Roger Clarke
email: roger.clarke@xamax.com.au

Big Data Prophylactics

Keywords: Data quality, decision quality, risk assessment, risk management, transparency, regulation, activism

Data mining has been re-branded as 'big data analytics'. The techniques involved harbour a substantial set of risks, many of which will be borne by individuals. This chapter argues that safeguards are needed, to protect individuals against the potentially harmful acts that organisations will take against them. Alternative forms of such 'big data prophylactics' are outlined.

Big Data

Big data is a fashion-item. It was an invention of marketers, as a means of breathing fresh life into the flagging booms in successively data mining and mash-ups. It has been given an aura of excitement because of the vast array of sources of data. There has been massive expropriation of social media profiles and traffic, and of wellness data from individuals' self-monitoring of their physiological states. A parallel development has been the open access movement in the public sector, which government agencies in various countries are utilising as an opportunity to break down both data silos and privacy protections. Other prospects that have been heralded include flows of streams of telemetry data (fast data), and from the Internet of Things.

In order to differentiate the big data concept, it was first proposed that its key characteristics were 'volume, velocity and variety' [START_REF] Laney | 3D Data Management: Controlling Data Volume, Velocity and Variety' Meta-Group[END_REF]. Subsequently, commentators added 'value', while a few have recently begun to add 'veracity' [START_REF] Schroeck | Analytics : The real world use of big data' IBM Institute for Business Value / Saïd Business School at the University of Oxford[END_REF]. Such vague formulations even find their way into the academic literature, in forms along the lines of 'data that's too big, too fast or too hard for existing tools to process'. A somewhat more useful characterisation is "the capacity to analyse a variety of mostly unstructured data sets from sources as diverse as web logs, social media, mobile communications, sensors and financial transactions" [25, p.12]. This reflects the widespread use of the term to encompass not only data collections, but also the processes applied to those collections. To overcome this ambiguity of scope, this paper distinguishes between 'big data' and 'big data analytics'.

Some commentators consider that big data bears a stronger resemblance to an ideology than to a science. At [5, p.663], boyd & Crawford depict 'big data' as "a cultural, technological, and scholarly phenomenon that rests on the interplay of [three elements]". Their first two elements correspond to 'big data' and 'big data analytics'. Their third element, on the other hand, emphasises the importance of "mythology: the widespread belief that large data sets offer a higher form of intelligence and knowledge that can generate insights that were previously impossible, with the aura of truth, objectivity, and accuracy".

The mythology, or as its proponents would prefer, the meme, has been spruiked in business school magazines [START_REF] Clarke | Quality Assurance for Security Applications of Big Data[END_REF], by business school academics [START_REF] Mcafee | Big Data: The Management Revolution[END_REF], and by academics in other disciplines who might have been expected to temper their enthusiasm [START_REF] Mayer-Schönberger | Big data: A revolution that will transform how we live, work, and think[END_REF]). The high level of enthusiasm, coupled with these authors' willing suspension of disbelief, has given rise to counter-arguments, and to accusations such as "There's a mania around big data ..." (David Searls, quoted in [START_REF]Big data is just a big, distracting bubble, soon to burst[END_REF][START_REF]Why big data evangelists should be sent to re-education camps[END_REF]).

Big Data Analytics

The term 'big data analytics' refers to the processes that are applied to big data collections. A substantial array of analytical tools pre-existed the big data era, and more are being developed.

One way to categorise big data analytics is according to the purpose for which the analysis is performed. In Table 1, two broad categories are first distinguished, according to whether the analysis is aiming to deliver insights into populations or about individual entities within those populations. Within each of these major categories, distinctly different kinds of problem-types can be addressed.

It would be reasonable to expect a highly-developed set of guidelines to exist, enabling analysts to recognise firstly which analytical techniques are suitable to which of those problem-categories (and, conversely, which are not); and secondly what attributes big data collections need in order to support each of those analytical techniques. However, it is very difficult to find such guidance. Despite the explosion in postgraduate degree offerings in the big data analytics area, the vague aura of art and craft has yet to be replaced by the clarity of science and engineering, the overtone of experimentation dressed up as innovation pervades, and the application of established expertise remains uncommon.

Risk Factors

A range of risks arise from the current spate of over-enthusiastic and uncritical adoption of the big data meme, and of its companion notions of open government data, social media exploitation, location and tracking of people and the devices that they use, and the Internet of Things.

Data

Tables 2 and3 show the results of prior research, which drew on the literature -particularly [19, 7 pp. 601-605, 36, 35, 29, 26] -in order to identify and briefly define relevant quality categories. Some are 'data quality' factors (which are capable of being assessed at the time the data is collected) whereas others are 'information quality' factors (which are not assessable until the data is used). Population Focus

• Hypothesis Testing

This approach evaluates whether propositions are supported by the available data. The propositions may be predictions from theory, existing heuristics, or hunches

• Population Inferencing

This approach draws inferences about the entire population of entities, or about subpopulations. In particular, correlations may be drawn among particular attributes

• Construction of Profiles

This approach identifies key characteristics of some category of entities. For example, attributes and behaviours of a target group, such as 'drug mules', sufferers from particular diseases, or children with particular aptitudes, may exhibit statistical consistencies Individual Focus

•

Discovery of Outliers

Statistical outliers are commonly disregarded, but this approach regards them instead as valuable needles in large haystacks, because they may herald a 'flex-point' or 'quantum shift'

• Discovery of Anomalies

This approach draws inferences about individual entities within the population. For example, a person may be inferred to have provided inconsistent information to two organisations, or to exhibit behaviour in one context inconsistent with behaviour in another

• Application of Profiles

A search can be conducted for individual entities that exhibit patterns associated with a particular, previously asserted or computed profile, thereby generating a set of suspect entities

In addition to quality, it is important that those conducting data analysis be clear about how data is to be interpreted. In syntactical terms, there must be clear answers to such questions as: Is each data-item mandatory or optional? What is the meaning of an empty (or 'null') field? What values may each field contain? At the level of semantics, the signification of each item must also be unambiguous, i.e. with which real-world attribute of which real-world entity does it correspond, and what does it say about that attribute?

During the earlier 'data mining' era, low data quality was recognised as a matter of real concern. Data was modified in a variety of ways, using a process that was referred to as 'data scrubbing' [START_REF] Widom | Research Problems in Data Warehousing[END_REF]. One focus of data scrubbing is on missing data -although finding an appropriate basis for interpolating appropriate values is fraught with difficulty. Modifications are mostly made on the basis of various heuristics, or after comparison with characteristics derived from the data-holdings as a whole. In only rare cases is it possible to check item-content against an external authority. There are limits to the improvements that can actually be achieved, and almost all scrubbing, by its nature, involves a proportion of false positives. Hence, while the process may achieve some improvements, there is inevitably also some worsening in quality through mistaken modifications.

By the 'big data' era, the honest term 'data scrubbing' had been replaced by 'data cleaning' and 'data cleansing' [START_REF] Müller | Problems, Methods and Challenges in Comprehensive Data Cleansing[END_REF]. These terms imply not only that an attempt has been made to achieve 'cleanliness', but also that cleanliness has been achieved. As is apparent from the use of heuristics and the inevitability of errors, the implication is false.

Problems with data quality and data meaning are major sources of risk. Of course, this applies to all forms of administrative data processing, whether or not the data-collection in question qualifies as 'big data'. The problems are compunded, however, when data from different sources, with different and potentially incompatible meanings, and with varying levels of quality, are consolidated, and then handled as though the melange of data constitutes a single, cohesive data-collection. • D1 Syntactical Validity Conformance of the data with the domain on which the data-item is defined

• D2

Appropriate Entity Association A high level of confidence that the data is associated with the particular realworld identity or entity whose attribute(s) it is intended to represent

• D3

Appropriate Attribute Association The absence of ambiguity about which real-world attribute(s) the data is intended to represent

• D4 Appropriate Attribute Signification

The absence of ambiguity about the state of the particular real-world attribute(s) that the data is intended to represent • D5 Accuracy A high degree of correspondence of the data with the real-world phenomenon that it is intended to represent, typically measured by a confidence interval, such as '±1°C'

• D6 Precision The level of detail at which the data is captured, reflecting the domain on which valid contents for that data-item are defined, such as 'whole numbers of degrees Celsius'

• D7

Temporal Applicability The absence of ambiguity about the date and time when, or the period of time during which, the data represents or represented a real-world phenomenon. This is important in the case of volatile data-items such as total rainfall for the last 12 months, marital status, fitness for work, age, and the period during which an income-figure was earned or a licence was applicable ) to make a difference to the process in which the data is to be used

• I2 Practical Relevance

Demonstrable capability of the content of a particular data-item (the content of a cell in a table) to make a difference to the process in which the data is to be used

• I3 Currency

The absence of a material lag between a real-world occurrence and the recording of the corresponding data

• I4 Completeness

The availability of sufficient contextual information that the data is not liable to be misinterpreted

• I5 Controls

The application of business processes that ensure that all data quality and information quality factors have been considered prior to the data's use

• I6 Auditability

The availability of metadata that evidences the data's provenance, and supports assertions relating to data semantics, data quality and information quality

Analytics

Reference has already been made to the issues arising from uncertainty about the appropriateness of analytical techniques to problem-categories, and uncertainty about the suitability of any particular data-collection for processing using any particular analytical technique.

Further inadequacies that give rise to risks are lack of transparency. This is needed in relation to the process whereby inferences are drawn, the basis on which particular data is considered to be relevant to the drawing of the inference, and the criteria that gave rise to any particular judgement.

Humans who make decisions can be called to account, and required to explain the basis on which they drew inferences, made decisions and took action. Computer systems programmed in algorithmic or procedural languages (as was the norm from the 1960s to the 1980s) embody explicit processes and criteria, and hence they are also capable of being interrogated.

Later forms of programming language, however, embody increasing layers of mystery and inscrutability [START_REF] Clarke | A Contingency Approach to the Software Generations[END_REF]. With so-called 'expert systems' approaches (which most commonly involve the expression of sets of rules), both the decision processes and the decision criteria are implicit. The most that can be re-constructed is that a particular set of rules 'fired', and that particular data was what caused them to be invoked. This is seldom a clear basis for justifying an action, and in any case many rule-based applications aren't designed to support the extraction of even this inadequate form of explanation.

The current vogue in software development can be reasonably described as 'empirical'. Neural nets and machine-learning 'algorithms' do not have anything that can sensibly be described as a decision process, and it is not feasible to extract or infer decision criteria. The issues are discussed in some depth in ss.2.1 and 2.2 of [START_REF] Clarke | What Drones Inherit from Their Ancestors[END_REF]. Use of these techniques represents blind faith, by the 'data scientists', by the organisations that apply them, by any regulator that attempts to review them, and ultimately by everyone affected by them.

Decision and Action

The challenges identified in the preceding sub-sections give rise to risk if they are not understood and managed. Where any such inadequacies carry forward into decisions made and actions taken, whether about resource allocation or about relationships between organisations and particular individuals, risks arise that the decisions may be unjustified, disproportionate, or just plain wrong.

It has been a fundamental tenet of democracy that dealings between government agencies and individuals must be subject to review and recourse. This principle has also found its way into consumer rights laws in many jurisdictions and many contexts, particularly the financial sector and health care. Purely empirical data analytic techniques are completely at odds with these public expectations. Yet these expectations are in some cases expressed as legal requirements. Hence many potential applications of AI and machine learning 'algorithms' are arguably in breach of existing laws.

Consequences

The proponents of big data analytics may rail against these restraints, and complain that conservative attitudes and slow-changing laws are stifling innovation. They may protest that human rights, anti-discrimination laws and privacy protections constrain the freedom of corporations, government agencies and 'data scientists' to act in economically efficient ways -which implies the scope to impose their will on people. They may even invoke the anti-humanitarian credo that 'logic is dead ... get over it'. Or to use their own words: "Faced with massive data, [the old] approach to science --hypothesize, model, test --is ... obsolete. Petabytes allow us to say: 'Correlation is enough'" [START_REF] Anderson | The End of Theory: The Data Deluge Makes the Scientific Method Obsolete[END_REF] and "Society will need to shed some of its obsession for causality in exchange for simple correlations: not knowing why but only what. Knowing why might be pleasant, but it's unimportant ..." [START_REF] Mayer-Schönberger | Big data: A revolution that will transform how we live, work, and think[END_REF].

Inherent in the boisterous claims of Anderson, Mayer-Schönberger and others is the abandonment of balance between the empirical and the rational, and its replacement by empiricism dominant and systemic explanations deprecated. This 'flight from reason' has consequences. The proponents of big data analytics focus on the value that they assert society can, and that they assert society will, extract from massive, low-quality data-collections using more or less ad hoc analytical techniques. Even if their assertions turned out to be right -and such positive evidence as exists to date is merely anecdotal and unaudited -the benefits would be accompanied by massive negative impacts -for some, if not for all.

The resource-allocation and administrative decision-making errors that follow on from poor-quality inferencing will produce losers. Review will be at least hampered, and where AI and machine-learning are involved, actually not possible. The losers will be forced to 'like it or lump it', without recourse.

But social equity cannot exist in a world in which rationality has been abandoned, and decisions are made mysteriously and enforced by the powerful. This breaches the social contract. The losers' natural reaction is to stop trusting the institutions that they deal with. Some are likely to become sullen non-compliers with the diktats of powerful organisations, while others will be more aggressive in their avoidance measures. There is then an unpleasant scale up through active falsification, via electronic forms of sabotage, to violence.

The consequences of the Anderson / Mayer-Schönberger thesis are the breakdown of social cohesion, and serious challenges to the social order on which economies and polities depend. That such seers fail to look ahead to the consequences of their wild enthusiasms is quite extraordinary.

Big Data Prophylactics

Many of the risks identified in the previous section will be borne by individuals rather than by the organisations that make big-data-originated mistakes. So if 'big data analytics' is to be more than just a passing fad, it needs to be accompanied by 'big data prophylactics', to provide people with protections against organisations' potentially harmful acts against them.

Evaluation

One of the most important forms of protection is the conduct of evaluations of big data initiatives prior to their implementation. These should identify in advance ideas whose potential benefits do not justify the negative impacts and risks, which should then lead to their substantial reworking or their abandonment.

An examination of business case preparation gives rise to serious doubts about its effectiveness as a means of protecting organisations against bad big data. Business cases evidence many variants, some disciplined and formalised, but most pragmatic and informal. Typically, they involve spreadsheet modelling, often with primarily financial data, and perhaps cost-benefit analysis, but internal-only. The focus is on payback / Return on Investment (RoI), or on alignment with corporate strategy. However, all such approaches are more or less explicitly designed to provide support for the proposal [START_REF] Humphrey | Justifying a Process Improvement Proposal' SEI Interactive[END_REF]. Business case preparation provides inadequate protection even for the organisations that conduct them; still less do such processes protect against unjustifiable negative impacts on other parties.

Previous research by this author has considered big data risks from the perspective of the organisations that conduct the analytics and/or rely on the inferences they lead to. Data quality assurance should in principle be the means whereby the risks to those organisations can be avoided, detected, investigated and managed. Further, risk assessment should be the means of tackling not only data quality issues, but also the risks arising from data semantics, non-relevance, inappropriate data analytic techniques, and lack of transparency [START_REF] Clarke | Quality Assurance for Security Applications of Big Data[END_REF].

In practice, however, a large proportion of the negative outcomes of poor-quality big data and poor-quality big data analytics arise in the form of 'externalities': rather than the relevant organisation suffering them, someone else will. Most commonly the entities bearing the harm will be those that lack institutional and market power, sometimes small business enterprises, but most commonly people.

A much broader form of evaluation is needed than that provided by organisation-internal risk assessment processes. However, the incentives are such that organisations are not going to perform them, at least not of their own volition. Market failure exists, so government intervention is necessary.

Regulation

Where a proposal harbours serious threats, the protection of parties other than the proposal's sponsor depends on the conduct of a form of evaluation that takes into account the interests of all stakeholders. A consolidation of mainstream 'meta-principles' in relation to the evaluation of potentially harmful initiatives is in [START_REF] Apf | Meta-Principles for Privacy Protection' Australian Privacy Foundation[END_REF].

Beyond an obligation to conduct an appropriate form of evaluation, an effective regulatory scheme needs to be in place, to ensure that the findings from the process are carried through into actions. A variety of forms of regulatory arrangement exist, commonly referred to as organisational self-regulation, industry self-regulation, co-regulation and formal regulation. Nomatter which approach is adopted, an effective regulatory scheme needs to satisfy a range of requirements. A consolidation of criteria found in the literature is in Table 2 of [START_REF] Moses | The Regulation of Civilian Drones' Impacts on Public Safety[END_REF].

Such industry and professional codes as exist in the big data analytics arena fail comprehensively when tested against criteria such as these.

Examples include [START_REF]Declaration of Professional Ethics' United Nations Statistical Division[END_REF], and a flurry of recent initiatives whose intention is quite bare-facedly to hold off demands for formal regulatory measures [START_REF] Asa | Ethical Guidelines for Statistical Practice[END_REF][START_REF] Dsa | Data Science Code Of Professional Conduct' Data Science Association[END_REF][START_REF] Ukco | Data Science Ethical Framework' UK Cabinet Office[END_REF]. For a scathing assessment of the UK 'ethical framework', see [START_REF] Raab | Inadequacies in the UK Cabinet Office's Data Science Ethical Framework[END_REF].

A specific regulatory mechanism that has been making considerable progress over the last two decades is Privacy Impact Assessment (PIA). A PIA is a systematic process, which identifies and evaluates from the perspectives of all stakeholders the potential effects on privacy of a project, initiative or proposed system or scheme and which includes a search for ways to avoid or mitigate negative privacy impacts [START_REF] Clarke | Privacy Impact Assessment: Its Origins and Development[END_REF][START_REF]Privacy Impact Assessments[END_REF]. Unfortunately, evidence also exists that PIAs are not being effective in exercising control over inappropriate initiatives, particularly in national security contexts [START_REF] Clarke | Privacy Impact Assessments as a Control Mechanism for Australian National Security Initiatives[END_REF].

A recent development that has raised some people's hopes is the 'Data Protection Impact Assessment' (DPIA) requirement within the EU's General Data Protection Regulation [START_REF]EU General Data Protection Regulation (EU-GDPR) Article 35 -Data protection impact assessment[END_REF], which is to come into force in 2018.

The provisions are, however, very weak. The trigger is limited to "high risks" (Art. 35.1-35.6). The impacts to be assessed are only those on the protection of personal data (35.1) -which is a poor proxy even for data privacy, and which excludes other dimensions of privacy. It appears inevitable that DPIA will be interpreted as a mere Data Protection Law Compliance Assessment. Moreover, seeking civil society's views is optional, and there is no requirement that they be reflected in the design (35.9). There is a complete exemption for authorised programs (35.10), rather than merely an exemption from the justification requirement. And it is far from clear whether any enforcement of design features will be feasible, and whether any review will ever be undertaken of the performance of schemes against the data used to justify them (35.7(d), 35.11).

A further factor that some argue to be a regulatory arrangement is the 'precautionary principle'. A strong form of this exists in some jurisdictions' environmental laws, along the lines of 'When human activities may lead to morally unacceptable harm that is scientifically plausible but uncertain, actions shall be taken to avoid or diminish that potential harm' [START_REF] Tvh | Telstra Corporation Limited v Hornsby Shire Council[END_REF]. However, no such strong form is applicable in human rights contexts. All that can be argued is that a moral responsibility exists, whereby, 'if an action or policy is suspected of causing harm, and scientific consensus that it is not harmful is lacking, the burden of proof arguably falls on those taking the action'.

These relevant and current examples all indicate that market failure is matched by regulatory failure. Neither parliaments nor regulatory agencies are providing effective restraints on organisational misbehaviour in the field of big data analytics.

Public Activism

In order to protect people's interests, public activism is needed. This could take the form of civil disobedience, in particular the obfuscation and falsification of data, traffic, location and identity. Further, pressure could be brought to bear on organisations, regulators and politicians, through coordinated actions focussed on a specific target, and the use of whatever communications channels are judged to have the greatest impact on that target at that particular point in time.

However, there are limited prospects of action by the general public, or even by the population segments most seriously affected by big data blunders. The issues are too complex and obscure for public discourse to cope with, and reduction to the simple slogans compatible with popular uprisings is very challenging. Another problem is that the regulatory failure noted in the previous section means that appropriate evaluation does not take place, and hence transparency is denied.

Rather than the general public, it appears more likely that the battle will be fought by advocacy organisations, called (originally in UN contexts) non-government organisations (NGOs), and collectively referred to as civil society. A review of privacy advocacy organisations around the world is in [START_REF] Bennett | The Privacy Advocates[END_REF]. These associations have very little direct power, and limited resources. However, they have a wide range of techniques at their disposal [START_REF] Bennett | The Privacy Advocates[END_REF][START_REF] Davies | Ideas for Change: Campaign principles that shift the world' The Privacy Surgeon[END_REF].

This author has previously proposed a further form that public activism can take. Civil society could abandon its half-hearted and ineffectual involvement in 'official' Standards processes conducted by industry and government. NGOs could develop, adopt, promulgate and promote their own series of Civil Society Standards [START_REF] Clarke | Civil Society Must Publish Standards Documents[END_REF]. Importantly in the big data context, these would specify principles and processes for evaluation, processes for quality assurance and audits, and checklists of mitigation measures and controls.

Conclusions

All uses of data involve issues with data semantics, data quality and information quality. However, the issues arising in conventional administrative systems are reasonably well-understood, and safeguards, controls, reviews, recourse and audit are factored into system designs.

Big data compounds the problems of data semantics, data quality and information quality. It merges data of uncertain and often low quality, and of often incompatible semantics, and it projects mystique rather than being founded on any real 'data science'.

Big data analytics then heaps further problems on the bonfire. One is the failure to provide a reliable way to identify appropriate techniques and to clearly specify the attributes that data must possess in order to justify processing in that manner. A second and very substantial problem is the lack of transparency inherent in contemporary analytical methods, whose rationality is not penetrable even by 'data science' specialists.

Inferences drawn by software that uses incomprehensible processes may be relied on to make decisions, and to take actions, variously affecting categories of people, particularly through resource allocation, and affecting individuals, particularly through administrative decison-making. These decisions are, quite fundamentally, unreviewable, because the rationale underlying them cannot be communicated -and a rationale, in the sense in which humans understand the notion, may not even exist.

The consequences of big data inferences, and decisions and actions based on them, will inevitably be negative for some entities, and some categories of entities. Those entities will mostly be normal human beings. They will be denied meaningful review and recourse processes, because no comprehensible information is available. Weak regulators will be cowed by the accusation of stultifying innovation. Social equity, the social contract, and ultimately social order, will be the victims.

Protections are needed, which I've referred to in this paper as 'prophylactics', to underline the fact that they are a counterpoint to 'analytics'. In the foreground are evaluation processes; but these are shown not to work, and hence market failure is evident. Various forms of regulatory mechanism should in principle come into play; but multiple examples show that market failure is matched by regulatory failure. All that remains is public activism. The conditions are not right for the general public, or even the affected segments of the public, to take decisive action. Civil society is likely to have to fill the void, if big data prophylactics are to arise, to protect the public from inappropriate applications of big data analytics.

Table 1 .

 1 Purposes of Big Data Analytics [after 12]

Table 2 .

 2 Data Quality Factors [after 11]

Table 3 :

 3 Information Quality Factors [after 11]

	Information Quality Factors
	•	I1 Theoretical Relevance
		Demonstrable capability of a category of data-item (a column in a table

Acknowledgements. This paper was developed from my opening keynote invited for the IFIP Summer School on Privacy and Identity Management, Karlstad, Sweden, 22 August 2016.


Hafida Boudjellaba
email: hboudjellaba@cs.laurentian.ca

Tewfik Sari
email: tewfik.sari@uha.fr

STABILITY LOSS DELAY IN A CLASS OF SLOW AND FAST ECOLOGICAL MODELS

Keywords: Asymptotic stability, Delayed loss of stability, Singular perturbations, Canards, Biological models AMS subject classifications. 34D15, 34E15, 92D25

We study a class of three-dimensional systems of harvesting two species. The dynamics of the harvesting effort is assumed to be slow comparatively to the dynamics of the species. By using singular perturbation theory and the stability loss delay phenomenon, we give precise conditions which guarantee the existence of a semi global asymptotically stable equilibrium. As an application, a well-known model considered by Clark will be discussed.

Introduction.

The main problem in the study of dynamical systems arising in applications is to determine the asymptotic behavior of the solutions. In mathematical ecology, this problem is related to the study of the persistence of the species. For instance, if there exists a globally asymptotically stable steady state of positive coordinates, then the system is persistent. On the other hand, in most applications the dynamics of different variables of the system are hierarchically scaled: for instance, in ecological models, often, the preys multiply much faster than the predators. Hence, the study and management of systems with various time scales were considered by many authors and remain a high point of interest from theoretical, computational and practical points of view [START_REF] Boudjellaba | Oscillations in prey-predator-superpredator systems[END_REF][START_REF]Stability Loss Delay in Harvesting Competing Populations[END_REF][START_REF] Kokotovic | Singular Perturbation Methods in Control: Analysis and Design[END_REF][START_REF] Lobry | Control theory and environmental problems: slow-fast models for management of renewable resources[END_REF][START_REF] Lobry | Singular Perturbation Methods in Control theory[END_REF][START_REF] Muratory | Low-and High-Frequency Oscillations in Three-Dimensional Food Chain Systems[END_REF][START_REF] Zagaris | Fast and slow dynamics for the computational singular perturbation method[END_REF].

In this paper, we investigate a class of slow and fast ecological models of particular interest for the applications and consider the question of the existence of a globally asymptotically stable persistent equilibrium. More precisely, we study the combined harvesting of two ecologically independent populations. Problems related to the exploitation of multispecies systems are more difficult than problems pertaining to the single-species model. For details and information, see Clark's book [START_REF] Clark | Mathematical Bioeconomics, The Optimal Management of Renewable Ressources[END_REF], or [START_REF][END_REF]. If each population is subject to logistic growth and E is the harvesting effort, we have the following equations (see [START_REF] Clark | Mathematical Bioeconomics, The Optimal Management of Renewable Ressources[END_REF], p. 311):

x ′ = rx(1 -x/K) -q 1 Ex, y ′ = sy(1 -y/L) -q 2 Ey.

(1.1)

In system (1.1), the parameters r and s are the intrinsic growth rates, q 1 and q 2 are the catchability coefficients and K and L are carrying capacities for populations x and y respectively. For net revenue, Clark used the following expression: π(x, y, E) = (p 1 q 1 x + p 2 q 2 y -c) E, where the prices p 1 and p 2 are constant and cE is the cost of fishing. Clark analyzed the system by finding the bionomic equilibrium, i.e. the equilibrium solutions x ′ = 0 and y ′ = 0 for equations (1.1) together with the condition π(x, y, E) = 0. He showed that, under some conditions, the bionomic equilibrium occurs, for a value E ∞ of the fishing effort, at a point (x ∞ , y ∞ ) where both x ∞ and y ∞ are positive. Clark suggested to extend system (1.1) by adding a dynamic reaction of the effort of fishing E of form E ′ = επ(x, y, E) where ε > 0. He obtained the system x ′ = rx(1 -x/K) -q 1 Ex, y ′ = sy(1 -y/L) -q 2 Ey, E ′ = ε (p 1 q 1 x + p 2 q 2 y -c) E.

(1.2) Then Clark stated, without proof, that in system (1.2) the equilibrium (x ∞ , y ∞ , E ∞ ) is approached asymptotically (see [START_REF] Clark | Mathematical Bioeconomics, The Optimal Management of Renewable Ressources[END_REF], p. 312). Our aim is to prove this result in the case where ε is a small parameter.

More precisely, we study a general model given by the following equations:

x ′ = xM (x, E), y ′ = yN (y, E), E ′ = εEP (x, y), (1.3)
where M , N and P are continuous functions. For the simplicity of the presentation, we assume that system (1.3) has a unique solution with prescribed initial conditions. We study this system in the positive octant of R 3 . We assume that ε is small , which means that reaction E is much slower than reactions x and y. Let us denote by τ the time in system (1.3). In terms of the slow time t = ετ , system (1.3) becomes

ε ẋ = xM (x, E), ε ẏ = yN (y, E), Ė = EP (x, y). (1.4)
Throughout the paper, the dot designates the derivatives with respect to time t and the prime designates the derivatives with respect to time τ . System (1.4) is a slow and fast vector field and its study belongs to singular perturbation theory or Tikhonov's theory (see Section A.4, see also [START_REF] Lobry | On Tikhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters multiplying the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF]). This system exhibits the phenomenon of delayed loss of stability or canard solution (see Section A.8). Our main aim is to give conditions on the functions M , N and P such that system (1.4) has an equilibrium (x ∞ , y ∞ , E ∞ ), with x ∞ > 0, y ∞ > 0 and E ∞ > 0, which is practically semiglobally asymptotically stable in the first octant, as ε → 0, that is, for any solution (x(t, ε), y(t, ε), E(t, ε)) of system (1.4) we have lim t→+∞,ε→0 (x(t, ε), y(t, ε), E(t, ε)) = (x ∞ , y ∞ , E ∞ ), (1.5) the limit being uniform with respect to the initial condition in any compact subset of the positive octant. The notion of practical semi-global asymptotic stability in systems depending on parameters is discussed in Section A.2. This notion, which is very important for applications, appeared first in control theory and is related to the problem of stabilization [START_REF] Byrnes | Bifurcation analysis of the zero dynamics and the practical stabilization of nonlinear minimum-phase systems[END_REF].

The paper is organized as follows. In Section 2, we apply Tikhonov's theory to system (1.4). In Section 3, we study the delayed loss of stability phenomenon in this system. In Section 4, the main results of the paper (theorems 4.6, 4.7 and 4.8) on the asymptotic stability of the equilibrium point (x ∞ , y ∞ , E ∞ ) of system (1.4) are stated. In Section 5, the results are illustrated on Clark's model (1.2) and by numerical simulations. In Appendix A, we give the main tools of singular perturbation theory which are disregarded in the literature. In this appendix, some stability results are obtained by using the decomposition of the dynamics of the system in a slow and a fast part.

Fast Dynamics and Slow Dynamics.

2.1. The fast equation. System (1.4) is a slow and fast system of the form (A.2) introduced in Section A.4, with u = E as the slow variable and v = (x, y) as the fast variable. For this system, the fast equation (A.4) is written as

x ′ = xM (x, E) y ′ = yN (y, E) (2.1)
where E is a constant parameter. We assume that equations x ′ = xM (x, E) and y ′ = yN (y, E) have unique solutions with prescribed initial conditions, for every E ≥ 0. The following assumptions are made. (A1) There exists a positive continuous decreasing function E → ξ(E) defined on [0, a], a > 0, such that ξ(a) = 0 and for all E ∈ [0, a] we have M (ξ(E), E) = 0 and M (x, E) > 0 for x < ξ(E) and M (x, E) < 0 for x > ξ(E). (A2) There exists a positive continuous decreasing function

E → η(E) defined on [0, b], b > a, such that η(b) = 0 and for all E ∈ [0, b] we have N (η(E), E) = 0
and N (y, E) > 0 for y < η(E), and N (y, E) < 0 for y > η(E). From assumptions (A1) and (A2), we deduce the following properties: when E ≥ b, (0, 0) is the only equilibrium of (2.1); when a ≤ E < b, (2.1) has two equilibria, (0, 0) and (0, η(E)); when 0 ≤ E < a, (2.1) has four equilibria, (0, 0), (ξ(E), 0), (0, η(E)) and (ξ(E), η(E)). The stability of these equilibria is summarized in the following lemma:

Lemma 2.1. When E ≥ b, (0, 0) is a stable node. When b > E ≥ a, (0, 0) is a saddle point and (0, η(E)) is a stable node. When a > E ≥ 0, (0, 0) is an unstable node, (ξ(E), 0) and (0, η(E)) are saddle points and (ξ(E), η(E)) is a stable node.

Proof. The behavior of equilibria results easily from our assumptions, since the flow (2.1) is the product of two one-dimensional flows.

The ecological interpretation of this lemma is as follows: when two populations are exploited jointly, with a harvesting effort E ≥ b, then both of them are driven to extinction. If the harvesting effort is such that a ≤ E < b, then one of the populations is driven to extinction, whereas the other population continues to support the fishery. If the harvesting effort is such that E < a, then both populations continue to support the fishery. The fast dynamics are illustrated in Fig. 2.1.

0 < E < a x ξ(E) η(E) y 0 T c T c E ' E ' a < E < b x η(E) y 0 © c T ' ' b < E x y 0 © c ' Fig. 2.1.
The fast dynamics of system (2.1). The equilibrium (0, 0) is attracting for E > b. The equilibrium (0, η(E)) is attracting for a < E < b. The equilibrium (ξ(E), η(E)) is attracting for 0 < E < a.

2.2. The slow equation. After a fast transition close to the trajectories of the fast equation (2.1), the solutions of (1.4) are approximated by those of the slow equation. A slow equation is defined on each component of the slow manifold. The slow manifold which is the set of equilibra of the fast equations consists of several components (see Fig. i) The E axis which is the set of equilibra (0, 0). This component is attracting for E > b. On this slow curve, the slow equation is given by

Ė = EP (0, 0), 0 < E < +∞. (2.2)
ii) The curve (x = 0, y = η(E)), 0 < E < b. This component is attracting for a < E < b. On this slow curve, the slow equation is given by

Ė = EP (0, η(E)), 0 < E < b. (2.3)
iii) The curve (x = ξ(E), y = η(E)), 0 < E < a. This component is attracting. On this slow curve, the slow equation is given by

Ė = EP (ξ(E), η(E)), 0 < E < a. (2.4)
iv) The curve (x = ξ(E), y = 0), 0 < E < a. This component is not attracting. On this slow curve, the slow equation is given by

Ė = EP (ξ(E), 0), 0 < E < a. (2.5)
We assume that the solution with prescribed initial conditions is unique for all equations (2.3), (2.4) and (2.5).

We add the following assumption (see Fig. 2.2, right): (A3) The subset Π = {(x, y) : P (x, y) = 0} is the graph of a decreasing function.

We have P (x, y) < 0 under Π, P (x, y) > 0 above Π, P (ξ(0), η(0)) > 0 and P (0, η(a)) < 0.

2.3. Equilibria of the system. By (A3) curves Γ = {(ξ(E), η(E)) : 0 < E < a} and Π intersect (see Fig. 2.2, right) at a unique point denoted by

(x ∞ , y ∞ ). Let E ∞ be defined by ξ(E ∞ ) = x ∞ and η(E ∞ ) = y ∞ . We have 0 < E ∞ < a and S = (x ∞ , y ∞ , E ∞ ) (2.6)
is an equilibrium point of (1.4). In fact (1.4) has at least three other equilibria, (0, 0, 0), (0, η(0), 0), (ξ(0), 0, 0), (2.7) which lie in the invariant xy-plane. In the case where P (0, η(0)) > 0, there exists a value c ∈]0, a[ such that P (0, η(c)) = 0. Thus (0, η(c), c) (2.8) is an equilibrium of (1.4) which lies in the invariant yE-plane. This equilibrium is unstable in the x-direction. In the case where P (ξ(0), 0) > 0, there exists a value d ∈]0, a[ such that P (ξ(d), 0) = 0. Thus (1.4) has another equilibrium (ξ(d), 0, d) (2.9) which lies in the invariant xE-plane. This equilibrium is unstable in the y-direction.

2.4. Application of Tikhonov's theory. For all E ∈ [0, a[, the equilibrium (ξ(E), η(E)) of (2.1) is Globally Asymptotically Stable (GAS) in the positive quadrant of R 2 . Moreover, (2.4) has an asymptotically stable equilibrium point E = E ∞ . Hence, Theorem A.7 applies and predicts that in the region 0 ≤ E < a, the solutions of (1.4) jump quickly near the slow curve (x, y) = (ξ(E), η(E)) and then move near this slow curve towards the equilibrium point S (see Proposition 4.1).

For all E ∈ [a, b[, the equilibrium (0, η(E)) of (2.1) is GAS in the positive quadrant of R 2 . Hence, Theorem A.6 applies and predicts that in the region a ≤ E < b, the solutions of (1.4) jump quickly near the slow curve (x, y) = (0, η(E)) and then move near this slow curve with decreasing E, until E reaches the value E = a at which this slow curve loses its stability (see Fig. 2.3, left). More precisely, we have the following result: The asymptotic behavior of the solution of (1.4) with initial condition m 0 = (x 0 , y 0 , E 0 ) when a < E 0 < b (on the left) and E 0 > b (on the right). The expected asymptotic behavior, when E crosses the value a (on the left) or the value b (on the right) is not correct. The correct behavior, showing the delayed loss of stability, is described in Fig. 3.2, 3.3, 3.4 or 3.5.

Lemma 2.2. Let (x(t, ε), y(t, ε), E(t, ε)) be the solution of (1.4) with initial condition x(0, ε) > 0, y(0, ε) > 0 and E(0, ε) = E 0 such that a ≤ E 0 < b. Let E 1 (t) be the solution of (2.3) with initial condition E 1 (0) = E 0 . Let t a > 0 be the instant of time for which E 1 (t a ) = a. Then, for any δ > 0, we have lim ε→0 E(t, ε) = E 1 (t), uniformly for t ∈ [0, t a -δ] and lim ε→0 x(t, ε) = 0, lim ε→0 y(t, ε) = η(E 1 (t)), uniformly for t ∈ [δ, t a -δ].

Proof. The solution E 1 (t) of (2.3) is decreasing towards the value E = a, since P (0, η(E)) < 0 for all a < E < b (see assumption (A3)). When E crosses this value, the slow curve (x, y) = (0, η(E)) becomes unstable. To conclude, we apply Theorem A.6 because its hypothesis are now satisfied for t ∈ [0, T ] where T = t a -δ.

For all E ≥ b, the equilibrium (0, 0) of (2.1) is GAS in the positive quadrant of R 2 . Hence, Theorem A.6 applies and predicts that in the region E > b, the solutions of (1.4) jump quickly near the slow curve (x, y) = (0, 0) and then move near this axis with decreasing E, until E reaches the value E = b at which this slow curve loses its stability (see Fig. 2.3, right). More precisely, we have the following result:

Lemma 2.3. Let (x(t, ε), y(t, ε), E(t, ε)) be the solution of (1.4) with initial condition x(0, ε) > 0, y(0, ε) > 0 and E(0, ε) = E 0 ≥ b. Let E 0 (t) be the solution of (2.2) with initial condition E 0 (0) = E 0 . Let t b > 0 be the instant of time for which E 0 (t b ) = b. Then, for any δ > 0, we have lim ε→0 E(t, ε) = E 0 (t), uniformly for t ∈ [0, t b -δ] and lim ε→0 x(t, ε) = 0, lim ε→0 y(t, ε) = 0, uniformly for t ∈ [δ, t b -δ].

Proof. The solution E 0 (t) of (2.2) is decreasing towards the value E = b since P (0, 0) < 0. When E crosses this value, the slow curve (x, y) = (0, 0) becomes unstable. Thus the hypothesis of Theorem A.6 are satisfied for t ∈ [0, T ] where T = t b -δ. The result follows from this theorem.

3. Delayed loss of stability. One might believe then, that the solution of (1.4) with initial condition E(0, ε) = E 0 > b described in Lemma 2.3 will move, for t > t b , near the attracting slow curve (x, y) = (0, η(E)), with decreasing E, until E reaches the value E = a (see Fig. 2.3, right). Similarly, one might believe then, that the solution of (1.4) with initial condition E(0, ε) = E 0 > b described in Lemma 2.2 will move, for t > t a , near the attracting slow curve (x, y) = (ξ(E), η(E)), towards the equilibrium point S (see Fig. 2

.3, left).

In fact, due to the delayed loss of stability phenomenon, this behavior is not the right one and a solution which jumps quickly near the slow curve (x, y) = (0, 0) with E 0 > b, will move near this axis, with decreasing E, until E reaches a value E 1 < b (see Fig. 3.3, 3.4 or 3.5, right). Similarly, a solution which jumps quickly near the slow curve (x, y) = (0, η(E)), with E 0 ∈]a, b[ will move near this slow curve, with decreasing E, until E reaches a value E 1 < a, (see Fig. 3.2, right). Now, we need to compute E 1 with respect to E 0 . The mapping E 0 → E 1 is called the entrance-exit function along the slow curve.

3.1. Entrance-exit functions. For c defined as in (2.8) when P (0, η(0)) > 0 and c = 0 when P (0, η(0)) ≤ 0, we have P (0, η(u)) < 0 for all u ∈]c, b], so the function

E ∈]c, b[ → f (E) ∈ [0, +∞[, f (E) = E a M (0, u) uP (0, η(u)) du,
is well defined. The function f reaches its minimum at a (see Fig.

3.1, left). It is decreasing from +∞ to 0 on ]c, a] and increasing on [a, b[. Definition 3.1. The function E ∈ [a, b[ → F (E) ∈]c, a], defined by F = f -1 + •f -, where f -and f + are the restrictions of f on [a, b[ and ]c, a] respectively, is called the entrance-exit function along the slow curve (x, y) = (0, η(E)). Since f (E) = f (F (E)) for all E ∈ [a, b[ we have F (E) E M (0, u) uP (0, η(u)) du = 0. (3.1)
Now to define the entrance-exit function near the slow curve (x, y) = (0, 0), let g the function

E ∈]0, +∞[ → g(E) ∈ [0, +∞[, g(E) = E b N (0, u) uP (0, 0) du,
that has a minimum at b (see Fig.

E ∈]0, +∞[ → f (E) ∈ [0, +∞[, h(E) = E a M (0, u) uP (0, 0) du.
This function has a minimum at a (see Fig. 3.1, right). It is decreasing from +∞ to 0 on ]0, a] and increasing on [a, +∞[. It defines a mapping

E ∈ [a, ∞[ → H(E) ∈]0, a], by H = h -1 + • h -, where h -and h + are the restrictions of h on [a, +∞[ and ]0, a] respectively. Let a * = G -1 (a). If E ∈ [b, a * ] then (see Fig. 3.1, right) H(E) < a ≤ G(E). If E > a * then, we either have H(E) < G(E) or H(E) > G(E). However, in (1.2), the property H(E) < G(E) is true for all E > a * (see Lemma 5.2). b a c E F (E) z z = f (E) b a a * E G(E) H(E) 0 z z = h(E) z = g(E)
3.2. The function E ∈ [b, ∞[ → K(E) = max(G(E), H(E)) ∈]0, b] is called the entrance-exit function along the slow curve (x, y) = (0, 0). If E ∈ [b, a * ] or E > a * and H(E) < G(E) then K(E) = G(E) and G(E) E N (0, u) uP (0, 0) du = 0. (3.2) If E > a * and H(E) > G(E) then K(E) = H(E) and H(E) E M (0, u) uP (0, 0) du = 0. (3.3) 3.2.
Behavior in the vicinity of the slow curve (0, η(E)). Let (x 0 , y 0 , E 0 ) be an initial condition such that a < E 0 < b. By Lemma 2.2, the corresponding trajectory of (1.4) will move quickly towards the equilibrium (0, η(E)) and then remains close to this equilibrium with E decreasing and until E reaches the value a where the equilibrium (0, η(E)) loses its stability. The departure of the trajectory from the neighborhood of the equilibrium takes place not immediately after a but rather after a time during which the effort E changes by a finite amount. There is a delayed loss of stability. More precisely, let x 0 < 1 positive and not too big. We consider the trajectory γ(t, ε) of (1.4) that starts at (x 0 , y 0 , E 0 ) with a < E 0 < b. If ε is small enough, according to Lemma 2.2, the trajectory remains between the planes x = x 0 and x = 0 and goes towards the equilibrium (0, η(E)) while E is decreasing as far as E > a. Denote now, the next intersection of this trajectory and the plane x = x 0 by (x 0 , y(t 1 , ε), E(t 1 , ε)) where t 1 = t 1 (x 0 , y 0 , E 0 , ε) depends on the initial condition (x 0 , y 0 , E 0 ) and on ε. The limit

E 1 = lim ε→0 E(t 1 (x 0 , y 0 , E 0 , ε), ε) (3.4)
exists and depends only on E 0 . This limit, as well as, a precise behavior of the trajectory are given by the following theorem:

Theorem 3.3. Let E 0 ∈]a, b[. Then, we have E 1 = F (E 0
) and for small ε > 0, the trajectory leaves the neighborhood of point (0, η(E 1 ), E 1 ) and jumps to the neighborhood of point (ξ(E 1 ), η(E 1 ), E 1 ) close to the unstable separatrix y = η(E 1 ) of the saddle point (0, η(E 1 )) of the fast dynamics (see Fig. 3

.2). X 0 E E0 E1 a X = α(E) x y E a b m0 S E0 E1 Fig. 3.2.
On the right: the asymptotic behavior of the solution of (1.4) with initial condition m 0 = (x 0 , y 0 , E 0 ), when a < E 0 < b, showing the delayed loss of stability when E crosses the value a. On the left: the asymptotic behavior in the coordinates (X, E) of (3.5). The function α is defined by (3.8).

Proof. The change of variable X = ε ln x maps the strip 0 < x < 1 into the half space X < 0. This change of variable transforms (1.4) into

Ẋ = M (exp (X/ε), E), ε ẏ = yN (y, E) Ė = EP (exp (X/ε), y) (3.5)
The initial condition becomes (ε ln x 0 , y 0 , E 0 ). System (3.5) is a slow and fast system, similar to (A.2) of Section A.4, with u = (X, E) as the slow variable and v = y as the fast variable. For this system, the fast equation (A.4) is written as

y ′ = yN (y, E) (3.6)
The equilibrium y = η(E) of (3.6) is attracting for all E ∈ [0, b[. We have lim ε→0 exp X/ε = 0 since X < 0. Thus, on the slow surface y = η(E), the slow equation is

Ẋ = M (0, E), Ė = EP (0, η(E)). (3.7)
According to Theorem A.6, y goes very quickly towards the stable equilibrium y = η(E). Then a slow transition develops near the surface y = η(E). This slow transition is approximated by the solution of (3.7) with initial condition X(0) = 0, E(0) = E 0 . This solution is given by E = E(t) and

X = α(E) := E E0 M (0, u) uP (0, η(u)) du, (3.8)
where E(t) is the solution of (2.3) such that E(0) = E 0 . Thus, according to (3.1), we have again (see Fig. 3.2, left) X = 0 for E 1 = F (E 0 ). Returning to the original variables, we see that the trajectory γ(t, ε) crosses again the plane x = x 0 when E is asymptotically equal to E 1 = F (E 0 ). Then (see Fig. 3.2, right), a fast transition brings the trajectory from the neighborhood of point (0, η(E 1 ), E 1 ) to the neighborhood of point (ξ(E 1 ), η(E 1 ), E 1 ) close to the unstable separatrix y = η(E 1 ) of the saddle point (0, η(E 1 )) of the fast dynamics.

3.3.

Behavior in the vicinity of the slow curve (0, 0). By analogy to the previous section, let (x 0 , y 0 , E 0 ) be an initial condition such that E 0 > b. By Lemma 2.3, the corresponding trajectory of system (1.4) will go quickly towards the origin (x, y) = (0, 0) and then remains close to this equilibrium while E is decreasing and until E reaches the value b where the origin loses its stability. The departure of the trajectory from the neighborhood of the equilibrium takes place not immediately after b. There is a delayed loss of stability and the trajectory will leave the neighborhood of the equilibrium (0, 0) when E is asymptotically equal to a value E 1 . More precisely, we have the following result: Proposition 3.4. For small ε > 0, the solution remains near the slow curve (x, y) = (0, 0) as long as

E 1 < E < E 0 where E 1 = K(E 0 ). a) If E 0 ∈ [b, a * ]
then, the solution leaves the neighborhood of point (0, 0, E 1 ) and jumps to the neighborhood of point (0, η(E 1 ), E 1 ) close to the separatrix x = 0 of the saddle point (0, 0) of the fast dynamics (see Fig. 3.3). b) If E 0 > a * and H(E 0 ) < G(E 0 ) then, the solution leaves the neighborhood of point (0, 0, E 1 ) and jumps to the neighborhood of point (0, η(E 1 ), E 1 ) close to the orbit x = 0 of the unstable node (0, 0) of the fast dynamics (see Fig. 3.4). c) If E 0 > a * and H(E 0 ) > G(E 0 ) then, the solution leaves the neighborhood of point (0, 0, E 1 ) and jumps to the neighborhood of point (ξ(E 1 ), 0, E 1 ) close to the orbit y = 0 of the unstable node (0, 0) of the fast dynamics (see Fig. 3.5).

Proof. Let x 0 < 1 and y 0 < 1 be positive and not too big. By Lemma 2.3, for small ε > 0 the solution remains between the planes x = x 0 , y = y 0 , x = 0 and y = 0. The change of variables X = ε ln x, Y = ε ln y maps the open set 0 < x < 1, 0 < y < 1 into the octant X < 0, Y < 0. This change of variables transforms (1.4) into

Ẋ = M (exp (X/ε), E), Ẏ = N (exp (Y /ε), E), Ė = EP (exp (X/ε), exp (Y /ε)). (3.9)
The initial condition (x 0 , y 0 , E 0 ) becomes (ε ln x 0 , ε ln y 0 , E 0 ). Since X < 0 and Y < 0, (3.9) is a regular perturbation of Ẋ = M (0, E),

Ẏ = N (0, E), Ė = EP (0, 0). (3.10)
The solution of (3.10) with initial condition

X(0) = 0, Y (0) = 0, E(0) = E 0 is X = β(E) := E E0 M (0, u) uP (0, 0) du, Y = γ(E) := E E0
N (0, u) uP (0, 0) du, (3.11) where E = E(t) := E 0 e P (0,0)t is the solution of (2.2) such that

E(0) = E 0 . Let E 1 = K(E 0 ) and t 1 = 1 P (0,0) ln E1 E0 (t 1 satisfies E(t 1 ) = E 1 ). Hence E(t 1 , ε) = E 1 + o(1)
. When E is asymptotically equal to E 1 , i.e. when t is asymptotically equal to t 1 , the solution jumps far from the neighborhood of the E-axis as shown below.

X, Y X1 0 E E0 E1 = G(E0) c E2 H(E0) ca b Y = γ(E) d d s X = β(E) X = δ(E) E x y E a b m0 S E0 E1 E2
a * Fig. 3.3. On the right: the asymptotic behavior of the solution of (1.4) with initial condition m 0 = (x 0 , y 0 , E 0 ), when b < E 0 < a * , showing the delayed loss of stability when E crosses values b and a. On the left: the asymptotic behavior in the coordinates (X, Y, E) of (3.9). The functions β (in blue), γ (in red) and δ (in green) are defined by (3.11) and (3.15) respectively.

Cases a and b)

If E 0 ∈ [b, a * ] (see Fig. 3.3, left), or E 0 > a * and H(E 0 ) < G(E 0 ) (see Fig. 3.4, left), then E 1 = G(E 0 ). Thus, according to (3.2) and h(G(E 0 )) < h(E 0 ), we have X 1 = E1 E0 M (0, u) uP (0, 0) du < 0, Y 1 = E1 E0 N (0, u) uP (0, 0) du = 0. (3.12) Since X(t 1 , ε) = X 1 + o(1) and Y (t 1 , ε) = o(1)
, the solution reaches again the plane y = y 0 asymptotically at time t 1 , and x(t 1 , ε) = exp((X 1 + o(1))/ε) is exponentially small. Thus, asymptotically at time t 1 , the solution jumps (see Fig. 3.3 or 3.4, right) from the neighborhood of point (0, 0, E 1 ) to the neighborhood of point (0, η(E 1 ), E 1 ) close to the orbit x = 0 of equilibrium point (0, 0) of the fast dynamics. According to Lemma 2.1, this equilibrium is a saddle point if

E 0 ∈ [b, a * ]. It is an unstable node if E 0 > a * . Case c) If E 0 > a * and H(E 0 ) > G(E 0 ) (see Fig. 3.5, left) then E 1 = H(E 0
). Thus, according to (3.3) and g(H(E 0 )) < g(E 0 ), we have 1), the solution reaches again the plane x = x 0 asymptotically at value E 1 and y(t 1 , ε) = exp((Y 1 + o(1))/ε) is exponentially small. Thus, asymptotically at time t 1 , the solution jumps (see Fig. 3.5, right) from the neighborhood of point (0, 0, E 1 ) to the neighborhood of point (ξ(E 1 ), 0, E 1 ) close to the orbit y = 0 of the unstable node (0, 0) of the fast dynamics.

X 1 = E1 E0 M (0, u) uP (0, 0) du = 0, Y 1 = E1 E0 N (0, u) uP (0, 0) du < 0. (3.13) Since X(t 1 , ε) = o(1) and Y (t 1 , ε) = Y 1 + o(
As stated in Proposition 3.4, the jump of the solution, far from the slow curve (0, 0), happens when E is asymptotically equal to E 1 = K(E 1 ). After this jump, the asymptotic behavior of the solution is given by Theorems 3.5 and 3.6 below.

Theorem 3.5.

Let E 0 ∈ [b, a * ].
For small ǫ > 0, the solution remains near the slow curve (x, y) = (0, η(E)) as long as E 2 < E < E 1 where E 1 = G(E 0 ) and

E 2 ∈]c, a] is given by E1 E0 M (0, E) EP (0, 0) dE + E2 E1 M (0, E) EP (0, η(E)) dE = 0. (3.14)
Afterwards, it jumps from the neighborhood of point (0, η(E 2 ), E 2 ) to the neighborhood of point (ξ(E 2 ), η(E 2 ), E 2 ) close to the unstable separatrix y = η(E 2 ) of the saddle point (0, η(E 2 )) of the fast dynamics (see Fig. 3.3). This result is also true in the case where E 0 > a * , H(E 0 ) < G(E 0 ) and P (0, η(0)) < 0 (see Fig. 3.4). In this case

E 2 ∈]0, E 1 [. X, Y X1 0 E E0 E1 = G(E0) c E2 H(E0) c a b Y = γ(E) X = β(E) X = δ(E) E x y E a b m0 S E0 E1 T E2 a * Fig. 3.4.
On the right: the asymptotic behavior of the solution of (1.4) with initial condition m 0 = (x 0 , y 0 , E 0 ), when E 0 > a * and H(E 0 ) < G(E 0 ), showing the delayed loss of stability when E crosses values b and a. On the left: the asymptotic behavior in the coordinates (X, Y, E) of (3.9). The functions β (in blue), γ (in red) and δ (in green) are defined by (3.11) and (3.15) respectively.

Proof. The asymptotic behavior of the solution for t ∈ [0, t 1 ] is described in cases (a) and (b) of Proposition 3.4. For t > t 1 we use, the same change of variable as in the proof of Theorem 3.3, X = ε ln x which maps the strip 0 < x < 1 into the half space X < 0. This change of variable transforms (1.4) into (3.5) with conditions X(t 1 , ε) = X 1 + o(1), Y (t 1 , ε) > 0, and E(t 1 , ε) = E 1 + o(1), where X 1 is defined by (3.12). According to Theorem A.6, y goes very quickly towards the stable equilibrium y = η(E). Then, a slow transition develops near the surface y = η(E). This slow transition is approximated by the solution of (3.7) with initial condition X(t 1 ) = X 1 , E(t 1 ) = E 1 . This solution is given by

X = δ(E) := X 1 + E E1 M (0, u) uP (0, η(u)) du, (3.15) where E = E(t) is the solution of (2.3) such that E(t 1 ) = E 1 . In case E 0 ∈ [b, a * ], the integral in (3.15) is well defined because E 1 ∈ [a, b] and P (0, η(u)) < 0 for all u ∈]c, b].
Thus, E 2 defined by (3.14) satisfies E 2 ∈]c, a[ (see Fig. 3.3, left). We have again X = 0 at value E 2 . Returning to the original variables, we notice that the trajectory γ(t, ε) crosses again the plane x = x 0 when E is asymptotically equal to E 2 . Then (see Fig. 3.3, right), a fast transition brings the trajectory from the neighborhood of point (0, η(E 2 ), E 2 ) to the neighborhood of point (ξ(E 2 ), η(E 2 ), E 2 ), close to the unstable separatrix y = η(E 2 ) of the saddle point (0, η(E 2 )) of the fast dynamics. In the case where E 0 > a * , H(E 0 ) < G(E 0 ) and P (0, η(0)) < 0 the proof is the same as in case E 0 ∈ [b, a * ], adapted to Fig. Note that the mapping E 1 → E 2 given by formula (3.14) is not equal to the entrance-exit function E 1 → F (E 1 ) of the slow curve (x, y) = (0, η(E)) as it was the case in Theorem 3.3. Indeed, in Theorem 3.5, the solution is exponentially close to the plane x = 0 before it arrives near the slow curve (x, y) = (0, η(E)). Recall that in Theorem 3.3 the solution arrived from a point (x 0 , y 0 , E 0 ) which was not very close to the plane x = 0. Theorem 3.6. Let E 0 > a * . If H(E 0 ) > G(E 0 ) and P (ξ(0), 0) < 0 then for small ε > 0, the solution remains near the slow curve (x, y) = (ξ(E), 0) as long as

E 2 < E < E 1 where E 1 = H(E 0 ) and E 2 ∈]0, E 1 [ is given by E1 E0 N (0, E) EP (0, 0) dE + E2 E1 M (0, E) EP (ξ(E), 0) dE = 0. (3.16)
Afterwards, it jumps from the neighborhood of point (ξ(E 2 ), 0, E 2 ) to the neighborhood of point (ξ(E 2 ), η(E 2 ), E 2 ) close to the unstable separatrix x = ξ(E 2 ) of the saddle point (ξ(E 2 ), 0) of the fast dynamics (see Fig. 3.5). If H(E 0 ) = G(E 0 ), P (0, η(0)) < 0 and P (ξ(0), 0) < 0 then for small ε > 0, the solution jumps from the neighborhood of point (0, 0, E 1 ), where E 1 = H(E 0 ), to the neighborhood of point (ξ(E 1 ), η(E 1 ), E 1 ) close to one of the trajectories of the fast dynamics connecting the unstable node (0, 0) to the stable node

(ξ(E 2 ), η(E 2 )) (see Fig. 2.1, left). X, Y Y1 0 E E0 E1 = H(E0) © E2 c G(E0) ¡ ¡ ¡ a b Y = γ(E) X = β(E) Y = ̺(E) E x y E a b m0 S E0 E1 T E2
a * Fig. 3.5. On the right: the asymptotic behavior of the solution of (1.4) with initial condition m 0 = (x 0 , y 0 , E 0 ), in the case where E 0 > a * and H(E 0 ) > G(E 0 ), showing the delayed loss of stability when E crosses values b and a. On the left: the asymptotic behavior in the coordinates (X, Y, E) of (3.9). The functions β (in blue), γ (in red) and ρ (in green) are defined by (3.11) and (3.20) respectively.

Proof. The asymptotic behavior of the solution for t ∈ [0, t 1 ] is described in case (c) of Proposition 3.4. For t > t 1 we use the change of variable Y = ε ln x which maps the strip 0 < y < 1 into the half space Y < 0. This change of variable transforms (1.4) into

ε ẋ = xM (x, E), Ẏ = N (exp (Y /ε), E), Ė = EP (x, exp (Y /ε)) (3.17) with conditions X(t 1 , ε) > 0, Y (t 1 , ε) = Y 1 + o(1)
, and

E(t 1 , ε) = E 1 + o(1)
, where Y 1 is defined by (3.13). System (3.17) is a slow and fast system as (A.2) of Section A.4, with u = (Y, E) as the slow variable and v = x as the fast variable. For this system, the fast equation (A.4) is written as

x ′ = xM (x, E) (3.18)
The equilibrium x = ξ(E) of (3.6) is attracting for all E ∈ [0, a[. We have lim ε→0 exp Y /ε = 0 since Y < 0. And on the slow surface x = ξ(E), the slow equation is

Ẏ = N (0, E), Ė = EP (ξ(E), 0). (3.19)
According to Theorem A.6, x goes very quickly towards the stable equilibrium x = ξ(E). Then a slow transition develops near the surface x = ξ(E). This slow transition is approximated by the solution of (3.19) with initial condition Y (t 1 ) = Y 1 , E(t 1 ) = E 1 . This solution is given by

Y = ̺(E) = Y 1 + E E1 N (0, u) uP (ξ(u), 0)
du, (3.20) where E = E(t) is the solution of (2.5) such that E(t 1 ) = E 1 . The integral in (3.20) is well defined because E 1 ∈]0, a[ and P (ξ(u), 0) < 0 for all u ∈]0, a]. Thus, E 2 given by (3.16) satisfies E 2 ∈]0, E 1 [ (see Fig. 3.3, left). We have again Y = 0 at value E 2 . Returning to the original variables, we see that the trajectory γ(t, ε) crosses again the plane y = y 0 when E is asymptotically equal to E 2 . Then (see Fig. 3.5, right), a fast transition brings the trajectory from the neighborhood of point (ξ(E 2 ), 0, E 2 ) to the neighborhood of point (ξ(E 2 ), η(E 2 ), E 2 ), close to the unstable separatrix x = ξ(E 2 ) of the saddle point (ξ(E 2 ), 0) of the fast dynamics. In the case where [START_REF] Kaper | An introduction to geometric methods and dynamical systems theory for singular perturbation problems[END_REF]). The fast transition from (0, 0, E 1 ) to (ξ(E 1 ), η(E 1 ), E 1 ) would hold close to one of the orbits of the fast dynamics. We exclude equilibrium points on (ξ(E), 0, E) and on (0, η(E), E) since P (0, η(0)) < 0 and P (ξ(0), 0) < 0. Thus, the trajectory does not stay near these slow curves and the jump happens when E is asymptotically equal to E 1 .

H(E 0 ) = G(E 0 ), E 2 = E 1 in (3.
In the case where H(E 0 ) = G(E 0 ), our analysis does not predict the orbit along which the fast transition from (0, 0, E 1 ) to (ξ(E 1 ), η(E 1 ), E 1 ) would hold. By continuous dependance, when E(0, ε) crosses a value E 0 such that H(E 0 ) = G(E 0 ), all orbits of the fast dynamics arise as transition orbits, since the transition holds close to x = 0 when H(E 0 ) < G(E 0 ), and close to y = 0 when H(E 0 ) < G(E 0 ). The geometric singular perturbation theory (see remark following Theorem A.8) could provide tools to address this question. However, in (1.2), the case H(E 0 ) = G(E 0 ) does not appear (see Lemma 5.2).

Remark. In the case where P (0, η(0)) > 0, (2.8) is an equilibrium of (1.4). Let c * > b be defined by g(c * ) = g(c). Assume that H(c * ) < G(c * ). Then, according to Theorem 3.4, the solution of (1.4) with initial condition x(0, ε) > 0, y(0, ε) > 0, E(0, ε) = c * will jump quickly near the slow curve (x, y) = (0, 0) and remains close to this curve, as long as c < E < c * . Then, the solution will jump quickly from the neighborhood of (0, 0, c) to the neighborhood of the equilibrium (0, η(c), c), so it could stay near this equilibrium for a long time. On the other hand, in the case where P (ξ(0), 0) > 0, (2.9) is an equilibrium of (1.4). Let d * > b be defined by h(d * ) = h(d). Assume that H(d * ) > G(d * ). Then, according to Proposition 3.4, the solution of (1.4) with initial condition x(0, ε) > 0, y(0, ε) > 0, E(0, ε) = d * will jump quickly near the slow curve (x, y) = (0, 0) and remains close to this curve, as long as d < E < d * . Then, the solution will jump quickly from the neighborhood of (0, 0, d) to the neighborhood of the equilibrium (ξ(d), 0, d), so it could remain near this equilibrium for a long time. To achieve a full description of the solution, as in Theorem 3.5 or 3.6, we must exclude the existence of equilibrium (2.8) or (2.9).

Persistence.

In this section γ(t, ǫ) = (x(t, ε), y(t, ε), E(t, ε)) is the solution of (1.4) with initial condition x(0, ε) > 0, y(0, ε) > 0 and E(0, ε) = E 0 > 0. Our aim is to show that the limit (1.5) is uniform with respect to the initial condition in any compact subset of the positive octant.

4.1. Asymptotic behavior for all t ≥ 0. We begin with the asymptotic behavior of the solutions when E 0 < a.

Proposition 4.1. Let E 0 ∈]0, a[. Let E(t) be the solution of (2.4) with initial condition E(0) = E 0 . Let x(t) = ξ(E(t)) and y(t) = η(E(t)). Then for any δ > 0, we have lim ε→0 (x(t, ε), y(t, ε)) = (x(t), y(t)), uniformly for t ∈ [δ, +∞[ and lim ε→0 E(t, ε) = E(t), uniformly for t ∈ [0, +∞[.

Proof. For all E ∈ [0, a] , the solution (ξ(E), η(E)) of (2.1) is GAS. Moreover, by assumption (A3), the solutions of (2.4) are converging towards E = E ∞ , which is an asymptotically stable equilibrium. The result follows from Theorem A. [START_REF] Clark | Mathematical Bioeconomics, The Optimal Management of Renewable Ressources[END_REF].

Assume now that E 0 ∈ [a, b]. From Theorem 3.3, we see that the solution γ(t, ε) reaches the neighborhood of point (ξ(E 1 ), η(E 1 ), E 1 ). Then, as shown in Proposition 4.1, it is approximated by a solution of (2.4). More precisely, we have the following result which completes, for t > t a , the one given by Lemma 2.2.

Proposition 4.2. let E 0 ∈ [a, b]. Let E 1 (t) be the solution of (2.3) with initial condition E 1 (0) = E 0 . Let E 1 = F (E 0 ). Let t 1 > 0 be the instant of time for which E(t 1 ) = E 1 . Let E 2 (t) be the solution of (2.4) with initial condition E 2 (t 1 ) = E 1 . Let E(t) = E 1 (t) for t ∈ [0, t 1 ], E 2 (t) for t ∈ [t 1 , +∞[, (4.1)
and

x(t) = 0 for t ∈]0, t 1 [, ξ(E 2 (t)) for t ∈]t 1 , +∞[ y(t) = η(E(t)) for t ∈]0, +∞[.
Then, for any δ > 0 we have lim ε→0 (x(t, ε), y(t, ε)) = (x(t), y(t)), uniformly for t ∈ [δ, t 1 -δ] ∪ [t 1 + δ, +∞[ and lim ε→0 E(t, ε) = E(t), uniformly for t ∈ [0, +∞[. Proof. The limit behavior of the solution for 0 ≤ t < t 1 is given by Lemma 2.2 and Theorem 3.3 and by Proposition 4.1 for t > t 1 (see Fig 3 .2).

Assume that E 0 > b. From Theorems 3.5 and 3.6 we see that the solution γ(t, ε) reaches the neighborhood of point (ξ(E 2 ), η(E 2 ), E 2 ). Then , as shown in Proposition 4.1, it is approximated by a solution of (2.4). In the next propositions, we set three results, in the case where H(E 0 ) < G(E 0 ), in the case where H(E 0 ) > G(E 0 ) and in the exceptional case where H(E 0 ) = G(E 0 ). These results complete, for t > t b , the one given in Lemma 2.3.

Proposition 4.3. Let E 0 > b. We assume that P (0, η(0)) < 0 and

H(E 0 ) < G(E 0 ). Let E 0 (t) be the solution of (2.2) with initial condition E 0 (0) = E 0 . Let E 1 = G(E 0 ). Let t 1 > 0 be the instant of time for which E 0 (t 1 ) = E 1 . Let E 1 (t) be the solution of (2.3) with initial condition E 1 (0) = E 0 . Let E 2 be defined by (3.14) and t 2 > t 1 be the instant of time for which E 1 (t 2 ) = E 2 . Let E 2 (t) be the solution of (2.4) with initial condition E 2 (t 2 ) = E 2 . Let E(t) =    E 0 (t) for t ∈ [0, t 1 ], E 1 (t) for t ∈ [t 1 , t 2 ], E 2 (t) for t ∈ [t 2 , +∞[, (4.2) and x(t) = 0 for t ∈]0, t 2 [, ξ(E(t)) for t ∈]t 2 , +∞[, y(t) = 0 for t ∈]0, t 1 [, η(E(t)) for t ∈]t 1 , +∞[, Then, for any δ > 0 we have lim ε→0 (x(t, ε), y(t, ε)) = (x(t), y(t)), uniformly for t ∈ [δ, t 1 -δ] ∪ [t 1 + δ, t 2 -δ] ∪ [t 2 + δ, +∞[ and lim ε→0 E(t, ε) = E(t), uniformly for t ∈ [0, +∞[.
Proof. The limit behavior of the solution for 0 ≤ t < t 2 is given by Lemma 2.3 and Theorem 3.5. The limit behavior of the solution for t 2 ≤ t is given in Proposition 4.1 (see Fig. 3.3 or 3.4).

Proposition 4.4. Let E 0 > a * . We assume that P (ξ(0), 0) < 0 and H(E 0 ) > G(E 0 ). Let E 0 (t) be the solution of (2.2) with initial condition E 0 (0) = E 0 . Let E 1 = H(E 0 ). Let t 1 > 0 be the instant of time for which E 0 (t 1 ) = E 1 . Let E 1 (t) be the solution of (2.5) with initial condition E 1 (0) = E 0 . Let E 2 be defined by (3.16) and t 2 > t 1 be the instant of time for which E 1 (t 2 ) = E 2 . Let E 2 (t) be the solution of (2.4) with initial condition E 2 (t 2 ) = E 2 . Let E(t) be defined by (4.2). Let

x(t) = 0 for t ∈]0, t 1 [, ξ(E(t)) for t ∈]t 1 , +∞[, y(t) = 0 for t ∈]0, t 2 [, η(E(t)) for t ∈]t 2 , +∞[, Then, for any δ > 0 we have lim ε→0 (x(t, ε), y(t, ε)) = (x(t), y(t)), uniformly for t ∈ [δ, t 1 -δ] ∪ [t 1 + δ, t 2 -δ] ∪ [t 2 + δ, +∞[ and lim ε→0 E(t, ε) = E(t), uniformly for t ∈ [0, +∞[.
Proof. The limit behavior of the solution for 0 ≤ t < t 2 is given by Lemma 2.3 and Theorem 3.6. The limit behavior of the solution for t 2 ≤ t is given in Proposition 4.1 (see Fig. 3.5).

Proposition 4.5. Let E 0 > a * . We assume that P (ξ(0), 0) < 0, P (0, η(0)) < 0 and H(E 0 ) = G(E 0 ). Let E 1 (t) be the solution of (2.2) with initial condition E 1 (0) = E 0 . Let E 1 = H(E 0 ). Let t 1 > 0 be the instant of time for which E(t 1 ) = E 1 . Let E 2 (t) be the solution of (2.4) with initial condition E 2 (t 1 ) = E 1 . Let E(t) be defined by (4.1). and

x(t) = 0 for t ∈]0, t 1 [, ξ(E(t)) for t ∈]t 1 , +∞[, y(t) = 0 for t ∈]0, t 1 [, η(E(t)) for t ∈]t 1 , +∞[, Then, for any δ > 0 we have lim ε→0 (x(t, ε), y(t, ε)) = (x(t), y(t)), uniformly for t ∈ [δ, t 1 -δ] ∪ [t 1 + δ, +∞[ and lim ε→0 E(t, ε) = E(t), uniformly for t ∈ [0, +∞[.
Proof. The limit behavior of the solution for 0 ≤ t < t 1 is given by Lemma 2.2 and Theorem 3.6 and by Proposition 4.1 for t > t 1 .

Stability results.

We make the following assumption: (A4) We have P (0, η(0)) < 0 and P (ξ(0), 0) < 0. For example, this last assumption is satisfied in Fig. 2.2, right. It guarantees that the conclusions of Propositions 4.3, 4.4 and 4.5 hold. Thus, as a consequence of the results given in Section 4.1, after some fast and slow transitions, every trajectory of (1.4) arrives near the slow curve (x, y) = (ξ(E), η(E)) and approaches the equilibrium S as t → ∞ and ε → 0. More precisely, we prove the following result: Theorem 4.6. Let assumptions (A1) to (A4) be satisfied. Then the equilibrium S is practically semi-globally asymptotically stable in the positive octant, as ε → 0, i.e. for any solution γ(t, ε) = (x(t, ε), y(t, ε), E(t, ε)) of (1.4) we have lim t→+∞,ε→0 γ(t, ε) = long time near the equilibrium (0, η(c), c), before leaving its neighborhood, close to its unstable separatrix and jumping to the neighborhood of (ξ(c), η(c), c) and then going towards the equilibrium S. Hence, it can stay for a long time near the equilibrium (2.8). In this case, the equilibrium S is not PSGAS because the limit given in Theorem 4.6 could be non uniform with respect to the initial conditions. It is not very easy to verify that condition H(E) < G(E) of assumption (A4') is satisfied, since functions H and G are not so easy to compute. This assumption was made to guarantee that a solution of (1.4) with initial condition x 0 > 0, y 0 > 0 and E 0 > a * leave the neighborhood of point (0, 0, E 1 ) and jumps to the neighborhood of point (0, η(E 1 ), E 1 ) close to the orbit x = 0 of the unstable node (0, 0). Now, this is what may happen if condition P (ξ(0), 0) < 0 of Assumption (A4) is relaxed. If P (ξ(0), 0) > 0, then (2.9) is an equilibrium point of (1.4). Even if this equilibrium is unstable in the y-direction, there are solutions of (1.4) with initial condition x 0 > 0, y 0 > 0 and E 0 > 0 that can approach the equilibrium (2.9), as ε → 0. Such solutions will stay for a very long time near the equilibrium (ξ(d), 0, d), before leaving its neighborhood, close to its unstable separatrix and junping to the neighborhood of (ξ(d), η(d), d) and then going towards the equilibrium S. Then the solution can stay for a long time near the equilibrium (2.9). In this case again, the equilibrium S is not PSGAS because the limit given in Theorem 4.6 could be non uniform with respect to the initial conditions.

5.

Applications to Clarks's model. Assume that all parameters of (1.2) are positive. We have

ξ(E) = K 1 - q 1 r E , η(E) = L 1 - q 2 s E , a = r q 1 , b = s q 2 .
The slow manifold is represented in Fig. 5.1. On the other hand, we have P (x, y) = p 1 q 1 x + p 2 q 2 y -c, so assumption (A3) holds if and only if (see Fig. 5.1, right)

sq 1 -rq 2 sq 1 p 2 q 2 L < c < p 1 q 1 K + p 2 q 2 L. The equilibrium S = (x ∞ , y ∞ , E ∞ ) is given by E ∞ = (p 1 q 1 K + p 2 q 2 L -c) rs ∆ , x ∞ = (sq 1 c -(sq 1 -rq 2 )p 2 q 2 L) K ∆ and y ∞ = (rq 2 c + (sq 1 -rq 2 )p 1 q 1 K) L ∆ ,
where ∆ = sp 1 q 2 1 K + rp 2 q 2 2 L. Also, Assumption (A4) holds if and only if p 2 q 2 L < c and p 1 q 1 K < c. (5.2)

By straightforward calculations, we can show that assumption also holds. As a corollary of Theorem 4.7, we deduce the following result:

Theorem 5.1. Assume that max(p 1 q 1 K, p 2 q 2 L) < c < p 1 q 1 K + p 2 q 2 L and sq 1 > rq 2 . Then the equilibrium S is semi-globally asymptotically stable in the positive octant, as ε → 0. Proof. We have

ψ(0) ϕ(0) v = ϕ(u) d d v = ψ(u) v = u E G(E) H(E) u v
h(E) = E a r -q 1 u -cu du = q 1 E -r ln E -q 1 a -r ln a c .
Let E > b, u = H(E) is defined by

q 1 u -r ln u = q 1 E -r ln E ⇔ u = ϕ(u), where ϕ(u) := Ee u-E a .
Hence H(E) is a fixed point of the function v = ϕ(u). We have

ϕ(0) > 0, ϕ(E) = E, ϕ ′ (E) = E a > 1 and ϕ ′′ (u) > 0. Hence the function v = ϕ(u) has a unique fixed point u = H(E) in ]0, E[. Similarly we have g(E) = E b s -q 2 u -cu du = q 2 E -s ln E -q 2 b -s ln b c . Thus u = G(E) is the unique fixed point, in ]0, E[, of the function v = ψ(u) := Ee u-E b . We have ψ(0) < ϕ(0), ψ(E) = ϕ(E) = E and 1 a > 1 b .
Hence the functions ϕ and ψ satisfy (see Fig. 5.2) ψ(u) < ϕ(u) for all u ∈]0, E[, so that H(E) < G(E).

From Lemma 5.2 we deduce that, if condition (5.1) holds, then assumption (A4') holds if and only if p 2 q 2 L < c and this condition is less restrictive than (5.2). As a consequence of Theorem 4.8, we have: Theorem 5.3. Assume that p 2 q 2 L < c < p 1 q 1 K +p 2 q 2 L and sq 1 > rq 2 . Then the equilibrium S is semi-globally asymptotically stable in the positive octant, as ε → 0.

The system (1.2) is dissipative, for any ε > 0. Thus any solution enters in a compact subset of the positive octant. Hence for any fixed ε > 0, the equilibrium S is GAS in the positive octant. Actually, we proved as ε → 0, the attractivity is uniform with respect to the initial condition in any compact subset of the positive octant.

Numerical simulations.

To illustrate our results, we carried some numerical experiments with ε = 0.01 and the following set of parameters:

K = 3, L = 3, r = 1, s = 2, p 1 = p 2 = q 1 = q 2 = 1, c = 4. (5.3)
For clarity, we draw the projections of the trajectories, related to two sets of initial values (see Fig. 5.3 and Fig. 5.4), on the planes (E, y), on the left, (E, x), in the center and (x, y), on the right. The behavior of the first trajectory with x 0 = 4, y 0 = 4 and E 0 = 3 < a * is in accordance with the results of Theorem 3.5 and Proposition 4.2. The behavior of the trajectory with x 0 = 0, y 0 = 0 and E 0 = 6 > a * supports the results of Theorem 3.5 and Proposition 4.3. Appendix A. Slow and fast vector fields. In this appendix, we give the main definitions and tools of singular perturbation theory which are used in this paper. For more details and information the reader may consult [START_REF] Hoppensteadt | Singular perturbations on the infinite interval[END_REF][START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Kokotovic | Singular Perturbation Methods in Control: Analysis and Design[END_REF][START_REF] O'malley | Singular Perturbation Methods for Ordinary Differential Equations[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF]. We follow here the presentation given in [START_REF] Lobry | Singular Perturbation Methods in Control theory[END_REF][START_REF] Lobry | On Tikhonov's theorem for convergence of solutions of slow and fast systems[END_REF].

A.1. Stability in systems with parameters. The notion of uniform stability in systems with parameters is crucial in the hypothesis of Tikhonov's theorem. Let

z ′ = Z(z, a), z ∈ Z ⊂ R d
be a system depending on the parameter a ∈ A ⊂ R m . We assume that there exists a mapping a ∈ A → h(a) ∈ Z such that for all a ∈ A we have Z(h(a), a) = 0, so that z = h(a) is an equilibrium point. Definition A.1. The equilibrium z = h(a) of equation z ′ = Z(z, a) is said to be uniformly asymptotically stable for a ∈ A if for all µ > 0, there exists δ > 0, such that for all a ∈ A and for all solution z(τ, a) z(0, a) -h(a) < δ ⇒ ∀τ > 0 z(τ, a) -h(a) < µ and lim

τ →+∞ z(τ, a) = h(a)
Definition A.2. The equilibrium z = h(a) of equation z ′ = Z(z, a) is said to be uniformly exponentially stable in a ∈ A if there exist k > 0, γ > 0 r > 0 such that for all a ∈ A any solution z(τ, a) with z(0, a) -h(a) ≤ r satisfies z(τ, a) ≤ k z(0, a) e -γτ for all τ ≥ 0.

Note that for C 1 systems, the uniform exponential stability holds if all eigenvalues of the linear part ∂Z ∂z (h(a), a) have negative real parts. A.2. Practical asymptotic stability. To introduce the notion of practical asymptotic stability, consider the system depending on a parameter ẋ = F (x, ε), (A.1) where x ∈ X ⊂ R d and ε > 0 is a real parameter.

Definition A.3. The point x = x * is practically asymptotically stable (PAS) when ε → 0 if there exists A > 0, such that for all r > 0 there exist ε 0 > 0 and T > 0 such that for all ε, for all solution x(t, ε) of equation (A.1) and for all time t if ε < ε 0

x(0, ε) -x * < A and t > T then, x(t, ε) -x * < r.

In the case where the solution x(t; x 0 , ε) of (A.1), with initial condition x(0; x 0 , ε) = x 0 , is unique the point x = x * is PAS if and only if lim t→+∞,ε→0

x(t, x 0 , ε) = x * , the limit being uniform for x 0 in some ball of radius A independent on ε.

A.3. Semi global asymptotic stability. Here, we define the notion of semi global asymptotic stability.

Definition A.4. The point x = x * is semi-globally asymptotically stable (SGAS) when ε → 0 if it is asymptotically stable for all ε ∈]0, 1] and for all compact K ⊂ X there exists ε 0 > 0 such that for all ε < ε 0 , the basin of attraction of x * contains K.

Definition A.5. The point x = x * is practically semi-globally asymptotically stable (PSGAS) when ε → 0 if for any compact K ⊂ X and r > 0 there exist ε 0 > 0 and T > 0 such that for all ε, for all solution x(t, ε) of equation (A.1) and for all time t ε < ε 0 , x(0, ε) ∈ K and t > T then, x(t, ε) -x * < r.

In the case where the solution x(t; x 0 , ε) of (A.1) with initial condition x(0, x 0 , ε) = x 0 is unique, the point x = x * is PSGAS if and only if lim t→+∞,ε→0

x(t, x 0 , ε) = x * , the limit being uniform for x 0 in any prescribed compact subset of X. A.7. Practical semi global asymptotic stability. In general, the attractivity of the full system is not global as shown by the following example:

Example. Consider the two dimensional slow fast system u = -u + u 2 v, ε v = -v. (A.8)

The fast equation v ′ = -v has v = 0 as a globally exponentially stable equilibrium. The corresponding slow equation u = -u has u = 0 as a globally exponentially stable equilibrium. By theorem A.8, the origin of (A.8) is exponentially stable for small ε. Actually, the origin is exponentially stable for all ε > 0, but the attractivity is not global since

d dt (uv) = uv(uv -1 -1/ε),
shows that the hyperbola uv = 1 + 1/ε is invariant. The basin of attraction of the origin is the set B := {(u, v) ∈ R 2 : uv < 1 + 1/ε}. Thus, the origin is not GAS for (A.8). However, the origin of (A.8) is semi-globally asymptotically stable when ε → 0. Let us give, now a result of practical semi global asymptotic stability in the slowfast system (A.2). We do not assume that U (u ∞ , v ∞ , ε) = 0 and V (u ∞ , v ∞ , ε) = 0 as in the previous section. Hence the point (u ∞ , v ∞ ) is not necessarily an equilibrium of (A.2).

Theorem A.9. Assume that U (u ∞ , v ∞ , 0) = 0 and V (u ∞ , v ∞ , 0) = 0, the equilibrium u = u ∞ is GAS for u = U (u, W (u), 0) and the equilibrium v = W (u) is GAS for v ′ = V (u, v, 0). Then the point (u ∞ , v ∞ ) is PSGAS for (A.2) as ε → 0.

Proof. See [START_REF] Lobry | Singular Perturbation Methods in Control theory[END_REF].

A.8. Canard Solutions and Delayed Loss of Stability. The results of the previous section do not apply easily, since in most examples met in applications, it is rare that the slow manifold has a unique component and that the equilibrium of the slow equation is GAS. In general, the slow manifold has several components and the solution jumps from one to the other. These jumps are often accompanied by the phenomenon of canard solutions or delayed loss of stability phenomenon. For details and complements see [START_REF] Arnold | Dynamical Systems V, Bifurcation Theory and Catastrophe Theory[END_REF], p. 179-192 and [START_REF] Benoît Ed | Dynamic Bifurcations[END_REF]. Stability loss delay in dynamical bifurcations is an important and newly discovered phenomenon [START_REF] Neishtadt | Asymptotic investigation of the loss of stability by an equilibrium as a pair of eigenvalues slowly crosses the imaginary axis[END_REF]. Consider again the slow-fast system, x ′ = ε, z ′ = Z(x, z, ε) (A.9) where x ∈ R and where system z ′ = Z(x, z, 0) has an equilibrium z = h(x) for each value of the parameter x. Suppose that there exists a bifurcation value x = x * of the parameter, at which the equilibrium loses stability, that is, z = h(x) is asymptotically stable for x < x * and unstable for x > x * . Assume that x(0, ε) = x 0 < x * . Then the solution of system (A.9) will go quickly near the equilibrium z = h(x 0 ) and then remains close to the curve z = h(x), until x reaches some value x 1 > x * . This means that the loss of stability which must occur at x = x * is delayed until x = x 1 . The general theory in [START_REF] Neishtadt | Asymptotic investigation of the loss of stability by an equilibrium as a pair of eigenvalues slowly crosses the imaginary axis[END_REF] does not apply in our system. This theory requires that a pair of eigenvalues crosses the imaginary axis, when in our case a real eigenvalue crosses zero. The stability loss delay phenomenon is closely related to the phenomenon of canard solutions. Canard solutions are special trajectories of slow and fast systems that first move near the stable part of the slow manifold, then move near the unstable part of it. These solutions were first discovered in the framework of Nonstandard Analysis. See [START_REF] Benoît | Chasse au canard[END_REF][START_REF] Diener | Ducks and rivers: three existence results[END_REF][START_REF] Zvonkin | Nonstandard Analysis and Singular Perturbations of Ordinary Differential Equations[END_REF] for historical comments and references. The study of canard solutions has also been made in the framework of classical asymptotic analysis [START_REF] Eckhaus | Relaxation oscillations including a standard chase on French ducks. in "Asymptotic analysis[END_REF] and also in the framework of GSPT, using center manifolds and blow-up. See [START_REF] Dumortier | Canard cycles and center manifolds[END_REF][START_REF] Wechselberger | Existence and Bifurcation of Canards in R 3 in the Case of a Folded Node[END_REF] for details and references.

Fig. 2 . 2 .

 22 Fig. 2.2. On the left, the slow manifold of system (1.4): attracting parts of the slow manifold are indicated by a bold line, non attracting parts of the slow manifold are indicated by a dashed line. On the right the relative positions of Γ and Π.

Fig. 2 .

 2 Fig. 2.3. The asymptotic behavior of the solution of (1.4) with initial condition m 0 = (x 0 , y 0 , E 0 ) when a < E 0 < b (on the left) and E 0 > b (on the right). The expected asymptotic behavior, when E crosses the value a (on the left) or the value b (on the right) is not correct. The correct behavior, showing the delayed loss of stability, is described in Fig. 3.2, 3.3, 3.4 or 3.5.

3 . 1 ,

 31 right). This function is decreasing from +∞ to 0 on ]0, b] and increasing on [b, +∞[. It defines a mapping E ∈ [b, ∞[ → G(E) ∈]0, b], by G = g -1 + • g -, where g -and g + are the restrictions of g on [b, +∞[ and ]0, b] respectively. Let h the function defined by

Fig. 3 . 1 .

 31 Fig. 3.1. On the left: the function f defining the entrance-exit function E → F (E) along the slow curve (x, y) = (0, η(E)). On the right: the functions g (in red) and h (in blue) defining the mappings E → G(E) and E → H(E) respectively.

Definition

3

 3

. 4 .

 4 Now E 1 ∈]0, a[ and c = 0. The integral in (3.15) is well defined since P (0, η(u)) < 0 for all u ∈ [0, a]. Hence E 2 , given by (3.14), satisfies E 2 ∈]0, E 1 [ (see Fig. 3.4, left).

Fig. 5 . 1 .

 51 Fig. 5.1. On the left: the slow manifold of system (1.2). Attracting parts of the slow manifold are indicated by a bold line, non attracting parts of the slow curve are indicated by a dashed line. On the right: the relative positions of Γ and Π. For the set of parameters (5.3) we have P (ξ(0), 0) > 0.

Fig. 5 . 2 .

 52 Fig. 5.2. The functions H and G corresponding to (1.2) satisfy H(E) < G(E) for all E > b. The result of Theorem 5.1 occurs for a larger range of parameters since H(E) < G(E) is always true for (1.2) as shown in the following lemma: Lemma 5.2. Under condition (5.1) we have H(E) < G(E), for all E > b.Proof. We have

Fig. 5 . 3 .Fig. 5 . 4 .

 5354 Fig. 5.3. Numerical solutions of (1.2) with x 0 = 4, y 0 = 4 and E 0 = 3 < a * .

S, the limit being uniform with respect to the initial condition in any compact subset of the positive octant.

Proof. Let t → e(t, E 0 ) = E(t) where E(t) is the function defined by Proposition 4.1 when E 0 < a, by Proposition 4.2 when a < E 0 < b and by Proposition 4.3, 4.4 or 4.5 when E 0 > b respectively. For all E 0 > 0, we have lim t→∞ e(t, E 0 ) = E ∞ . Let r > 0 and let T (E 0 ) = inf{t > 0 : |e(t, E 0 ) -E ∞ | ≤ r/2}. Let K be a compact subset of the positive octant and let (x 0 , y 0 , E 0 ) ∈ K. Let T = max{T (E 0 ) : (x 0 , y 0 , E 0 ) ∈ K}. There exists ε 0 > 0 such that for all t > T and all ε ∈]0, ε 0 ], we have γ(t, ε) -S ≤ r. Thus S is PSGAS as ε → 0.

In order to reach an asymptotic stability instead of a practical asymptotic stability we need the supplementary assumption: (A5) The functions M , N and P are of class C 1 and satisfy ∂M ∂x (ξ(E), E) < 0, for all 0 < E < a and ∂N ∂y (η(E), E) < 0, for all 0 < E < b. We also have

Theorem 4.7. Let assumptions (A1) to (A5) satisfied. Then the equilibrium S is semi-globally asymptotically stable in the positive octant, as ε → 0, that is, it is asymptotically stable and for any compact subset K of the positive octant there exists ε such that, for all ε < ε 0 , K is in the basin of attraction of S.

Proof. From assumption (A5), the equilibrium (ξ(E), η(E)) is uniformly exponentially stable for a < E < b. On the other hand, we have

where p(E) = EP (ξ(E), η(E)). Hence the equilibrium E = E ∞ of the slow equation (2.4) is exponentially stable. Thus, Theorem A.8 applies and there exists ε 0 > 0 such that for all 0 < ε < ε 0 , the equilibrium S is exponentially stable for system (1.4). Using Theorem 4.6, we see that the attractivity is semi global. Hence S is SGAS as ε → 0. Instead of Assumption (A4), we make now the following assumption: (A4') We have P (0, η(0)) < 0 and for all E > a * , H(E) < G(E). Under this assumption, we always have H(E 0 )) < G(E 0 ) and the asymptotic behavior of all solutions for which E 0 > a * is described by Proposition 4.3. We do not need the condition P (ξ(0), 0) < 0, as it is the case in Fig. 5.1, right. So we have the following result:

Theorem 4.8. The conclusions of Theorems 4.6 and 4.7 hold if assumption (A4) is replaced by assumption (A4').

Remarks.

In this section, we give some comments on the assumptions (A4) and (A4'). In Section 2.3 we have seen that (1.4) has at least four equilibria (2.6) and (2.7). The equilibria (2.7) lie in the invariant xy-plane. From the description of the solutions in Section 4.1, we see that there are no solution of (1.4) with initial condition x 0 > 0, y 0 > 0 and E 0 > 0 that can approach the equilibria (2.7) as ε → 0.

Let us examine what may happen if condition P (0, η(0)) < 0 of Assumption (A4') is relaxed. In the case where P (0, η(0)) > 0, (2.8) is an equilibrium of (1.4). Even if this equilibrium is unstable in the x-direction, there are solutions of (1.4) with initial condition x 0 > 0, y 0 > 0 and E 0 > 0 that can approach the equilibrium (2.8), as ε → 0 (see remark following Theorem 3.6). Such solutions will stay for a very

dt and ε is a real parameter. We assume that U and V are continuous functions and that problem (A.2) has a unique solution. We look at the solutions behavior when ε → 0 and t ∈ [0, T ]. The small parameter is multiplying the derivative and so the usual theory of continuous dependence of the solutions with respect to the parameters cannot be applied. If we write (A.2) at time scale τ = t/ε we obtain

where u ′ = du dτ and v ′ = dv dτ . The equation

where u a constant parameter, is called the fast equation. We assume that it has a unique solution with prescribed initial condition. The manifold L of equation

is called the slow manifold: it is the set of equilibria of (A.4). We make the following assumptions which are listed by the letter (H) below.

(H1) For all u in some compact domain K ⊂ R n , all solutions of (A.4) tend towards an equilibrium v = W (u), where v = W (u) is an isolated root of equation (A.5). The equilibrium v = W (u) is uniformly asymptotically stable for u ∈ K, (see definition in Appendix A.1).

(H2) We have α 0 ∈ K and β 0 belongs to the basin of attraction of W (α 0 ). On the component v = W (u) we define the slow equation

We assume that this equation has a unique solution with prescribed initial condition. Let u(t) be the solution of (A.6) with initial condition u(0) = α 0 .

(H3) We assume that u(t) is defined for 0 ≤ t ≤ T . Theorem A.6. Under hypothesis (H1) to (H3), the solution (u(t, ε), v(t, ε)) of (A.2) is defined at least on [0, T ]. For all δ > 0, it satisfies lim ε→0 u(t, ε) = u(t) uniformly on [0, T ] and

Proof. See [START_REF] Lobry | On Tikhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters multiplying the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF].

The limit for the fast variable v does not hold at 0 since there is a boundary layer in z. More precisely, let ũ(τ ) be the solution of the boundary layer equation

This solution is defined for all τ ≥ 0 and tends to W (α 0 ). We have:

A.5. Approximation for all t ≥ 0. To obtain approximations on the infinite interval, we need a supplementary assumption.

(H4) The slow equation (A.6) has an asymptotically stable equilibrium u = u ∞ and the initial condition α 0 is in its basin of attraction.

Theorem A.7. Under hypothesis (H1) to (H4), the solution (u(t, ε), v(t, ε)) of (A.2) is defined for all t ≥ 0. For all δ > 0, it satisfies lim ε→0 u(t, ε) = u(t) for all t ≥ 0 and lim ε→0 v(t, ε) = W (u(t)) for all t ≥ δ.

Proof. See [START_REF] Hoppensteadt | Singular perturbations on the infinite interval[END_REF][START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Lobry | On Tikhonov's theorem for convergence of solutions of slow and fast systems[END_REF]. From theorem A.7 we deduce that lim t→+∞,ε→0

where v ∞ = W (u ∞ ). This result does not imply that (u ∞ , v ∞ ) is an asymptotically stable equilibrium of (A.2) as it is illustrated by the following example:

Example. Consider the planar slow-fast system

The slow manifold is v = 0 and v = 0 is GAS for the fast equation v ′ = -v. The equilibrium u = 0 is GAS for the reduced equation u = -u 3 . However, the equilibrium (0, 0) of (A.7) is unstable.

A.6. Stability results. To obtain stability results of the equilibrium (u ∞ , v ∞ ), we need supplementary conditions on the system. Assume that for all ε > 0, we have U (u ∞ , v ∞ , ε) = 0 and V (u ∞ , v ∞ , ε) = 0, so that, for all ε > 0, (u ∞ , v ∞ ) is an equilibrium of system (A.2). We assume that asymptotic stability is replaced by exponential stability in assumptions (H1) and (H4). We have the following result:

Theorem A.8. Let hypothesis (H1) to (H4) be satisfied, where asymptotic stability is replaced by exponential stability, that is, v = W (u) is an uniformly exponentially stable point for v ′ = V (u, v, 0) on u ∈ K and u ∞ is an exponentially stable point for u = U (u, W (u), 0). Then, there exists ε * > 0 such that for all 0 < ε < ε * , (u ∞ , W (u ∞ )) is exponentially stable for system (A.2).

Proof. See [START_REF] Khalil | Nonlinear Systems[END_REF], Section 9.4 or [START_REF] Kokotovic | Singular Perturbation Methods in Control: Analysis and Design[END_REF], Section 7.5. Exponential stability cannot be replaced by asymptotic stability as shown by (A.7). In this example x = 0 is GAS but not exponentially stable for the reduced equation ẋ = -x 3 and the origin is unstable for the complete system.

Remark. Similar results was obtained using geometric singular perturbation theory (GSPT) (see [START_REF] Isidori | Nonlinear Control Systems[END_REF], Section B.3). In GSPT [START_REF] Fenichel | Geometric singular perturbation theory for ordinary differential equations[END_REF][START_REF] Jones | Geometric singular perturbation theory[END_REF][START_REF] Kaper | An introduction to geometric methods and dynamical systems theory for singular perturbation problems[END_REF], system (A.2) is called the slow system and system (A.3) is called the fast system. In this paper we adopted the terminology of the classical singular perturbation theory [START_REF] Lobry | On Tikhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters multiplying the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] and we refer to (A.4), which is the limit of (A.3) when ε → 0, as the fast equation and to (A.6), which is the limit of (A.2) when ε → 0, as the slow equation. Note that in GSPT, the slow manifold is not necessarily attracting, as in Tikhonov's theorem. In GSPT the results hold for the more general case of slow and fast systems for which the slow manifold is normally hyperbolic, i.e., the real part of the eigenvalues of ∂V ∂v (u, W (u), 0) are non 0, possibly positive or negative. However, in GSPT, the functions U and V in (A.2) must be smooth, not only continuous, as it is the case in Tikhonov's theory. GSPT shows that for small ε > 0, (A.2) has a locally invariant manifold which is O(ε)-close to the slow manifold.