility which is accelerating their aging. While only China has a deliberate one-child policy, fertility involuntarily seems to be lower than in China in both South Korea and Japan. Since below-replacement fertility will speed up population aging, this is ultimately not a sustainable equilibrium. Sooner or later the current financial support for the elderly provided by the economically active population will become insufficient. That may well take a long time though and in the meantime resources can be shifted from reproduction to support for the elderly.

Recently it has been put forward, e.g. [START_REF] Lutz | The low fertility trap hypothesis: Forces that may lead to further postponement and fewer births in Europe[END_REF], that countries with very low fertility (below say 1.5) may be caught in a low fertility trap (the Low Fertility Trap Hypothesis, LFTH). This builds on the observation by inter alia [START_REF] Mcdonald | Low fertility in Singapore: Causes, consequences and policies[END_REF] that a recovery towards replacement levels seems to become increasingly difficult to achieve in countries with very low fertility. The mechanism suggested by Lutz et al. builds on three different feedbacks. One is the demographic inertia of a baby bust where the small cohorts will have fewer babies simply because there are fewer people of reproductive age. The second feedback is hypothesized to work through social norms which make fewer children more desirable as the prevalence of children in society becomes rarer. This social interaction and its effect on the ideal family size work as a negative feedback reinforcing low fertility standards. Thirdly, Lutz et al. add the socio-economic relative income effect of [START_REF] Easterlin | The American Baby Boom in Historical Perspective[END_REF], where the aspired standard of living is determined by consumption standards in the parental home acting as a constraint on the number of desired children when it becomes harder to achieve these aspirations (also cf [START_REF] Macunovich | Fertility and the Easterlin hypothesis: An assessment of the literature[END_REF]. Such social interaction mechanisms are hard to capture in a standard economic model framework. For this reason we propose to use an agent-based simulation model (ABM) to investigate the economic consequences of low fertility and its feedback on the fertility decision.

We use a simple agent-based framework model as an experimental tool to investigate under what circumstances a low fertility trap may be likely to appear. The model is built around the tension between a long education interfering with the prime fertility period of females and the need for such education in order to satisfy consumption aspirations arising from the relative income effect.

The crucial elements in an agent-based simulation model are the rules of thumb that agents use for making decisions. These rules can incorporate both an economic dimension and a social interaction dimension, both of which are important for the low fertility trap. Individual micro behavior results in a macro outcome which in turns feeds back into individual decisions, and hence a micro-macro interdependence is obtained which cannot be modeled in more traditional microsimulation models.

By calibration to Swedish micro and macro data2 the simulation model offers an experimental laboratory to test different theoretical mechanisms and their implications for the balance between current benefits from low cohort fertility and the losses in terms of future shrinkage of the tax base and of growth potential. Thus our base scenario in this paper approximately reproduces the natural reproduction features of the Swedish population during the 20 th century.

We then experiment with alternative scenarios focused on the relative income mechanism in order to spring a low fertility trap. It turns out that introducing a mechanism that increases the relative cost of children versus consumption has the potential to set off a low fertility trap. The growth rate in GDP per capita increases substantially and for a long time before the elderly dependency burden makes the system fiscally unsustainable. It takes a very large benefit that is permanently increased to counteract this and even to delay the low fertility trap. The crucial element of the fertility trap turns out to be the social norms system. When the social norm mechanism is disabled the increase in relative costs of children does decrease fertility somewhat but does not provoke a fertility trap.

For reasons of brevity, this paper does not go into detail about the model, and a more comprehensive description is available in Žamac et al. (2008). In the next section we first discuss the background and theoretical starting point more completely. This is followed by a short account of the model structure and the crucial mechanisms in the modeling context. In the third section, we analyze the results of alternative scenarios and discuss the potential implications. Finally we conclude our argument by discussing how the results can be used to focus further work around growth and fertility.

Background and theoretical starting point

Negative momentum from low birth rates that decreases the future fertile population is an obvious demographic feedback from low fertility. Ideal family size as measured in attitude surveys shows a downward trend in many countries. Whereas ideal family size is still higher than actual total fertility rates (TFR) in Europe, there are some countries, e.g. Austria and Germany, where ideal family size is now well below replacement rates. There seems to be a downward trend of relative income in many industrialized countries (as far as it can be correctly measured) where it is claimed that the next generation will fare worse than their parents. All this seems to coincide with downward trends in cohort fertility and definitely with postponement of childbearing. But will these factors combine into a mutually reinforcing feedback circle as the LFTH suggests? There are three apparent escape routes to consider. First, extensive immigration may increase the fertile population and reverse the negative momentum. Second, ideological pressure could be mobilized to favor ideals of replacement fertility. Third, social policies can be implemented to relieve the relative income pressure. Our focus is on the third escape route, although it turns out that the second is more crucial for creating a trap. The first with immigration cannot be implemented in the current version of the model.

The social policy frameworks differ substantially across the countries that are now experiencing negative fertility trends. It has been argued that the Scandinavian emphasis on policies favoring dual-earner families explains why the downward fertility trend is much less pronounced in these countries [START_REF] Ferrarini | Parental Leave Institutions in Eighteen Post War Welfare States[END_REF]. The Swedish case is thus very relevant to study as a potential model that refutes the inevitability of the LFTH. Sweden has had a very stable cohort fertility rate (CFR), at around 2 for all cohorts born in the 20th century where we can observe completed fertility, i.e. up to the early 1960s. The TFR, however, has fluctuated between 2.1 and 1.5 in the postwar period. [START_REF] Björklund | Does family policy affect fertility[END_REF] thinks younger cohorts in Sweden will make up for the lag in their CFR, and indeed birth rates have been picking up and may be headed for a new baby boom. It nonetheless seems to be a fact that the relative income of young adults in Sweden has been falling over the same period. Norms regarding the desired family size have been very stable, at around 2 children, as measured in attitude surveys. At the same time the average age of the mother at first birth has risen steadily (24 years in 1970, 29 years in 2005) substantially increasing the probability of not achieving desired fertility. Thus, one may hypothesize that family policy is a crucial factor interfering with the mechanisms of the LFTH, both by relieving or amplifying economic pressures and maybe also by a strong influence on social norms (or family policy designed to conform to social norms). However, as the South Korean and Japanese examples show, the design of pro-natalist social policies may be hard to implement efficiently in order to actually achieve any results. 3There is so far little consensus on whether the demographic transition will end up maintaining world population at approximately stable levels or will result in a future shrinking population. A generally accepted theory to generate predictions on fertility is missing today. The LFTH suggested by [START_REF] Lutz | The low fertility trap hypothesis: Forces that may lead to further postponement and fewer births in Europe[END_REF] offers a framework to structure further research around the micro-macro feedbacks. There are difficulties, however, since we cannot in general observe these feedbacks in isolation. We are dealing with very long-range processes taking place in a quickly changing social environment. Although OLG models following [START_REF] Allais | Economie et Intérêt[END_REF], [START_REF] Samuelson | An Exact Consumption-Loan Model of Interest with or Without the Social Contrivance of Money[END_REF] and [START_REF] Diamond | National Debt in a Neo-Classical Growth Model[END_REF] have become a standard tool for economists analyzing intergenerational issues as well as general macroeconomics, it is still the case that these models quickly become intractable when the population structure is non-stationary. This has led to numerous attempts to use simulation in order to investigate their properties under more realistic assumptions (e.g. [START_REF] Kotlikoff | The Coming Generational Storm[END_REF]. Increased realism, however, comes at the price of increasingly non-transparent calibration and assumptions that are not readily verifiable.

Agent-based modeling offers an alternative where we can observe the aggregate effects of decentralized decision making without very strong assumptions on individual behavior and still maintain a degree of transparency and the opportunity to experiment with different mechanisms. Traditional microsimulation models building on estimated micro relations [START_REF] Klevmarken | Statistical inference in micro-simulation models: incorporating external information[END_REF] have reached a point where severe problems have arisen concerning their ability to actually improve our understanding of behavioral mechanisms and their repercussions on the economy at large. Agent-based simulation modeling has recently been increasingly explored as a more flexible alternative focusing on actual behavior rather than the optimal behavior of individuals, in recognition of the fact that even if agents do behave rationally under their respective information sets, statistical methods will not allow us to evaluate the full heterogeneity of individual behavior [START_REF] Richiardi | A Common Protocol for Agent-Based Social Simulation[END_REF].

While the flexibility of ABM allows us to use the model as a laboratory to experiment over a wide range of issues, its drawback is that it easily tempts the researcher to do too much, to keep too many options open, to start playing a Sims computer game instead of investigating real issues. It is therefore of paramount importance to define the focus of each study rather narrowly. The basic model that we use has been developed at the Institute for Futures Studies using JAVA programming. The variant used in this paper is adapted to the specific issue of fertility and the mechanisms of the LFTH.

In brief, it exploits the interaction of education choice, mating and the fertility decision to generate a self-propagating population embedded in an endogenous growth model. This base version of the model is adapted here to the low fertility issue by introducing a relative income mechanism.

What Is Agent-Based Modeling?

ABM starts from the premise that the "real world" is hardly the work of a central planner, built to conform to rational rules. Rather, the real world is characterized by decentralized, simultaneous interactions between a very large number of different agents, whose decision-making is based on limited rationality, imperfect information and habits, and where the local relational context also contributes to those agents' strategies and behaviors.

There is a growing interest among economists in modeling micro-macro linkages between individual and aggregate level variables. Most recent attempts have consisted of combining economy-wide Computable General Equilibrium Models (CGE) with microsimulation models (see [START_REF] Davies | Microsimulation, SGE and Macro Modelling for Transition and Developing Economies[END_REF], for a review). They rely on the classical assumptions (e.g. rationality, perfect foresight, competitive markets, perfect information, market clearing, etc.), in order to find an optimal solution or "equilibrium" for aggregate level variables such as total output. CGE cannot account for heterogeneity across households, preferences or technologies; only a few types of representative agents are assigned the same production or utility functions. This is clearly a simplification which overlooks important variations at the micro level, and more generally makes distributional analysis unfeasible (i.e. how total output and consumption are actually distributed between different agents and what drives these differences).

Microsimulation models on the other hand, are mostly used to study distributional effects e.g. of tax and benefit systems, at the micro level, including (in the case of dynamic microsimulation models) projections over the individual agent's entire life cycle (including behavioral responses e.g. labor supply, fertility choice, education, etc). Usually built on household survey data (or other micro-level data), they allow access to detailed information e.g. about individuals' income sources, areas of residence, past employment history etc., but they cannot deal with modeling the monetary side of the economy or with the inclusion of structural macro features and aggregate feedbacks, which therefore have to be assumed as exogenous.

In practice, integrated macro-micro models are difficult to implement, mostly due to a tradeoff between adding model complexity and finding solutions which can be handled by standard computational tools. ABMs represent a further step in the development of dynamic macro-microsimulation modeling. They can incorporate modern computing developments (e.g. object-oriented programming languages) to simulate complex interactions simultaneously, and to simulate how these interactions evolve in time through the accumulation of new information, with no need to have two separate converging models (e.g. one micro and one macro), nor to have convergence to an equilibrium solution at all.

The principle behind ABMs is that of multiple interacting agents who are goal directed (e.g. preserving a certain aspired consumption level in our case), and who try to control their environment, in a decentralized (i.e. non-coordinated, non-centrally planned) system. ABMs do not assume rationality or the existence of a pre-defined equilibrium outcome. Agents might behave in sub-optimal ways, but they can gradually learn from their experiences and adjust their behavior to the neighboring environment.

The first attempt to apply ABM to the social sciences is considered to be T. Schelling's "Models of segregation" [START_REF] Schelling | Models of Segregation[END_REF]. Using JAVA or similar object-oriented programming languages, ABM agents are usually implemented in software as objects i.e. computational entities that have initial states (e.g. se x, age), are able to perform some pre-specified action or method, can communicate or share information with others, pass on or even inherit characteristics or behavioral rules.

Running an ABM simply means instantiating an agent population, and letting it run forward in time, i.e. executing it, rather than solving it. The outcomes of agents' interactions can be observed at any given time by the modeler who only needs to specify some initial behavioral algorithms (the equivalent of classical preferences) and initial conditions for his agents and their environment. These agents or objects can represent people (say consumers, sellers, or voters), but they can also represent social groupings such as families, firms, communities, government agencies and nations.

In our application we view the model as an experimental laboratory for testing mechanisms in a more complex setting than analytical modeling, allowing yet more transparency, and subject to experimental control. This makes inference possible regarding causal mechanisms that cannot be gleaned from econometric estimation on real world data. This does not replace other modes of analysis but is a complement for testing the logic of economic mechanisms in a more complex setting that can, to some extent, be validated against real world data.

The IFSIM model

The IFSIM model, as we call it, consists of a small number of interacting modules. Due to space considerations we only give a brief overview of some features that are important for this paper.

4.1.

Demographic module

The starting population comes directly from the initial dataset. 4 There is no migration in the model so the population evolves according to fertility and mortality rates. Since we focus on the fertility decision, the mortality rates are exogenously set in line with the Swedish rates in 2006. The maximum age an individual can reach is 110.

Before a woman can give birth she needs to find a partner and move out of her parental home. Starting from age 18, individuals living with their parents may start to leave the parental household and set up a household of their own. The decision to leave home is modeled as an exponential probability function depending on age and on the share of youngsters living with their mother within the network. We check that leaving home and mating occurs at relatively early ages so that it does not impose a restriction on fertility. By the age of 21 every individual has left the parental household.

The mating process is assortative such that pairs with similar human capital and belonging to the same network are more likely to create a new married household.

Fertility

There are several variables that affect the fertility outcome. First we have fecundity, which is beyond individual control. We roughly capture this biological res triction by calibrating the probability of conceiving to medical studies. This yields a fast declining probability from age 30 onward. The upper limit for conceiving is set to age 40 since only few births are recorded in Swedish data after that. We also have a lower starting age set to 20 below which it is not possible to give birth. There are few teenage pregnancies in Sweden today.

Second, we model a fertility function that allows us to capture the main elements of the LFTH. The LFTH has a social dimension and an economic dimension. The social dimension refers to how our desire to acquire children is influenced by the number of children around us. We first start by assuming that the norm of how many children one wants is set during youth. We call this the desired number of children. Every individual has a desired number of children that is determined by the number of siblings that he/she has. For the couple, we use the average of the man's and the woman's desired number of children, weighted with a random number. The couple strives to reach its desired number, but, due to their economic outcome and the social influence of their network group, it is not sure that they will reach this target. Given that the woman is of fertile age and that the number of desired children is higher than the actual number, she will give birth if the two following conditions hold:

, income median r n DISP  , * 10 ASPC PROJC or SocialFact
where DISP is the disposable household income, n is the number of individuals in the household and r is the cost of a new child. In the baseline scenario we fix this cost r to 1. However, in the alternative scenarios, when we simulate an increase in the cost of having a new child, we model this by increasing r. The square root captures the economies of scale of large families. The first condition says that the household's equivalent income (adjusted to include one extra child) must exceed the median individual income, which implies that today's economic conditions are very important in the fertility choice. The median income is considered as a sort of "minimum" income for affording a child.

The second condition says that the social factor in combination with expected future income, PROJC 10 , must exceed the aspired consumption level, denoted ASPC. We define the social factor according to: The economic dimension consists of determining if the couple can afford a new child or not in the future given their consumption aspirations. We follow the LFTH and state that a couple aspires to a certain consumption level based on their previous experience.

They will not acquire a child unless they can reach this aspired consumption level. As postulated by Easterlin, this aspired consumption level is a norm that is formed during youth and which has the consumption level of the home of origin as the reference point.

We model this aspired consumption, ASPC, according to:

C ASPC ,
where

  n DISP C
/ is the average equivalent disposable income today and

    10 10 10 10 ) 1 ( PROJC C C C at at at  
. C is the equivalent disposable income for an individual in the household. The subscript at10 indicates that the aspirations were set when the individual was 10 years old. The first term states that the position in the consumption distribution at the age of 10 affects the aspired consumption level. The idea is that children should not obtain a worse position in the consumption distribution. The second factor captures the idea that parents do not want their children to have a lower consumption level than they themselves had when young, adjusted for economic growth. We also apply a weighting factor, )

, currently set at 0.5, for the two different reference points for aspired consumption. Since the reference point was set when the parents were 10 years old it is natural to compare the new child's consumption level at the age of 10. For this reason they project expected future income according to:

r n DISP PROJC  10 10
, Where 10 DISP is the disposable income ten years from today if they choose to have one additional child. The household's disposable income ten years into the future is estimated from the sample of individuals in the model that today have ten more years of labor market experience. As mentioned earlier, in the base scenario, r is equal to unity, but varies between 1 and 3 in alternative scenarios, depending on economic growth in society.

Once a child is born, the mother is on parental leave for three years before returning to her previous labor market status. This exaggerates the actual Swedish paid parental leave (today it is 13 months at maximum), but is a way to avoid complications in taking into account other benefits, such as the right to part-time work until the child is 8 years old and the right to paid leave to care for sick children up to age 12. Doing a sensitivity analysis with parental leave reduced to just one year results in only a minor tendency for higher fertility but the dynamic behavior of the model remains very close to what we present below.

Social networks

Newborn individuals inherit some of their parents' characteristics. From birth they will belong to a network group, the first one being determined by their parents' characteristics. The networks are segmented first by age groups and second, within each age group, by a spatial dimension. A network group consists of all those individuals to whom the individual is close. We follow Billari et al. (2006) in defining social "closeness" as a spatial area representing the individual's scope of interaction, by age group. Specifically, agents are arranged along the surface of an imaginary cylinder, whose height is subdivided into as many segments as there are age groups in the model (at present there are 8 age groups, from age 0 to 110). Individuals will migrate between network groups as they age, and two individuals that belong to the same network group at one age may belong to different groups at later ages.

Educational module

When reaching the age of 7, all individuals are universally enrolled in basic schooling until the age of 16, corresponding to 9 years of compulsory education. If they enter high school, they will stay in school another 3 years. The choice to apply to university is determined by individuals' prospective earnings compared to their aspired consumption level, and preferred number of children. If secondary education is enough to reach the same per capita equivalent income as their parents, given the preferred number of children, they will not apply for university. Hence, the educational choice does not depend directly on fertility choices. However, it does so indirectly since if individuals estimate that they can reach their aspired consumption level given their preferred number of children without having to invest in education, then they will not choose university.

The university applicants are ranked according to accumulated human capital (see below) such that the ones with the highest human capital are actually accepted by the university. The number of available positions at the university is set to a fixed proportion of the current number of individuals aged 19 to 30. If attending university, the student will be entitled to a student allowance for the duration of the course (5 years), amounting to a fixed proportion of average earnings.

4.4.

Modules for human capital formation, the labor market and production of consumption goods

We postulate a production technology that only depends on human capital (i.e. there are no savings into other types of productive assets). Production depends on low-skilled and high-skilled workers according to:

H L t H t L t H H Q , ,
where 1  H L , and t L H , and t H H , are the aggregate human capital for the nonuniversity and university degree groups, respectively. There are no monetary values in the model so earnings are represented by the share of total production output going to each worker. Allocation of the produced goods to workers is separated into two steps. First, the total produced goods are allocated to the two production factors (non-university and university degree individuals) such that the shares reflect each group's marginal product. Then, within each group, the consumption good units are allocated proportionally to the human capital of the individual. If the supply of university degree individuals is reduced, their marginal product will increase and thereby increase their share of the produced goods. As a result, the education premium also increases. This will be observed by young individuals who will be more likely to choose university and thus increase the future supply of university degree individuals.

As human capital is of paramount importance in the model, we will describe in detail how it is modeled. There are four main inputs that determine the human capital of each individual: (i) innate individual ability; (ii) ability acquired from parental influence and parental own human capital levels; (iii) ability acquired through formal education; (iv) skills and expertise acquired through training on the job.

At birth, an individual is immediately assigned a native human capital stock which captures the average native human capital stock of the parents (i.e. a purely genetically inherited feature) plus a random number. Subsequently, from the first year of life, human capital evolves every year depending on events during the year. We allow for three different functions for human capital updating depending on which life phase the individual is in. The three phases that we consider are: pre-school, in-school, and postschool.

Pre-school

Consider an individual with innate human capital 0 i h . The discrete time evolution of the human capital in the pre-school phase is modeled as

  parents f m own t i t i it it h h h h   1
where indicates first differences and it h is the human capital of individual i in period t.

mf ii hh  denote the human capital of individual i 's mother and father in period t. How the two input factors, the sum of parents' human capital and the child's own capital, are combined is determined by the parameters own and parents which are specific to the pre-school period and set exogenously.

In-school

During schooling a third input factor affects the production of human capital, namely the ratio between the aggregate level of human capital among the teachers and the number of students. Since teachers are randomly drawn from the labor force there will be a spillover effect when the overall human capital increases. However, the teacherstudent ratio is also important so we account for the number of students. To formalize, the human capital production during school periods is defined as: where the superscript s indicates that the parameter value depends on the level of education (primary, secondary, and tertiary), and s teach indicates how the new input factor is combined. teach t H and t stud denote aggregate teacher human capital and number of students, respectively. The policy makers can thus directly influence the production of human capital by allocating more resources to the educational sector (employing more teachers). The ratio of teachers per student is fixed at 0.1.

Post-school

In standard wage equations, labor market experience approximates human capital production at work. In our setup, we can use a pure "at-work" human capital production function similar to the in-school production function. The human capital increases for every work year but proportionately less for each additional year up to age 55. We extend the production function to incorporate a deterioration of human capital in periods of parental leave. Formally, the human capital production at work is :

1 55 10 it it k it age h edu h  
where edu k is a factor that is dependent on the level of education: basic, secondary, and university. The rate of increase accelerates as level of education rises, thus generating steeper wage profiles over life for the higher educated. During periods out of the labor market, notably on parental leave, the human capital is set to depreciate at a yearly rate of 0.015.

4.5.

State, tax and benefit systems

Beside individual agents, the model includes an institutional object acting as an agent that represents "the State" who collects and redistributes resources. First, the State calculates the total expenditure bill, by aggregating the costs of the education, teacher salaries and student allowances, parental leave subsidies, child allowances and pensions.5 Once total expenditures are calculated, the State will adapt the tax system so as to raise sufficient revenues to balance the budget (no debt is allowed). The tax system comprises a state tax and a local tax. The state tax is paid by the top 20 percent of the income distribution and amounts to 20 percent. The local tax paid by everybody with positive earnings is a flat-rate tax on earnings. It is residually derived to cover the part of total expenditure not covered by the state tax. The individual income tax will therefore be a combination of the state tax (if eligible) and the local tax. The individual disposable income is therefore the sum of any earnings, pensions, student or parental allowances, minus the income tax.

Model scenarios

Given the basic model setup above, both the demographic and the social norm mechanisms are included. The relative income mechanism has been implemented by defining an aspired consumption level based on the household income in the parental home at 10 years of age. See Žamac et al. (2008) for more details. For a fertility decision to take place, a match must first have been achieved with a partner in the local network. In the next step a decision is taken whether to have a child. First it is determined whether a child can be afforded at the current income level. If so, and if future income expectations allow a new child, then procreation is initiated. This is the base scenario which we then subject to a cost shock in order to provoke a fertility trap.

5.1.

Base scenario

In the base scenario the simulation runs for more than 300 years. The influence of initial conditions takes about 100 years to vanish. An initialization period of 150 years is therefore disregarded in the analysis. After this period the model stabilizes and roughly reproduces 20 th century demographic behavior in Sweden.

Poor economic circumstances at fertile ages with respect to the aspired income make potential parents postpone childbearing, in hope of better economic conditions in the future. As the returns of education increase with education level, highly educated parents increase their chance of getting their preferred number of children. On the other hand, education takes valuable time from their fertile years. Since fecundity decreases with age, this increases the risk of not reaching the desired number of children.

Below we briefly present some main features of the base scenario. In Figure 1 we depict the population development in the last 160 years of the simulation (for convenience we label model years starting from the year 0 but note that this is arbitrary). There are short-term periods of positive and negative growth, but the long-term trend is positive population growth. Over the study period there is an increase from about 11,000 to 15,000 inhabitants.

Figure 1 Population development in base scenario

The age composition of the population is very stable over time; see Figure 2 below, which shows the shares of young (0-20), prime-aged (20-64), retired (65+) and oldestold (80+) by model year. There is an oscillatory pattern (suggesting a saw-toothed age distribution) that reflects influential baby boom cohorts and which, to some extent, actually resembles the Swedish demography over the 20 th century. In the 65+ group these oscillations cancel out. There is variation in the CFR over time that explains this pattern. For the whole period CFR is on average above replacement level at 2.15 children per woman. We observe a rather steady pattern but there are some temporary swings in CFR, as shown in Figure 3. We do not have any mechanism that leads to intensified efforts to procreate as fecundity declines. Intuitively there is no "biological clock" that makes individuals try to catch up in their 30s. Therefore the swings in period fertility due to changing economic conditions mostly carry over to cohort fertility. As can be seen from Table 1, the age of the mother in general increases with education. Mothers are on average about 27 years old when they have their first child if they have only high school or basic education, while mothers with a university degree are on average about 29 when they have their first child. Note that university studies are spread over a rather long period in young adulthood as some actually enter at a late age. The typical age at university graduation is 24 (i.e. the youngest possible graduation age), but the rest, about 17 percent, graduate between ages 25-40. These delays in achieving a diploma will push child birth later into the fertile period. There is, however, a small chance that students will have a baby during education enrolment. Such events will delay university graduation even further for female students (by three years per baby). The larger effect on the high educated may reflect the increased difficulty of combining education and children. At the aggregate level, the base scenario data show that rising enrolment rates in higher education are linked to lower birth rates. Once enrolment rates drop, individuals obtain their diploma, and birth rates increase.

The development of earnings over time, presented for the years 60-125 in Figure 4, shows strong period effects. These are due to a shortage of one education class relative to the other, creating a wage drift upwards for the scarce education category. In comparison, most of the period effects are not reflected in the human capital stock, shown in Figure 5 for the same period. The ratio of earnings to human capital hence fluctuates over time.

Alternative scenarios with a higher child cost

In three alternative scenarios we modified the function that determines the cost of children in order to provoke a fertility trap. In the base scenario, the expected cost of having a new child is a fixed share of current median income. In the alternative scenarios, when the economy grows faster than usual there will be a relative price shift making children costlier in terms of consumption. This will force parents to invest a higher share of their income in the child, if they decide to have one. One rationalization for this upward drift in the relative cost of children is that the opportunity cost of parental time is increasing. Since everybody works full time in our model we cannot implement that mechanism directly through labor supply.

Below we describe the three scenarios.

1. No policy: our first alternative scenario, nothing is done to counteract the change in cost of children. 2. Increasing benefits : the second alternative scenario implements a child benefit immediately. In this scenario, the benefit is given by 11 0.1 /

t it t i b W n  % where 1  it
W is the wage earnings for person i, 1  t n is the number of children eligible for the benefit in the population, both measured in the previous period. In a situation where few children are born, the benefit t b ~ will rise to counteract the upward drift in the relative cost of children. 3. Fixed social norms: this is the same as the "no policy" scenario 1 except that the social norm mechanism for adapting to the local network and siblings is turned off.

In sum, alternative 1 lets the cost shock have full impact on behavior, while alternative 2 implements active policies immediately. Alternative 3 includes no economic policy responses but prevent social norms from adapting and thus cuts out one element of negative feedback.

The relative cost shock mechanism is introduced into the model year 60.6 This is also the year when the social norm mechanism is turned off in the third scenario. One can clearly see that fertility is affected by the introduction of the changed child cost. In the no policy scenario, and with some delay also in the increasing benefits scenario, population actually declines at a fast rate (Figure 6) while population in the fixed social norms case still grows but more slowly. As is obvious in Figure 7, the CFR declines and two decades after the introduction of the relative cost mechanism the CFR only rarely exceeds replacement level. Increasing benefits are successful in blocking the drop in fertility about 40 years into the future, but this policy cannot turn the tide as fertility rates fall to low levels below 1.5. The relative size of the working-age population age 20-64 initially increases in all alternative scenarios (Figure 8). This gives rise boosted GDP growth and lower taxes since less spending is needed in the educational system. But eventually the lowest-low fertility scenarios enter into a phase with a decreasing active population. In the period before the child cost change occurs, mothers are about 28-29 years old when they have their first child, see upper part of Table 2. When the change is introduced in the model, births are postponed in all alternative scenarios. As time goes by, the fixed norm scenario stabilizes the year at first birth while in the other two it keeps on increasing. This is because aspired income is more difficult to reach with the same number of children as before. As a consequence, women postpone births to an extent that finally lowers the desired number of children and leads to decreasing fertility in both the no policy and increasing benefit scenarios. In Figure 9 the share of high skilled in the working population suggests that the human capital stock per capita grows faster in both the no policy and increasing benefits scenarios than in the base scenario. This is a self-enforcing process. Once education has started to increase, the increase is sustained by a higher level of human capital in the economy making education more efficient. Agents also need more education in order to reach an aspired consumption level which is increasing likewise. This also contributes to further postponement of births.

Last, the growth of the economy in the scenarios, depicted as per capita GDP and shown in Figure 10, exhibits the expected pattern. The alternatives with lower fertility at first allow much faster growth. As mentioned, the relative cost mechanism is set off by the GDP growth. High GDP growth scenarios therefore create a self-generating process of high child costs, further reducing fertility. The high per capita growth compared to the base scenario lasts in the increasing benefits case for almost 100 years (but note that GDP does not grow for that long in overall terms, due to shrinking population). This raises the issue of how far into the future the altruism of the current generations will persist. By remaining childless they can improve their material well-being but only at the expense of as yet unborn generations in the future.

Figure 11 depicts the increase in tax rates as per capita growth declines. Tax ratios in the no policy case quickly increase to a hundred percent of GDP. 7 The increasing benefit scenario, which temporarily stops the drop in fertility, implies a very high cost requiring higher tax rates -10 percent or morein all future periods. Implementing this policy would hardly be politically possible unless median voters had a very high degree of altruism towards future generations. In the end, however, high tax increases like these also fail to prevent fiscally unsustainable tax levels, so tax hikes to provide high child benefits are not sufficient in themselves to reverse the trap. Introducing saving and capital investment might change this conclusion (see [START_REF] Mason | Transfers, Capital and Consumption over the Demographic Transition[END_REF], about the second demographic dividend) but its implementation in the model would substantially shift the focus of the study. In contrast to the increasing benefit scenario, there is no dramatic population decline in the fixed norm scenario, but instead a mild increase, although smaller than in the base scenario. Cohort fertility is a little lower than the base but still above replacement levels. As could be seen from the previous figures, this scenario also bears a close resemblance to the base scenario in its other dimensions, such as age dependency ratios, per capita GDP growth and tax rates. Although a new child will be viewed as more expensive, this affects outcomes more like an idiosyncratic disturbance that can be adapted to. It seems thus that the social norms as we have modeled them have a major impact on development, reinforcing small imbalances and continually allowing them to spread both through the population and over time. The economic mechanisms, on the other hand, allow the agents to adapt in order to reach their desired number of children. Thus the adaptation of social norms is the key to creating a fertility trap.

Conclusions

Our simulation model is capable of reproducing a fertility trap without crashing or exhibiting unexpected behavior, such as overly strong reactions to alternative policies, over a secular period of time. The increasing benefit scenario shows that, compared to doing nothing, getting out of the trap requires very determined and persistent policy measures with high temporary growth costs. Temporary in this context implies a period longer than the expected remaining life time of the currently active population. Even so, our experiments indicate that in the long run benefits will have to increase even further than the policy we implement. Once downturns in fertility start the social norm mechanism, the system quickly gets into a negative spiral where new policies cannot be financed. Without a fair degree of altruism such measures would not be politically feasible anyway. Selfish individuals without consideration for their descendants far into the future would delay action until recovery is impossible because the tax base has grown too weak to support the necessary transfers.

Removing the social norm dynamics is, in itself, sufficient to avoid getting trapped into low fertility; the economic mechanisms can adjust to restore balance in spite of the disturbances. It is the social norm dynamics which entrench new, ultimately unsustainable, behavior in the population.

Of course, our conclusions are only strictly valid for our virtual world. In the real world there is a lot more scope for adaptation. Our virtual world is, however, reasonably complex and still reproduces features of demographic dynamics in Sweden that we had not expected to model originally. Nor had we expected the results we found. We therefore conclude from this study the following.

1. Within an isolated system of intergenerational transfers where relative costs of children are increasing a low fertility trap can form. 2. When this process has been entrenched in social norms it is very costly, and may ultimately be impossible, to reverse the trend. 3. Unless voters are fairly altruistic this will not be politically feasible in a democracy. 4. Dissolution of the social norm mechanism seems the most important policy to pursue.

An obvious question is whether there is a policy that can actually achieve dissolution of strong social norms. Increasing social mobility via increased income redistribution might not be efficient, but in our model framework it would be a step in the right direction. More likely to meet with success is a more proactive policy that prevents fertility rates from falling to levels where the social norms become entrenched. Given the failure of proactive policy information on future consequences for society, immigration and tax policies more directly tailored to compensating for the rising relative costs of children might be effective in a real-world setting. Future empirical research should pay more attention to measuring the relative costs of children and whether variation in this variable can be causally tied to actual fertility behavior.

In simulations it is important to test whether a savings mechanism would modify the conclusions. There are a number of reasons why capital markets may circumvent the set of vicious circles we have set in motion in our virtual world. First, capital investment may to some extent substitute labor. Second, the "second demographic dividend" of net saving for old age may circumvent the aspired income mechanism by providing sufficient returns to ensure that aspired consumption levels are achieved without foregoing either education or children. Third, a capital market would allow public borrowing to invest in children without further diminishing the current income available for consumption. Other extensions could be to introduce migration and ethnic barriers in the networks.

  is the number of children belonging to the Members N members in the network group. Here is a parameter that controls the strength of social pressure, currently set to 0.92. If many individuals within a couple's network have children this will positively affect the couple's fertility decision.

Figure 2

 2 Figure 2 Age structure in the base scenario

Figure 3

 3 Figure 3 Completed fertility, by birth cohort

Figure 4

 4 Figure 4 Earnings in model time 60-125 by education, base scenario

Figure 6

 6 Figure 6 Population in the base and alternative scenarios

Figure 7 Figure 8

 78 Figure 7 Completed cohort fertility rate (CFR) in base and alte rnative scenarios, by model time

1 .

 1 Figure 9 Proportion of high skilled (university educated) in the age group 20-64

Table 1 Average age of mothers when giving birth in the base scenario, by birth order of child and education level

 1

	Birth order of child Basic+High school University	Total
	1 st	27.4	29.0	28.1
	2 nd	30.3	31.2	30.7
	3 rd	33.2	33.9	33.5
	4 th	34.9	35.5	35.2
	5 th	36.2	36.6	36.4
	6 th	36.8	37.4	37.1
	Total	30.7	31.7	31.2

Table 2 Average age of mothers at first birth in the base scenario compared to the different alte rnative scenarios, by model time

 2

	Model year	Base	No policy Increasing	Fixed norms
				benefits
	0-9	29.2	29.2	29.2	29.2
	10-19	27.9	27.9	27.9	27.9
	20-29	27.4	27.4	27.4	27.4
	30-39	28.5	28.5	28.5	28.5
	40-49	28.2	28.2	28.2	28.2
	50-59	28.1	28.1	28.1	28.1
	60-69 (decade of change)	28.2	30.2	28.6	29.5
	70-79	28.1	30.4	30.4	29.7
	80-89	27.7	30.8	31.2	29.7
	90-99	28.6	32.4	32.0	31.1
	100-109	27.9	32.2	32.6	30.9
	110-119	27.5	33.0	32.9	30.6

[START_REF] Myrskylä | Human development and low fertility[END_REF] find a U-shaped relation between a Hu man Develop ment Index and fertility. While an HDI weighs in GDP per capita as one factor, this is not the same as GDP gro wth.

We have also used some data fro m the Swed ish National Transfer Accounts . See[START_REF] Lee | A Research Plan for the Macroeconomic Demography of Intergenerational Transfers[END_REF] about National Transfer Accounts that track intergenerational flows in an economy.

The efficiency of social policies in increasing fert ility is still an open issue. While Fey rer et al. (2008),[START_REF] Björklund | Does family policy affect fertility[END_REF] and others have estimated large effects there are other studies like OECD (2003) and[START_REF] Grant | Low Fertility and Population Ageing. Causes, Consequences,and Policy Options[END_REF] wh ich find that income -related policies have limited effects on fertility.

The data comes fro m the Swed ish micro data set HUS(See Klev marken and[START_REF] Klevmarken | Household Market and Nonmarket Activities, Procedures and Codes 1984-1991[END_REF][START_REF] Flood | Household Market and Nonmarket Activities (HUS), Volumes III-VI[END_REF], which is a representative sample of about 3000 individuals wh ich we scale to obtain our desired number of indiv iduals in the init ial year of simulat ion. In the first period of the model we also introduce some init ial aggregate variables (like, e.g., mo rtality rates) fro m Statistics Sweden.

The pension system is modelled according to the new Swedish pension system except that we have a fixed ret irement age at

and we do not have any funded part. We only model the pay -as-you-go component which in reality co mprises about 87 percent of the total public coverage .

All alternative scenarios are programmed in such a way that perfect replication of the base scenario is attained, up to a point when a change of arbitrary choice (like the change in child cost) is brought into play. This means that the initial random co mponents in the model are exactly identical in all scenarios. Thus the alternatives can be interpreted as counter-factual to the base scenario.

This is possible since we have no labor supply choice and people do not die of starvation when their income is confiscated. Also note that the taxes are redistributed. In practice the 100 percent tax rate should be interpreted as the collapse of the system.

Acknowledge ments

We greatly appreciate comments from two anonymous referees as well as Vegard Skirbekk and other participants in The International Conference on the Economic Consequences of Low Fertility in St Gallen April 2008. We are very grateful to Matias Eklöf, Gustav Öberg and Elisa Baroni who have programmed the simulation model IFSIM and with great patience implemented the experiments this paper builds on. Funding from the Swedish Research Council is gratefully acknowledged.


M Perreira
email: mperreir@univ-lr.fr

Da Silva

V Courboulay

A Prigent
email: aprigent@univ-lr.fr

P Estraillier

EVALUATION OF PREYS / PREDATORS SYSTEMS FOR VISUAL ATTENTION SIMULATION

Keywords: Visual attention, preys / predators, evaluation

 visual attention model. Sixteen persons have participated in a qualitative evaluation protocol on a database of 48 images. Six different methods were evaluated, including a random fixations generation model. A real time conspicuity maps generation algorithm is also described. Evaluations show that this algorithm allows fast maps generation while improving saliency maps accuracy. The results of this study reveal that preys / predators systems can help modelling visual attention. The relatively good performances of our centrally biased random model also show the importance of the central preference in attentional models.

Introduction

In [START_REF] Da Silva | Adaptativité et interactivité -vers un système de vision comportemental[END_REF], we have presented a behavioural vision architecture designed to be an elementary block for a virtual companion. In order to be interactive, this companion needs a real time vision system. A clever way to reduce allocated computer resource is to focus its attention on the most salient scene elements. Many visual attention computer models have been developed [START_REF] Tsotsos | Attending to visual motion[END_REF], [START_REF] Bruce | Saliency, attention, and visual search: An information theoretic approach[END_REF], [START_REF] Frintrop | VOCUS: A Visual Attention System for Object Detection and Goal-Directed Search[END_REF], [START_REF] Mancas | Computational Attention: Towards attentive computers[END_REF], [START_REF] Ouerhani | Visual Attention: From Bio-Inspired Modeling to Real-Time Implementation[END_REF], [START_REF] Le Meur | Attention sélective en visualisation d'images fixes et animées affichées sur écran : Modèles et évaluation de performances -Applications[END_REF] or [START_REF] Hamker | The emergence of attention by population-based inference and its role in distributed processing and cognitive control of vision[END_REF], nevertheless, they are often too complex for a real time execution. Besides, time evolution of the focus of attention is another weakness of many models, due to unsuitable decision systems.

In this article, we propose a new method which allows studying the temporal evolution of the visual focus of attention. We have modified the classical algorithm proposed by Itti in [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF], in which first part of his architecture relies on extraction on three conspicuity maps based on low level computation. These three conspicuity maps are representative of the three main human perceptual channels: colour, intensity and orientation. In our architecture, these low level computations are inspired from works pre-sented in [START_REF] Frintrop | VOCUS: A Visual Attention System for Object Detection and Goal-Directed Search[END_REF] [START_REF] Frintrop | A realtime visual attention system using integral images[END_REF], actually the way to accelerate computation, (i.e. the use of integral images [START_REF] Viola | Robust real-time face detection[END_REF]) is reused and extended to all maps. This real time computation is described in section 2.

The second part of Itti's architecture proposes a medium level system which allows merging conspicuity maps and then simulates a visual attention path on the observed scene. The focus is determined by a winner takes all and an inhibition of return algorithms. We propose to substitute this second part by a preys / predators system, in order to introduce a temporal parameter, which allows generating saccades, fixations and more realistic paths (figure 1).

Prey / predators equations are particularly well adapted for such a task. The main reasons are:

• prey predator systems are dynamic, they include intrinsically time evolution of their activities. Thus, visual attention focus, seen as a predator, could evolve dynamically;

• without any objective (top down information or pregnancy), choosing a method for conspicuity maps fusion is hard. A solution consists in developing a competition between conspicuity maps [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF].

and waiting for a natural balance in the preys / predators system, reflecting the competition between emergence and inhibition of elements that engage or not our attention;

• discrete dynamic systems can have a chaotic behaviour. Despite the fact that this property is not often interesting, it is an important one for us. Actually, it allows the emergence of original paths and exploration of visual scene, even in non salient areas, reflecting something like curiosity.

Finally, we present the results of experiments designed to validate the relevance of these different improvements. We have decided to compare different models, included those of [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF], our improvements, and two random models (with or without central bias).

In the following section, we present the approach we have used to generate efficient and real time conspicuity maps.

Real-time generation of conspicuity maps

Our solution is derived from work done in [START_REF] Frintrop | VOCUS: A Visual Attention System for Object Detection and Goal-Directed Search[END_REF] and [START_REF] Frintrop | A realtime visual attention system using integral images[END_REF]. The author uses integral images [START_REF] Viola | Robust real-time face detection[END_REF] in order to rapidly create conspicuity maps. Nevertheless, she explains that these optimisations are only applied to intensity and colour maps. Furthermore, she uses many integral images (one for each multi-resolution pyramid level of each conspicuity map), even if this approach is sub-optimal it was chosen in order not to change the original structure of their algorithm. Lastly, integral image were not used for computing the orientation map because results would have been less accurate than using Gabor filters, but also because is not trivial to compute oriented filters with angle different from 0 or 90 deg with integral images.

To reach optimal processing times, we have decided to use integral images for all the conspicuity maps. As a consequence, Gabor filters were replaced by simpler Haar like oriented band pass filter. Thus, for all levels of On and Off intensity channel, R/G and B/Y colour channels and 0 and 90 deg oriented filtered, we use integral images. For 45 deg and 135 deg maps, an oriented integral image is computed using [START_REF] Barczak | Toward an efficient implementation of a rotation invariant detector using haar-like features[END_REF] method.

All the information needed by multi-resolution analysis is finally processed from only four integral images.

The following results were obtained on a Compaq nc8430 computer with 2Go of memory and an Intel dual-core T2400 CPU 1.83 Ghz, using a C# implementation: Accordingly, it is possible to stay in real time for 320x240 images. Nevertheless, these results are difficult to compare with those of [START_REF] Frintrop | A realtime visual attention system using integral images[END_REF] mainly because:

• configurations are different (experiments and hardware);

• programming languages (C# vs C++);

• levels of resolution is more numerous in our system.

Regarding the last point, [START_REF] Frintrop | A realtime visual attention system using integral images[END_REF] as [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF] computes five levels, and only three of them are used, those of lower resolution. In our approach, we use all the resolution levels until a size of 8x8 pixels, ensuring that a maximum amount of information is taken into account.

By generalizing the use of integral images technique it is possible to compute more information in reasonable computation time. Experimental results presented in section 4 shows that out approach is more efficient since our global system performance is clearly better when we use our conspicuity maps.

A fast and simple retinal blur model

We have also studied the impact of the retinal blur model on our system behaviour. Actually, since scenes we see have a variable resolution (the closer from the fovea we are, the sharper scenes are), it could impact attentional models. Besides, we could benefit of this gradual lack of details to compute our model more quickly. Thus we do not compute a multi-resolution pyramid, but a multi-resolution column. For example if we define a size of 16x16 for representing the details of the finest resolution, we will generate a Nx16 pixels column (N being the number of resolution levels we need). Figure 3 allows to graphically comparing the two previous approaches (pyramid vs column). The following table presents computation times with this new method. Figure 4 presents its influence on conspicuity maps. Using the retinal blur model improves computation time by a 33% factor. However, benefit is not higher than expected due to more complex interpolations implied by the fusion of the different resolution images. Nevertheless, an improvement generated by this retinal blur is that the temporal focus of attention is modified. Our visual attention focus is more stable, so potentially more credible knowing the fixation time of human being (approximately 100 to 500 ms). 3 Real-time simulation of the temporal evolution of visual attention focus

This section presents a fusion method used to mix the conspicuity maps described above; interested reader may refer to [START_REF] Da Silva | Attention visuelle et systèmes proies / prédateurs[END_REF] for more details on preys / predators system. We focus the following sections on describing the details necessary to understand the results of the evaluation protocol used to compare different models of attention.

Preys / predators systems and visual attention analysis

Preys / predators systems are defined by a set of equations whose objective is to simulate the evolution and the interactions of some colonies of preys and predators. Interested reader can find more details about these systems in [START_REF] Murray | Mathematical Biology: I. An Introduction[END_REF]. For our system, we have based our work on [START_REF] Lesser | Mind as a dynamical system: Implications for autism[END_REF]) so as to represent the time evolution of interest (or focus of attention) linked to, in a first time (see section 6), a static image.

Traditionally, the evolution of preys / predators systems is governed by a small set of simple rules, inspired from Volterra-Lotka [START_REF] Murray | Mathematical Biology: I. An Introduction[END_REF] equations: 1. the growth rate of preys is proportional to their population C and to a growth factor b;

2. the growth rate of predators is proportional to their predation rate CI (rate a which preys and predators encounter) and to a predation factor s;

3. preys and predators spread using a classical diffusion rule, proportional to their Laplacian C and a diffusion factor f ; 4. the mortality rate of predators m I is proportional to their population I;

5. the mortality rate of preys mC is proportional to their population, plus an additional mortality proportional to the predation rate CI.

In order to simulate the time evolution of the focus of attention, we propose a preys / predators system (as described above) with the following features:

• the system is comprised of three types of preys and one type of predators;

• these three types of preys represent the spacial distribution of the curiosity generated by our three types of conspicuity maps (intensity, colour, orientation);

• the predators represent the interest generated by the consumption of curiosity (preys) associated to the different conspicuity maps;

• the global maximum of the predators maps (interest) represents the focus of attention at time t.

The equations described in subsection 3.2 were obtained by building a preys / predators system which integrates the above cited features.

Simulating the evolution of the attentional focus with a Preys / predators system

For each of the three conspicuity maps (colour, intensity and orientation), the preys population C evolution is governed by the following equation : and

d C n x,y dt = hC * n x,y + h f C * n x,y -m C C n x,
h = b(1 -g + gG)(a * R + (1 -a) * S)(1 -e)
C represents the curiosity generated by the image's intrinsic conspicuity. It is produced by a sum h of four factors:

• the image's conspicuity S is generated using Laurent Itti's Neuromorphic Visual Toolkit [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF], or our real time algorithm. Its contribution is inversely proportional to a;

• a source of random noise R simulates the high level of noise that can be measured when monitoring our brain activity [START_REF] Fox | Intrinsic fluctuations within cortical systems account for intertrial variability in human behavior[END_REF]. Its importance is proportional to a;

• a Gaussian map G which simulates the central bias generally observed during psycho-visual experiments [START_REF] Tatler | The central fixation bias in scene viewing: Selecting an optimal viewing position independently of motor biases and image feature distributions[END_REF]. The importance of this map is modulated by g;

• the entropy e of the conspicuity map (colour, intensity or orientation). This map is normalized between 0 and 1. C is modulated by 1e in order to favour maps with a small number of local minimums. Explained in terms of preys / predators system, we favour the growth of the most organized populations (grouped in a small number of sites).

Like in [START_REF] Lesser | Mind as a dynamical system: Implications for autism[END_REF], a quadratic term (modulated by w) has been added to the classical Volterra-Lotka equations. This term was added to simulate non-linearitiy (positive feedback) in our system. It enforces the system dynamics and facilitates the emergence of chaotic behaviours by speeding up saturation in some areas of the maps. Lastly, please note that curiosity C is consumed by interest I, and that the maximum of the interest map I at time t is the location of the focus of attention.

During the experiments presented in section 4, the following (empirically determined) parameters were used: a b g w m C m I s f 0.3 0.005 0.01 0.001 0.3 0.5 0.025 0.2 These parameters represent reasonable values that can be used to obtain a system at equilibrium. Other combinations are nevertheless possible. In particular, these values can be varied within a range of plus or minus ±50% without compromising the system's stability. Our system is thus quite robust to its parameters variation.

Experiments

We ran a series of experiments with a double objective. The first was to check that the simplifications we have introduced in order to generate the conspicuity maps in real time does not impact the predictive performances of our model. The second was to estimate the performances of our dynamical, preys / predators based, conspicuity maps fusion system.

We compared the six following visual attention models:

• Itti's reference model [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF] with default parameters (including the iterative normalization). We used the open-source NVT implementation of these algorithms provided by the iLab team. The tests were ran on still images. The conspicuity maps used were the intensity, colour and orientation ones.

• Our model based on preys / predators fusion applied on Itti's conspicuity maps. We used the model's default parameters except for the prey's birth rate which was dropped to 0.05 (instead of 0.1) because of the more contrasted maps generated by Itti's model. In order to avoid any loss of information, the maps were not normalized.

• Our model based on preys / predators fusion applied on our standard real time conspicuity maps.

• Our model based on preys / predators fusion applied on our retinal blurred real time conspicuity maps.

• A centrally biased random fixations generation model. Actually this model is our preys / predators system with a equal to 1 (which mean that the prey's birth rate is driven only by randomness).

• A random fixations generation model without central bias. This model is the same as cited above but with a null g term.

Image database

We collected a set of 48 images on the Flickr image sharing website. The images were published under the Attribution creative commons licence. We organized the database into 6 categories of 8 images:

• Abstract • Animals • City • Flowers • Landscapes • Portraits
This categorization, allowed us to study the link between the image categories and the performance of the different visual attention models.

Methods

Usually visual attention models performances are evaluated by comparing the saliency maps generated by these models with heat-maps processed from eye-tracking experiments. However, this evaluation method is complex and suffers from known biases.

One of them is the semantic bias. This bias is due to the fact that we need to display the images a few seconds in order to capture enough eye fixations to build a statistically valid heat-map. During that time our brain starts analysing the meaning of the image which affects the way we look at the image. Since that bias can't be avoided, we have decided to use an alternative evaluation method which does not require the use of an eye tracker. That way, anybody can reproduce our experiments. The subjects were asked to watch a set of 288 image couples. Each couple is composed of a reference image, randomly selected from the experiment database, and a visual attention map generated by one of the six algorithms evaluated. For each of these couples the subject had to rate the potential of the attention map to represent the different parts of the image that attracts the attention of most people.

The rate scale was 0 (very bad) to 3 (very good). No time limit was given, but the subjects were advised to spend two or three seconds on each image so that the experiment last no more than fifteen minutes. In order to let the subjects become familiar with the experiment without impacting or biasing the final results, we choose to remove the ten first results from the final analysis.

Participants

Sixteen participants (eleven men and five women) took part in the experiment. They were aged from 21 to 57 years old.

Results

Since the participant ratings were quite heterogeneous, we normalized each of their 288 marks so that they have a constant mean and standard deviation (with respective values 1.5 and 1). That way, we had a more consistent dataset. Figure 7 shows the mean results obtained by each of the algorithms over the whole database. We notice that our preys / predators fusion system globally improves the plausibility of the attention maps. One can also note that the use of our real time conspicuity maps, despite its numerous simplifications and approximations, also improves the overall predictive performances of the attention models. However, our (very simple) retina simulation algorithm does not show and significant performance improvement. The overall good performance of the central biased random model is also quite surprising, but actually these results are very dependent of the image category (see figure 8). Lastly and unsurprisingly, the full random model obtains the worst performances. The results of the different algorithms for the six categories of our experiments database (figure 8) are quite variable:

• The centrally biased random model obtains the best performances on the portraits and abstract categories. This surprising phenomenon is partly due to the photographer bias. Indeed, any photographer tends to centre his subject in the image. This model also performs quite well on the abstract category. It seems that defining the area s of interest for this category is a very challenging task; as a consequence, participants have accepted the central choice as a least bad solution.

• Images from the abstract category are very hard to rate. As a consequence, all models seem to perform equally. For this kind of images, random model are as good as other more sophisticated models. • Fusing the conspicuity maps with a preys / predators dynamical system significantly improves the performances of Itti's model for all categories, except for landscapes were Itti's algorithm already performs well. In this case, the performances seem superior to Itti's model, but we can't assert this, since the 95% confidence interval for these data is quite high.

Discussions

This article shows that preys / predators based conspicuity maps fusion can be used to improve the plausibility of visual attention models. Moreover, the results of the experiments performed by 16 participants on a database of 48 images show the importance of taking into account the central bias when building new visual attention models. Finally, we have demonstrated the good performances our method for real time generation of conspicuity maps (with or without simple retinal blur). The results presented in this article are promising, but they need to be confirmed by eye tracking experimentation. Although eye tracking is far from being perfect for visual attention analysis (see [START_REF] Da Silva | Attention visuelle et systèmes proies / prédateurs[END_REF]), it allows collecting the temporal evolution of eye positions, which cannot be obtained by another mean. Our future researches will focus on this kind of experimentation, so as to validate the dynamical behaviour of our model.

Figure 1 :

 1 Figure 1: Architecture of our visual attention model. This diagram has been adapted from[START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF].

Figure 2 :

 2 Figure 2: Sample real-time conspicuity maps : (a) Original image; (b) Intensity conspicuity map; (c) colour conspicuity map; (c) Orientation conspicuity map.

Figure 3 :

 3 Figure 3: Left: a classical multi-resolutions pyramid. Right: Multi-resolutions column with a width and height of 4 pixels.This kind of structure allows keeping only a part of the initial information of the finest resolutions.

Figure 4 :

 4 Figure 4: Sample real time conspicuity maps with retinal blur simulation: (a) Original image. The green circle represents the focus of attention; (b) Intensity conspicuity map; (c) colour conspicuity map; (c) Orientation conspicuity map.

Figure 6 :

 6 Figure 6: Sample images from each of the six categories of the experiment database

Figure 7 :

 7 Figure 7: Global results of the experiment for each evaluated attention model. Black bars represent the 95% confidence interval

Figure 5 :Figure 8 :

 58 Figure 5: Sample image from the landscape category of the test database and associated results for different algorithms