domain: shs math sdv.ee qfin

text:
H G O Me S

C R A I N E R

L Doyen
email: luc.doyen@cnrs.fr

M D Smith

U R Sumaila

G Zaccour

I Ekeland

P Cury

C Lett

O Thebaud

J.-C Poggiale

A Moussaoui

J.-M Fromentin

S Gourguet

P Guillotreau

H Gomes

P Courtois

R Schaap

F Blanchard

C Rainer

T Villain

F Menard

T Sari

Mathematical Bio-Economics 2.0 for Sustainable Fisheries

Keywords: Bio-economics, Fisheries, Marine Biodiversity, Marine Ecosystems, Ecosystem services, Modeling, Scenarios, Management, Public policy, Dynamic systems, Control theory, Game theory, Sustainability, Resilience, Governance

Reconciling food security, economic development and biodiversity conservation in the face of global changes is a major challenge. The sustainable uses of marine biodiversity in the context of climate change, invasive species, water pollution and demographic growth is an example of this bio-economic challenge. There is a need for quantitative methods, models, scenarios and indicators to support policies addressing this challenge. Although bio-economic models for marine resources date back to the 1950s and are still used in fisheries management and policy design, they need major improvements, extensions and breakthroughs. This paper proposes to design a Mathematical Bio-Economics 2.0 for Sustainable Fisheries in order to advance the development of bio-economic models and scenarios for the management of fisheries and marine ecosystems confronted with unprecedented global change. These models and scenarios should make both ecological and economic senses while being well-posed mathematically and numerically. To achieve this, we propose to base the Mathematical Bio-Economics 2.0 framework for Sustainable Fisheries on four research axes regarding the mathematics and modeling of: (i) ecosystem-based fisheries management; (ii) criteria of sustainability; (iii) criteria of resilience; (iv) governance and strategic interactions. The associated methodology of Mathematical Bio-Economics 2.0 draws mainly on dynamic systems theory, optimal and viable controls of systems, game theory and stochastic approaches. Our analysis, which is based on these four axes, allows us to identify the main methodological gaps to fill compared to current models for fisheries management.

Introduction

Balancing biodiversity conservation with food security and the preservation of a broader set of ecosystem services (ESs), in a context of ecological transition and climate change, is one of the greatest challenges of the century [START_REF] Godfray | Food security: The challenge of feeding 9 billion people[END_REF][START_REF] Rice | Fisheries, food security, climate change, and biodiversity: characteristics of the sector and perspectives on emerging issues[END_REF][START_REF] Crist | The interaction of human population, food production, and biodiversity protection[END_REF]. The creation and development of the IPBES (International Platform for Biodiversity and Ecosystem Services) at the interface of decisionsupport and scientific knowledge is in direct line with these concerns [START_REF] Díaz | The ipbes conceptual framework -connecting nature and people[END_REF]. Implementing this bio-economic perspective is particularly important and challenging in the case of fisheries and marine ecosystems. Marine fisheries indeed employ 120 million people directly and indirectly around the world, and account for nearly 500 million people who depend on it for their livelihood (FAO, Duke University and Worldfish, 2023). Moreover, fisheries and marine ecosystems are experiencing accelerating changes affecting species, communities and trophic webs, sometimes with alarming trends [START_REF] Butchart | Global biodiversity: Indicators of recent declines[END_REF][START_REF] Österblom | Where and how to prioritize fishery reform[END_REF][START_REF] Smith | Sustainability and global seafood[END_REF]IPBES, 2019). An important part of these changes is due to past and current unsustainable fishing pressures and practices, raising key questions in terms of food security, in particular for developing countries in the tropics facing high human population growth and a growing middle class demand for seafood [START_REF] Asche | Aquaculture: Externalities and policy options[END_REF][START_REF] Cojocaru | The "seafood" system: Aquatic foods, food security, and the global south[END_REF]. Climate change exacerbates the issues and iniquities between high-income and low-income countries by inducing new, or intensifying, risks and vulnerabilities [START_REF] Cury | Ecosystem oceanography for global change in fisheries[END_REF][START_REF] Sumaila | Climate change impacts on the biophysics and economics of world fisheries[END_REF][START_REF] Fromentin | Summary for policymakers of the thematic assessment of the sustainable use of wild species of the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES)[END_REF][START_REF] Barange | Impacts of climate change on fisheries and aquaculture: synthesis of current knowledge, adaptation and mitigation options. Food & Agriculture Organization of the United Nations (FAO)[END_REF]) through e.g., changes in primary production and fish distribution, thus potentially affecting yields [START_REF] Kroetz | The future of wild-caught fisheries: Expanding the scope of management[END_REF].

Ensuring the long-term sustainability of marine fisheries while preserving the marine biodiversity and ecosystems functioning that support them, have become a major issue for national and international agencies (UNSDG; IPBES). However, the agencies and other management institutions require more quantitative methods, models, scenarios and indicators to support policies addressing jointly food security, economic development, and conservation of biodiversity and ESs. Fortunately, with respect to marine fisheries, we are not starting from scratch. Mathematical models applied to fisheries have been used for many decades building, in particular, on the seminal works of [START_REF] Gordon | The Economic Theory of a Common-Property Resource: The Fishery[END_REF] and [START_REF] Schaefer | Some aspects of the dynamics of populations important to the management of the commercial marine fisheries[END_REF] combining equilibrium and optimality approaches. These key initial bio-economic contributions have been extended and generalized by [START_REF] Smith | On models of commercial fishing[END_REF]; [START_REF] Clark | The economics of overexploitation[END_REF] and many others using the more dynamic frameworks of optimal control and capital theory. Some of the proposed quantitative and modeling concepts such as Maximum Sustainable Yield (MSY) or Maximum Economic Yield (MEY) are still widely used worldwide to support management, public policy design and development for fisheries [START_REF] Nielsen | Integrated ecological-economic fisheries models-evaluation, review and challenges for implementation[END_REF]Thébaud et al., 2023a). In that regard, important successes have been obtained applying MSY or MEY, including recovering stocks in Europe [START_REF] Froese | Status and rebuilding of European fisheries[END_REF], Australia [START_REF] Dichmont | On implementing maximum economic yield in commercial fisheries[END_REF], Canada [START_REF] Teh | Contribution of marine fisheries to worldwide employment[END_REF] and at the global scale [START_REF] Sumaila | Benefits of rebuilding global marine fisheries outweigh costs[END_REF][START_REF] Bank | The Sunken Billions Revisited: Progress and Challenges in Global Marine Fisheries[END_REF].

Despite successes of these key bio-economic models, major improvements, extensions or breakthroughs are required in the face of the global marine biodiversity erosion, the stagnation of capture fishery landings, and the growing global demand for seafood (Food and Agriculture Organization, 2020). New bio-economic models and mathematics are needed to sustainably balance food security, economic development of fisheries and marine biodiversity conservation in the context of global changes. In particular, these new bio-economic models and mathematics need to explore broader conceptualizations of sustainability, more comprehensive ecosystem dynamics and more participative and adaptive governance. We argue here that such scientific progress require new contributions and involvements of Mathematicians and Modelers (De [START_REF] Lara | Sustainable management of natural resources: mathematical models and methods[END_REF][START_REF] Thébaud | Building ecological-economic models and scenarios of marine resource systems: Workshop report[END_REF][START_REF] Doyen | Ecological-economic modelling for the sustainable management of biodiversity[END_REF][START_REF] Doyen | Mathematics for Scenarios of Biodiversity and Ecosystem Services[END_REF][START_REF] Maury | An overview of apecosm, a spatialized mass balanced "apex predators ecosystem model" to study physiologically structured tuna population dynamics in their ecosystem[END_REF]. Thus, we advocate a new 'Mathematical Bio-economics for Sustainable Fisheries' to advance the development of models and scenarios for the sustainable management of fisheries and marine ecosystems. In particular, these models and scenarios should take into account the full complexity of these systems, including their ecological and socio-economic dimensions, while being well-posed mathematically and numerically. To achieve this, we propose to split the 'Mathematical Bio-economics 2.0 for Fisheries' into four main research challenges and axes, that are the mathematics and modeling of:

(i) Ecosystem-based fisheries management and scenarios;

(ii) Criteria of sustainability and conservation in fisheries;

(iii) Criteria of resilience in fisheries;

(iv) Governance, instruments and strategic interactions for fisheries.

These four axes and challenges aim at bridging the gap between theory and management practice for fisheries. Challenge (i) refers mostly to the ecological complexity brought about by moving from single-species models exemplified by MSY or MEY towards multi-species and trophic web approaches. Challenge (ii) relates to both intergenerational equity and the multi-criteria perspectives underlying the definition of fisheries sustainability. Challenge (iii) essentially accounts for the numerous uncertainties underpinning the bio-economic dynamics for fisheries, including climate change. Challenge (iv) stresses the decisional complexity involved in fisheries management in particular with regard to the governance of natural resources used in common. Our paper detail the motivations, goals as well as the methodological content of these four axes. We also synthesize the merits and shortcomings of the different classes of current models for fisheries along with the key gaps to fill for achieving a successful Mathematical Bio-economics 2.0 for sustainable uses of of marine biodiversity.

The rest of the paper is organized as follows: a preliminary section 2 reminds the numerous intersecting interests of bio-economic models, mathematics and theory. Then, in Section 3, we elaborate on the mathematics of ecosystem-based fisheries management and scenarios. The mathematics for criteria of sustainability and of resilience in fisheries are then discussed in Section 4 and 5 respectively. Section 6 gives insights into the modeling and mathematical challenges emerging from the governance, instruments and strategic interactions in fisheries. We conclude, in particular, through Table 1, by identifying the major methodological gaps for the design of a successful Mathematical Bio-economics 2.0 for Sustainable Fisheries.

Why are bio-economic mathematical models and theory needed ?

A mathematical model is an abstract description of a concrete system using mathematical concepts and language. Mathematical models are used in many disciplines ranging from life sciences, physics to social sciences as well as engineering disciplines. Bio-economic models are models coupling biological and/or ecological systems with models of human and social systems including economic activities [START_REF] Gordon | The Economic Theory of a Common-Property Resource: The Fishery[END_REF][START_REF] Schaefer | Some aspects of the dynamics of populations important to the management of the commercial marine fisheries[END_REF][START_REF] Clark | The economics of overexploitation[END_REF][START_REF] Wilen | Chapter 2 bioeconomics of renewable resource use[END_REF][START_REF] Seijo | of the United Nations. Fisheries Bioeconomics: Theory, Modelling and Management[END_REF][START_REF] Béné | A viability analysis for a bio-economic model[END_REF][START_REF] Lara | Sustainable management of natural resources: mathematical models and methods[END_REF][START_REF] Smith | Bioeconometrics: Empirical modeling of bioeconomic systems[END_REF]. Bio-economic models can thus be applied to many environmental issues arising in forestry, agriculture, water use, fisheries management and, more broadly, to ecosystems and biodiversity management. Bio-economic models are thus used to study social-ecological systems [START_REF] Ostrom | A General Framework for Analyzing Sustainability of Social-Ecological Systems[END_REF], and are considered as early examples of modeling coupled human-natural systems [START_REF] Abbott | Common property resources and the dynamics of overexploitation: The case of the north pacific fur seal-a 42-year legacy[END_REF][START_REF] Ferraro | Causal inference in coupled human and natural systems[END_REF]. In this preliminary section, we first question the interest of such mathematical models for fisheries. It turns out that the role and purpose of bio-economic models are multiple in nature, in particular, when dealing with fisheries and marine ecosystems. We hereafter distinguish between five key roles of models useful for Mathematical Bio-Economics 2.0 for Fisheries: (i) models to understand the underlying systems; (ii) models for interdisciplinarity; (iii) models to fit the data; (iv) models to evaluate systems; (v) models for decision making.

Firstly, mathematical models are needed to understand the social-ecological systems involved in fisheries and marine biodiversity. This is because mathematics, in particular, mechanistic models, simplify the processes at play and describe them using equations. In other words, a model helps to explain a system, to disentangle its different components and to study the effects of these different components. In that regard, dealing with marine biodiversity is a challenge because of its intrinsic complexity [START_REF] Doyen | Mathematics for Scenarios of Biodiversity and Ecosystem Services[END_REF]. In particular, processes involved in fisheries are numerous and nonlinear. Sections 3 and 6 pay particular attention to such complexity, both ecological and economic.

Secondly, mathematical models are needed for interdisciplinarity. Mathematics constitute an interdisciplinary or even transdisciplinary language. This is crucial for bio-economics and, in particular, for fisheries where there is a need to articulate the dynamics of ecological and economic systems and to manage their interactions, especially in a context of global change [START_REF] Abbott | Common property resources and the dynamics of overexploitation: The case of the north pacific fur seal-a 42-year legacy[END_REF][START_REF] Sumaila | Climate change impacts on the biophysics and economics of world fisheries[END_REF]Thébaud et al., 2023a). The various ways that sustainability sciences refer to systems -coupled human and natural, social-ecological, or nature-society -all highlight their inherent interdisciplinary content. Bio-economics have been interdisciplinary since day 1, with models that depict the fish stock (the natural system) together with the fishing activity (the human system). The transdisciplinary virtue of models and in particular bio-economic models is beneficial to participatory approaches that aim to bring together different actors, despite their heterogeneity. Such a feature is elaborated in Section 4 where we discuss sustainability and the underlying multi-criteria requirements. This feature is also discussed in Section 6 where strategic interactions between stakeholders are considered.

Thirdly, mathematical models are needed to calibrate and validate models with data. Said differently, mathematical models are useful to help confront our (conceptual) understanding of fisheries systems with observations of these systems' properties. What is meant here is that the mathematics and methods underlying statistics, machine learning or econometrics play a pivotal role in rigorously fitting the data to models to understand system properties [START_REF] Smith | Bioeconometrics: Empirical modeling of bioeconomic systems[END_REF]. For bio-economic models in fisheries, this can mean relying on spatio-temporal data about abundance (or biomass) of species, diet (stomach) information, data on sea surface temperature, ocean acidification as well as data on fishing catches, efforts, prices or seafood consumptions. Such data are clearly non-homogeneous. Moreover, these data can also be of poor quality, as many countries lack the institutional capacity to collect high-quality data and track large fleets in small-scale fisheries [START_REF] Cojocaru | The "seafood" system: Aquatic foods, food security, and the global south[END_REF].

Fourthly, mathematical models can be used to evaluate marine social-ecological systems. Models can also synthetize, summarize or highlight key information on the system performance and how it relates to reference values. Bio-economic indicators and metrics include biodiversity metrics, fishing fleet profits, demand of consumers, to mention a few [START_REF] Seijo | Uncertainty in bio-economic reference points and indicators of marine fisheries[END_REF]; and increasingly there is interest in triple bottom line indicators (people, planet, profit) for fisheries [START_REF] Péreau | The triple bottom line: Meeting ecological, economic and social goals with individual transferable quotas[END_REF][START_REF] Anderson | The fishery performance indicators: A management tool for triple bottom line outcomes[END_REF]. Such bio-economic indicators and criteria are necessary for relevant empirical analysis of fisheries data. Still, such indicators are clearly non-homogeneous and difficult to aggregate and integrate, and some simple indicators based on catches alone can be misleading about the status of the fishery [START_REF] Li | Fishery Collapse Revisited[END_REF]. In that regard, of interest for bio-economic models are assessments relating to the good health, the safety, the sustainability and viability of systems based on tolerable or desirable thresholds [START_REF] Bruckner | Climate change decision-support and the tolerable windows approach[END_REF][START_REF] Béné | A viability analysis for a bio-economic model[END_REF][START_REF] Cury | Viability theory for an ecosystem approach to fisheries[END_REF][START_REF] Rockström | A safe operating space for humanity[END_REF][START_REF] Baumgärtner | Ecological-economic viability as a criterion of strong sustainability under uncertainty[END_REF][START_REF] Oubraham | A survey of applications of viability theory to the sustainable exploitation of renewable resources[END_REF].

Fifthly, mathematical models are useful for producing insights and results in support of decision making and management. In particular model-based scenarios to derive projections about the future from a model are very informative tools for management and public policies (IPBES, 2016) in fisheries as illustrated in Figure 1. As stressed by IPBES (2016) or [START_REF] Doyen | Mathematics for Scenarios of Biodiversity and Ecosystem Services[END_REF], model-based scenarios are paths-trajectories relying on a mathematical or numerical model that is consistent with historical or observed evolutions, dynamics and data. This consistency is usually obtained through the calibration of models. In other words, the estimation of the value of parameters underpinning the model is achieved by fitting the observations and data to the outputs induced by the model. Once calibrated or estimated, bio-economic models can be used to conduct ex post policy evaluation, i.e. to evaluate the causal effects of a policy that was put in place [START_REF] Ferraro | Causal inference in coupled human and natural systems[END_REF] or the gaps between observed outcomes of such a policy and those theoretically expected. Calibrated bio-economic models can also be used to conduct ex ante policy evaluation by projecting the effects of a policy intervention and incentives through scenarios. At this stage, it is useful to distinguish between predictive, exploratory and normative scenarios [START_REF] Börjeson | Scenario types and techniques: towards a user's guide[END_REF], as described in Figure 1. Predictive scenarios, such as forecasts, can respond to the question 'What is likely to happen?' Predictive scenarios include status quo or business as usual, baseline or most likely scenarios. Exploratory scenarios describe other alternatives of the future and intend to respond to the question 'What could happen if. . . ?' [START_REF] Maury | From shared socio-economic pathways (ssps) to oceanic system pathways (osps): Building policy-relevant scenarios for global oceanic ecosystems and fisheries[END_REF]. They aid in the decision support process to investigate the outcomes of specific management strategies or drivers including economic, social or technological factors, but also climate change. The four scenarios of MEA (2005) including 'Global Orchestration', 'Adapting Mosaic', 'TechnoGarden' and 'Order from Strength' exemplify such explorative scenarios. By contrast, target-seeking scenarios or normative scenarios, deal with the question 'What should happen to reach. . . ?' and represent an agreed-upon future goals and scenarios that provide alternative pathways for reaching such an objective. In that normative context, determining sustainability scenarios for marine biodiversity and ecosystem services constitutes an important challenge.

In the perspective of model-based scenarios, there is a debate between mechanistic and statistical causal inference models. Mechanistic models are based on laws, functional responses, dynamics and processes; as such, mechanistic models are valid far beyond the calibration space if they are well parameterized empirically. Such flexibility and ubiquity is limited with causal inference statistical models, which typically rely on fewer assumptions but are fundamentally about evaluating past interventions (e.g, the effects of catch shares on the race to fish as in Birkenbach et al. ( 2017)). In practice, one can easily change process setup and parameters in mechanistic models to generate process understanding, but they are only as good as their underlying assumptions and empirical grounding. Increasingly, there are efforts to combine bio-economic mechanistic models with policy evaluation to prevent us from drawing spurious inferences and thereby guarding against bad policy decisions [START_REF] Smith | Seafood prices reveal impacts of a major ecological disturbance[END_REF][START_REF] Ferraro | Causal inference in coupled human and natural systems[END_REF]Li and Smith, 2021a). Implicit in developing predictive, normative, and exploratory scenarios is a parameterization of causal mechanisms, which creates a dialog between bio-economic model development and empirical causal inference studies in applied economics and other disciplines [START_REF] Ferraro | Causal inference in coupled human and natural systems[END_REF][START_REF] Schlüter | Unraveling complex causal processes that affect sustainability requires more integration between empirical and modeling approaches[END_REF].

Mathematics of the ecosystem-based management approach

This section outlines the ecological complexity to consider in a Mathematical Bio-economics 2.0 Program for Sustainable Fisheries. Many scientists and stakeholders indeed argue that the current shortcomings of public policies for the management of biodiversity and ecosystems can be explained by an insufficient account for complexity in existing models. Typically in fisheries, they advocate an ecosystem-based fishery management (EBFM) [START_REF] Pikitch | Ecosystem-based fishery management[END_REF]ICES;NOAA). Operationalizing such a framework is however a difficult challenge that entails moving from single-species models such as those underlying MSY or MEY towards multi-species, trophic interactions and spatially explicit models [START_REF] Fulton | Ecosystems say good management pays off[END_REF].

As a preliminary step before multi-species and ecosystem complexities, the account of life cycles in species dynamics can be considered as a first stage of complexity [START_REF] Quinn | Quantitative fish dynamics[END_REF]. Recruitment of new individuals into a single population is indeed already "a complex process in the sense that it is determined by many factors operating and interacting on multiple time and spatial scales in various environments" [START_REF] Pineda | Complexity and simplification in understanding recruitment in benthic populations[END_REF]. Indeed, most marine organisms change across life stages while they disperse into the ocean from spawning to settlement areas into which they recruit, being successively eggs, then larvae carrying their own food reserve, then larvae able to feed externally, and finally juveniles with swimming capacities.

Interspecific interactions constitute key elements of ecological complexity and EBFM as captured by Figure 2. Interspecific interactions include trophic (predator-prey), competition and mutualism processes between species, species groups or families. Understanding the interactions and feedback mechanisms among species in marine communities, food webs and food chains is essential to the conservation of marine biodiversity and management of fisheries. Such interactions can indeed mediate the distribution, abundance, and diversity of species within communities and across habitats, food-webs and ecosystems. Including dispersal and spatial connectivity among sites is also a crucial issue of EBFM as emphasized by the major role played by marine protected areas to foster marine biodiversity and ecosystems. Spatially explicit management of invasive species constitutes another challenging issue [START_REF] Courtois | Spatial priorities for invasive alien species control in protected areas[END_REF]. Dispersal in marine environments of suspended spores and larvae is mainly governed by ocean circulation, although vertical motility and differential mortality matter [START_REF] Cowen | Larval dispersal and marine population connectivity[END_REF]. Analysis of dispersal network topologies, e.g. using graph theory, and metapopulation dynamics are informative mathematical tools in that regard. The influence of environmental drivers such as climate and habitat changes is also pivotal for marine ecosystems and thus fisheries management. In particular, climate change impact primary production and fish distribution and thus potentially affect fishing yields [START_REF] Cury | Ecosystem oceanography for global change in fisheries[END_REF][START_REF] Sumaila | Climate change impacts on the biophysics and economics of world fisheries[END_REF][START_REF] Fromentin | Summary for policymakers of the thematic assessment of the sustainable use of wild species of the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES)[END_REF][START_REF] Barange | Impacts of climate change on fisheries and aquaculture: synthesis of current knowledge, adaptation and mitigation options. Food & Agriculture Organization of the United Nations (FAO)[END_REF][START_REF] Kroetz | The future of wild-caught fisheries: Expanding the scope of management[END_REF]. The key role of habitat for EBFM is exemplified by mangroves as nurseries for fishes or by coral reef as refuge for prey facing piscivors. Mangroves and coral reefs are under pressure worldwide.

Going further into the integration of complexities for EBFM thus adopting a very holistic viewpoint, FAO (2003); [START_REF] Pitcher | An evaluation of progress in implementing ecosystem-based management of fisheries in 33 countries[END_REF]; [START_REF] Plaganyi | Models for an ecosystem approach to fisheries[END_REF]; [START_REF] Link | Keeping humans in the ecosystem[END_REF] also argue that EBFM should integrate social, economic and human well-being dimensions of complexity. However, while the general idea of EBFM is widely accepted [START_REF] Spence | A general framework for combining ecosystem models[END_REF], it remains largely unused in bio-economic models, and therefore in supporting decision making. The curse of dimensionality (De [START_REF] Lara | Sustainable management of natural resources: mathematical models and methods[END_REF] entailed by EBFM is a major methodological limitation for the use of operations research, mathematics of decision making and management sciences for EBFM. Thus operationalizing the EBFM approach requires new models or, at the minimum, the expansion and adaptation of existing ones to make them fit for purpose.

In that regard, after early developments in bio-economics based on single stocks typically through MSY-MEY approaches, a number of generalizations have extended models to include key features of ecosystems and more realistic depictions of fish population dynamics, such as age-structure [START_REF] Tahvonen | Economics of harvesting age-structured fish populations[END_REF][START_REF] Smith | Econometric modeling of fisheries with complex life histories: Avoiding biological management failures[END_REF], multiple species interactions [START_REF] Hannesson | Optimal harvesting of ecologically interdependent fish species[END_REF][START_REF] Ragozin | Harvest policies and nonmarket valuation in a predator -prey system[END_REF], habitat dependence [START_REF] Barbier | Habitat-fishery linkages and mangrove loss in thailand[END_REF][START_REF] Smith | Generating Value in Habitat-Dependent Fisheries: The Importance of Fishery Management Institutions[END_REF][START_REF] Jean-Marie | Dynamic fishing with endogenous habitat damage[END_REF], and spatial-dynamics of a metapopulation [START_REF] Sanchirico | Bioeconomics of spatial exploitation in a patchy environment[END_REF][START_REF] Smith | The economics of spatial-dynamic processes: Applications to renewable resources[END_REF]. Many of these models share a common mathematical structure. For example, a spatial-dynamic bio-economic model with continuous time and discrete (patchy) space [START_REF] Sanchirico | Bioeconomics of spatial exploitation in a patchy environment[END_REF] resembles a predator-prey or multispecies bio-economic model in continuous time (e.g, [START_REF] Hannesson | Optimal harvesting of ecologically interdependent fish species[END_REF]) if one attaches different interpretations to the species or spatial interaction terms [START_REF] Smith | A large population of king crabs in Palmer Deep on the west Antarctic Peninsula shelf and potential invasive impacts[END_REF]. Some attempts to deal with the dimensionality and EBFM also used stylized approaches, such as financial portfolio theory [START_REF] Sanchirico | An empirical approach to ecosystem-based fishery management[END_REF], but are limited by losing the mechanistic content of the underlying bio-economic system. Other areas in which bioeconomic modeling is expected to continue developing include the effects of environmental changes on tradeoffs associated with managing a fishery, such as the expected gradual impacts of warming on fish population dynamics (e.g. 2023)), and interactions between fisheries and other uses of marine areas and resources (e.g. [START_REF] Boncoeur | Fish, fishers, seals and tourists: Economic consequences of creating a marine reserve in a multi-species, multi-activity context[END_REF]; [START_REF] Scheld | The Atlantic surfclam fishery and offshore wind energy development: 2. Assessing economic impacts[END_REF]; see also section 5 below).

Anyway, more globally, there is a methodological debate when operationalizing the EBFM between 'Whole of Ecosystem' (e.g. end-to-end models), such as Atlantis, Osmose or Apecosm [START_REF] Shin | Using an individual-based model of fish assemblages to study the response of size spectra to changes in fishing[END_REF][START_REF] Fulton | Human behaviour: the key source of uncertainty in fisheries management[END_REF][START_REF] Maury | From individuals to populations to communities: A dynamic energy budget model of marine ecosystem size-spectrum including life history diversity[END_REF] and 'Models of Intermediate Complexity for Ecosystem' (MICE) [START_REF] Hannah | The case for marine ecosystem models of intermediate complexity[END_REF][START_REF] Plagányi | Multispecies fisheries management and conservation: tactical applications using models of intermediate complexity[END_REF][START_REF] Gomes | The Major Roles of Climate Warming and Ecological Competition in the Small-scale Coastal Fishery in French Guiana[END_REF][START_REF] Cuilleret | Resilience management for coastal fisheries facing with global changes and uncertainties[END_REF]. 'Whole of Ecosystem' or end-to-end models adopt a very holistic approach articulating many system components. Sometimes such models integrate a fleet dynamics and other economic components (see FISHMIP community). In contrast, the use of MICE aims to maintain the simplicity of stylized mathemaical models and the ability to use statistical methods for their calibration, while also accounting for ecosystem dynamics, in relation to a limited number of management goals. We here argue that 'Whole of Ecosystem' models and MICE or stylized models are not contradictory but complementary and belong to a hierarchy of models that make sense in both ecological and economic terms while being well-posed mathematically and numerically. From that viewpoint, the theory and tools of nonlinear and complex dynamic systems, both in discrete and continuous time, will play a major role. In particular, the description of ecosystems, and more broadly social-ecological systems, in terms of states, controls, parameters, disturbances and observations allows for relevant integrated modeling taking into account the complex dynamics, the multiplicity of drivers (external, direct, indirect), decisions and uncertainties underlying scenarios of biodiversity and ecosystem services [START_REF] Doyen | Mathematics for Scenarios of Biodiversity and Ecosystem Services[END_REF]. Such a mathematical approach can thus represent in a synthetic way multi-species, multi-drivers and multi-scale dynamics while also capturing various sources of uncertainty [START_REF] Sanchirico | An empirical approach to ecosystem-based fishery management[END_REF].

At this stage, a methodological and mathematical question that arises is how to deal with complexity and ideally simplify it. Dynamical systems theory provides us with several methods for dealing with the complexity of models and model reduction. For example, when a system has components exhibiting characteristic time scales of different orders of magnitude, there are methods allowing to analyze the system by splitting the system in several parts [START_REF] Auger | Aggregation and emergence in systems of ordinary differential equations[END_REF]Boudjellaba andSari, 1999, 2009;[START_REF] Poggiale | Analysis of a predator-prey model with specific time scales: a geometrical approach proving the occurrence of canard solutions[END_REF]. In a first step, it allows us to simplify the analysis because we can deal with subsystems which are simpler than the whole system. Furthermore, the theory allows to make the link between the dynamics of the subcomponents and that of the whole system. For example, one such method is the separation of system's dynamics into fast and slow components. This approach is particularly useful when modeling systems with multiple time scales of behavior, where the fast dynamics can be approximated using steady-state assumptions, while the slow dynamics require more detailed modeling. There are several references in the literature that discuss the use of fast-slow dynamics and model reduction techniques. For instance, Moussaoui and Auger (2021); [START_REF] Moussaoui | A model of a fishery with fish storage and variable price involving delay equations[END_REF]; Bravo de la Parra et al. ( 2023) discuss the use of aggregation methods in model reduction in ecological and fishery systems. Some promising extensions of the method would be very useful to deal with more complex systems [START_REF] Poggiale | Analysis of a predator-prey model with specific time scales: a geometrical approach proving the occurrence of canard solutions[END_REF] exhibiting complex dynamics, like rapid population collapses after a smooth and small perturbation for instance. Of interest is also the strategy to reduce the complexity in MICE consisting in a focus on questions related to management and public policies.

In the words of [START_REF] Plagányi | Multispecies fisheries management and conservation: tactical applications using models of intermediate complexity[END_REF], MICE 'limit complexity by restricting the focus to those components of the ecosystem needed to address the main effects of the management question under consideration'.

The first rows of Table 1 lists the merits and shortcomings of the different classes of models regarding EBFM and thus highlights the gaps that need to be addressed in Mathematical Bio-economics 2.0 for Fisheries. The different classes of models include 'classical stylized bio-economic models' in line with seminal works of Gordon -Schaefer -Clark, MICE models and 'whole of ecosystem' models such as EwE or Atlantis models. The comparison on the first rows relies on different key items derived from EBFM challenges and the complexity of the ecosystem. We here first focus on the content of models in terms of dynamics, multi-species, multi-fleet, spatiality, habitat quality, climate. Notation '+' means a moderate account of the item in the model class while symbol '++' stands for a strong focus of the model class. Such a qualitative evaluation of model classes is not based on any statistical analysis but on the multidisciplinary knowledge and scientific expertise of the numerous authors of this paper.

Mathematics for sustainable fisheries

Operationalizing sustainability for fisheries is a major challenge in terms of criteria, standards and strategies.

In this section, special attention is paid to both intergenerational equity and multi-criteria issues underpinning sustainability. The assessment of sustainability is crucial for a 'Mathematical Bio-economics 2.0 for Fisheries' program. It is achieved through the evaluation of bio-economic management, policies and model-based scenarios whether they are predictive, exploratory or normative (goal seeking) (IPBES, 2016). Again, we are not starting from scratch here since the word 'sustainable' is central in the very definition of MSY (maximum sustainable yield) but restricted to the sole 'ecological' dimension of sustainability. In other terms, MSY does not take into account key social and economic aspects of sustainability, such as the equitable distribution of access and costs and benefits from the use of marine biodiversity (IPBES 2022). More broadly, the different mathematical approaches to characterize and design sustainable fisheries draw on the theory of controlled dynamic systems, in particular, steady-state, optimal and viable controls. Below we discuss the merits of these different approaches in that framework.

Following the Brundtland Report, many quantitative methods, metrics and criteria have been proposed to operationalize sustainability [START_REF] Cairns | Sustained development of a society with a renewable resource[END_REF][START_REF] Asheim | Resource conservation across generations in a ramsey-chichilnisky model[END_REF]Doyen and Martinet, 2012a;[START_REF] Fleurbaey | On sustainability and social welfare[END_REF] in particular in the bio-economics context. The challenge for a 'Mathematical Bio-economics 2.0' program is to go beyond the seminal approach of MSY [START_REF] Gordon | The Economic Theory of a Common-Property Resource: The Fishery[END_REF][START_REF] Schaefer | Some aspects of the dynamics of populations important to the management of the commercial marine fisheries[END_REF] based on equilibrium and steady state controls by adopting a more dynamic viewpoint. In that perspective, the results obtained by [START_REF] Clark | The economics of overexploitation[END_REF] from the optimal control approach in terms of transient and generalization of MSY-MEY (with the discounted or dynamic MEY versions) are pivotal and very fruitful. However, the discounted approach underlying the usual optimal control approach in economics, qualified as a 'dictatorship of the present' is criticized because this criterion over discounts long-run payoffs, entailing unsustainable trajectories [START_REF] Sumaila | Intergenerational cost-benefit analysis and marine ecosystem restoration[END_REF] and 'optimal' extinctions as stressed by [START_REF] Clark | The economics of overexploitation[END_REF]; [START_REF] Swanson | The economics of extinction revisited and revised: a generalised framework for the analysis of the problems of endangered species and biodiversity losses[END_REF]. As an alternative criterion, the maximin [START_REF] Cairns | Maximin: a direct approach to sustainability[END_REF], defined as the highest utility level that can be sustained over time, promotes intergenerational equity. In addition, [START_REF] Sumaila | Intergenerational cost-benefit analysis and marine ecosystem restoration[END_REF]; [START_REF] Sumaila | Making future generations count: Comment on "remembering the future[END_REF] included future generations, alongside the present one, in the current welfare function, with suitable Pareto weights. This overlapping generations approach leads to more practical recommendations for the sustainable management of fish stocks [START_REF] Ekeland | Equilibrium resource management with altruistic overlapping generations[END_REF][START_REF] Sumaila | Infinity Fish; Economics and the Future of Fish and Fisheries[END_REF].

Nevertheless, the use of optimization methods to quantify sustainability, including the maximin criterion or overlapping generations, is globally criticized in [START_REF] Howarth | Sustainability under uncertainty: A deontological approach[END_REF] who argues that sustainability conditions need to be imposed prior to the maximization of any social welfare. In that regard, the account for safety or conservation constraints to fulfill over time emerges as a crucial issue [START_REF] Rockström | A safe operating space for humanity[END_REF]. In addition, it may prove difficult in practice to identify a unique measure of system performance that is considered adequate to support management decision-making. Multiple, non aggregable evaluation criteria may often coexist and serve as the basis for assessing whether a fishery is meeting sustainability objectives. If the constraints induced by reference points, thresholds, standards and tipping points have to be satisfied over time, such sustainability problems can be formulated into the mathematical framework of viable control [START_REF] Aubin | Viability kernel of control systems[END_REF][START_REF] Béné | A viability analysis for a bio-economic model[END_REF][START_REF] Cury | Viability theory for an ecosystem approach to fisheries[END_REF][START_REF] Eisenack | Viability analysis of management frameworks for fisheries[END_REF][START_REF] Martinet | Defining viable recovery paths toward sustainable fisheries[END_REF][START_REF] Krawczyk | Satisficing solutions to a monetary policy problem: A viability theory approach[END_REF]Doyen and Martinet, 2012a;[START_REF] Schuhbauer | Economic viability and small-scale fisheries-A review[END_REF][START_REF] Oubraham | A survey of applications of viability theory to the sustainable exploitation of renewable resources[END_REF][START_REF] Gajardo | Methods for the sustainable rebuilding of overexploited natural resources[END_REF]. In this framework, reference and tipping points not to exceed for ecological, economic, or social indicators stand for sustainable management objectives in line with the triple bottom line of sustainability illustrated by Figure 3. In fisheries, typical examples of such thresholds are given by ICES (International Council for the Exploration of the Sea), which defined, in the frame of the precautionary approach, spawning stock biomass limits named B lim , or B pa [START_REF] Kell | Evaluation of multiple management objectives for Northeast Atlantic flatfish stocks: sustainability vs. stability of yield[END_REF]. But thresholds may also pertain to economic and social sustainability criteria [START_REF] Krawczyk | Satisficing solutions to a monetary policy problem: A viability theory approach[END_REF][START_REF] Maynou | Coviability analysis of Western Mediterranean fisheries under MSY scenarios for 2020[END_REF][START_REF] Cissé | Ecoviability for small-scale fisheries in the context of food security constraints[END_REF][START_REF] Doyen | Ecoviability for ecosystem-based fisheries management[END_REF][START_REF] Briton | Providing Integrated Total Catch Advice for the Management of Mixed Fisheries with an Eco-viability Approach[END_REF]. Among other approaches close to the viability approach and based on safety thresholds, one can mention the concept of Safe Minimum Standards (SMS) [START_REF] Margolis | Climate change decision-support and the tolerable windows approach[END_REF] where tipping thresholds and risky areas are introduced (do [START_REF] Val | Fishery management under poorly known dynamics[END_REF], or the Tolerable Windows Approach (TWA) [START_REF] Bruckner | Climate change decision-support and the tolerable windows approach[END_REF], based on safe boundaries and feasibility regions. Similarly, [START_REF] Rockström | A safe operating space for humanity[END_REF] developed a framework based on boundaries that delineate the safe operating space (SOS) for humanity, associated with the planet's biophysical subsystems or processes. The viability approach has been applied by numerous authors to the sustainable management of renewable resources and fisheries, as recently reviewed in [START_REF] Schuhbauer | Economic viability and small-scale fisheries-A review[END_REF]; [START_REF] Oubraham | A survey of applications of viability theory to the sustainable exploitation of renewable resources[END_REF]. The approach has also been proposed in support of identifying operational management objectives in the context of Management Strategy Evaluations involving stakeholder participation [START_REF] Thébaud | Building ecological-economic models and scenarios of marine resource systems: Workshop report[END_REF]. Furthermore, [START_REF] Baumgärtner | Ecological-economic viability as a criterion of strong sustainability under uncertainty[END_REF]; Doyen and Martinet (2012b) point out the strong sustainability content of the viable control approach as opposed to the weak sustainability content underlying the optimal control framework [START_REF] Neumayer | Weak versus Strong Sustainability: Exploring the Limits of Two Opposing Paradigms[END_REF]. Weak sustainability indeed allows for substitutability between the economic, social and ecological metrics and scores for instance through monetary values, social welfare or aggregated scores. In contrast, by clearly distinguishing between ecological, economic and social metrics, thresholds and constraints, viable control brings important insights into strong sustainability. Such non-substitutability occurs for instance in fisheries management since biodiversity and some ecosystem services (e.g. carbon sequestration, generation of oxygen and protection from sea level rise, recreational values) may not have clear and agreed upon monetary values.

Interestingly, it turns out that maximin and viability approaches are strongly connected since the maximin emerges as a 'maximal viability' [START_REF] Martinet | Defining viable recovery paths toward sustainable fisheries[END_REF]Doyen and Martinet, 2012b). More specifically, Doyen and Martinet (2012b) proved that the value function of the maximin problem is the solution of a static optimization problem involving the viability kernel as state constraint. Such a result has been generalized in the context of strong sustainability using Pareto optimality in [START_REF] Doyen | Sustainability standards, multicriteria maximin, and viability[END_REF]. Consequently, maximin trajectories or controls are specific and extreme viable trajectories or controls and thus inherit viability properties. Moreover, [START_REF] Doyen | Sustainability standards, multicriteria maximin, and viability[END_REF] proved that MSY-MEY play key roles for the identification of multi-criteria maximin solutions. In other words, optimal and viable control frameworks are not contradictory and should be articulated through maximin criteria or optimization under constraints. Therefore, for the 'Mathematical Bio-economics 2.0 for Fisheries' program, we suggest that the scientific works inspired by maximin and viability criteria be intensified, for instance by integrating social indicators.

Again, Table 1 captures the pros and cons of the different classes of models for fisheries regarding the operationalization of sustainability. In this table, the comparison about sustainability focuses on the content of models in terms of intergenerational equity and multi-criteria, equilibrium, optimality and viability.

Mathematics of resilience for fisheries

Operationalizing resilience, i.e. the ability to cope with shocks and uncertainties, is also a key challenge for fisheries in terms of criteria, standards and strategies [START_REF] Fromentin | The spectre of uncertainty in management of exploited fish stocks: The illustrative case of atlantic bluefin tuna[END_REF][START_REF] Grafton | Towards resilience-based management of marine capture fisheries[END_REF]. Resilience influences many decisions and policies including how to tackle risk management in the private sector [START_REF] Sheffi | The power of resilience: How the best companies manage the unexpected[END_REF], development and finance investments (OECD), and management objectives of influential multilateral and United Nations agencies (e.g. FAO; World Bank). As a result, resilience is now included in several Sustainable Development Goals (SDGs): SDG 1 (No Poverty); SDG 2 (Zero Hunger); SDG 9 (Industry, Innovation and Infrastructure); SDG 11 (Sustainable Cities and Communities); SDG 13 (Climate Action); and SDG 14 (Life Below Water). This rising popularity of resilience contrasts, however, with a lack of clarity over the concept and how to implement it in practice [START_REF] Downes | How do we know about resilience? An analysis of empirical research on resilience, and implications for interdisciplinary praxis[END_REF][START_REF] Quinlan | Measuring and assessing resilience: broadening understanding through multiple disciplinary perspectives[END_REF][START_REF] Grafton | Risks, resilience, and natural resource management: Lessons from selected findings[END_REF][START_REF] Béné | From Resistance to Transformation: A Generic Metric of Resilience Through Viability[END_REF]. For several decades now, leading scientists from different disciplines ranging from ecology, engineering sciences, psychology to economics have highlighted the significance of resilience and, importantly, the need for much better inclusion of resilience management into decision-making [START_REF] Levin | Resilience in natural and socioeconomic systems[END_REF]. In particular, practical guidance and modeling about how to operationalize resilience management are still required for fisheries and marine ecosystems.

To address uncertainties, it is very common to complete the study of the bioeconomic models with parameter sensitivity analysis. Such analysis informs the manager about the uncertainty of the model outcomes with respect to the variability of the parameters. Less common but very important is the structural sensitivity [START_REF] Cordoleani | Structural sensitivity of biological models revisited[END_REF][START_REF] Aldebert | Structural sensitivity and resilience in a predator-prey model with density-dependent mortality[END_REF]. This approach completes the sensitivity analysis by considering the importance of functional choices made in the model formulation. Think of the population dynamics (logistic, Ricker, Beverton-Holt, Gomperz, . . . ) or production functions (Shaefer, Cobb-Douglass, CES, . . . )

Figure 4: Illustration of the 3Rs of resilience: recovery, resistance and robustness with three contrasted trajectories in blue, purple and red.

underlying stylized bio-economic models. The model can indeed provide very different outputs even with qualitatively similar functional forms.

Beyond parameter and structural sensitivity analysis, there is an abundant literature about uncertainties and stochasticities in fisheries modeling, starting with, among others [START_REF] Reed | Optimal escapement levels in stochastic and deterministic harvesting models[END_REF][START_REF] Clark | On uncertain renewable resource stocks: Optimal harvest policies and the value of stock surveys[END_REF] and later [START_REF] Sethi | Fishery management under multiple uncertainty[END_REF]. The accounting of uncertainties in fisheries management has been also put forward for decades with the concept of Adaptive Management and Feedback Control (Holling, 1978;Walters, 1986). Adaptive management is a strategy for considering unpredictable changes in the ecosystem, such as recruitment failure, as major causes of stock collapse. Recent works identifying adaptive fishing strategies based on risk aversion and diversification include [START_REF] Gourguet | Risk versus economic performance in a mixed fishery[END_REF][START_REF] Tromeur | Risk averse policies foster bioeconomic sustainability in mixed fisheries[END_REF]. Moreover, all fishers learn from their successes and failures. A fisher will try a new fishing method, for example, monitor the results and see how the results compare to what was predicted to happen. Thus, adaptive management is an iterative process that consists of an integrated progression of learning by doing. Such an iterative process also relate to 'Management Strategy Evaluation' [START_REF] Butterworth | Why a management procedure approach? Some positives and negatives[END_REF][START_REF] Thébaud | Building ecological-economic models and scenarios of marine resource systems: Workshop report[END_REF]. Kalman Filter can be a relevant mathematical tool to deal with adaptive management and management strategy evaluation. We argue that resilience-based management should constitute an extension of adaptive management in the face of uncertainties.

Recently, [START_REF] Béné | From Resistance to Transformation: A Generic Metric of Resilience Through Viability[END_REF]; [START_REF] Grafton | Realizing resilience for decisionmaking[END_REF]; [START_REF] Cuilleret | Resilience management for coastal fisheries facing with global changes and uncertainties[END_REF]; [START_REF] Grafton | Towards resilience-based management of marine capture fisheries[END_REF] have made significant progress in the definitions and objectives of resilience-based management for environmental issues. They postulate that resilience-based management needs to be first defined with respect to desired and normative situations for the system at play (resilience with respect to what ?). For fisheries, marine biodiversity and bio-economic issues, we postulate that such desired states correspond to safe or sustainable states of fisheries as investigated in the previous section 4 about sustainability. Such a sustainability target for resilience expands the usual approaches associating resilience mainly with the stability of equilibria [START_REF] Levin | Resilience, Robustness, and Marine Ecosystem-based Management[END_REF][START_REF] Derissen | The relationship between resilience and sustainability of ecological-economic systems[END_REF]. At this stage, it is of interest to consider and evaluate the bio-economic resilience of natural living resources through the so-called 3R's of resilience-recovery, resistance and robustness. The 3Rs are complementary quantitative ingredients of resilience as exemplified by Figure 4. Recovery relates to the time necessary for a system to bounce back to safe, sustainable or viable situations after shocks, perturbations or adverse events [START_REF] Holling | Resilience and stability of ecological systems[END_REF][START_REF] Pimm | The complexity and stability of ecosystems[END_REF][START_REF] Martin | The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication[END_REF]. Resistance refers to the magnitude of shocks, perturbations and uncertainties that can be withstood by a system to remain in safe, sustainable or viable systems [START_REF] Holling | Resilience and stability of ecological systems[END_REF][START_REF] Béné | From Resistance to Transformation: A Generic Metric of Resilience Through Viability[END_REF]. Robustness, or reliability, refers to the probability of coping with the shocks and uncertainties with respect to a sustainable system [START_REF] Baumgärtner | The economic insurance value of ecosystem resilience[END_REF]. Said differently, recovery highlights the temporal dimension of resilience in line with well-known minimal time problem [START_REF] Cardaliaguet | Optimal times for constrained nonlinear control problems without local controllability[END_REF][START_REF] Cannarsa | Convexity properties of the minimum time function[END_REF] in control theory. Resistance gives insights on the 'room for manoeuvre' for resilience in line with basin of attraction for stability issues or capture basin for target issues. And robustness (or reliability) sheds light on the probability of resilience, in line with risk management (Rockafellar, 2010), the so-called chance constraints and the value at risk. In that regard, accounting for extreme event such as heatwaves [START_REF] Cheung | Marine high temperature extremes amplify the impacts of climate change on fish and fisheries[END_REF] within the probability distributions is a key challenge because shocks relate to uncertain events with low probability and high impact. More globally, by integrating recovery, resistance and robustness issues, the 3R-based management is a way to reconcile risk and crisis management. Such a 3R-based framework has recently been applied to fisheries management by [START_REF] Cuilleret | Resilience management for coastal fisheries facing with global changes and uncertainties[END_REF] in the case of small-scale fisheries in French Guiana.

More generally, we argue that the mathematics related to stable control, decision under uncertainty, risk management and stochastic control will play major roles for the resilience-based management of fisheries and the Mathematical Bio-economics 2.0 program for Fisheries. Table 1 synthesizes the pros and cons of different classes of models for fisheries with respect to the operationalization of resilience. In this table, the comparison focuses on the content of models in terms of stochasticity, risk and stability.

Mathematics of the governance for bio-economic public policies

The heterogeneity of actors involved in fisheries also contributes to the difficulty of management and public decisions. These actors, which include stakeholders, fishermen, scientists, NGOs, consumers, tourists, conservation agencies and regulating agencies, can indeed differ in their preferences, strategies, levels of information and inputs into the marine ecosystems. Furthermore, agents do not consider all the external consequences (externalities) of their actions. A catastrophic outcome of such a process is the resource collapse as stressed by the tragedy of the commons and fish war, a typical example of deficient governance of natural resources used by different individuals, groups or countries [START_REF] Munro | The optimal management of transboundary renewable resources[END_REF][START_REF] Levhari | The great fish war: An example using a dynamic cournot-nash solution[END_REF][START_REF] Hannesson | Fishing as a supergame[END_REF][START_REF] Sumaila | A review of game-theoretic models of fishing[END_REF][START_REF] Vallee | Nash versus stackelberg equilibria in a revisited fish war[END_REF][START_REF] Bailey | Application of game theory to fisheries over three decades[END_REF][START_REF] Breton | A great fish war model with asymmetric players[END_REF][START_REF] Grønbaek | Game theory and fisheries[END_REF][START_REF] Doyen | The tragedy of open ecosystems[END_REF][START_REF] Breton | Equilibria in a two-species fishery[END_REF][START_REF] Dahmouni | A fair and time-consistent sharing of the joint exploitation payoff of a fishery[END_REF][START_REF] Dahmouni | Great fish war with moratorium[END_REF]. Moreover, the goals of the different agents or groups are often contradictory and public policies may result in trade-offs [START_REF] Bellanger | Cross-sectoral externalities related to natural resources and ecosystem services[END_REF]. Bio-economics is thus required to design instruments entailing consensus, coordination and participation among agents towards ecosystem-based, sustainable and resilient fisheries. Existing cooperative structures for fisheries vary with scales, ranging from RFMO (Regional Fishery Management Organization) at the international scale to local fishing committees at national scales. At these different scales, there are many instruments for fisheries management including:

• Total allowable catch (TAC) or output controls (often based on MSY reference points),

• Fishing effort limits, access controls (i.e., licenses, limited number of boats or gear; restrictions on the number of trips) or input controls;

• Marine protected area (MPA), spatial and time closures, Territorial Use Rights for Fishing (TURFs) and other effective area-based conservation measures (OECMs);

• Selectivity or technology constraints: restrictions on the size of fish that can be caught or retained;

• Monetary instruments: Taxes, subsidies on fishing catches, efforts;

• Individual (or community) Transferable quotas (ITQs); and more globally allocation of shares in a fishery that can be trade on a market with a price;

• Information instruments for consumers, such as the Marine Stewardship Council (MSC) eco-label.

The merits of these different instruments are discussed in Clark (1990); Thébaud et al. (2023b) to mention a few. TAC, licenses and MPAs are among the most applied or studied instruments. We argue that the role of MPAs, of market-based solutions including ITQs [START_REF] Costello | Can catch shares prevent fisheries collapse?[END_REF][START_REF] Péreau | The triple bottom line: Meeting ecological, economic and social goals with individual transferable quotas[END_REF] as well as eco-labels need to be especially explored for incorporation in a Mathematical Bio-economics Program 2.0 for fisheries. We discuss below in more details the strengths and drawbacks of these instruments.

Restricting access to fisheries and allocating shares of a Total Allowable Catch (TAC) is a key and often successful tool for regulating fisheries [START_REF] Hilborn | Effective fisheries management instrumental in improving fish stock status[END_REF]. With costs and fishing abilities varying among fishers, the addition of transferability of individual quotas (ITQs) allows fishers to choose between continuing to fish or transferring (by sale or lease) their quota holdings to other fishers. ITQs thus offer a decentralized method of allocating catch possibilities within fisheries that should promote efficient resource use [START_REF] Grafton | Private property and economic efficiency: A study of a commonpool resource[END_REF]. Recent reviews of the experience with ITQs in fisheries show that they are increasingly being adopted, and that this has been associated with improved status of fish stocks and levels of catches [START_REF] Newell | Fishing quota markets[END_REF][START_REF] Thébaud | From anecdotes to scientific evidence? A review of recent literature on catch share systems in marine fisheries[END_REF]. In contexts where excess capacity in the fishery exists, introducing ITQs should lead to a decrease in fishing capacity as catch privileges are transferred to the more efficient fishers. Although this is an impact that could be expected, and to some extent desired, it has turned out to be one of the key points of debate on the opportunity and effectiveness of introducing ITQs as a management instrument in fisheries. The social consequences of these reductions have in some cases been considered important enough that they may outweigh the expected ecological and economic benefits of the regulations, leading to questions about their acceptability and feasibility [START_REF] Péreau | The triple bottom line: Meeting ecological, economic and social goals with individual transferable quotas[END_REF]. These criticisms demonstrate once again that no single management instrument is a panacea [START_REF] Ostrom | Going beyond panaceas[END_REF]. Some authors (e.g., [START_REF] Sumaila | How to make individual transferable quotas work economically, socially, and environmentally[END_REF]) have tried to identify design considerations for ITQs that can mitigate some of their shortcomings.

MPAs -areas where fishing is prohibited or restricted -are a multipurpose fishery management and conservation tool in line with sustainability [START_REF] Sumaila | Marine protected area performance in a model of the fishery[END_REF]. They are particularly useful to protect habitat, biodiversity, and species of particular concern, and they may be a useful hedge against uncertainty and thus promote resilience, provided that effective and efficient control of access to the MPA be in place [START_REF] Grafton | The bioeconomics of marine reserves: A selected review with policy implications[END_REF][START_REF] Doyen | Sustainability of exploited marine ecosystems through protected areas: A viability model and a coral reef case study[END_REF][START_REF] Reithe | Marine protected areas in a welfare-based perspective[END_REF]. Their bio-economic relevance however strongly depends on the fish spillovers between the conservation areas and the surrounding fished areas whenever these MPA have fixed locations. Rotational MPAs constitute another option [START_REF] Chen | Advantages and disadvantages of rotating spatial closures for managing fisheries[END_REF].

Regarding eco-labels, they relate to consumer choices which are based on a wide variety of motivations, ranging from social responsibility to specific individual needs. Interesting in this perspective is that consumers, worldwide, are more and more aware of the necessity of ensuring that the environment is protected and that their individual choices and actions can contribute towards achieving this goal [START_REF] Vermeir | Sustainable food consumption: Exploring the consumer "attitude -behavioral intention" gap[END_REF].

In that context, eco-labels [START_REF] Salladarré | The demand for seafood eco-labels in france[END_REF][START_REF] Jonell | Eco-labeled seafood: Determinants for (blue) green consumption[END_REF][START_REF] Giacomarra | The ability of fish ecolabels to promote a change in the sustainability awareness[END_REF][START_REF] Bronnmann | Measuring motivations for choosing ecolabeled seafood: Environmental concerns and warm glow[END_REF] can play a major role in increasing the awareness of consumers regarding sustainability issues.

Since their introduction in the late 1990s, fishery sustainability certification has become a major ingredient of marine conservation strategies. Many of these programs emerged largely from increased concerns within civil society that current stock management and policies have failed in ensuring the sustainability of fisheries [START_REF] Sainsbury | Review of ecolabelling schemes for fish and fishery products from capture fisheries. 5[END_REF]. The key function of these programs is to differentiate fisheries through a set of standards relating to stock status, management practices, and ecosystem impacts.

To assess the ability of these different instruments in supporting sustainability of uses of marine biodiversity and resilience of marine ecosystems, the conceptual tools of game theory, from both cooperative and non cooperative branches, are extremely useful [START_REF] Munro | The optimal management of transboundary renewable resources[END_REF][START_REF] Sumaila | A review of game-theoretic models of fishing[END_REF][START_REF] Kaitala | Game Theoretic Applications to Fisheries[END_REF][START_REF] Bailey | Application of game theory to fisheries over three decades[END_REF][START_REF] Breton | A great fish war model with asymmetric players[END_REF][START_REF] Grønbaek | Game theory and fisheries[END_REF][START_REF] Dahmouni | A fair and time-consistent sharing of the joint exploitation payoff of a fishery[END_REF]. Cooperation is indeed crucial for the sustainable use of renewable resources, exploited ecosystems and biodiversity as stressed by the well-known tragedy of the commons. Game theory is a particularly relevant modeling tool to study such issues because it provides important quantitative and qualitative insights into the strategic interactions between users exploiting a common renewable resource (see e.g. [START_REF] Kaitala | The economic management of high seas fishery resources: Some game theoretic aspects[END_REF]; [START_REF] Hannesson | Fishing as a supergame[END_REF]; [START_REF] Finus | Game theory and international environmental cooperation[END_REF]). It can also be used to study policy interventions that incompletely control the behaviors of fishery participants [START_REF] Huang | The dynamic efficiency costs of common-pool resource exploitation[END_REF]. In the extensive game theory literature applied to fisheries, the dynamic model of [START_REF] Levhari | The great fish war: An example using a dynamic cournot-nash solution[END_REF] provides a solid framework for analyzing the consequences of users' strategies on the resource in open-access fisheries. Using a dynamic Cournot-Nash solution, these authors show that the non-cooperative equilibrium yields a higher harvest fraction and a smaller steady-state stock than the cooperative equilibrium1 . The result of [START_REF] Levhari | The great fish war: An example using a dynamic cournot-nash solution[END_REF] illustrates the famous tragedy of over-exploitation of resources in open access. Between these two extreme cases, full cooperation and no cooperation, the sustainability of partial cooperation has been recently studied by [START_REF] Breton | A great fish war model with asymmetric players[END_REF]; [START_REF] Doyen | Sustainable coalitions in the commons[END_REF]. It also turns out such sustainability of partial cooperation can be enhanced by the closeness to bioeconomic tipping points and critical limits [START_REF] Lindahl | Potential disasters can turn the tragedy into success[END_REF][START_REF] Flaaten | On the bioeconomics of predator and prey fishing[END_REF].

However, as pointed out by [START_REF] Bailey | Application of game theory to fisheries over three decades[END_REF], the majority of game-theoretic models have been applied to single stocks. Notable exceptions exist such as the study of predator-prey models [START_REF] Mesterton-Gibbons | A technique for finding optimal two-species harvesting policies[END_REF][START_REF] Flaaten | On the bioeconomics of predator and prey fishing[END_REF], meta-populations distributed over connected areas [START_REF] Costello | Spatial renewable resource extraction under possible regime shift[END_REF], or the works of [START_REF] Fischer | The compleat fish wars: Biological and dynamic interactions[END_REF]; [START_REF] Breton | Equilibria in a two-species fishery[END_REF]; [START_REF] Doyen | The tragedy of open ecosystems[END_REF] which account for the interaction between different species of fish, including prey-predator relations, symbiotic interactions and mutual competition.

A computational application of a predator-prey model can be found in [START_REF] Sumaila | Strategic dynamic interaction: The case of barents sea fisheries[END_REF] in which a game theoretic cod-capelin predator-prey model was developed. [START_REF] Herrera | High-seas fish wars generate marine reserves[END_REF] also showed that MPAs may also play a role in ensuring sustainable fisheries on the high seas, where a limited number of countries compete in a noncooperative fishing game. Nevertheless, the use of game theory in broader ecosystem-based context remains an open research field. Said differently, the generalization of the dynamic game approach to multi-species and spatial frameworks is an important challenge in the area of ecosystem-based and biodiversity management.

More generally, we argue that the mathematics related to game theory and strategic interactions will play major roles for a relevant governance towards resilience and sustainability-based management of fisheries and 'Mathematical Bio-economics 2.0'. They should help reduce the gap between the theoretical insights of Mathematical Bio-economics 2.0 and practical fisheries management that incorporates the behavior and interests of stakeholders. Last rows of Table 1 summarizes the merits of different classes of models for fisheries with respect to such governance issues, in particular, in terms of instruments and strategic interactions.

Conclusions

Balancing marine biodiversity conservation with food security, the conservation of marine ecosystem functioning and services, and the economic viability of fisheries in a complex and dynamic context of global change, is among the greatest challenges of the century. Dealing with such a challenge implies the development of model-based scenarios of biodiversity and ecosystems that make sense economically, ecologically, biologically, and that are well-posed mathematically and numerically. In that vein, our perspective paper proposes to design a 'Mathematical Bio-Economics 2.0 Program for Sustainable Fisheries'. The paper addresses the 'Mathematical Bio-Economics 2.0' Program for Fisheries with four general challenges: (i) the mathematics and modeling of ecosystem-based fisheries management (ii) the mathematics and modeling of sustainability criteria and strategies; (iii) the mathematics and modeling of resilience criteria and strategies; (iv) the mathematics and modeling of governance and strategic interactions. Our analysis based on those four axes makes it possible to elicit guidelines and more specific key mathematical and modeling gaps to fill up for a relevant 'Mathematical and Modeling Bio-Economics 2.0 Program for Sustainable Fisheries' when compared to models currently used ranging from stylized bio-economic models (e.g. MSY-MEY) to 'Whole of Ecosystem' models.

To this end, the present paper provides modeling and mathematical insights in line with IPBES Chapter [START_REF] Ferrier | Summary for policymakers of the methodological assessment of scenarios and models of biodiversity and ecosystem services of the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services[END_REF] devoted to 'Methodological assessment of scenarios and models of biodiversity and ecosystem services'. We argue that complex dynamic systems combined with control theory, including stochastic control together with game theory, is relevant to design predictive, exploratory and normative model-based scenarios [START_REF] Börjeson | Scenario types and techniques: towards a user's guide[END_REF][START_REF] Doyen | Mathematics for Scenarios of Biodiversity and Ecosystem Services[END_REF] of marine biodiversity, ecosystem and the services they provide. Such relevance to design bio-economic model-based scenarios arises first because these domains of applied mathematics constitute a transdisciplinary language potentially articulating the ecological, social and economic dimensions of these systems and handling their interactions. These mathematical frameworks for the design of bio-economic model-based scenarios make it possible to operationalize and quantify both the ecosystem approach, sustainability, resilience and governance of fisheries. Indeed, challenge (i) addresses the capability of this general modeling framework to account within model-based scenarios for the various degrees of complexity occurring in ecological processes and at their interfaces with social and economic processes as in bio-economics. In particular, the description of natural and social marine systems in terms of states, controls, parameters, disturbances and observations allows for relevant integrated modeling to consider the complex dynamics, the multiplicity of drivers, decisions and uncertainties underlying scenarios of biodiversity and ecosystem services. Such a mathematical approach can thus represent in a synthetic way multi-species, multi-fishing fleets, multi-drivers, spatially explicit and multi-scale dynamics while also capturing various sources of uncertainty. Furthermore, challenge (ii) about sustainability modeling and metrics addresses multi-criteria and multi-target issues underpinning the evaluation of scenarios of marine biodiversity and ecosystem services. The multiplicity of metrics needed to address sustainability issues for fisheries management is exemplified by the numerous indicators of biodiversity as well as the numerous marine ecosystem services. The ability of the optimal and viable control approach to balance such various ecological and economic scores and to promote sustainability through scenarios [START_REF] Clark | The economics of fishing and modern capital theory: a simplified approach[END_REF][START_REF] Doyen | Mathematics for Scenarios of Biodiversity and Ecosystem Services[END_REF][START_REF] Oubraham | A survey of applications of viability theory to the sustainable exploitation of renewable resources[END_REF] shows the relevance of the general framework based on (dynamic) control theory advocated in this paper. The sustainability approach of challenge (ii) addresses through normative scenarios the ability of such modeling framework to reconcile short and long terms and promoting intergenerational equity (Doyen and Martinet, 2012a;[START_REF] Ekeland | Equilibrium resource management with altruistic overlapping generations[END_REF]. Challenge (iii) focusing on resilience stresses that the control theory of dynamics systems under uncertainty combined with bio-economic viability objectives is well-suited for the mitigation of bio-economic risks and vulnerabilities by considering uncertainties, shocks within management strategies and scenarios, including endogenous uncertainty. The governance issues constitute another major challenge as pointed out in [START_REF] Bailey | Application of game theory to fisheries over three decades[END_REF]. Facing the heterogeneity of stakeholders involved in biodiversity dynamics and including consumers, farmers, fishers, ONGs, or regulating agencies, there is an obvious need for decision makers to coordinate them or to limit the failures of cooperation and the tragedy of open-access. In that regard, the dynamic framework put forward in our paper can also be easily expanded by using tools and concepts from dynamic game theory to address the challenges relating to strategic interactions and the relevance of regulatory instruments in terms of sustainability and resilience. On all five axes, bioeconomic models have the potential to help mediate discussions relating to collective choices, as part of the decision-making processes which structure policy design and implementation regarding sustainable ocean uses. In particular, MathBioEco 2.0 should support stakeholder engagement in model-based approaches in line with the purpose of science-policy interfaces (see e.g. Gail et al., 2022).

More specifically, among the main gaps emerging from Table 1 for a Mathematical and Modeling Bio-Economics 2.0 for Sustainable Fisheries, we would like to stress the following cutting-edge research and challenges:

• Definition and quantification of an ecosystem-based MSY (EBMSY) and MEY (EBMEY);

• Highlight bioeconomic resilience gains of 'EBMEY' when compared to 'EBMSY';

• Application of multi-criteria, maximin and viable control approach for sustainable and resilient fisheries;

• Account of extreme events in resilience-based management for fisheries through flat tail random distribution;

• Application and analyze bioeconomic sustainability (and resilience) gains vs. loss of market-based instruments for the decentralization of public policies for fisheries;

• Propose and analyse MICE models of sustainable seafood systems (From Fish to Fork).

Advancing these more specific issues should allow for bridging the divide between theoretical mathematical bioeconomics 2.0, model-based scenarios and the practical quantitative management of fisheries and seafood systems.

CEE-M Working Papers

Figure 1 :

 1 Figure 1: Description and use of predictive, exploratory and normative scenarios. Solid black lines correspond to past observed trajectories, and dashed color lines represent projected pathways. The figure is inspired by IPBES (2016).

Figure 2 :

 2 Figure 2: An illustration of marine ecosystem interactions and dynamics.

Table 1 :

 1 Comparative analysis of the merits of the different classes of models for fisheries along with the gaps for Mathematical Bioeconomics 2.0 for Fisheries. Notation '+' means a moderate account of the item in the model class while symbol '++' stands for a strong focus of the model class.

Figure 3 :

 3 Figure 3: The triple bottom line of sustainability stressing the need of multi-criteria approaches.

  1 -2024 WP 2024-01 Ivric Valaire Yatat-Djeumen, Luc Doyen, Jean Jules Tewa & Bapan Ghosh « Bioeconomic Sustainability and Resilience of Savanna » M. D. Smith, U. R. Sumaila, G. Zaccour, I. Ekeland, P. Cury, C. Lett, O.Thebaud, J.-C. Poggiale, A. Moussaoui, J.-M. Fromentin, S. Gourguet, P. Guillotreau, H. Gomes, Pierre. Courtois, Robert Schaap, F. Blanchard, C. Rainer, T. Villain, F. Menard, & T. Sari. « Mathematical Bio-Economics 2.0 for Sustainable Fisheries »

	WP 2024-02	Mathieu Cuilleret, Luc Doyen & Fabian Blanchard « Reducing IUU for Bioeconomic Resilience of Fisheries: Necessary but Not Sufficient »
	WP 2024-03	Luc. Doyen,

1 CEE-M Working Papers / Contact : laurent.garnier@inrae.fr  RePEc https://ideas.repec.org/s/hal/wpceem.html  HAL https://halshs.archives-ouvertes.fr/CEE-M-WP/

The non-cooperative situation refers to a framework in which each user maximizes its own intertemporal payoff without taking into account other users' scores[START_REF] Sumaila | A review of game-theoretic models of fishing[END_REF]. In contrast, in the cooperative case, users jointly define a harvesting strategy.

Acknowledgments. This work has been carried out with the financial support of the CNRS and IMPT (Institut des Mathématiques de la Planète Terre) through the network MESSH (Mathematics for bio-Economics and Sustainability of fiSHeries). In particular, this paper is a consequence of the international and interdisciplinary Symposium named '3Days MESSH' held in Sète, France on January 2023. The support of the research centers MARBEC, CEREMADE, CEEM and AMURE was decisive. The role of the research projects COVPATH (Belmont Forum) and ENTROPIC (CNRS PRIME80) has also to be mentioned.