European Schoolnet

Dr Patrick Felicia
email: pfelicia@wit.ie

Caroline Kearney

Paul Gerhard

Alexa Joyce

Marie Le

Boniec Design

Nathalie Scheeck

José Luis

Diez Lerma

Sara Crimi

Hofi Studio

Derek Robertson

Ella Myhring

Farm Frenzy), IPRASE Alawar Games

1 Introduction

The purpose of this handbook

This handbook has been written in the framework of European Schoolnet's Games in Schools project which began in January 2008 and ended in June 2009. The project's aim was to analyse the current situation in eight countries (Austria, Denmark, France, Italy, Lithuania, the Netherlands, Spain and UK) with regard to games based learning. It is composed of various elements, including a synthesis and final report based on the results of a Europe-wide teachers' survey, case studies and interviews with relevant national policy makers, researchers and experts, as well as an online community of practice and finally the present teachers' handbook. The Games in Schools blog provides more information regarding European Schoolnet's project, and also provides links to the study's reports (http://games.eun.org). This handbook is intended for teachers interested in using digital games in their lessons. It provides the necessary information to understand the educational benefits of digital games and to learn how to use them as educational and motivational resources. After reading this handbook, you should be able to make informed decisions on the choice and use of digital games in the classroom, and obtain all the benefits digital games can offer. This handbook is a practical guide aimed at providing you with theoretical and practical information. It is an introduction to the educational use of digital games and it provides references to useful resources such as articles, websites and books where you can find additional information.

Learning objectives

After reading this handbook, you should be able to:

• Understand the benefits of game-based learning. • Be aware of successful use of digital games as a learning resource. • Know digital games that you can use as a learning resource in the classroom and their expected learning benefits. • Understand the differences between digital game genres and their learning benefits. • Understand the requirements for the successful use of digital games for learning. • Understand how to assess if a digital game is suitable for the classroom. • Understand digital game ratings and standards.

• Know how to organize the classroom for a play session. • Understand how to promote and apply healthy and safe gaming habits. • Understand how to run a debriefing session after playing the game. • Understand how to maximize the knowledge transfer so that information acquired in the game can be re-used at a later stage by pupils. • Understand how to use digital games as a starting point to discuss sensitive issues.

Why use digital games for learning?

2 Why use digital games for learning?

Change in the perception of digital games

Until recently, digital games were associated with many stereotypes and alleged to have negative effects on gamers' physical and mental health. However, later studies have shown that while digital games, played beyond a reasonable length of time, may have negative effects like any other excess, if good gaming habits are followed (eg. appropriate time, environment, moderation of online games, etc.) they can be considered a safe and fulfilling activity. The recent success of the Nintendo Wii Fit and Nintendo DS (e.g. Brain Training) illustrates some of the numerous ways in which digital games can impact positively on children's health and general well-being.

Lately, a new movement identified as Serious Games has emerged. This movement aims to use new gaming technologies for educational or training purposes. It investigates the educational, therapeutic and social impact of digital games built with or without learning outcomes in mind. This movement has emerged to meet the needs of a new generation of learners, often referred to as the digital natives, whose distinctive characteristics should be acknowledged in order to ensure successful learning outcomes and motivation on their part. This generation of learners was born after the 1970s and has been familiarized with digital technology from a very young age. They use digital devices frequently and ICT (Information and Communications Technology) has almost become their native language, a language through which they communicate, express themselves and understand the world around them. Digital natives also play digital games extensively and are passionately engaged in social networks, sometimes in the form of virtual worlds (Second Life, There, etc.). They usually engage in activities that reward them for their perseverance, and as a result they expect the same level of reward from learning activities. On the other hand, some instructors and teachers have found it difficult to engage and motivate this generation to take part in traditional learning activities, perhaps because the format used for formal teaching has failed to match pupils' needs, preferences and expectations.

Digital natives' characteristics have been acknowledged by some educators who have understood that digital games could play an important part in education or training. These instructors have been using immersive environments and gaming technology in order to reach their students.

The benefits of digital games

Digital games include several implicit educational benefits. They can develop cognitive, spatial and motor skills and help improve ICT skills. They can be used to teach facts (e.g. knowledge, recall, rote learning or memorization), principles (e.g. cause and effect relationship) and complex problem solving, to increase creativity or to provide practical examples of concepts and rules that would be difficult to illustrate in the real world. They can be particularly useful to perform experiments that could be dangerous in real life such as the use of hazardous chemicals. Despite their instructive features, not all digital games are built with learning objectives in mind; however, they all possess intrinsic learning qualities that challenge and foster learners' Playing is learning and one of the main advantages of digital games is their ability to let players learn in a challenging environment, where they can make mistakes and learn by doing. These types of environments might be particularly suited to pupils who are pragmatically minded. These pupils might prefer to go through the process of experimenting, instead of regurgitating information. Such an experience could help them to better understand some concepts that they might otherwise perceive as complicated or boring. frustration or triumph. This succession of emotions tends to keep the players immersed. Furthermore, studies have shown that emotions can help the memorization process, especially if the emotional content or tone of the material to be learned matches the emotions of the learner. Because they induce different emotions in the players, digital games might help them to remember facts more vividly and so assist the cognitive process. Digital games can increase players' self-confidence. This is achieved by keeping success within their reach (i.e. winnability), making sure that the learning curve is suitable to their skills and providing feedback on their progress.

From CBT (Computer-Based Training) to digital games for learning

Since the emergence of the first CBT system, computer technologies have evolved significantly, and so have learning theories. The design of educational systems has been greatly influenced by trends in educational psychology and instructional design. For instance, when the first CBT systems were released, they were based on a number of questions and predefined answers allowing for little interaction with the user. Subsequent systems have included more flexibility by incorporating tutoring systems. Tutoring systems added the ability to monitor learners' progress and to adapt pedagogical strategies dynamically. Although these systems were effective, their development costs were sometimes perceived as prohibitive, partly because they were based on advanced Artificial Intelligence (AI). Later on, the use of simulation matched the need for learners to experiment with their mistakes, through a constructivist approach to learning, an approach that prioritizes learning by doing. Recent learning environments are based on virtual reality and built with collaboration in mind. They allow participants to learn from their experience but also to learn from their peers. Recent popular digital games such as MMORPGs or virtual communities (e.g. There III , Second Life) inherently include these aspects. Because collaborative learning occurs naturally in these environments, they have been considered to support traditional teaching methodologies.

Digital games and the cognitive process

Educational theories and Instructional Design can help to create learning material and to ensure that the learning objectives are achieved by the learners. Such theories have been used to design school curricula and training programmes. Amongst existing learning theories, different approaches can be applied to ensure successful learning outcomes. Most educational theories fall into one of the following categories: cognitivist, behaviourist and constructivist. In behaviourist III There is an online virtual environment where the user can take part in social activities. It is available at: http://www.there.com 2 approaches, subjects are not directly responsible for their learning activities; instead, they are conditioned to react to a stimulus. In cognitivist theories, subjects possess an internal map (knowledge), which external events will require them to update. In these theories the emphasis is on the underlying cognitive process. Several wellknown theories have been established under the cognitivist movement, such as transfer effect, whereby learning can be affected by previous knowledge. Finally, in constructivist theories, subjects learn by interacting with their environment and peers. This involves a process of trial and error and the learners' ability to interpret their past and present experiences to update their knowledge.

Because they were initially designed for entertainment, digital games are not all built based on Instructional Design theories. However, some of them intrinsically implement some well-known pedagogical concepts. For example, digital games usually include high intensity of interaction, specific goals, a continuous feeling of challenge, and a sense of engagement; these concepts have been associated with successful learning environments by Norman (1993 1 ). To some extent, digital games feature all approaches -behaviourist, cognitivist and constructivist. However, whereas early educational software placed an emphasis on the first two theories, recent digital games, due to their complexity and their open-ended and collaborative nature, encourage a constructivist approach to learning. In digital games, players can elaborate new theories and hypotheses, test them and readjust their knowledge and skills accordingly. New digital games, which include a 3D environment, advanced AI and realistic physics engines, offer a simulation-like environment that reacts to players' actions in a highly realistic manner. and back to active experiments. To some extent, the cycle of learning events experienced in digital games can be compared to Kolb's learning cycle: the players experience something discordant or a failure (failure to win); they subsequently need to reflect and identify the cause of the failure. Following this analysis, they formulate hypotheses on the cause(s) of the failure, plan actions that might help them to overcome the problem and then test and assess their hypotheses. Likewise, according to Vygotsky's Zone of Proximal Development (ZPD), learners should be assisted with scaffoldings and be progressively made more and more independent. As they improve their skills, less and less help is provided. Learners' independence and meta-cognitive skills are improved gradually. This principle can also be found in digital games, which offer an easy learning curve and are usually forgiving in the first levels, so that players can progressively familiarize themselves with the game's mechanics and become proficient. However, players need to learn new skills to succeed, and, to some extent, to take responsibility for learning. This ability of digital games to engage children and motivate them to learn in order to succeed is by far the most interesting feature that could help instructors to make learning a more engaging and motivating activity.

Digital games and motivation

One of the foremost qualities of digital games is their capacity to motivate, to engage and to immerse players. Digital games include a rich variety of auditory, tactile, visual and intellectual stimuli that make them both enjoyable and, to some extent, addictive. During play, users are immersed in a state of flow (Csíkszentmihályi, 1990 6 ), a state in which they can forget about their surroundings and become totally engaged and focused on the task in hand. In this state, provided that they possess sufficient skills, players will strive to achieve their goal, regardless of the challenges

Why use digital games for learning?

encountered. Players' motivation can be triggered or hindered by many different factors such as the game play, the graphics, the interface or the game genre. Players' behaviours in digital games depend on their personality and aspirations. Therefore, the relevance of the reward offered in the game might differ across players. Whereas some players value exploration, other might prefer highly complex scenarios, where more strategic skills are required; some of them will enjoy very simple games with a linear scenario that necessitates a short time of play to succeed. Cultural background and gender might also influence the motivation to play games.

Successful applications of digital games

Digital games have already been used for training, educational or therapeutic purposes. Some of the most common uses of serious games are listed below.

Military and firefighter training

In the military sector, MMORPGs are used not only to recruit soldiers but also to teach them strategic skills. For example, America's Army IV has been used to recruit soldiers and as a training resource by the American government. It is now available as a commercial digital game, which has been successful.

Fitness, mental and physical health

The technology available for 3D-based digital games has made it possible to create highly realistic environments and simulations. This remarkable level of detail has been used to cure post-traumatic stress or phobias (Emmelkamp et al., 2001 7 ). When they are immersed in a safe but realistic environment, patients can learn to cope with their fears while feeling in control. Digital games have also been used to relax patients before surgery so as to reduce their apprehension V . Nintendo Wii and Wii Fit have recently been employed and assessed for their potential to improve health and fitness and it was shown that, when used as part of a balanced lifestyle, these consoles can help to improve fitness and general health (ISSA, 2007 8 ). Digital games have also been employed successfully for adults with chronic illnesses.

Learning by creating digital games

Experiments have been conducted in which students were required to create their own digital games using a game engine. These initiatives made it possible for students to understand programming principles, while collaborating with their peers. For example, the game engine Game Maker is used in several schools. It consists of 2

Why use digital games for learning?

intuitive drag and drop tools to create 2D and 3D digital games which can then be uploaded onto the publisher's website VI and made available for download. This type of development tool benefits technical and collaborative skills and the creation of online communities.

Mobile development and augmented reality

Using augmented reality, it is possible to obtain relevant information on one's surroundings through a digital device (e.g. helmet or mobile phone). Due to their popularity, handheld devices have been combined with augmented reality in projects where mobility and location were significant factors for learning activities. For example, in Savannah VII , a strategy-based adventure game, children use their mobile phones, and have to successfully adopt strategies followed by lions to survive VIII . Similarly, in the Museum Augmented Reality Quest (MARK), players are equipped with a handheld game console. They visit a museum and are asked to complete tasks and challenges. The handheld console is used to provide them with additional information on artifacts found in the museum and to interact virtually with them (Schmalstieg and Dieter, 2007 9 ).

Raising awareness

Digital games have been adopted to raise awareness and to help discuss topics that can be taboo, such as pollution, environmental threats, sexual health or bullying. For example, in the recently developed digital game Global Conflicts: Latin America IX , the player impersonates a journalist who conducts an investigation to uncover the causes and consequences of industrial pollution in South America. Likewise, in the digital game Darfur is Dying X , players impersonate refugees in Darfur camps. Through their journey, players are led to understand the horror of genocide.

Learning history

Digital games have been used in primary and secondary schools to illustrate concepts in science, history or geography. For example, the digital game Civilization III was used in American schools to teach history (Squire, 2004 10 ). Similar experiments were carried out in Danish schools (Egenfeldt-Nielsen, 2007 11 ) using Europa Universalis II.

Choosing the appropriate game

Choice of games 3

Choosing the appropriate game

Digital games can be delivered in various formats including CD-ROMS, DVDs, or online. The following sections will help you as a teacher to understand the different game genres, their key features and educational benefits.

Taxonomy of digital games and associated benefits

Before choosing a game, it is important to understand the different genres available and how to identify them. The following list classifies and describes different game genres. Their potential learning applications and benefits are underlined.

• Shooters (and shoot 'em ups): In these games, players have to win by shooting their opponents. Shooters can be static or scrolling.

In static shooters, the battlefield is restricted to the size of the screen, whereas in scrolling shooters, only part of the battlefield is displayed, and the screen scrolls horizontally or vertically. Beat 'em ups also belong to this category. In these games the players need to fight opponents. Shooter games place emphasis on reflexes and coordination, rather than strategy. Recent digital games based on this genre, such as Re-Mission XI or Immune Attack XII , have been used to introduce basic concepts of immunology or cancer treatment.

• Bat and ball games: In these games, players use a bat to hit a ball. One of the first digital games based on this principle was Pong XIII ; since then, many variations have flourished, including Breakout, a game in which players have to hit a ball that travels across the screen and strikes and destroys bricks. Some educational games have been based on this genre, including 10 Finger BreakOut XIV , a game that teaches typing skills.

• Platformers: In these games, players move through an environment where they need to progress to platforms (hence the name Before choosing a game for the classroom, you should check the specifications of the computer room and try to answer the following questions:

• Which operating software is required for the game? • How much RAM (Random Access Memory) is recommended for the game to function properly? • How much hard drive space is needed to install the game? • Does the game need to be played over a network or an Internet connection? • What type of input device is needed to interact with the game (joystick, keyboard or mouse)?

Answering all these questions will help you to assess if the digital game you wish to use is suitable for the school's computers. Not taking these features into consideration at an early stage could result in a frustrating experience for the pupils (e.g. lack of responsiveness or inability to go further in the game). The type of input used in the game should be assessed based on pupils' age and skills. For instance, answering questions by selecting the correct option with the mouse might be more suitable than typing the solution. The latter option requires a higher level of proficiency in spelling. Pupils' home machine specifications should also be considered. Enabling pupils to play the game at home in their spare time will help them to become more familiar with the interface and improve their skills. In this case, you will need to make sure that the technology required for the digital game is available in most computers used by pupils at home. Choosing the appropriate game

Understanding digital game rating and standards

Beyond practical considerations, you should determine if the content of the digital game is suitable for the pupils in terms of age and content. This choice can be guided by existing rating standards. For example, PEGI XXVII (Pan European Game Information) is a European digital game rating system. It is supported by most game publishers and developers in Europe. This rating system helps to ensure that the content of a game is suitable for the target audience. It is a voluntary system used in 32 European countries, but is only enforced in two (Finland and Norway).

PEGI consists of two levels of information to guide the consumer. First, a logo that represents the minimum recommended age (the age categories are: 3, 7, 12, 16 and 18).

A series of game content descriptors, designed as icons, are displayed on the back of the game box and indicate, where required, the nature of the content.

There are 8 icons, depending on the type of content.

More information on PEGI is available on the official website XXVIII . This website makes it possible to find a game XXIX according to the country in which it is published, the age rating, the game genre or the publisher. Gambling: game encourages or teaches gambling.

Online: game can be played online

What to look for in a digital game: Testing the game

Once you have found a suitable game to support your lessons, you should play-test the game to check if the content is appropriate for the pupils and suitable for the topic taught. Also, be sure to complete all training tutorials and read relevant material beforehand. The following points should be observed:

Technical considerations:

User interface: The interface should be clear, intuitive and easy to use. Some children might stop enjoying the digital game because they find it 3

Choosing the appropriate game difficult to perform common tasks such as navigating through the menus or moving the main character.

Saving and loading the game:

The game should enable players to save the current level and to resume the game at a later stage. This will be particularly useful if the computer rooms are only available for short periods of time.

Audio: If the game features sounds, a mute button should be available.

Customization: You should check if it is possible to customize the game (e.g. character, colour, scenario or level of difficulty). Customization makes the gaming experience more personal and keeps players challenged (e.g. by level of difficulty).

Contextual considerations:

Age group: The activities and type of skills required for the game should be suitable for the targeted age group. Language: The level of the language used should be adequate for the age group. Time: You need to estimate the time required to complete challenges to ensure that pupils will have enough time to finish the game levels and benefit from their educational features. Completion time will vary across games. Whereas mini-games available on the Internet are usually designed to be completed relatively quickly, adventure games or RPGs might require several hours or days to be completed. For the latter, it is good practice to plan the playing session over a week or more, so that pupils can learn at their own pace and become familiar with the mechanics of the game.

In addition, games can also be used after school as homework.

Taking account of people with disabilities:

You should check if the digital game accommodates people with disabilities (see next chapter for more information on digital games designed for this audience).

Network games: If the game is played over the Internet and involves interaction between participants (e.g. text or chat), care should be taken to prevent bullying and to use mechanisms to detect and condemn such behaviours. It is very important for children to feel safe while playing digital games, and to know that they can report unacceptable behaviour if it is encountered.

Pedagogical considerations:

Learning curve: The game should use an easy learning curve, allowing players to make mistakes at the start.

Educational content:

The game content should illustrate the topics taught. Even if the content is not closely related to the curriculum, it should provide a clear and simplified representation of some of the concepts taught. Clear objectives: Teacher should ensure that the objectives of the game are stated clearly, so that children know exactly what they are required to do. Frustrating situations can arise from vague instructions. In this case, children might feel stuck, because they don't know how to progress further in the game.

Choosing the appropriate game

Clear progression: The teacher should check whether the progression of the player is displayed at all times in the form of a score or progression bar. This will help children to have a positive attitude to their performance, and show them that their actions have an influence on their progression. This should motivate players to take responsibility for their learning activities.

Feedback: Feedback provided to players should be gentle. Verbal guidance or hints can help to maintain their focus.

Opportunities for collaboration and group work:

It is good to use digital games that enable players to take part in collaborative activities.

Assessment and follow-up: Software that tracks pupils' progress enables you to analyse areas that are misunderstood and where more work is required. Not all software includes this feature, but some of it is SCORM-compliant (Sharable Content Object Reference Model) and can be integrated into a Learning Management System (LMS). A Learning Management System will make it possible for you to follow the progress of your pupils and to identify areas where more attention and work is required.

Opportunities for creativity: You should check whether the material included in the digital game can encourage pupils' creativity by enabling them to make and share objects. Help: A Help section should be available and comprehensive. If possible, you should print it and have it ready for pupils before they start playing the digital game.

Taking account of children with disabilities

Games for visually impaired people

Some recent games accommodate visual impairment, and can be mainstream digital games and audio games. XXX AudioGames.net XXXI and Blindstick.com XXXII provide information on audio games and blind-accessible games. XXXIII Educational digital games suitable for people with impaired vision include Terraformers, XXXIV Talking Typing Teacher XXXV (typing skills), Braille Twister XXXVI (learn Braille) and Quality Quizz XXXVII (trivia game). When using mainstream digital games, you need to check whether buttons can be read aloud and if other information can be delivered through audio cues. To help reading onscreen information in the digital game, screen readers and screen magnifiers can be used.

Games for people with an auditory disability

To

Games for people with a learning disability

Although players with a mild disability will be able to play most mainstream digital games, some of these games might not be suited to specific disabilities. For example, people with dyscalculia XXXIX might find it problematic to play games that involve basic mathematics (e.g. counting). Likewise, people with dyslexia have difficulties playing digital games where the information is delivered essentially through text. People with ADD/ADHD XL might find it difficult to play games with no immediate feedback or games that require prolonged periods of play. You should therefore test the game and identify issues that might arise depending on pupils' learning disabilities. Some digital games have been specifically designed to help particular learning disabilities, and might be used in the classroom. For example, Brigadoon XLI is a digital game based on Second Life. It is designed for people with autism and Asperger syndrome to help them to socialize and to interact with other people.

Games for people with a physical disability

For people with physical disabilities, digital games can be made accessible by choosing a dedicated controller and also thanks to mechanisms provided in the game itself or by the operating system. For example, slow-motion gameplay enables players with a physical disability to go through part of the game where the action is intense. It is important for players to be able to customize the controls and to use external controllers. One-switch digital games are suitable for players with severe physical disability because they only require one or two buttons XLII . Children should be encouraged to exercise regularly in addition to playing digital games, except of course if these digital games include exercising. In addition, it is important to communicate with them and take the time to listen to their success in the adventure they have undertaken. They should also be made aware of potential issues and risks such as bullying and be informed of the resources they can contact to complain.

Finally, information should be provided to your teacher colleagues, parents and course-board members on the pedagogical process, the preparation and the learning objectives of gamebased-learning. This information should be given before and while using the game.

Evaluating and strengthening pupils' knowledge through a debriefing session

General considerations to conduct a debriefing session

Digital games can promote many different cognitive skills; however, some of the aptitudes needed to make a connection between the game and the curriculum, such as reflection, observation, prediction or theory-building, are not always built into the digital game. It is therefore necessary to include a debriefing session through which pupils will have an opportunity to reflect on the content of the game, and to share the knowledge that they have acquired. The debriefing session can also be used to ask children to explain how they feel about the digital game and describe their achievements or frustrations. Debriefing sessions are a way to make a link between the game and the learning outcomes. The strategy for the debriefing session is summarized below:

• Review learning objectives.

• Ask pupils to answer questions stated at the start of the session. • Ask pupils to relate to their experience. • Initiate discussions.

• Make the link between the game experience and the learning objectives. • Summarize findings.

• During formal classes, refer to findings made while playing the game.

General understanding of the game

After playing a digital game, pupils should have a good understanding of its goal and objectives. They need to understand the purpose and challenges raised by the game. Asking pupils the following questions will help you to assess their general understanding of the game:

• Where and when does the scene take place?

• What is the goal of the game?

General understanding of the issues raised by the game

When you have ensured that pupils have understood the general goals of the game, it is time to check that they understand the underlying ideas or topics introduced by the game so that a link can be established between the game and the topic taught. For this purpose, you can ask the following questions:

• What is the main topic of the game?

• Do the events that take place in the game remind you of something you know, or something you have heard of from your friends, family or on TV? • Why do you think this topic is important? • What did you learn from this game? 32 4

Linking game concepts to real life

While pupils might understand the game's goals, it is particularly important that they understand key principles in the digital game that mirror real life concepts. You might especially ask them to explain key actions in the game and their consequences. For example, if you have chosen to use the game Global Conflicts: Latin America to explain industrial pollution in South America, you could ask the following questions:

• What are the causes of pollution in South America? • What are the effects of industrial pollution on people living near maquiladoras? • What would happen if all maquiladoras were closed-down?

Factual information

Before starting the game, you should give a list of factual information that pupils need to gather while playing. After the game, they will need to show that they have remembered important locations, names or dates from the game.

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Conducting a play session 5. Frequently Asked Questions

5

Frequently Asked Questions

Are digital games bad for pupils' health?

If children are taught healthy and safe habits and if a balance is maintained between play and other activities, digital games can be a fulfilling activity with little or no risk.

How can I make sure that the digital game I have chosen is appropriate for my pupils?

You need to check the game rating and to play the game yourself to assess whether it is appropriate in terms of language, violence and educational content.

How can I make sure children will learn from the game?

Not all digital games are built with educational objectives in mind. However, some can prove successful in teaching or in introducing basic principles. The most important part is the debriefing session, because it enables you to make a link between the game content and the learning objectives.

Our machines are very old and can't run games supplied on CD-ROMS. What are our options?

If your machines' specifications are inadequate for games supplied on a CD-ROM, you might try Java or Flash-based games. These games (especially those using Flash) are lightweight and not so processor-intensive.

We have a very low budget for digital games. How can we obtain good-value games for learning?

Whereas COTS (Commercial-Off-The-Shelf) games can be expensive, many other digital games with educational benefits can be used for free. Alternatively, digital games can be rented.

How can I know whether children will enjoy the game?

It is very difficult to know exactly what game children will or won't like. However, a good starting point is to read the reviews provided in online forums and magazines.

Frequently Asked Questions

Can digital games replace traditional classes?

No. Digital games cannot replace traditional classes. However, they can considerably help to involve and to motivate children. Some children might find some topics very difficult to understand until they experience them through play, and digital games can provide them with this opportunity. In this regard, digital games do not replace traditional classes but instead provide an additional way to reach children and enable them to understand topics that they might otherwise find too complicated.

Is there any evidence that digital games can help learning?

Yes. Several reports have shown that digital games can be used effectively in schools to improve both learning and motivation. The resource section lists some of these reports.

Resources for teachers

2

  Why use digital games for learning?In term of cognition and learning process, digital games can be analysed through well-known models such as Carroll's minimalist theory (1990 2 , 1998 3 ), Vygotsky's Zone of Proximal Development (ZPD)(1978 4 ) or Kolb's basic learning model(1975 5 ). For example, the basic learning model illustrates the process of accretation whereby students modify their internal map (or knowledge) based on the information and feedback obtained from their previous actions. They successively go through active experiments, concrete experiences, reflective observations, abstract conceptualizations 2 Why use digital games for learning?

  XXVhttp://ww.adobe.com/flash/ XXVI http://www.java.sun.com/ 3

  XXVIII http://www.pegi.info/en/index/id/33/ XXIX http://www.pegi.info/en/index/global_id/505/ XXVII http://www.pegi.info Violence: game contains depictions of violence Bad Language: game contains bad language Fear: game might be frightening or scary for young children Sex: game depicts nudity and/or sexual behaviour or sexual references Drugs: game refers to or depicts the use of drugs (including alcohol and tabaco) Discrimination: game contains depiction of or material which may encourage discrimination.

  Teachers and parents need to regulate the amount of time during which children play. Children might be asked to set the length of the play session themselves. Some children may find it hard to stop playing but this could be due to the fact that they are trying to avoid boredom. It is good practice to encourage children to play different types of game genres to challenge them in different ways. For example, children might alternate between action games, strategy games and puzzles.

  During a debriefing session, you should encourage pupils to relate to their experience and to discuss what they have learned while playing. You might ask students to compare the different methods or techniques presented in the digital

4

  game, or to describe the different solutions they have tried, and identify the ones that have worked for them. If the game is used to discuss sensitive issues, the characters of the story can be used as the starting point for discussion. Pupils might be asked to comment on the behaviour of some characters, the consequences of their acts and possible actions to solve the issues. After running the debriefing session, you should summarize the different points brought forward in the discussion 4Conducting a play sessionConducting a play sessionand the learning objectives of the session. You should also ask pupils to summarize what they have learned from the digital game. This will help to formalize and memorize the concepts and ideas introduced. The issues raised during the debriefing session can then be referred to during formal classes.

2 Why use digital games for learning? Why use digital games for learning?

	Digital games promote genuine collaboration
	between users, and are, to some extent, similar to
	collaborative learning environments or
	collaborative working environments, where
	participants share information and learn from each
	other (e.g. Computer Supported Collaborative
	Working environments). Multiplayer digital games
	develop both competition and collaboration, by
	motivating players to join teams (or guilds) and to
	compete against other teams. For example in
	MMORPGs (Massive Multiplayer Online Role-
	Playing Games), players can create teams, share
	information through text and voice and learn by
	observing other players. Beginners can learn from
	their peers and improve their skills.
	Digital games can have an emotional impact on
	players, increasing their self-esteem (provided that
	proper monitoring is applied) and enabling them to
	engage in social activities. Playing can have
	a calming effect on participants who, as in other
	entertaining activities (e.g. watching films), feel
	a wide range of emotions, but in a safe and
	controlled environment. Emotions experienced
	during play can vary through joy, empathy, anger,

the appropriate game 3.2 Understanding digital games' technical requirements

  In these games, players have to solve a puzzle to progress further in the game. The gameplay usually occurs within a static screen. Tetris is one of the most popular puzzle games. Puzzle games are essentially based on strategy. In these games, players have to navigate through a maze and are chased by enemies that they have to avoid. These games usually offer a view from above and require strategy, forward planning skills and reflexes. Pacman is one of the most successful Maze games. An educational version of Pacman called PacWriter XVII has been created to improve typing skills.

		Squire (2003 13 ), it was shown how Civilization III, • MMORPG (Massive Multiple Online
		a popular RTS game, could be used in schools Role-Playing Games):
		to help pupils understand geography and MMORPGs are a variation of RPGs, in which
	It is difficult for teachers to find software that is	history. a large number of players interact in an online
	suited to their computer lab and to pupils' home	virtual world. These digital games are
	• Sport Games: These games simulate popular sports such as football, golf or basketball. They are available in 2D or 3D. They require coordination and strategy, especially if the player has to manage a team. • Racing Games: In these games, players take part in a race, driving a car, a motorbike or a spaceship. Racing Academy XVIII is based on this genre and is used to increase learners' familiarity with engineering concepts. Research has shown that this digital game has helped pupils to understand engineering notions computers. High computer technical specifications, such as processor speed, graphic card or screen resolution, can sometimes prevent the use of digital games in schools where computers are not up-to-date. Whereas commercial off-the-shelf digital games may require high technical specifications, those available on the Internet in the form of Flash XXV or Java XXVI Applets may impose fewer constraints, other than the installation of plug-ins (which are often installed by default on the computer). Digital games available on the Internet can be played within the browser window. They usually offer a very intuitive interface and require less processing power. They are generally available in the form of mini-games, with challenges that can be solved quickly. They also use relatively simple (Sandford and Williamson, 2004 12 ). interaction techniques, and are therefore suitable	• RPG (Role Playing Games): In these games, players impersonate a fictional character. This character has several features that can evolve through play, such as health, strength or other skills. For example, SimCity XIX is a popular RPG game that makes it possible for players to build cities and to understand urban management. • FPS (First Person Shooters): In these digital games, players see the world through the eyes of the character they are impersonating (first person view) and need to eliminate enemies in order to progress further. These games can be played individually or collectively. They generally include violent content. However, if the game is played within a team, these games can promote a valuable basis for collaborative and exploratory activities. • Educational versions of existing board games or TV game shows have also been released. For example, Stu's Double Jeopardy XXI is based on the well-known TV show Jeopardy and can be used in classrooms to teach any topic in an interactive manner. Players navigate through a complex world, collecting objects and overcoming challenges until they reach the final goal. This game genre, initially based on text, has evolved to include 2D (e.g. King's Quest XXII ) and 3D graphics (e.g. collaboration. Mods of these games have been EverQuest XXIII ).	Choosing the appropriate game
	for people with little or no experience of digital	created for educational purposes. For example,
	• RTS (Real Time Strategy): These games games.	DimensionM XX is an FPS developed to help and The following table includes possible
	are essentially based on strategy. The players	encourage the learning of mathematics. educational uses of existing commercial digital
	control both the economic and military aspects	It has been shown that this game had games.
	of an army or population and have to take fast strategic decisions. In a study by Jenkins and	Educational versions of puzzle games have been developed to teach mathematics (e.g. PrimeTime Adventure or Rocky's Boots XVI ). a significant positive impact on learners' academic performance and motivation (Kebritchi et al., 2008).	3

). Mario was one of the most popular platformer games of its generation. Platformers are usually based on hand-eye coordination, and educational versions of these games have been developed to teach geography (e.g. Mario is Missing XV ), reading (e.g. Mario's Early Years: Fun with Letters) or typing skills (e.g. Mario Teaches Typing).

• Puzzles:

• Mazes: • Adventure games: In these digital games, the gameplay is based on story telling. 18 3 Choosing the appropriate game XXI http://jeopardygame.wordpress.com/ XXII http://www.vintage-sierra.com/kingsquest.php XXIII http://everquest.station.sony.com/ Choosing

Play session Conducting a play session 4.1.2 Promoting good, safe and healthy gaming habits

	Like any other form of activity, digital games
	should be used wisely and appropriately.
	Therefore, you as teachers, and also parents and
	pupils, should be informed of good gaming habits.
	Following these guidelines should help to ensure
	that digital games do not negatively affect
	children's physical and mental health.
	4. Conducting
	a play session

XXXVIII These games also have an educational benefit. XXXIX People who have difficulties in handling everyday maths functions. XL Attention Deficit [Hyperactivity] Disorder. XLI http://braintalk.blogs.com/brigadoon/2005/01/about_brigadoon.html XLII Several one-switch digital games can be found on http://www.oneswitch.org.uk

  • Who is the main character? • What are the main challenges faced by the main character? • What do you need to do to be successful in this game? • What items do you use to help you pursue your quest? • What prevents you from achieving your goal? (e.g. characters, events, etc.) • What are the main issues raised by the game? • Who are the main protagonists?

http://www.americasarmy.com V http://www.msnbc.msn.com/id/6687019/

Choosing the appropriate game

Choosing the appropriate game

Table 1: List of some popular commercial games and their learning benefits

XXIV The definition of 'serious games' can vary

Whereas the previous games were not initially built with educational objectives, other games, referred to as serious games, XXIV were built to teach, train or raise awareness. They can be news games (journalistic games reporting on current events), simulation games (games using strategy and simulating several aspects of reality) or organizational-dynamic games (games used to teach the dynamics of organizations). Once you have tested the game and decided to use it as an educational resource, you will need to identify the scenario that you think will be most beneficial to support your students. Several key points should be considered:

Technical and contextual points:

• If the digital game uses sound and if the activities are not of a collaborative nature, you should ensure that pupils use headphones. • Depending on the type of game and on the learning objectives, you may need to create groups. This will be very effective to engage children in discussions or collaborative activities.

• Take breaks: it is good practice to encourage children to take regular five-minute breaks every 45 minutes. • Make sure that the light in the room is sufficient and adequate. This can avoid occurrences of epilepsy seizures.

Pedagogical points:

Before the lesson:

• Identify learning objectives.

• Identify what part of the game or levels might best support the learning objectives. • Print the list of objectives and help section (or booklet) of the game to give to the students.

During the lesson:

• Begin by explaining the objectives of the session. • Demonstrate the game and explain how common tasks are performed (e.g. accessing Help, navigating through menus or moving the character).

Glossary of terms

Glossary of terms

This handbook has been written in the framework of European Schoolnet's Games in Schools project which began in January 2008 and ended in June 2009. The project's aim was to analyse the current situation in eight countries (Austria, Denmark, France, Italy, Lithuania, the Netherlands, Spain and UK) with regard to games based learning.

It is composed of various elements, including a synthesis and final report based on the results of a Europe-wide teachers' survey, case studies and interviews with relevant national policy makers, researchers and experts, as well as an online community of practice and finally the present teachers' handbook.

This handbook is intended for teachers interested in using digital games in their lessons. It provides the necessary information to understand the educational benefits of digital games and to learn how to use them as educational and motivational resources. After reading this handbook, teachers should be able to make informed decisions on the choice and use of digital games in the classroom, and obtain all the benefits digital games can offer.

European Schoolnet (EUN -www.europeanschoolnet.org) is a network of 31 Ministries of Education in Europe and beyond. EUN was created more than 10 years ago to bring about innovation in teaching and learning for its key stakeholders: Ministries of Education, schools, teachers and researchers.

The Interactive Software Federation of Europe (ISFE -www.isfe-eu.org) was established in 1998 to represent the interests of the interactive software sector vis-à-vis the EU and international institutions. Thirteen major publishers of interactive software and thirteen interactive software trade associations throughout Europe have joined ISFE.


Hans ) Metz

Viet Chi Tran

J A J Metz

C Tran

Daphnias: from the individual based model to the large population equation

Keywords: Birth and death process, age and size-structured populations, stochastic interacting particle systems, piecewise deterministic motion, large population limits. AMS codes: 92D40, 60J80, 60K35, 60F99

a long history going back to Nisbet and Gurney (Theor Pop Biol 23: 114-135, 1983) and Diekmann et al. (Nieuw Archief voor Wiskunde 4: 82-109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and throught their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et al., l.c.).

Introduction

The theory of physiologically structured populations as developed in works by Diekmann and co-authors [START_REF] Metz | The dynamics of physiologically structured populations[END_REF][START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models I Linear theory[END_REF][START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models. II. Nonlinear Theory[END_REF][START_REF] Diekmann | Steady state analysis of structured population models[END_REF][START_REF] Diekmann | a) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example[END_REF] derives its motivation from its supposed ability to link population level phenomena to specific mechanisms in and around individual organisms. Yet, those individuals do not figure as such in the models, which treat the spatial concentrations of those individuals as continua. The supposition has always been that the proposed deterministic frameworks would in principle be derivable from individualbased stochastic models (c.f. [START_REF] Metz | The role of physiologically structured population models within a general individualbased modeling perspective[END_REF]), but full derivations of this sort so far have only been done for the special cases of finite i-state models (i from individual; leading to ODEs in the large number limit, e.g. [START_REF] Kurtz | Solutions of ordinary differential equations as limits of pure jump Markov processes[END_REF][START_REF] Kurtz | Approximation of population Processes[END_REF]) and age-based models (e.g. [START_REF] Oelschläger | Limit theorem for age-structured populations[END_REF][START_REF] Tran | Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques[END_REF][START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF][START_REF] Ferrière | Stochastic and deterministic models for age-structured populations with genetically variable traits[END_REF][START_REF] Jagers | Population-size-dependent and age-dependent branching processes[END_REF][START_REF] Jagers | Population-size-dependent, age-structured branching processes linger around their carrying capacity[END_REF]). Moreover, it is possible to reinterpret the results in [START_REF] Champagnat | Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models[END_REF] as pertaining to general structured models with only jump transitions. Although some age-structured models are phrased in terms of i-state variables that change with time in a fixed manner, uninfluenced by the environment, these can truly represent but a small fraction of the rich variety of physiological mechanisms seen in nature (e.g. [START_REF] Diekmann | How to lift a model for individual behaviour to the population level?[END_REF][START_REF] Metz | The dynamics of physiologically structured populations[END_REF]chapters I and III]). For example, for the majority of species, size with a growth rate that depends on the environment is a far more important determinant of an individual's population dynamical behaviour than is age. Except in the physiologically well buffered homeotherms (on which we anthropocentrically are inclined to focus) reproduction tends to be under a strong influence of past food availabilities, with reproductive size being reached far earlier when an individual has encountered good than when it has encountered bad feeding conditions. In, for example, arthropods or fishes, maturing to the reproductive states often largely depends on how much food they have encountered but little on their age. However, the toolbox for proving the appropriate law of large number results that was developed in [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF][START_REF] Tran | Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques[END_REF][START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF][START_REF] Ferrière | Stochastic and deterministic models for age-structured populations with genetically variable traits[END_REF] needs to be extended a bit to deal with size-based models, referred to as 'Daphnia' models by [START_REF] Diekmann | a) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example[END_REF], let alone to the even more general models considered in [START_REF] Metz | The dynamics of physiologically structured populations[END_REF][START_REF] Metz | The role of physiologically structured population models within a general individualbased modeling perspective[END_REF][START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models I Linear theory[END_REF][START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models. II. Nonlinear Theory[END_REF][START_REF] Diekmann | Steady state analysis of structured population models[END_REF]. In this paper we, as a birthday present to Odo Diekmann, intend to remedy this lack, although so far only for the 'Daphnia' models that appear as central example in his work (c.f. [START_REF] Diekmann | a) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example[END_REF]).

Diekmann et al. [START_REF] Diekmann | a) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example[END_REF] lay out a general framework for investigating the interaction between a general age or size-structured consumer population (referred to as 'Daphnia') and an unstructured resource (referred to as 'algae'), a class of problems special forms of which were first considered in [START_REF] Nisbet | The systematic formulation of populationmodels with dynamically varying instar duration[END_REF][START_REF] Gurney | Fluctuation periodicity, generation separation, and the expression of larval competition[END_REF] and [START_REF] Diekmann | Continuum population dynamics with an application to Daphnia magna[END_REF][START_REF] De Roos | A size dependent predator-prey interaction: who pursues whom?[END_REF]. Diekmann et al. (l.c.) show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations that are analysed using results in [START_REF] Diekmann | Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars[END_REF][START_REF] Diekmann | Equations with infinite delay: Blending the abstract and the concrete[END_REF]. In this note, we derive their model from a microscopic description: starting from a stochastic, age-and size-structured, individual based model (IBM) for the Daphnia population we recover the equations of [START_REF] Diekmann | a) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example[END_REF]. In our study, the Daphnia population is discrete and stochastic while the algal population is continuous. We treat the Daphnia population as a point measure on a space spanned by size and age and from this platform generalize the microscopic construction given in [START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF] to arrive at a stochastic differential equation (SDE in the sequel) driven by Poisson point processes that gives a pathwise description of the population evolution. The main novelty is that the individual growth rate also depends on the population as a whole, instead of only on an individuals' characteristics, since the algal concentration and thus each individuals' resource access is affected by the entire Daphnia population. We then, in the wake of [START_REF] Oelschläger | Limit theorem for age-structured populations[END_REF][START_REF] Tran | Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques[END_REF][START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF][START_REF] Ferrière | Stochastic and deterministic models for age-structured populations with genetically variable traits[END_REF][START_REF] Jagers | Population-size-dependent and age-dependent branching processes[END_REF][START_REF] Jagers | Population-size-dependent, age-structured branching processes linger around their carrying capacity[END_REF], provide a law of large numbers that allows approximating the dynamics of the individual-based size-and age-structured model by means of a partial differential equation (PDE) when the volume containing the Daphnia and algae as well as the population sizes are large. In [START_REF] Méléard | Trait substitution sequence process and canonical equation for age-structured populations[END_REF] similar limit theorems are used as a basis for obtaining adaptive dynamics approximations for the evolution of hereditary traits in age-structured populations, while [START_REF] Durinx | Adaptive dynamics for physiologically structured models[END_REF] gives informed conjectures about such approximations for the more encompassing model classes treated in [START_REF] Diekmann | Steady state analysis of structured population models[END_REF]. Finally we study the limiting equations to recover the macroscopic (deterministic) system considered in [START_REF] Diekmann | a) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example[END_REF].

Notation: We will denote the set of finite measures on R 2 + endowed with the weak convergence topology as M F (R 2 + ). For µ ∈ M F (R 2 + ) and a real measurable bounded function f on this space, we set µ, f = R 2 + f (ξ, a)µ(dξ, da). The set of bounded real functions of class C 1 on E with bounded derivatives is denoted as C 1 b (E, R). If X is a process indexed by time, then we will denote the value of X at time t as either X(t) or X t (the latter to avoid formulas becoming cluttered with too many brackets).

2 Individual-based Daphnia model

Model specification

Our Daphnia population consists of discrete individuals living in continuous time, differentiated by an age a ∈ R + and a size ξ ∈ R + . Individuals are given labels i ∈ N * = {1, 2, . . .}, with the individuals present in the population at t = 0 bearing numbers in an order opposite to that of their ages, followed by the individuals born after t = 0 in the order of their appearance in the population. We denote as I t ∈ N * the total number of individuals that were present at time 0 or were born between time 0 and time t. V t ⊂ {1, . . . , I t } denotes the set of individual alive at time t. We can then represent the Daphnia population as a point measure on the state space

N * × R + × R + : Zt = i∈Vt δ (i,ξ i t ,a i t ) , (1)
where ξ i t and a i t are the size and age of individual i at time t. We denote as Z t (dξ, da) = Zt (N * × dξ × da) the marginal measure of Zt on R 2 + ; V t equals the support of the marginal measure of Zt on N * . Size and age are related as follows. An individual's age is equal to tt 0 where t is the current time and t 0 its birth time. Individuals grow up in an environment that is characterized at time t by the algal concentration S(t). All individuals are born with the same size ξ 0 (to keep things simple) and an individual with size ξ at time t grows at speed g(ξ, S(t)), so that the size of an individual aged a born at time t 0 is: ξ(a; t 0 ) = ξ 0 + a 0 g(ξ(α; t 0 ), S(t 0 + α))dα.

(

The growth rates depend on the other individuals in the population. The latter point is the novelty of the IBM presented here. (We could have taken hereditary traits as well as further age-like i-state variables on board as in [START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF], but have decided not do this in order not to unduly complicate the story). Reproduction is asexual. The birth and death rates of an individual with size ξ and age a at time t are β(ξ, a, S(t)) respectively µ(ξ, a, S(t)).

An individual with size ξ depletes the food density at rate 1 K γ(ξ, S(t)) and the food density replenishes with rate f (S(t)) so that the food concentration S(t) evolves according to

dS dt (t) =f (S t ) - 1 K γ(ξ, S t )Z t (dξ, da) = f (S t ) - 1 K i∈Vt γ(ξ i t , S t ). (3)
For the biological justification think of the Daphnia population as living in a container of size K, so that the Daphnia density is |Zt| K , with |Z t | := Card(V t ) = Z t , 1 the number of individuals. Assumption 2.1. In the sequel, we assume that the growth speed g(ξ, S), the rates f (S) and γ(ξ, S) are continuous bounded functions and that (i) the birth rate β(ξ, a, S) is piecewise continuous and bounded by β.

(ii) the death rate µ(ξ, a, S) is continuous and there exists a function µ(a) and a constant A ∈ (0, +∞] such that ∀(ξ, a, S) ∈ R 3 + , µ(ξ, a, S) ≥ µ(a) and A 0 µ(a)da = +∞. (iii) g is Lipschitz continuous with respect to ξ, uniformly in S on compact intervals of R + , and bounded by ḡ. (iv) f and γ are uniformly Lipschitz continuous with respect to S uniformly in ξ on compact intervals of R + .

The assumption (i) on the birth rate ensures that in a short time interval a single individual can not beget too many young: intervals between births are stochastically lower bounded by exponential random variables with rate β. The assumption (ii) on the death rate implies that individuals a.s. have lifetimes bounded by A. Finally the assumptions (iii) and (iv) ensure that there exist unique continuous solutions to (2) and (3) as long as the number of individuals |Z t | remains finite, the latter being guaranteed by the fact that |Z t | is stochastically bounded by a pure birth process with birth rate β.

For Φ ∈ C 1 b (R 2 , R) and ϕ ∈ C 1 b (R 2 + , R), we denote by Φ φ the function on M F (R 2 + ) × R + defined by Φ ϕ (Z, S) = Φ( Z, ϕ , S). From the description of the population dynamics, it follows that the process (Z(t), S(t)) t∈R+ is characterized by the infinitesimal generator L operating on the functions Φ ϕ :

LΦ ϕ (Z, S) = ∂ 1 Φ ϕ (Z, S) Z, g(., S), ∂ ξ f (., .) + ∂ a f (., .) + Z, µ(., ., S) Φ Z, ϕ -ϕ(., .), S -Φ ϕ Z, S + Z, β(., ., S) Φ Z, ϕ + ϕ(ξ 0 , 0), S -Φ ϕ Z, S +∂ 2 Φ ϕ (Z, S) f (S) -Z, 1 K γ(., S) (4)
The first term describes the aging and growth of the living individuals of the population. The second and third terms represent the demography of the population (deaths and births). The fourth term corresponds to the variation of the food.

In the next Subsection we introduce the pathwise construction of an IBM with the described dynamics and give an SDE driven by a Poisson point process that admits (4) as infinitesimal generator. This is useful for simulations and for deriving moment conditions and large population approximations (e.g. [START_REF] Champagnat | Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models[END_REF]). The evolution is piecewise deterministic: The size of the population is modified at birth or death events. Between these, conditionally on the structure of the population after the last event, the growth of the individuals and the food dynamics are deterministic.

Construction of the IBM and a useful SDE

Let us start with some heuristics. Consider at time t a population given by Zt and food concentration S(t). If no birth or death event occurs between time t and t + s, then V t+s = V t . The sizes ξ i (t + s) for i ∈ V t at time t + s and the food concentration S(t + s) are obtained by solving:

ξ i (t + s) = ξ i (t) + t+s t g(ξ i (τ ), S(τ ))dτ (5)
S(t + s) = S(t) + t+s t f (S(τ )) - 1 K i∈Vt γ(ξ i (τ ), S(τ )) dτ.
Under Assumptions 2.1, this system has a unique solution, which we denote as (Ξ i (t + s; t, Zt , S t ), Σ(t + s; t, Zt , S t ) ; s ∈ R + , i ∈ V t ). We will denote the coordinate of the flow corresponding to ( 5) for an individual with initial condition ξ as Ξ(t + .; t, ξ, Zt , S t ), so that in particular, Ξ i (t + s; t, Zt , S t ) = Ξ(t + s; t, ξ i t , Zt , S t ). In the sequel, we will also use that, if no births or deaths occur, for all 0 ≤ s ≤ t:

Ξ(t; s, Ξ(s; 0, ξ 0 , Z0 , S 0 ), Zt , S t ) = Ξ(t; 0, ξ 0 , Z0 , S 0 ). ( 6
)
After a birth or a death the process is restarted with appropriately adapted initial conditions at that instant.

The above description suggests a simple direct way for simulating the IBM. Starting from a birth or death event, first generate a standard exponentially distributed random number τ , and then run the differential equations for the states of all Daphnia individuals and for the algae. Simultaneously integrate the sum of the birth and death rates of the Daphnia, starting from zero. When this integral reaches τ , one of the Daphnia dies or gives birth. Which individual is the culprit and whether the event is a birth or a death is then decided from a single multinomial draw with probabilities proportional to the contributions of all the different events to the total event rate at that time. For the SDE that describes the process ( Z(t), S(t)) t∈R+ , we proceed as in [START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF], following a construction introduced by [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF] for the case without age or size, while accounting for the additional difficulty that the growth rate now depends on the rest of the population. To this end we again use that between two birth or death events the evolution of the population, conditionally on its state at the last event, is deterministic. As the integral form in which we present the SDE looks back at the end result of the events happening over a time interval, we also have to look in a retrospective manner at the resetting of the initial conditions at the moments that a birth or death occurs. Assume that the initial condition Z0 and the initial food concentration S 0 are given. The idea is that to construct the population at time t, we can proceed as follows:

• If no event happens during [0, t], then it is sufficient to consider the predicted sizes (Ξ i (t; 0, Z0 , S 0 ) ; i ∈ V 0 ) of individuals at t. The population at time t is

Zt = i∈V0 δ (i,Ξ i (t;0, Z0,S0),a i 0 +t)
and the food concentration is S t = Σ(t; 0, Z0 , S 0 ).

• If a birth event occurs at time s ∈ [0, t], then V s = V s-∪ {I s-+ 1}, where I s-is the number of labels already used so that the new individual gets labelled with the first available number j = I s-+ 1. The predicted sizes at time t, (Ξ i (t; s, Zs-, S s-) ; i ∈ V s-), that we had for the individuals i ∈ V s-before the event on the supposition that it were not to occur, are replaced by (Ξ i (t; s, Zs-+ δ (j,ξ0,0) , S s-) ; i ∈ V s-); see Fig. 1. Moreover for the new individual with label j, we add a Dirac mass at (j, Ξ j (t; s, Zs-+ δ (j,ξ0,0) , S s-), ts).

• If the individual j ∈ V s-dies at time s ∈ [0, t], then V s = V s-\ {j} and the predicted sizes at time t, (Ξ i (t; s, Zs-, S s-) ; i ∈ V s ), that we had for the individuals i ∈ V s before the event on the supposition that it were not to occur, are replaced by (Ξ i (t; s, Zs-δ (j,ξ j s -,a j s -) , S s-) ; i ∈ V s ); see Fig. 1. Additionally, we delete the Dirac mass at (j, Ξ j (t; s, Zs-, S s-), a ) . At time s1, a second particle is born. Just before birth, the population is Zs 1-= δ (1,Ξ(s 1 ;0,ξ ′ , Z0 ,S 0 ),a ′ +s 1 ) . After the birth, the size expected for the first particle at time t is changed from ξ1 to ξ2 = Ξ(t; s1, Ξ(s1; 0, ξ ′ , Z0, S0), Zs 1-+ δ (2,ξ 0 ,0) , Ss 1 -) since there will be less resources for the two particles. At time s2, the second particle dies. The size expected for the first particle at t is changed again from ξ2 to ξ3 as there is now more resources for the first particles's growth. The PPM provides possible times of events. For each time, we draw the label i of the individual who may reproduce or die. The parameter θ allows to define whether a birth of death occurs. Assume also that the initial condition are Z0 , S 0 and V 0 = {1, . . . , I 0 }, with E( Z 0 , 1 ) < +∞ a.s. Then Zt = i∈V0 δ (i,Ξ i (t;0, Z0,S0),a i 0 +t) + t 0 N * ×R+ Q(ds, di, dθ) 1l i∈Vs - δ (Is -+1,Ξ Is -+1 (t;s, Zs -+δ (Is -+1,ξ 0 ,0) ,Ss),t-s) + j∈Vs - δ (j,Ξ j (t;s, Zs -+δ (Is -+1,ξ 0 ,0) ,Ss),a j s +(t-s))δ (j,Ξ j (t;s, Zs -,Ss),a j s +(t-s))

× 1l θ≤m1(i,s-, Zs -,Ss)

+ -δ (i,Ξ i (t;s, Zs -,Ss),a i s +(t-s)) + j∈Vs - δ (j,Ξ j (t;s, Zs --δ (i,ξ i s - ,a i s -
) ,Ss),a j s +(t-s))δ (j,Ξ j (t;s, Zs -,Ss),a j s +(t-s))

× 1l m1(i,s-, Zs -,Ss)<θ≤m2(i,s-, Zs -,Ss)

where:

m 1 (i, s -, Zs-, S s ) =β(ξ i s-, a i s-, S s-) (8)
m 2 (i, s -, Zs-, S s ) =m 1 (i, s -, Zs-, S s-) + µ(ξ s , a i s-, S s-). (9)
In this definition the first of the two terms in the square brackets corresponds to the births and the second to the deaths. The first term starts with the addition of one new individual, followed by the corresponding updating of the future course of the trajectories of the other individuals in order to eventually get the right outcome at the final time t. The first part of the second term, dealing with the deaths, starts with removing an individual, followed by a corresponding updating of the growth trajectories Moment estimates obtained from [START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models. II. Nonlinear Theory[END_REF] are very important for the proofs. By adapting the proofs in [START_REF] Champagnat | Individual-based probabilistic models of adaptive evolution and various scaling approximations[END_REF] (Th. 2.5) (see also [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF]) where the main ingredient is the boundedness of the birth rate, we can show that: Lemma 2.3. Under Assumptions 2.1 and if E( Z 0 , 1 ) < +∞ as in Def. 2.2, then: (i) For any T > 0, E sup t∈[0,T ] Z t , 1 ) < E( Z 0 , 1 )e βT < +∞, (ii) If we have additionally that E( Z 0 , 1 p ) < +∞, then

E sup t∈[0,T ] Z t , 1 p ) < +∞. ( 10
)
As a consequence we obtain that (see [START_REF] Champagnat | Individual-based probabilistic models of adaptive evolution and various scaling approximations[END_REF][START_REF] Tran | Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques[END_REF]):

Proposition 2.4. The process (Z t , S t ) t∈R+ is well and uniquely defined on [0, T ] for any initial condition such that E( Z 0 , 1 ) < +∞ and for any Poisson point measure Q. Moreover, the infinitesimal generator of (Z t ) t∈R+ is (4).

Martingale problem

We end this section with a martingale problem that will be useful to derive the large population limits. Heuristically, the decomposition of the process Z t , f , for any test function f ∈ C 1 , into a predictable finite variation process and a square integrable martingale can be viewed as a description of the paths as solutions of the evolution equation associated with the generator L (predictable finite variation part) plus noise (martingale part). The proof is given in Appendix A.

Proposition 2.5. Let us assume that E( Z 0 , 1 p ) < +∞ for p ≥ 2. Let us consider a test function f (t, ξ, a) of class C 1 . Then:

M f t = Z t , f (t, ., .) -Z 0 , f (0, ., .) - t 0 R 2 + ∂f ∂s (s, ξ, a) + ∂f ∂a (s, ξ, a) + g(ξ, S s ) ∂f ∂ξ (s, ξ, a) + f (s, ξ 0 , 0)β(ξ, a, S s ) -f (s, ξ, a)µ(ξ, a, S s ) Z s (dξ, da)ds, (11)
is a square integrable martingale with predictable quadratic variation:

M f t = t 0 R 2 + f 2 (s, ξ 0 , 0)β(ξ, a, S s ) + f 2 (s, ξ, a)µ(ξ, a, S s ) Z s (dξ, da) ds. ( 12
)
3 Large populations

We now focus on large populations of Daphnia. To that end we consider a sequence of processes describing the evolution of the population through time when starting with an initial condition of size proportional to the integer parameter K (3) that we let increase to infinity. As already indicated, we may think of this K as the volume in which the population and its food live. We moreover scale the population with 1 K , i.e., we transform from population size to population density. We thus consider a sequence ZK of populations such that:

ZK t (dj, dξ, da) = 1 K i∈V K t δ (i,ξ i t ,a i t ) (dj, dξ, da) (13)
where V K t is the set of individuals alive at time t. Again,

Z K t (dξ, da) = ZK t (N * × dξ × da) is the marginal on R 2 + of ZK t .
We also consider (S K t ) t∈R+ the sequence of associated food concentrations, also indexed by K and satisfying:

dS K t dt = f (S K t ) - N * ×R 2 + γ(ξ, S K t ) ZK t (dj, dξ, da), (14)
with initial conditions (S K 0 ) K∈N * that converge in probability to S 0 ∈ R + . For each K ∈ N * , K ZK has the dynamics of the process introduced in Definition 2.2 with initial conditions K ZK 0 for which we assume that sup

K∈N * E Z K 0 , 1 2 < +∞. ( 15
)
) 16
d̺ dt = f (̺(t)) - R 2 + γ(ξ, ̺(t))ζ t (dξ, da). ( 17
)
The proof of Proposition 3.1 is given in Appendix B.

We conclude by showing that equations ( 16)-( 17) allow us to recover the equations of [START_REF] Diekmann | a) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example[END_REF]. We first establish a precise form of the solution ζ t . That this measure is not absolutely continuous with respect to the Lebesgue measure on R 2 + , even if the initial condition ζ 0 is, was already noticed in e.g. [START_REF] Tran | Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques[END_REF][START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF]. There it was stated that age and size where both parameterized by time, but no precise form for the measure for these cases was given.

Corollary 3.2. Assume that the initial condition has a marginal measure in age that is absolutely continuous with respect to the Lebesgue measure on R + so that ζ 0 (dξ, da) = q 0 (a, dξ)da, where q 0 (a, dξ) is a transition measure on R + . We denote by ν a,t (dξ) the image measure of q 0 (at, dξ) through the application ξ → Ξ(t; 0, ξ, ζ 0 , ̺ 0 ). For any t ∈ R + , the marginal ζ t at time t of the solution of ( 16) is a.s. given by:

ζ t (dξ, da) =1l a<t b(t -a)F (a, t -a, ̺ [0,t] )δ Ξ(t;t-a,0,ζt-a,̺t-a) (dξ) da +1l a≥t F ′ (t, a -t, ξ ′ , ̺ [0,t] )1l ξ=Ξ(t;0,ξ ′ ,ζ0,̺0) ν a,t (dξ) da ( 18
)
where b(t

) = R 2 + β(ξ, a, ̺ t )ζ t (dξ, da)
is the total birth rate at time t, where ̺ [0,t] = (̺ s ) s∈[0,t] and where for a < t,

F (a, t 0 , ̺ [0,t0+a] ) = exp - a 0 µ Ξ(t 0 + α; t 0 , ξ 0 , ζ t0 , ̺ t0 ), α, ̺ t0+α dα ( 19
)
is the probability that an individual born at t 0 survives until age a when the food environment is given by (̺ s ) s∈[0,t0+a] . For a ≥ t,

F ′ (t, a 0 , ξ ′ , ̺ [0,t] ) = exp - t 0 µ Ξ(s; 0, ξ ′ , ζ 0 , ̺ 0 ), a 0 + s, ̺ s ds (20)
is the probability that an individual alive at t = 0 with age a 0 and size ξ ′ survives until time t in an environment ̺ [0,t] .

With Corollary 3.2, we recover the equations of [START_REF] Diekmann | a) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example[END_REF]. Equation ( 2) provides the deterministic differential equation describing the growth of Daphnias, represented by the distribution ζ t (dξ, da) (see [START_REF] Jagers | Population-size-dependent, age-structured branching processes linger around their carrying capacity[END_REF]):

dξ da (a) = g ξ(a), ̺(t 0 + a) ; ξ(0) = ξ 0 .
If we consider an individual born at time t 0 > 0 and follow the survival probability through time a → F (a, t 0 , ̺ [0,t0+a] ), Equation [START_REF] Joffe | Weak convergence of sequences of semimartingales with applications to multitype branching processes[END_REF] gives the decay of the survival probability of an individual of age a at time t:

dF da (a, t 0 , ̺ [0,t0+a] ) = -µ Ξ(t 0 + a; t 0 , ξ 0 , ζ t0 , ̺ t0 ), a, ̺ t0+a F (a, t 0 , ̺ [0,t0+a] ).
From [START_REF] Jagers | Population-size-dependent, age-structured branching processes linger around their carrying capacity[END_REF], we have the Daphnia population birth rate at time t:

b(t) = t 0 β(Ξ(t; t -a, 0, ζ t-a , ̺ t-a ), a, ̺ t )b(t -a)F (a, t -a, ̺ [0,t] )da + R 2 + β(Ξ(t; 0, ξ, ζ 0 , ̺ 0 ), a + t, ̺ t )F ′ (t, a, ξ, ̺ [0,t] )ζ 0 (dξ, da)
The first term represents the contributions of individuals born after time 0, while the second term corresponds to individuals who where present initially. From this, we can deduce the algal concentration:

d̺ dt (t) = f (̺(t)) - t 0 γ(Ξ(t; t -a, 0, ζ t-a , ̺ t-a ), ̺(t))b(t -a)F (a, t -a, ̺ [0,t] )da + R 2 + γ(Ξ(t; 0, ξ, ζ 0 , ̺ 0 ), ̺(t))F ′ (t, a, ξ, ̺ [0,t] )ζ 0 (dξ, da).
Proof. of Corollary 3.2 For the proof, we start by showing that ζ t (dξ, da) admits a density m(ξ, a, t) w.r.t. a dominating measure underlying [START_REF] Jagers | Population-size-dependent, age-structured branching processes linger around their carrying capacity[END_REF]. The equations satisfied by m(ξ, a, t) are then derived by separating the domain into R + × {a ≥ t} and R + × {a < t}, which corresponds to first studying the individuals born before and after initial time. First, recall that there is a unique solution ̺ to [START_REF] Jagers | Population-size-dependent and age-dependent branching processes[END_REF].

Let ϕ ∈ C 1 b (R 2 + , R
), let t ∈ R + and consider the associated test function:

f (s, ξ, a) = ϕ Ξ(t; s, ξ, ζ s , ̺ s ), a + t -s .
This function f is the unique solution of:

∂f ∂s + ∂f ∂a + g ∂f ∂ξ (s, ξ, a) = 0, f (t, ξ, a) = ϕ(ξ, a)
(e.g. [START_REF] Evans | Partial Differential Equations[END_REF]). As a consequence, using this test function f in ( 16):

ζ t , ϕ = ζ 0 , ϕ(Ξ(t; 0, ξ, ζ 0 , ̺ 0 ), . + t) + t 0 ϕ(Ξ(t; s, ξ 0 , ζ s , ̺ s ), t -s) ζ s , β(., ., ̺ s ) - R 2 + ϕ Ξ(t; s, ξ, ζ s , ̺ s ), a + t -s µ(ξ, a, ̺ s )ζ s (dξ, da) ds. (21)
The first term is related to individuals that are alive at time 0. The second integral relates to births between time 0 and time t. The third term corresponds to the deaths between time 0 and time t.

If we consider positive functions ϕ, then, neglecting the non-positive terms in [START_REF] Kurtz | Approximation of population Processes[END_REF]:

0 ≤ ζ t , ϕ ≤ +∞ t R+ ϕ(Ξ(t; 0, ξ, ζ 0 , ̺ 0 ), a)q 0 (a -t, dξ) da + t 0 b(t -a)ϕ(Ξ(t; t -a, ξ 0 , ζ t-a , ̺ t-a ), a)da. (22)
Notice that the population is naturally divided into two sets. Since the aging velocity is where there exists a unique ξ ′ such that ξ = Ξ(t; 0, ξ ′ , ζ 0 , ̺ 0 ) under Assumptions 2.1. Notice that at is the age of the individual at time 0 and is not a real function of time. Thus, we recognize an ordinary differential equation of degree 1 for s → m(Ξ(s; 0, ξ ′ , ζ 0 , ̺ 0 ), at + s, s) from which m(ξ, a, t) = exp -t 0 µ(Ξ(s; 0, ξ ′ , ζ 0 , ̺ 0 ), at + s, ̺ s )ds 1l {ξ=Ξ(t;0,ξ ′ ,ζ0,̺0)} .

This yields the second part of [START_REF] Jagers | Population-size-dependent, age-structured branching processes linger around their carrying capacity[END_REF]. Choosing ϕ with support in R + × {a < t}, (21) yields: Notice that ta is the time of birth of the individual and is not a real function of age. Then we recognize again an ordinary differential equation of order 1 from which

ζ t , ϕ = t 0 da b(t -a)ϕ Ξ(t; t -a, ξ 0 , ζ t-a , ̺ t-a ), a - t 0 da ϕ(Ξ(t; t -a, ξ 0 , ζ t-a , ρ t-a ), a) × a 0 µ Ξ(u + t -a; t -a, ξ 0 , ζ t-a , ρ t-a ), u, ρ u+t-a m Ξ(u + t -a; t -a, ξ 0 , ζ t-a , ρ t-a ), u, u + t -a du = t 0 da b(t -a)ϕ Ξ(t; t -a, ξ 0 , ζ t-a , ̺ t-a ), a - t 0 da ϕ(Ξ(t; t -a, ξ 0 , ζ t-a , ρ t-a ),
m(ξ, a, t) =b(t -a) exp - a 0 µ Ξ(u + t -a; t -a, ξ 0 , ζ t-a , ρ t-a ), u, ρ u+t-a du .
This ends the proof of the announced result (18) for a < t.

Conclusion

In this paper we rigorously underpin the long suspected individual-based nature of physiological structured population models as studied by the team of researchers surrounding Odo Diekmann. Such results were already known for purely age-based models (e.g. [START_REF] Oelschläger | Limit theorem for age-structured populations[END_REF][START_REF] Tran | Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques[END_REF][START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF][START_REF] Ferrière | Stochastic and deterministic models for age-structured populations with genetically variable traits[END_REF][START_REF] Jagers | Population-size-dependent and age-dependent branching processes[END_REF][START_REF] Jagers | Population-size-dependent, age-structured branching processes linger around their carrying capacity[END_REF]), and our present result still applies only to a subclass of the models considered by Odo and his co-workers, although a paradigmatic one, to wit their so-called "Daphnia" models e.g. [START_REF] Diekmann | a) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example[END_REF][START_REF] Diekmann | How to lift a model for individual behaviour to the population level?[END_REF]. The i-state variables of these models move in a continuous deterministic fashion, dependent on their own value and the environment (food: "algae"), and births occur to a single fixed birth state at a rate that depends on the i-state and the environment (and are not e.g. coupled to specific i-state transitions). Even for this restricted class of physiologically structured population models some twiddling of the existing probabilistic toolbox was in order. Yet, overall the toolbox proved its mettle, and the long standing assumption of a solid individual-based foundation of the theory of physiologically structured populations was duly vindicated. The longer term goal is to extend this vindication to the full class of models put forward in [START_REF] Metz | The dynamics of physiologically structured populations[END_REF][START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models I Linear theory[END_REF][START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models. II. Nonlinear Theory[END_REF][START_REF] Diekmann | Steady state analysis of structured population models[END_REF].

Acknowledgements: This work benefitted from the support from the "Chaire Modélisation Mathématique et Biodiversité of Veolia Environnement -Ecole Polytechnique -Museum National d'Histoire Naturelle -Fondation X".

Appendices

A Proof of Proposition 2.5

Let f (t, ξ, a) be a function of class C 1 . From [START_REF] Diekmann | On the formulation and analysis of general deterministic structured population models. II. Nonlinear Theory[END_REF], we obtain Recall that we denoted by T k , k ≥ 1 the birth and death events in the population. By convention, we let T 0 = 0. Let us consider an individual i. Let t 0 ∈ {T k , k ≥ 0} be the birth time of the individual (or 0 if the individual is alive at time 0) and a i t0 be its age at time t 0 (0 if t 0 is the birth time). The sum of the terms in the r.h.s. of ( 23) associated with individual i is equal to:

Z t , f = i∈V0 f (t, Ξ i (t; 0, Z0 , S 0 ), a i 0 + t) + t 0 N * ×R+ Q(ds, di, dθ) 1l i∈Vs - 1l θ≤m1(i,s-, Zs -,Ss) f (t, Ξ(t; s, 0, Zs + δ (Is -+1,ξ0,0) , S s ), t -s) + j∈Vs - f (t, Ξ j (t; s, Zs-+ δ (Is -+1,ξ0,0) , S s ), a j s + (t -s)) -f (t, Ξ j (t; s, Zs-, S s ), a j s + (t -s)) +1l m1(i,s-, Zs -,Ss)<θ≤m2(i,s-, Zs -,Ss) -f (t, Ξ i (t; s, Zs-, S s ), a i s + (t -s)) + j∈Vs - f (t, Ξ j (t; s, Zs--δ (i,ξ i s - ,a i s - ) , S s ), a j s + (t -s)) -f (t, Ξ j (t; s, Zs-, S s ), a j s + (t -s)) . (23)
f (t 0 , ξ i t0 , a i t0 ) + k≥0 t∧T k+1 ∨t0 t∧T k ∨t0 ∂f ∂u + ∂f ∂a (s, Ξ i (s; T k , ZT k , S T k ), a i t0 + s -t 0 ) + g(s, Ξ i (s; T k , ZT k , S T k ), S s ) ∂f ∂x (s, Ξ i (s; T k , ZT k , S T k ), a i t0 + s -t 0 ) ds - t 0 N * ×R+ 1l j=i;i∈Vs -1l m1(i,s-, Zs -,Ss)<θ≤m2(i,s-, Zs -,Ss) f (s, ξ i s-, a i s-) dQ.
The last integral correspond to the death term when individual i is dead before t. Thus, [START_REF] Méléard | Trait substitution sequence process and canonical equation for age-structured populations[END_REF] gives: -f (s, ξ i s-, a)1l m1(i,s-, Zs -,Ss)<θ≤m2(i,s-, Zs -,Ss) ,

Z t , f (t, ., .) = i∈V0 f (0, ξ i 0 , a i 0 ) + k≥0 t∧T k+1 t∧T k ∂f ∂u + ∂f ∂a (s, Ξ i (s; T k , ZT k , S T k ), a i 0 + s) + g(s, Ξ i (s; T k , ZT k , S T k ), S s ) ∂f ∂x (s, Ξ i (s; T k , ZT k , S T k ), a i 0 + s) ds - t 0 N * ×R+ Q(ds,
where the first bracket corresponds to individuals alive at time 0 and where the second bracket correspond to individuals born after time 0. For s < u i∈Vs - δ (Ξ i (u;s, Zs,Ss),a i s +u-s) (dξ, da) = Z u (dξ, da)

provided there has been no jumps between s and u. Thus, we have: f (s, ξ 0 , 0)β(ξ, a, S s )f (s, ξ, a)µ(ξ, a, S s ) Z s (dξ, da)

+ t 0 N * ×R+
1l i∈Vs -f (s, ξ 0 , 0)1l θ≤m1(i,s-, Zs -,Ss) -f (s, ξ i s-, a)1l m1(i,s-, Zs -,Ss)<θ≤m2(i,s-, Zs -,Ss) Q(ds, di, dθ),

where Q(ds, di, dθ) = Q(ds, di, dθ)ds ⊗ n(di) ⊗ dθ is the compensated Poisson point measure associated with Q. The integral with respect to (w.r.t.) Q(ds, di, dθ) provides the martingale M f . This achieves the proof. 2

B Sketch of the proof of Proposition 3.1

When starting from [START_REF] Durinx | Adaptive dynamics for physiologically structured models[END_REF] and using controls of moments as in [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF], the proof is similar to the one in [START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF][START_REF] Tran | Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques[END_REF].

Step 1 We start by noticing that under the Assumption (15), we have the following estimate (e.g. [START_REF] Champagnat | Individual-based probabilistic models of adaptive evolution and various scaling approximations[END_REF]):

sup n∈N * E sup t∈[0,T ] Z n t , 1 2 < +∞. ( 24
)
Moreover, from Assumptions 2.1 and ( 5), the size of any individual is bounded on [0, T ] by ξ = ξ 0 + ḡT and there exists for every ε a non random constant Sε such that:

sup n∈N * P sup t∈[0,T ] S n t > Sε < ε. (25)
From these estimates and Assumption 2.1 (ii), there exists a constant A ε ∈ (0, A) such that:

sup n∈N * P sup t∈[0,T ] Z n t [ξ 0 , ξ] × [0, A ε ] > ε < ε. ( 26
)
Step 2 It is easy to see that the limiting values of (Z n , S n ) n∈N * are necessary continuous. Let us check the C-tightness (e.g. [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]) of (Z n , S n ) n∈N * in D([0, T ], M F (R 2 + )×R + ). Using a criterion by [START_REF] Méléard | Sur les convergences étroite ou vague de processus à valeurs mesures[END_REF] and given the compact containment that follows from Step 1, it is sufficient to prove the tightness of (S n ) n∈N * and of the predictable finite variation part and martingale part of ( Z n , f ) n∈N * for f in C 1 b (R 3 + , R) (which contains the constant function equal to 1). This is obtained by using Aldous-Rebolledo criteria (e.g. [START_REF] Joffe | Weak convergence of sequences of semimartingales with applications to multitype branching processes[END_REF]) and adapting for instance [START_REF] Champagnat | Individual-based probabilistic models of adaptive evolution and various scaling approximations[END_REF][START_REF] Tran | Large population limit and time behaviour of a stochastic particle model describing an age-structured population[END_REF] with the estimates of Step 1.

Step 3 The identification of the martingale problem satisfied by the limiting values provides ( 16)- [START_REF] Jagers | Population-size-dependent and age-dependent branching processes[END_REF]. Uniqueness of the solution of ( 16)-( 17) stems from the Assumptions 2.1. As a consequence, there is a unique limiting value and we have convergence in distribution of (Z n , S n ) n∈N * to the solution (ζ, ̺). Since the latter is deterministic, the convergence is also a convergence in distribution.

Definition 2 . 2 .

 22 Let Q(ds, di, dθ) be a Poisson point measure (PPM) on R + × N * × R + with intensity ds ⊗ n(di) ⊗ dθ where ds and dθ are Lebesgue measures and where n(di) is the counting measure on N * .

  u, ρ u+t-a ζ u+t-a (dξ, du) Thus, on {a < t}, the density m(ξ, a, t) of ζ t w.r.t. δ Ξ(t;t-a,ξ0,ζt-a,̺t-a) (dξ) da satisfies m(ξ, a, t) =b(ta) -a 0 R+ µ ξ ′ , u, ρ u+t-a ζ u+t-a (dξ ′ , du).

Z

  t , f (t, ., .) = Z 0 , f (0, ., .) +

  At time 0 there is a single particle of size ξ ′ and age a ′ , which is expected to have size ξ1 =

	time ξ 1 t ξ 2 ξ 3 t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s 2 × . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . ξ 0 ξ ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

j s + (ts)) that corresponds to the predicted size and age at time t of the dead individual. size Figure 1: Ξ(t; 0, ξ ′ , Z0, S0) at time t, where Z0 = δ (1,ξ ′ ,a ′

  Proposition 3.1. Let T > 0. Under the Assumptions 2.1, the sequence (Z K , S K ) n≥1 introduced in this section converges in probability in D([0, T ], M F (R 2 + ) × R + ) to the unique continuous solution (ζ, ̺) of the following deterministic equations, characterized for any function f (t, ξ, a) of class C 1 by: ζ t , f (t, ., .) = ζ 0 , f (0, ., .) +

		0	t	ζ s ,	∂f (s, ., .) ∂s	+	∂f (s, ., .) ∂a	+ g	∂f (s, ., .) ∂ξ	ds
	t
	+	ζ s , f (s, ξ 0 , 0)β(., ., ̺
	0

s )f (s, ., .)µ(., ., ̺ s ) ds. (

  1, the individuals who were alive at initial time are of age greater than t at time t. Individuals born after time 0 are of age smaller than t. So, if the function ϕ has support included in the set R + × {a < t}, then we see that on R + × {a < t}, ζ t (dξ, da) is absolutely continuous with respect to δ Ξ(t;t-a,ξ0,ζt-a,̺t-a) (dξ) da. Similarly, on the set R + × {a ≥ t}, ζ t (dξ, da) admits a density with respect to ν a,t (dξ) da. Denote by m(ξ, a, t) the density of ζ t with respect to the measure 1l a<t δ Ξ(t;t-a,ξ0,ζt-a,̺t-a) (dξ) da + 1l a≥t ν a,t (dξ) da. Substituting this density in the third term of[START_REF] Kurtz | Approximation of population Processes[END_REF] gives, for ϕ with support in R + × {a ≥ t}:By identification, the density m(ξ, a, t) of ζ t , on {a ≥ t}, w.r.t. ν a,t (dξ)da satisfies: ′ , ζ 0 , ̺ 0 ), at + s), ̺ s ) m(Ξ(s; 0, ξ ′ , ζ 0 , ̺ 0 ), at + s, s) ds 1l {ξ=Ξ(t;0,ξ ′ ,ζ0,̺0)}

	t
	ζ t , ϕ = -µ(Ξ(s; 0, ξ, ζ 0 , ̺ m(ξ, a, t) =1 -+∞ t R+ ϕ(Ξ(t; 0, ξ, ζ 0 , ̺ 0 ), a)q 0 (a -t, dξ)da +∞ t da R+ q 0 (a -t, dξ) ϕ Ξ(t; 0, ξ, ζ 0 , ̺ 0 ), a × t 0 µ(Ξ(s; 0, ξ
	0

0 ), at + s), ̺ s ) m(Ξ(s; 0, ξ, ζ 0 , ̺ 0 ), at + s, s) ds

  dj, dθ)1l j=i 1l i∈Vs -1l m1(i,s-, Zs -,Ss)<θ≤m2(i,s-, Zs -,Ss) f (s,ξ i Is -+1 (u; T k , ZT k , S T k ), us) +g(u, Ξ Is -+1 (u; T k , ZT k , S T k ), S u ) ∂f ∂x (u, Ξ Is -+1 (u; T k , ZT k , S T k ), us) du1l θ≤m1(i,s-, Zs -,Ss)

								s-, a i s-)
		t
	+	0	N * ×R+	Q(ds, di, dθ) 1l i∈Vs -\V0
		f (s, ξ 0 , 0) +	k≥0	t∧T k+1 ∨s t∧T k ∨s	∂f ∂u	+	∂f ∂a	(u, Ξ