Idriss Mazari
email: mazari@ceremade.dauphine.fr

Domènec Ruiz-Balet
email: d.ruiz-i-balet@imperial.ac.uk

Spatial ecology, optimal control and game theoretical fishing problems

Keywords: diffusive logistic equation, optimal control, bilinear optimal control, calculus of variations, Nash equilibria, game theory AMS classification: 35Q92, 49J99, 34B15, 49N90, 91A05

Of paramount importance in both ecological systems and economic policies are the problems of harvesting of natural resources. A paradigmatic situation where this question is raised is that of fishing strategies. Indeed, overfishing is a well-known problem in the management of live-stocks, as being too greedy may lead to an overall dramatic depletion of the population we are harvesting. A closely related topic is that of Nash equilibria in the context of fishing policies. Namely, two players being in competition for the same pool of resources, is it possible for them to find an equilibrium situation?

The goal of this paper is to provide a detailed analysis of these two queries (i.e optimal fishing strategies for single-player models and study of Nash equilibria for multiple players games) by using a basic yet instructive mathematical model, the logistic-diffusive equation. In this framework, the underlying model simply reads -µ∆θ = θ(K(x) -α(x) -θ) where K accounts for natural resources, θ for the density of the population that is being harvested and α = α(x) encodes either the single player fishing strategy or, when dealing with Nash equilibria, a combination of the fishing strategies of both players. This article consists of two main parts. The first one gives a very fine characterisation of the optimisers for the singleplayer game where one aims at solving sup α ´Ω αθ, under L ∞ and L 1 constraints on the fishing strategies α. In particular, we show that, depending on the value of these constraints, this optimal control problem may behave like a convex or, conversely, concave problem. We also provide a detailed analysis of the large diffusivity limit of this problem. In the case where two players are involved, we rather write α as α1 + α2 where αi, the fishing strategy of the i-th player, also satisfies L ∞ and L 1 constraints. Defining I1 := ´Ω αiθ we aim at finding a Nash equilibrium. We prove the existence of Nash equilibria in several different regimes and investigate several related qualitative queries, for instance providing examples of the wellknown tragedy of commons.

Our study is completed by a variety of numerical simulations that illustrate our results and allow us to formulate open questions and conjectures.

Introduction 1.Scope of the paper & summary of the models

In this paper, we study an optimal harvesting problem motivated by the ecological management of wild fisheries. One of the main ecological threats we currently face is the depletion of fish populations in oceans [START_REF]Global fish stocks are exploited or depleted to such an extent that without urgent measures we may be the last generation to catch food from the oceans[END_REF][START_REF] Davies | Extinction risk and overfishing: reconciling conservation and fisheries perspectives on the status of marine fishes[END_REF][START_REF] Pinsky | Unexpected patterns of fisheries collapse in the world's oceans[END_REF]. While many factors can be held accountable for this situation, one of the overarching ones is overfishing and, more generally, the poor management of fisheries. The resulting very high strain that is exerted on fishing stocks puts at risk the biomass [START_REF] Costello | Status and solutions for the world's unassessed fisheries[END_REF][START_REF] Pikitch | The risks of overfishing[END_REF]. While it is clear that this overfishing problem may arise when only one population of fishermen is present, the situation can be more dramatic when several populations of fishermen are competing for the same pool of resources. This is an example of the ubiquitous tragedy of commons [START_REF] Hardin | The tragedy of the commons[END_REF]: the competition over finite common resources may lead to the extinction of said resources. But not only does this affect the fish population, it also endangers the fishing-based economies of several societies [START_REF] Hamilton | Outport adaptations: Social indicators through newfoundland's cod crisis[END_REF]. Consequently, the future of fisheries and the study of optimal fishing strategies is now a central topic both in the scientific community and in society [START_REF]only 50 years left' for sea fish[END_REF][START_REF] Costello | Status and solutions for the world's unassessed fisheries[END_REF][START_REF] Worm | The future of fish[END_REF][START_REF]Fisheries waste 'costs billions[END_REF].

In the present work, we aim at providing an in-depth analysis of a paradigmatic model of such (over)fishing problems from the perspective of optimal control of spatial ecology models and game theory. Using, as a basic building block, the logistic-diffusive equation, we offer several qualitative results that exemplify the intricate and rich qualitative behaviours of such queries, and provide theoretical illustrations of the aforementioned concepts in the management of fisheries (in particular, of the tragedy of commons).

Summary of the models Since the introduction is long let us for the sake of convenience summarise here the models and questions we investigate. In general, the fishes' population will be described using the standard logistic-diffusive equation (see section 1.2 for more details): θ being the population density, we assume that θ solves where α described the fishing rate, µ > 0 is the diffusivity of the population and K(x) accounts for the natural resources available in the environment. The optimisation problem we seek to understand is the maximisation of the fishing outcome:

max α∈L ∞ (Ω) ˆΩ α(x)θ α (x)dx,
where by θ α we indicate the dependence of θ with the variable α in (1.1). Of course we would need to specify which constraints we enforce on α. We shall make this precise in section 1.2. In certain cases, this problem can be solved explicitly; this is the case when K is a constant, see Remark 2. However, when we consider a general capacity K(x) the study becomes more intricate. The first part of this article is devoted to the study of this optimal fishing problem.

The second part is devoted to understanding a related game-theoretical problem. In this model, two populations are fishing in the same pool of natural resources. Considering two players, the state equation becomes:

     -µ∆θ = θ(K(x) -θ) -α 1 θ Player 1 -α 2 θ Player 2
in Ω,

∂θ ∂ν = 0 on ∂Ω, (1.2)
where each player wants to optimise their fishing output I 1 (α 1 , α 2 ) = ˆΩ α 1 θ α1,α2 dx, I 2 (α 1 , α 2 ) = ˆΩ α 2 θ α1,α2 dx, the outcome of one player depends on the strategy of the other player, since, both players have an impact on the total population θ through equation (1.2). A pair of strategies (α * 1 , α * 2 ) is said to be a Nash equilibria if

α * 1 ∈ arg max α1 I 1 (α 1 , α * 2 ), α * 2 ∈ arg max α2 I 2 (α * 1 , α 2 ). (1.3)
In general, Nash equilibria do not necessary exist, and obtaining their existence is a core point of this article. Additionally, we shall give some qualitative contributions to the study of the impact of competition on the total outcome: is it better, when two fishers' population are fishing, to be competing or cooperating? In particular, we will see that competition is sometimes detrimental to the total fishing outcome. Furthermore, when considering n players we will see a more devastating effect. We will see that as the number of players increase, there exist a Nash equilibrium such that the total harvested amount of fishes goes to 0 as the number of players increases. Again, this result validates the principle known as tragedy of the commons [START_REF] Hardin | The tragedy of the commons[END_REF]: if one increases the number of players in the harvesting game, the total amount harvested may decrease dramatically.

In order to avoid overfishing, typically governments impose regulations on the fishing capacity of the players. Furthermore, players themselves might have limited fishing ability. In this paper we will model this by imposing an integral constraint on α, ´Ω α(x)dx V 0 or ´Ω α(x)dx = V 0 .

Throughout this study, we shall also cover several aspects of optimal control problems that are interesting in their own right, and that belong to a currently very active field of research devoted to the understanding of spatial heterogeneity in population dynamics and, more generally, in the study of spatial ecologoy [START_REF] Bai | An optimization problem and its application in population dynamics[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model : I -species persistence[END_REF][START_REF] Deangelis | Carrying capacity of a population diffusing in a heterogeneous environment[END_REF][START_REF] Heo | On the ratio of biomass to total carrying capacity in high dimensions[END_REF][START_REF] Inoue | On the unboundedness of the ratio of species and resources for the diffusive logistic equation[END_REF][START_REF] Kao | Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains[END_REF][START_REF] Lam | Selected topics on reaction-diffusion-advection models from spatial ecology[END_REF][START_REF] Lamboley | Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions[END_REF][START_REF] Liang | On the dependence of population size upon random dispersal rate[END_REF][START_REF] Liang | The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration[END_REF][START_REF] Lou | On the effects of migration and spatial heterogeneity on single and multiple species[END_REF][START_REF] Lou | Some challenging mathematical problems in evolution of dispersal and population dynamics[END_REF][START_REF] Lou | Maximizing the total population with logistic growth in a patchy environment[END_REF][START_REF] Lou | Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics[END_REF][START_REF] Mazari | Shape optimization and spatial heterogeneity in reaction-diffusion equations[END_REF][START_REF] Mazari | Optimization of a two-phase, weighted eigenvalue with dirichlet boundary conditions[END_REF][START_REF] Mazari | Handbook of optimal control and numerical analysis, chapter Some challenging optimisation problems for logistic diffusive equations and numerical issues[END_REF][START_REF] Mazari | A fragmentation phenomenon for a non-energetic optimal control problem: optimisation of the total population size in logistic diffusive models[END_REF][START_REF] Su | Effects of nonlocal dispersal and spatial heterogeneity on total biomass[END_REF]. Let us give a more mathematical point of view on our contributions:

From the applied mathematics perspective In this paper we investigate several optimal fishing problems in spatial ecology. The first class of problem corresponds to a single fisher problem, while the other two deal with multiple players problems. In the single fisher case, we mostly investigate the influence of the total fishing capacity on the qualitative features of optimal fishing strategies, while in the other problems we provide some contribution to the existence of Nash equilibria. For multiple player games, we mostly consider the case of two players. Our approach can also be used for analysing games with more players. Our theoretical analysis is illustrated by several detailed numerical solutions.

From the optimal control perspective Another outlook on the results of this paper is to notice that we are investigating a non-monotonic bilinear optimal control problem. By this we mean the following in the case of a single fisher problem: the population of fishes being modelled by its density θ and a fishing strategy being accounted for by a certain function α, the equation features a loss term -αθ, while the player tries to optimise a criterion of the form ffl αθ. Then it is clear that overfishing will be detrimental to the fisher, as it is going to be detrimental for the overall population. In this paper, we exemplify the shift this creates in the qualitative analysis; for instance, maximisers can saturate certain constraints, or not at all depending on the values of the parameters of the problem.

For further references and discussion, we refer to section 1.6 of the introduction.

= 0 on ∂Ω , θ K,α,µ 0 , = 0.

( 1.4) We refer to [START_REF] Mazari | Shape optimization and spatial heterogeneity in reaction-diffusion equations[END_REF]Introduction] and the references therein for more details on the modelling. The question of existence and uniqueness of solutions of (1.4) can be tedious. It is known [START_REF] Berestycki | Analysis of the periodically fragmented environment model : I -species persistence[END_REF] that for fixed α , K there exists a unique solution to (1.4) if and only if the first eigenvalue of the operator -µ∆ -(K -α) is negative. Since we work on optimisation problems, it is easier to ensure the existence and uniqueness of the solution for any control. As the first eigenvalue is bounded from above [START_REF] Dockery | The evolution of slow dispersal rates: a reaction diffusion model[END_REF] by ffl Ω (α -K) we will simply work with controls α satisfying 0

< Ω α < Ω K. (1.5)
Under these conditions, classical results from [START_REF] Berestycki | Analysis of the periodically fragmented environment model : I -species persistence[END_REF][START_REF] Cantrell | Permanence in ecological systems with spatial heterogeneity[END_REF] guarantee the existence and uniqueness of a solution of (1.4). We introduce a parameter K 0 ∈ (0; 1) and always assume that K ∈ K(Ω), where K(Ω) is defined as

K(Ω) := K ∈ L ∞ (Ω) , 0 K 1 , Ω K = K 0 , (1.6)
where, for any f ∈ L 1 (Ω) we use the notation

Ω f = 1 Vol(Ω) ˆΩ f.
Single player functional The functional to optimise in the single player case is the total fishing output

J µ : α → Ω αθ α,µ ,
and the relevant optimisation problem is

sup α J µ (α).
Of course, we need to specify which admissible fishing strategies α we consider.

Admissible controls Beyond the integral condition (1.5), we enforce a pointwise bound 0 α κ, where κ > 0 is a fixed parameter: a single player has a limited fishing capacity at any given spots. Second, we need to implement a global, L 1 constraint (the player has a globally limited fishing ability); in order to still satisfy (1.5), we fixe a parameter V 0 ∈ (0; K 0 ) and we assume that either all controls satisfy Ω α V 0 (Inequality constraint) or, on the other hand, that

Ω α = V 0 (equality constraint).
Overall, we thus define, for these two fixed paramers κ , V 0 , the two admissible classes of controls

M (κ, V 0 ) = α ∈ L ∞ (Ω) , 0 α κ a.e., Ω α V 0 (1.7)
and

M = (κ, V 0 ) = α ∈ L ∞ (Ω) , 0 α κ a.e., Ω α = V 0 (1.8)
Working in one or the other of these admissible classes changes the features of the problem drastically. This is related to the problem of overfishing: as we shall see throughout the proofs, depending on the value of V 0 , the functional J µ may be increasing (in the sense that α 1 α 2 ⇒ J µ (α 1 ) J µ (α 2 )), in which case optimisers for the problem sup m∈M=(κ,v0) J µ (α) are also optimisers for the problem sup m∈M (κ,v0) J µ (α), or loose this monotonicity, in which case the optimisers for the inequality case are strictly better than optimisers for the equality constraint: sup m∈M=(κ,v0) J µ (α) < sup m∈M (κ,v0) J µ (α). This is a first major difference between between the fishing problem and the problem of optimisation of the total population size, where the monotonicity of the functional is a stepping stone for further qualitative analysis of optimisers, see section 1.6.

The main problem Thus, the first two optimisation problems to be considered here and that are the main foci of the present contribution are:

sup α∈M (κ,V0) J µ (α) (P single ,V0 ) and sup α∈M=(κ,V0) J µ (α) (P single =,V0 )
For these two problems, we can provide a fine analysis in the case of low fishing abilities (V 0 1) orin the large diffusivity asymptotic regime µ → ∞. In particular, we will show that, in general (i.e. for a fixed diffusivity), if V 0 1, J µ is increasing on M (κ, V 0 ) (Theorem I), and also concave (Theorem III) while, in the large diffusivity case µ → ∞, we can attain an explicit description of optimal strategies (Proposition 4, Theorem V).

A "large fishing ability" model to showcase the complexity of fishing problems To exemplify, however, the breadth of behaviours such fishing problems can display, we also propose a deep exploration of another asymptotic case, that of large fishing abilities.

Let us make this more precise. What we mean here is that the fishing strategy is going to be a small perturbation of the resources distribution K, i.e. that any fishing strategy writes α = K +δm for a small parameter δ > 0.

This leads us to introduce the auxiliary classes

N (Ω) := m ∈ L ∞ (Ω) , m L ∞ (Ω) 1 , -m 1 Ω m = -m 0 and N = (Ω) := m ∈ L ∞ (Ω) , m L ∞ (Ω) 1 , Ω m = -m 0
where m 0 is a fixed volume constraint, m 1 > -1 and we define, for any m ∈ N (Ω) and any δ > 0, the fishing strategy α δ,m := K + δm.

The parameter δ is destined to be small, so we are essentially, through this reparameterisation, assuming that fishing strategies are close to natural resources distribution, and essentially lead to killing all the population off.

Remark 1. For any m ∈ N (Ω), the zones {m < 0} correspond to zones where we are not exhausting the natural resources modelled by K.

We define, for any δ > 0, the map

J δ,µ : N (Ω) m → Ω α δ,m θ α δm ,µ .
The related optimisation problems are sup

m∈N (Ω) J δ,µ (m) (Q single ,δ )
and sup

m∈N=(Ω) J δ,µ (m) (Q single =,δ )
While these two problems seem extremely related to our original formulations (P single ,V0 )-(P single =,V0 ) the qualitative behaviours of (Q single ,δ )-(Q single =,δ ) are very different. For instance, we show in Theorem I that when δ 1 the functional J δ,µ is not monotonic, and that it even behaves like a convex function, in the sense that its maximisers are extreme points of the admissible set (see Theorem IV).

Structure of the statement of the results for single fisher models While it would seem natural to divide our presentation of the results in two batches, one devoted to (P single =,V0 )-(P single ,V0 ) and another to (Q single =,δ )-(Q single ,δ ), the coherence of the methods of proofs used prompts us to rather present them in the following order:

1. Monotonicity properties: in the first two theorems, Theorems I and II, we investigate the monotonicity of the functionals J µ and J δ,µ . In Theorem I we show that (P single ,V0 ) and (P single =,V0 ) coincide when V 0 1. In Theorem II we prove that when δ 1 the problems (Q single ,δ ) and (Q single =,δ ) do not coincide. While such results can be obtained in a very straightforward manner when we consider the case of a constant resources distribution K (see in particular Remark 2), it is not immediate at all in the case of varying K. The interest of Theorem III is twofold: first, it exemplifies the qualitative change of behaviour of the functional J µ when the volume constraint is perturbed. Second, it is an essential building block to obtain concavity properties for the functional and, therefore, to derive the existence of Nash equilibria when we will, in the second part of the paper, study multiple players games.

Concavity and convexity properties:

In Theorems III-IV, we focus on the problems with equality constraints (P single =,V0 )-(Q single =,δ ). We first show in Theorem III that, if V 0 is small enough and if Ω is one-dimensional then, regardless of the resources distribution K, J µ is a concave functional, and we identify the maximising controls for particular values of V 0 or for particular resources distribution K. This relies on very fine properties of the one-dimensional logistic diffusive equation previously investigated in [START_REF] Bai | An optimization problem and its application in population dynamics[END_REF]. We prove the same result in higher dimensions, provided K remains close to a constant. We show in particular that if K is constant, then the maximising controls are constant as well. Then, in Theorem IV we show that, if δ > 0 is small enough, the functional J δ,µ behaves, conversely, like a convex function from the point of view of optimisation in N = (Ω): all solutions of (Q single =,δ ) are extreme points of the admissible sets and so they write m * = κ1 E * for some suitable subset E * of Ω.

3. Precised behaviour in asymptotic regimes: finally, to conclude the theoretical contributions to single player games, we offer an in-depth analysis of the large diffusivity limit µ → ∞ of the optimisation problem (P single =,V0 ). Building on techniques of [START_REF] Mazari | Optimal location of resources maximizing the total population size in logistic models[END_REF], we give explicit maximisers in the one-dimensional case; we refer to Theorem V. Similarly, this result will be used to exhibit Nash equilibria in two-players games.

All these results are gathered in subsection 1.3. In Section 5, we present and comment several numerical simulations.

Remark on the techniques used Throughout this first part of the paper, especially for Theorems I-III-IV one of the key ingredient is the second-order technique introduced in [START_REF] Mazari | Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate[END_REF] to tackle the problem of optimising the total population size. While this method proved fruitful in a variety of other situations [START_REF] Mazari | The bang-bang property in some parabolic bilinear optimal control problems via two-scale asymptotic expansions[END_REF][START_REF] Mazari | Qualitative analysis of optimisation problems with respect to nonconstant Robin coefficients[END_REF], it is here impossible to apply directly, and it needs to be coupled with some fine analytical study of the functions at hand. The characterisation of optimisers in the large diffusivity limit is on the obtained using rearrangement-like arguments and Talenti inequalities. Specifically, we shall use some results of [START_REF] Langford | Comparison Theorems in Elliptic Partial Differential Equations with Neumann Boundary Conditions[END_REF] and of [START_REF] Mazari | Optimal location of resources maximizing the total population size in logistic models[END_REF], the latter being used solely to derive the limit model.

Terminology: bang-bang functions We shall often refer in this paper to "bang-bang" functions. They are simply admissible controls that write α = κ1 E .

Such bang-bang functions are known to be important in the optimal control of reaction-diffusion equations (see in particular section 1.6 of this introduction), and, geometrically, are extreme points of the convex set M = (κ, V 0 ).

Qualitative properties for single player games: general diffusivities

Monotonicity of the fishing output We begin with the monotonicity of the fishing output functional and explain how the volume constraint may have an influence on the increasing character of J µ . Of course, this is a theoretical, optimal control formulation of the overfishing problem. Before we state our result, let us explain in the following remark that such a result is very much expected when working in homogeneous environments (K ≡ 1) where explicit computations allow for an explicit characterisation of maximisers; this shows that monotonicity is not the general rule.

Remark 2 (A standard example with loss of monotonicity). A simple yet instructive case to exemplify the loss of monotonicity is given by the case K ≡ K 0 . In this case, for any strategy

α ∈ M (κ, V 0 ), θ α,µ solves -µ∆θ α,µ -θ α,µ (K 0 -θ α,µ ) = αθ α,µ .
As θ α,µ satisfies Neumann boundary conditions, this entails

J µ (α) = Ω αθ α,µ = Ω θ α,µ (K 0 -θ α,µ ).
Besides, if we assume that κ < 2, so that 1 -α L ∞ 1, the maximum principle implies θ α,µ 1 almost everywhere. As the maximiser of ϕ :

x → x(K 0 -x) on [0; K 0 ] is reached at x = K0 2 it follows that J µ (α) ϕ 1 2 ,
with equality if, and only if, θ α,µ ≡ K0 2 . However, θ α,µ = K0 2 if and only if α ≡ K0 2 . We thus obtain the following conclusion: for any

V 0 K0 2 , α * ≡ K0
2 is the unique maximiser of J µ on M (κ, V 0 ). In particular, if V 0 > K0 2 , the volume constraint is not saturated in (P single ,V0 ). We now state our main theorem: Theorem I. Let κ > 0 be fixed. There exists ε 1 > 0 such that, if V 0 ∈ (0; ε 1 ), the map α → J µ (α) is monotonic on M (κ, V 0 ):

α 1 α 2 ⇒ J µ (α 1 ) J µ (α 2 ).
As a consequence, any solution α * of (P single ,V0 ) satisfies

Ω α * = V 0 .
Our second theorem deals with (Q single ,δ )-(Q single =,δ ):

Theorem II. There exists δ 1 > 0 such that, for any δ ∈ (0; δ 1 ), the functional J δ,µ is not increasing on N (Ω); furthermore, for any solution α * of (Q single ,δ ), there holds

Ω α * < V 0 .
As was explained for example in [START_REF] Mazari | Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate[END_REF], the monotonicity is intimately linked to pointwise properties of optimisers. In [START_REF] Mazari | Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate[END_REF][START_REF] Mazari | The bang-bang property in some parabolic bilinear optimal control problems via two-scale asymptotic expansions[END_REF] it is shown that for certain bilinear control problems, the monotonocity of the functional entails that optimisers are extreme points of the convex set under consideration, the aforementioned "bang-bang" functions. Here, we show related results, in that we obtain concavity and convexity-like properties. The first theorem deals with the "low fishing capacity" limit.

Theorem III.

1. Assume Ω = (0; 1) i.e. that we are working in the one-dimensional case. There exists ε 2 > 0 such that, for any V 0 ∈ (0; ε 2 ), the map J µ is strictly concave on M (κ, V 0 ). If K is constant, and if V 0 ∈ (0; ε 2 ), the solution of (P single =,V0 ) and of (P single ,V0 ) is α ≡ V 0 .

2. In any dimension d, there exists ε 2 > 0 and ε 3 > 0 such that for any V 0 ∈ (0; ε 2 ) and for any K ∈ K(Ω) such that, defining K := K 0 ,

K -K L 1 (Ω) ε 3
then the map J µ is strictly concave on M (κ, V 0 ). If K is constant, and if V 0 ∈ (0; ε 2 ), the solution of (P single =,V0 ) and of (P single ,V0 ) is α ≡ V 0 .

Theorem IV. There exists δ 2 > 0 such that, for any 0 < δ < δ 2 , any solution m * of (Q single =,δ ) is a bang-bang function: there exists a subset E * ⊂ Ω such that

m * = -1 E * .
As mentioned before we stated the Theorem, the parameters δ 1 , δ 2 are linked to the monotonicity of the functional and it will be shown through the proof that

δ k ε k (k = 1, 2).
Remark 3. In Theorems II and IV we have interpreted "large fishing capacity limit" in an L ∞ sense, by requiring that the L ∞ distance from K to any fishing strategy be small. Another possibility would be to require that the L 1 distance of K to the admissible controls is small.

Comment on the proofs The proofs of the three theorems above rely on the computation of first and second-orde Gateaux derivatives of the map J µ . The first order Gateaux-derivative of J µ will be denoted by Jµ . These computations can be used to determine whether or not certain configurations can be optimal, by checking whether or not they satisfy first order optimality conditions.

The large diffusivity limit for single player games: precised change of convexity All the information above can be made much more precise in certain asymptotic limits. In this section, we analyse in depth the behaviour, as µ → ∞, of the optimisation problems (P single ,V0 )-(P single =,V0 ). This interest of this part is two fold: first, it allows to make the change of regime of the functional J µ , from concave to convex, much more precise and, second, as the problem is linearised, this allows to gain a full characterisation of certain optimal configurations; this will be used at length in the section devoted to the analysis of Nash equilibria in two player games.

It should be noted that this approach is natural in the context of the spatial ecology: as the intricate nature of the problems at hand makes them hard to solve explicitly, it is hoped that such large diffusivity limits may provide meaningful simplifications of the problem at hand. For instance, we refer to [START_REF] He | Global dynamics of the lotka-volterra competition-diffusion system: Diffusion and spatial heterogeneity I[END_REF][START_REF] He | Global dynamics of the lotka-volterra competition-diffusion system with equal amount of total resources[END_REF][START_REF] He | Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources[END_REF][START_REF] Mazari | Optimal location of resources maximizing the total population size in logistic models[END_REF], where such asymptotic regimes are used to tackle both the optimisation of the total population size and the study of stability of certain equilibria in Lotka-Volterra systems.

Recall from [START_REF] He | Global dynamics of the lotka-volterra competition-diffusion system: Diffusion and spatial heterogeneity I[END_REF][START_REF] He | Global dynamics of the lotka-volterra competition-diffusion system with equal amount of total resources[END_REF][START_REF] He | Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources[END_REF][START_REF] Mazari | Optimal location of resources maximizing the total population size in logistic models[END_REF] that uniformly in α ∈ M (κ, V 0 )(Ω) there holds, in the W 1,2 (Ω) sense,

θ α,µ = K 0 - Ω α =:Mα + v α µ + O µ→∞ 1 µ 2 where      -∆v α -M α (K -α -M α ) = 0 in Ω , ∂vα ∂ν = 0 , ffl Ω v α = 1 M 2 α ffl Ω |∇v α | 2 .
(1.9) Also note that as we wish to investigate the monotonicity of the functional with respect to α in order to analyse whether or not the two formulations (P single ,V0 ) and (P single =,V0 ) are equivalent, we keep ffl Ω α and do not replace it with V 0 .

In particular we can already see the influence of the total fishing capacity on the first order asymptotic expansion of the functional: as in [START_REF] Mazari | Optimal location of resources maximizing the total population size in logistic models[END_REF], we obtain, uniformly in α ∈ M (κ, V 0 )(Ω), the expansion

J µ (α) = J 0 (α) + 1 µ where J 0 : α → Ω α K 0 - Ω α ,
and it is natural to invest the two asymptotic problems sup α∈M (κ,V0)

J 0 (α) (P ,single,µ→∞,0 ) and sup α∈M=(κ,V0)

J 0 (α) (P =,single,µ→∞,0 )

Of course the particularly simple shape of the limit functional J 0 makes it amenable to an easy analysis and we have the following Proposition:

Proposition 4. 1. If V 0 < K0 2 then sup α∈M=(κ,V0) J 0 (α) = sup α∈M (κ,V0) J 0 (α).
In particular the two problems (P ,single,µ→∞,0 ) and (P =,single,µ→∞,0 ) coincide.

If

V 0 > K0 2 then sup α∈M=(κ,V0) J 0 (α) < sup α∈M (κ,V0)
J 0 (α).

In particular the two problems (P ,single,µ→∞,0 ) and (P =,single,µ→∞,0 ) do not coincide.

The content of this proposition is that at the first order the asymptotic expansion of the functional selects an optimal fishing ability. However, it characterises neither its pointwise nor its geometric properties. This information is carried by the next order of this asymptotic expansion, and we will only work with equality constraints. To make this more precise we define the functional

J 1 : α → Ω αv α where      -∆v α -M α (K -α -M α ) = 0 in Ω , ∂vα ∂ν = 0 , ffl Ω v α = 1 M 2 α ffl Ω |∇v α | 2 .
where we recall that M α is defined in (1.9) and, similarly to [START_REF] Mazari | Optimal location of resources maximizing the total population size in logistic models[END_REF], we obtain, uniformly in α ∈ M (κ, V 0 ),

J µ (α) = J 0 (α) + J 1 (α) µ + O µ→∞ 1 µ 2
so that the next order optimisation problem is sup α∈M=(κ,V0)

J 1 (α). (P =,single,µ→∞,1 )
We have a fairly good understanding of this optimisation problem, as showcased by the following theorem:

Theorem V. We have the following results:

1. Concavity for low fishing abilities: if V 0 < K0 2 , the functional J 1 is strictly concave on M = (κ, V 0 ).

Convexity for large fishing abilities

: if V 0 > K0 2 , the functional J 1 is strictly convex on M = (κ, V 0 )
. Consequently the solutions of (P =,single,µ→∞,1 ) are bang-bang functions.

Characterisation in one dimension

: if Ω = (0; 1), if V 0 > K0
2 , if K is non-increasing and non constant, the optimal fishing strategy α * is equal to

α * = κ1 [1-;1] with κ = V 0 .
The proof of this theorem relies on a rewriting of the functional J 1 and, for the characterisation of optimisers in the one-dimensional case, we use Talenti inequalities [START_REF] Langford | Comparison Theorems in Elliptic Partial Differential Equations with Neumann Boundary Conditions[END_REF].

With the elements that will be used in the proof of Theorem V we also derive the following result that shows the particular role of the volume constraint V 0 = K0 3 . Before we state it, let us simply recall that a critical point of J 1 is simply a fishing strategy α ∈ M = (κ, V 0 ) such that the Gateaux-derivative of J 1 at α in any admissible direction is zero.

Proposition 5. Consider the constant fishing strategy α ≡ V 0 . Then α is a critical point of J 1 on M = (κ, V 0 ) if, and only if, one of the following is satisfied:

Either K is constant or V 0 = K0 3 .
In particular, for any K, if V 0 = K0 3 , the only solution of (P =,single,µ→∞,1 ) is α.

Qualitative analysis of Nash equilibria for two-player games: general diffusivities

In this section, we present the second facet of the fishing problems we laid out in the introduction, namely, the problem of existence and equilibria of Nash equilibria for multiple player games. For the sake of simplicity, we will only work on two-players games.

Set-up and definitions We consider two players; the first player plays a fishing strategy α 1 and the second player uses a fishing strategy α 2 . We assume that the fish population still accesses resources modelled by the function K : Ω → IR, with K 0 = ffl Ω K, and that there exists constant

κ i , V i (i = 1, 2) such that for i = 1, 2 , α i ∈ M = (κ i , V i ).
Let us note that here we work with equality constraints. We refer to Remark 7 for additional comments about the constraints but simply note here that this simplifies our presentation. If we assume that

V 1 + V 2 < K 0
then we can define θ α1,α2,µ as the unique solution of

     -µ∆θ α1,α2,µ -θ α1,α2,µ (K -α 1 -α 2 -θ α1,α2,µ ) = 0 in Ω , ∂θα 1 ,α 2 ,µ ∂ν = 0 on ∂Ω , θ α1,α2,µ 0 , θ α1,α2,µ = 0. (1.10)
It should be noted that throughout this section we once again changed the subscript defining the solution θ α1,α2,µ in order to emphasise that our optimisation variables are α 1 , α 2 .

For the i-th player (i = 1, 2) the fishing output is given by the functional

I i,µ : (α 1 , α 2 ) → Ω α i θ α1,α2,µ .
Each player wants to maximise its fishing outcome, so that we are typically in a situation where we want to investigate the existence of Nash equilibria, defined as follows: Definition 6. A Nash equilibrium for our two-players game is a couple of fishing strategies

(α * 1 , α * 2 ) such that    I 1,µ (α * 1 , α * 2 ) = max α1∈M=(κ1,V1) I 1,µ (α 1 , α * 2 ) , I 2,µ (α * 1 , α * 2 ) = max α2∈M=(κ2,V2) I 2,µ (α * 1 , α 2 ).
Remark 7. [Equality vs. Inequality constraints] Of course, the same type of results that we obtained in the single player case (Theorem I) could be derived in the case of multiple players games when considering the influence of an (in)equality constraint in the set of admissible fishing strategies; as the results would be extremely similar, we do not detail the influence of an inequality constraint and this is why we work with an equality constraint.

Our main research question here is:

Does there exist a Nash equilibrium for the two-players game described above?

Let us note here that, in general, establishing the existence of Nash equilibria is a delicate matter, that can usually be achieved using concavity or convexity properties of the functionals at hand [START_REF] González-Díaz | An Introductory Course on Mathematical Game Theory[END_REF].

Our first theorem shows that whenever the fishing abilities of both players are small enough, a Nash equilibrium exists.

Theorem VI. In the one-dimensional case Ω = (0; 1), the constants κ 1 , κ 2 , µ being fixed, there exists δ = δ(κ 1 , κ 2 , µ) > 0 such that, if

V 1 + V 2 δ
there exists a Nash equilibrium.

In any dimension d, the constants κ 1 , κ 2 , µ being fixed, there exists

δ 1 > 0 , δ 2 > 0 such that, if V 1 + V 2 δ 1 , K -K L 1 (Ω) δ 2
there exists a Nash equilibrium.

Of course this result is linked to Theorem III above, as, since the seminal paper [START_REF] Nash | Non-cooperative games[END_REF], the concavity of the cost functionals is known to be of paramount importance to obtain the existence of equilibria. Nonetheless, the proof is not immediate.

The standard results do not enable us to obtain the existence of a Nash equilibrium when, on the other hand, V 1 + V 2 is close to K 0 , and we can not conclude in the general case. We can, however, pursuing our investigation of asymptotic regimes, show that, even in this case, in which the cost functionals can behave, from the point of view of optimal control, as convex functions, (see Theorem IV above), there exists a Nash equilibrium when the diffusivity µ is large enough.

Existence of Nash equilibria when the cost functionals are convex: asymptotic analysis

Our final result deals with a slightly more complicated case, that of convex functionals. Here, we provide a result for the asymptotic expansion of the fishing functionals, and in the case where K ≡ K is constant. This problem corresponds to taking the limit µ → ∞. Following the analysis that was succinctly presented when introducing the problem (P =,single,µ→∞,1 ) we define the two limiting functionals (in what follows,

M = K 0 -V 1 -V 2 ) I 1 i : M = (κ i , V i ) α i → Ω α i v α1,α2 where v α1,α2 solves      -∆v α1,α2 -M (K 0 -α 1 -α 2 -M ) = 0 in Ω , ∂vα 1 ,α 2 ∂ν = 0 on ∂Ω , ffl Ω v α1,α2 = 1 M 2 ffl Ω |∇v α1,α2 | 2 .
An asymptotic Nash equilibrium is then defined as follows: Definition 8. An asymptotic Nash equilibrium for our two-players game is a couple of fishing strategies (α * 1 , α * 2 ) such that

I 1 1 (α * 1 , α * 2 ) = max α1∈M=(κ1,V1) I 1 1 (α 1 , α * 2 ) , I 1 2 (α * 1 , α * 2 ) = max α2∈M=(κ2,V2) I 1 2 (α * 1 , α 2 ). Theorem VII. Assume V 1 , V 2 > K0 4 , assume K is constant and let α * i = κ i 1 [0; i] with κ i i = V i (i = 1, 2). (α * 1 , α * 2
) is a Nash equilibrium in the sense of Definition 8.

Regarding "the price of anarchy" and the uniqueness of Nash equilibria We conclude with two remarks about Theorem VII. First, regarding the uniqueness of Nash equilibria, we can conclude that it does not hold in general. Indeed, consider the conclusion of Theorem VII and then compare it with the following analysis: if we assume that

K 0 = 1 , V 1 = V 2 = 1 3 , κ 1 = κ 2 = 1
and if we let α 1 = α 2 ≡ 1 3 then it is readily checked that (α 1 , α 2 ) is also a Nash equilibrium: indeed, this follows from the consideration of Remark 2 and the fact that with these definitions we have α 1 = K0-α2 2 whence the conclusion. We are thus left with two different Nash equilibria, the one given by Theorem VII and the constant one (α 1 , α 2 ). In particular, we can not expect the uniqueness of Nash equilibria to hold.

Second, we can use this particular example to illustrate a concept known, in economics, as the "price of anarchy". As we sketched briefly in the introduction to our paper, the price of anarchy quantifies the insufficiency of selfish strategies when compared to cooperative strategies. In other words, is it true that, in general, the two players would be better off collaborating and then sharing the common fishing output rather than competing in a selfish manner? Consider once again the Nash equilibrium (α 1 , α 2 ) and now assume that, instead of competing against each other, the two players united their strength, and decided to solve

max α∈M ( 1 2 , 2 3 ) J 1 (α). (1.11)
In the end they would simply split the total fishing outcome associated with an optimal strategy α * . However, from Remark 2, the unique solution of (1.11) is α = 1 2 = α * 1 + α * 2 . Thus,

I 1 1 (α 1 , α 2 ) + I 1 2 (α 1 , α 2 ) < J 1 (α) :
the total fishing output is worse than if the players had convened a strategy before playing.

Competition and cooperation: a drastic example of the "tragedy of commons" situation We can actually prove something stronger when the number of players goes to ∞. If we consider a game between n players, we can construct a sequence of Nash equilibria when K is a constant. Assuming that K ≡ 1, we can adapt the arguments above to observe that the configuration where all players have the same strategy, namely when

∀i ∈ {1, . . . n} , α * i := α * = 1 n + 1 , then (α * i ) i=1,.
..,n is a Nash equilibrium. Defining α := (α * i ) i=1,...,n , the associated steady state is

θ α = 1 - n n + 1 -→ n→+∞ 0.
We hence conclude that all µ > 0, there exist a Nash equilibria

-→ α * = (α * 1 , α * 2 , ..., α * n ) such that: 1 4 = max - → α ˆΩ n i=1 α i θ-→ α dx > ˆΩ n i=1 α * i θ-→ α * dx -→ n→+∞ 0
In particular, for this sequence of Nash equilibrium, the total harvested amount goes to zero as the number of player goes to ∞: cooperation would have been better than competition. For further discussion of this concept of "price of anarchy", we refer to [START_REF] Johari | The price of anarchy and the design of scalable resource allocation mechanisms[END_REF][START_REF] Roughgarden | Introduction to the inefficiency of equilibria[END_REF].

Bibliographical references

As there are several bodies of literature the present work fits in, we split the detailed presentation of our references accordingly.

Optimisation problem in spatial ecology Over the past two decades, a wide range of efforts have been devoted to provide a better mathematical understanding of spatially heterogeneous phenomena. Indeed, after the pioneering works of Fisher, Kolmogorov, Petrovski and Piskunoff [START_REF] Fisher | The wave of advances of advantageous genes[END_REF][START_REF] Kolmogorov | étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF], a wide body of literature was produced in an attempt to grasp fine propagation or invasion phenomena but, more recently, a new line of research has emerged that strongly emphasises the influence of heterogeneous reaction terms. After the works of Shigesada and Kawasaki, which provided a first qualitative insight into the influence of the geometry of environments [START_REF] Shigesada | Biological Invasions: Theory and Practice[END_REF] on the survival of populations, and several results of Cantrell and Cosner [START_REF] Cantrell | Diffusive logistic equations with indefinite weights: Population models in disrupted environments II[END_REF][START_REF] Cantrell | Permanence in ecological systems with spatial heterogeneity[END_REF][START_REF] Cantrell | Spatial Ecology via Reaction-Diffusion Equations[END_REF][START_REF] Cantrell | The effects of spatial heterogeneity in population dynamics[END_REF], optimising the spatial heterogeneity became a fruitful point of view. In other words: which is the optimal heterogeneity from the point of population dynamics? Of course, we need to specify which criteria are considered when using the word "optimal", but let us point out that this way of looking at the question brought forth combinations of PDE or ODE techniques and of optimal control theory. Let us also, on the topic of optimal control of biological models, point to the monograph [START_REF] Lenhart | Optimal control applied to biological models[END_REF]. A typical instance of optimal control problem of the type under study in the present paper is that of the optimal survival ability. A spectral optimisation problem, it has sparked a wealth of scientific articles devoted to its understanding and is by now fairly well understood [START_REF] Berestycki | Analysis of the periodically fragmented environment model : I -species persistence[END_REF][START_REF] Cantrell | Permanence in ecological systems with spatial heterogeneity[END_REF][START_REF] Henrot | Extremum problems for eigenvalues of elliptic operators[END_REF][START_REF] Kao | Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains[END_REF][START_REF] Lamboley | Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions[END_REF][START_REF] Lou | Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics[END_REF][START_REF] Mazari | Optimization of a two-phase, weighted eigenvalue with dirichlet boundary conditions[END_REF].

Let us point out that, in studying this problem, [START_REF] Berestycki | Analysis of the periodically fragmented environment model : I -species persistence[END_REF] features what is, to the best of our knowledge, the first use of rearrangement techniques and isoperimetric inequalities to spatial ecology problems. More recently, a new question that has drawn a lot of attention from the mathematical community is that of the optimisation of the total population size. In other words, how should we spread resources in logistic-diffusive models in order to maximise the total population size? Originating in the works of Lou [START_REF] Lou | On the effects of migration and spatial heterogeneity on single and multiple species[END_REF][START_REF] Lou | Some challenging mathematical problems in evolution of dispersal and population dynamics[END_REF] this question was then explored in details in a series of works [START_REF] Heo | On the ratio of biomass to total carrying capacity in high dimensions[END_REF][START_REF] Inoue | On the unboundedness of the ratio of species and resources for the diffusive logistic equation[END_REF][START_REF] Liang | On the dependence of population size upon random dispersal rate[END_REF][START_REF] Liang | The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration[END_REF][START_REF] Lou | Maximizing the total population with logistic growth in a patchy environment[END_REF][START_REF] Mazari | A fragmentation phenomenon for a non-energetic optimal control problem: optimisation of the total population size in logistic diffusive models[END_REF][START_REF] Mazari | Optimal location of resources maximizing the total population size in logistic models[END_REF][START_REF] Mazari | Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate[END_REF][START_REF] Nagahara | Maximization of the total population in a reaction-diffusion model with logistic growth[END_REF]. Of particular relevance in the context of the total population size was the bang-bang property: are optimisers for the total population size bang-bang functions? After several partial results [START_REF] Mazari | Optimal location of resources maximizing the total population size in logistic models[END_REF][START_REF] Nagahara | Maximization of the total population in a reaction-diffusion model with logistic growth[END_REF] the answer was proved to be yes in [START_REF] Mazari | Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate[END_REF]. It should be noted that in the proof of Theorem IV we build on the techniques of [START_REF] Mazari | Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate[END_REF] to prove a bang-bang property for optimal fishing strategies.

Optimal fishing problems Of course, all the problems we described in the previous paragraphs describe, in a way, "nice" problems, in the following sense: since we are trying to optimise a criterion with respect to resources, it is expected that adding resources will prove beneficial. One of the conclusion of [START_REF] Mazari | Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate[END_REF] is indeed that, for monotonic bilinear functionals (i.e. that increasing the resources increases the criterion) the bang-bang property holds. However, the case under study in this paper is quite different since, as we already touched upon, the problem of overfishing makes it so that the functional we are considering is no longer monotonic: it makes no sense to fish as much as we can for we may risk killing all the population. In that regard, our paper can be seen as a first detailed analysis of an optimal control problem for spatially heterogeneous fishing problems.

Of course, several authors have considered many different aspects of optimal fishing problems before. While it is impossible to list all these contributions here, let us single out [START_REF] Cooke | One-dimensional linear and logistic harvesting models[END_REF], where a survey of the early works (e.g. one-dimensional harvesting models, stochastic harvesting models...) is presented and [START_REF] Braverman | Optimal harvesting of diffusive models in a nonhomogeneous environment[END_REF] where several types of models are considered, including the logistic diffusive models, but where the diffusion operator would be (if we were to adopt our notations) ∆ • α , which changes the qualitative behaviour of the optimisation problem dramatically. Notably, it is not possible to lift their results to the case of non-regular fishing strategies α (that may be discontinuous for instance).

Nash equilibria in optimal control theory Several recent contributions deal with the existence and computation of Nash equilibria in optimal control theory. Let us single out two of these works, namely, [START_REF] Carvalho | On the computation of nash and pareto equilibria for some bi-objective control problems[END_REF][START_REF] Fernández-Cara | Bi-objective optimal control of some PDEs: Nash equilibria and quasi-equilibria[END_REF] . In these works, the functionals one seeks Nash equilibria for are of tracking-type (in the sense that we seek to minimise the distance to certain objective functions) and, very importantly, consider linearly controlled PDEs with L 2 penalisations of the constraints. This changes the features of the optimisation problem drastically. In [START_REF] Campana | Nash equilibria and bargaining solutions of differential bilinear games[END_REF] on the other hand, the question of existence and computation of Nash equilibria in bilinear problems, but for ODE models. Our paper is, to the best of our knowledge, a first contribution to the qualitative analysis of L ∞ -L 1 constrained bilinear optimal control problems with a cost function that is not of tracking-type.

Plan of the paper

The proofs of the theorems of the paper are grouped by the tools used in their proof. In section 2 we give the proof of Theorems I, II, III and IV as they all rely strongly on the computation of first and second-order Gateaux derivatives of the functional. In section 3, the proofs of the asymptotic behaviours described in Theorems V are presented. Finally, we gatherd in section 4 the proofs of those results dealing with multiple player games, Theorems VI and VII.

Proofs of Theorems I, II, III and IV

Notational simplifications Throughout this section we investigate the influence of the range of the parameter V 0 on the optimisation problems (P single ,V0 )-(P single =,V0 ), and we thus drop the diffusivity µ from all subscripts. Henceforth, θ α denotes the solution of (1.4), and we set

J : α → Ω αθ α .
Furthermore, the proofs of the three theorems under scrutiny derive from the computations of the first and second order Gateaux-derivatives of the functional J. We recall that, for an admissible fishing strategy α, an admissible perturbation h at α is a function h ∈ L 2 (Ω) such that there exists two sequences

{h n } n∈IN ∈ L 2 (Ω) IN , {ε n } n∈IN ∈ (IR + \{0})
IN satisfying:

ε n → n→∞ 0 , h n → n→∞ h in L 2 (Ω) and, for any n ∈ IN , α + ε n h n is admissible.
Whenever ffl Ω α < K 0 , we can adapt in a straightforward manner the proof of [23, Lemma 4.1] and prove that the functional J and the map α → θ α are twice Gateaux-differentiable. In the first part of this section we give these Gateaux-derivatives in expanded form, and analyse their specific features when proving our results.

Computations of the first and second-order Gateaux-derivatives of the functional J µ

We fix an admissible fishing strategy α and an admissible perturbation h at α. At this point, since we do not specify in which admissible set we work, an admissible perturbation is any h ∈ L 2 (Ω).

From the computations of [23, Lemma 4.1], the first, respectively second, order Gateaux-derivative of α → θ α at α in the direction is the unique solution θα of the equation

-µ∆ θα -θα (K -α -2θ α ) = -hθ α in Ω , ∂ θα ∂ν = 0, (2.1)
respectively the unique solution θα of the equation Similarly the first, respectively second, order Gateaux-derivative of the map J at α in the direction h is given by the expression

-µ∆ θα -θα (K -α -2θ α ) = -2h θα -2 θ2 α in Ω , ∂ θα ∂ν = 0 on ∂Ω. ( 2
J(α)[h] = Ω hθ α + Ω α θα , (2.3) respectively by J(α)[h, h] = 2 Ω h θα + Ω α θα . (2.4)
We need to introduce an adjoint state in order to give equations (2.3)-(2.4) tractable expressions. We introduce the unique solution p α of

-µ∆p α -p α (K -α -2θ α ) = α in Ω ∂pα ∂ν = 0 on ∂Ω. (2.5)
The following properties are obtained by adapting the reasoning of [59, Lemma 13], which simply relies on the aforementioned Remark 9 that the first eigenvalue of -µ∆ -(K -α -2θ α ) is positive:

Lemma 10. There exists a unique solution p α of (2.5). Furthermore, if α 0 , α = 0, inf

Ω p α > 0.
Now, if we multiply (2.5) by θα and integrate by parts, and, similarly, multiply (2.1) by p α and integrate by parts we derive the equality

Ω α θα = µ Ω ∇ θα , ∇p α - Ω p α θα (K -α -2θ α ) = - Ω hp α θ α , so that J(α)[h] = Ω (1 -p α )θ α h.
Similarly, we obtain

1 2 J(α)[h] = Ω h θα - Ω p α h θα - Ω p α θ2 α .
We have thus proved the following lemma:

Lemma 11. The first and second order Gateaux-derivative of the functional admit the following expressions:

J(α)[h] = Ω (1 -p α )θ α h, (2.6
)

and 1 2 J(α)[h, h] = Ω (1 -p α )h θα - Ω p α θ2 α . (2.7)
2.2 Computation of the first and second-order Gateaux-derivatives of J δ,µ

We can adapt the proofs of the previous section to J δ,µ . Similar to the notational conventions we adopted above, we now denote by θ K+δm the solution of (1.4) with α = K + δm. We define, for any m ∈ N (Ω), the unique solution q δ,m of

-µ∆q δ,m -q δ,m (-δm -2θ K+δm ) = K + δm in Ω , ∂qm ∂ν = 0 on ∂Ω. (2.8)
Similarly to Lemma 11 we obtain the following expression:

Lemma 12. The first and second order Gateaux-derivative of the functional admit the following expressions:

Jδ,µ (m)[h] = δ Ω (1 -q δ,m )θ K+δm h, (2.9
)

and 1 2 Jδ,µ (m)[h, h] = δ Ω (1 -q δ,m )h θK+δm - Ω q δ,m θ2 m , (2.10)
where θK+δm satisfies

-µ∆ θK+δm -θK+δm (-δm -2θ K+δm ) = -δhθ K+δm in Ω , ∂ θK+δm ∂ν = 0 on ∂Ω.
(2.11)

Proof of Theorems I-II: monotonicity of the functionals

Plan of the proofs We recall that monotonicity, for instance for J µ , means that

∀α 1 , α 2 ∈ M (κ, V 0 ) , α 1 α 2 a.e. ⇒ J µ (α 1 ) J µ (α 2 ).
However, by the mean value theorem, we know that, for any

α 1 , α 2 ∈ M (κ, V 0 ), there exists ξ ∈ [0; 1] such that J µ (α 2 ) -J µ (α 1 ) = Jµ (α 1 + ξ(α 2 -α 1 ))[α 2 -α 1 ] (2.12)
and thus Lemma 11 yields the existence of ξ ∈ [0; 1] such that

J µ (α 2 ) -J µ (α 1 ) = Ω (1 -p α1+ξ(α2-α1) )θ α1+ξ(α2-α1) (α 2 -α 1 ). (2.13)
We can thus read the monotonicity of J µ on (2.13): if α 1 α 2 almost everywhere, and since θ α1+ξ(α2-α1) is positive on Ω for any ξ ∈ [0; 1], obtaining the monotonicity of the functional boils down to deriving the sign of 1 -p α1+ξ(α2-α1) . Thus the proof of Theorem I is simply to show that under certain volume constraints we have p α < 1.

Similarly, using Lemma 12, to show the non-monotonicity of J δ,µ it suffices to prove that, for δ > 0 small enough, q δ,m > 1 for any m ∈ N = (Ω). This will imply that the optimal values of the two problems (Q single ,δ )-(Q single =,δ ) differ. Proof of Theorem I. Following the general idea explained in the plan of the proof, it suffices to prove the following lemma: Proposition 13. The two constants κ, µ being fixed, we have the following property: for any δ > 0, there exists ε 1 = ε 1 (Ω, κ, µ, δ) > 0 such that, for any V 0 ∈ (0; ε 1 ), for any α ∈ M (κ, V 0 ), we have

0 < inf Ω p α sup Ω p α 1 -δ.
With this result at hand it is easy to obtain the monotonicity property: fixing δ > 0 and choosing the ε 1 given by Proposition 13, we obtain

∀V 0 ∈ (0; ε 1 ) , ∀α ∈ M (κ, V 0 ) , inf Ω (Ψ α := (1 -p α )θ α ) δ inf Ω θ α > 0
so that (2.13) implies the conclusion: with δ = 1 2 , the functional is monotonically increasing if V 0 < ε 1 where ε 1 is given by Proposition 13. This implies that any solution α * of (P single ,V0 ) satisfies

Ω α * = V 0
and thus that α * is a solution of (P single ,V0 ). Indeed, should we have ffl Ω α * < V 0 we simply take any positive function h ∈ L ∞ (Ω) such that α * + h ∈ M = (κ, V 0 ). By monotonicity of the functional,

J µ (α * + h) > J µ (α * ), a contradiction.
It remains to prove Proposition 13.

Proof of Proposition 13. Let us note that, as κ is fixed and as K ∈ L ∞ (Ω), a classical application of the maximum principle there holds

∀µ > 0 , θ α,µ L ∞ (Ω) K L ∞ (Ω) + α L ∞ (Ω) =: M 0 . (2.14)
We will prove that lim

V0→0 + sup α∈M (κ,V0) p α L ∞ (Ω) = 0.
To control the L ∞ norm of any p α , we need to use the first eigenvalue

λ(K -α -2θ α ) of the operator -µ∆ -(K -α -2θ α )
endowed with Neumann boundary conditions. As this operator is symmetric, we know that

λ(K -α -2θ α ) = inf u∈W 1,2 (Ω) , ffl Ω u 2 =1 µ Ω |∇u| 2 - Ω u 2 (K -α -2θ α ) . (2.15)
As in [49, Proof of Lemma 2.1] (see also Remark 9 above) we know that for any V 0 ∈ (0; K 0 ) and any α ∈ M (κ, V 0 ) there holds λ(K -α -2θ α ) > 0.

To obtain uniform regularity estimates on p α we need to obtain a uniform lower bound on λ(Kα -2θ α ) as V 0 → 0 + . Lemma 14. There exists a 0 , ε 0 > 0 such that

∀V 0 ∈ (0; ε 0 ) , ∀α ∈ M (κ, V 0 ) , λ(K -α -2θ α ) a 0 .
Proof of Lemma 14. We observe that from (2.14) and standard L p elliptic regularity theory, for any p ∈ [1; +∞) there exists a constant M p = M p (µ, Ω) > 0 such that uniformly in V 0 and uniformly in α ∈ M (κ, V 0 ) there holds θ α W 2,p (Ω) M p .

Using Sobolev embeddings this implies that for any s ∈ [0; 1) there exists a constant C s such that uniformly in V 0 and uniformly in α ∈ M (κ, V 0 ) there holds

θ α C 1,s (Ω) C s . (2.16)
It is expected that as V 0 → 0 we should have θ α → θ where θ is the solution of

     -µ∆θ -θ K -θ = 0 in Ω , ∂θ ∂ν = 0 on ∂Ω , θ 0 , θ = 0.
(2.17) Let us show that this convergence is uniform in the following sense: lim

V0→0 + sup α∈M (κ,V0) θ α -θ C 0 (Ω) = 0.
(2.18)

Argue by contradiction and assume there exists a sequence {V k } k∈IN , c 1 > 0 and, such that, for any k ∈ IN, there exists

α k ∈ M (κ, V k ) such that θ α k -θ C 0 (Ω) c 1 .
From (2.16), we extract from {θ α k } k∈IN a C 1 converging subsequence, still labeled {θ α k } k∈IN and its C 1 limit θ ∞ . From [49, Equation (2.4)] there exists a constant c 0 uniform in V 0 such that

θ α k -θ L 1 (Ω) c 0 α k 1 3 L 1 (Ω) .
We thus conclude that θ ∞ = θ, a contradiction. From this uniform convergence and the simplicity of the first eigenvalue λ(K -α -2θ α ), we deduce that lim

V0→0 + inf α∈M (κ,V0) λ(K -α -2θ α ) = λ(K -2θ) > 0 (2.19)
where the last inequality comes from the aforementioned [49, Proof of Lemma 2.1]. The proof of (2. [START_REF] Cooke | One-dimensional linear and logistic harvesting models[END_REF]) is standard and we postpone it to appendix A.1. Lemma 14 is proved.

We can go back to the proof of Proposition 13. We argue via a standard bootstrap method, as follows: using p α as a test function in (2.5) we obtain

µ Ω |∇p α | 2 - Ω p 2 α (K -α -2θ α ) Vol(Ω) -1 α L 2 (Ω) p α L 2 (Ω) Vol(Ω) -1 κV 0 p α L 2 (Ω) .
From the Rayleigh quotient formulation of eigenvalue (2.15) and the lower estimate of Lemma 14 we deduce that

p α L 2 (Ω) √ κV 0 a 0 Vol(Ω) ,
which in turn yields a uniform W 1,2 (Ω) bound on the family {p α }. Using Sobolev embeddings, the bootstrap method implies the following uniform bound: there exists ε 0 > 0 such that, for any p ∈ [1; +∞), for any V 0 ∈ (0; ε 0 ), there exists M p such that for any α ∈ M (κ, V 0 ), there holds

p α W 2,p (Ω) M p .
It is then clear that for any sequence {V k } k∈IN converging to zero and for any

{α k } k∈IN ∈ k∈IN M (κ, V k ), the sequence {p α k } k∈IN converges in C 1 (Ω) to the solution p of -µ∆p -p(K -2θ) = 0 in Ω , ∂p ∂ν = 0 on ∂Ω.
As λ(K -2θ) > 0, p = 0. Adapting the arguments of the proof of Lemma 14, it is easily shown that such convergence is uniform and that lim V0→0 sup α∈M (κ,V0)

p α C 1 (Ω) = 0. (2.20)
The proof of Proposition 13 is finished.

Thus, as we explained how Proposition 13 implied Theorem I, the proof of Theorem I is complete.

Proof of Theorem II. For large fishing abilities, on the contrary, we will prove that

∀η > 0 , ∃δ 1 > 0 , ∀0 < δ < δ 1 , ∀m ∈ N (Ω) , inf Ω q δ,m 1 η .
To do so, we need to investigate the asymptotic behaviour of q δ,m as δ → 0. Given that q δ,m solves (2.8), this requires a knowledge of the behaviour of θ K+δm as δ → 0. This is the object of the following proposition:

Proposition 15. Uniformly in m ∈ N (Ω) the following asymptotic expansion holds in C 1 (Ω):

θ K+δm = δm 0 + O δ→0 (δ 2 ).
Proof of Proposition 15. We set z δ,m := θ K+δm δ . Direct computations show that z δ,m solves

     -µ δ ∆z δ,m -z δ,m (-m -z δ,m ) = 0 in Ω , ∂z δ,m ∂ν = 0 on ∂Ω , z δ,m 0 , z δ,m = 0.
(2.21) Thus the large fishing ability limit corresponds to a large-diffusivity limit for a standard logistic diffusive equation. We can now apply [58, Appendix A-Convergence of the series]. Let us simply recall the main steps: first it is proved that, uniformly in m, the asymptotic expansion

z δ,m = m 0 + O µ→∞ (δ) (2.22)
holds in W 1,2 (Ω). We then use a standard bootstrap argument, to obtain that (2.22) holds in any W 2,p (Ω). From the definition of z δ,m we infer that θ K+δm admits the expansion

θ K+δm = δm 0 + O µ→∞ δ 2 (2.23)
in C 1 (Ω). This concludes the proof.

From this proposition we obtain an asymptotic expansion of the adjoint state q δ,m :

Proposition 16. Uniformly in m ∈ N (Ω), the following asymptotic expansion holds in C 1 (Ω):

q δ,m = 1 δ • K 0 m 0 + o δ→0 1 δ .
Proof of Proposition 16. From Proposition 15 the function q m,δ solves

-µ∆q m,δ -δq m,δ -m -2m 0 + O δ→0 (δ) = K + δm. (2.24)
We set z δ,m := δq δ,m . Direct computations show that z δ,m solves

   -µ δ ∆z δ,m -z δ,m -m -2m 0 + O δ→0 (δ) = K + δm in Ω , ∂z δ,m ∂ν = 0 on ∂Ω.
(2.25)

We can apply exactly the same reasoning as in [START_REF] Mazari | Optimal location of resources maximizing the total population size in logistic models[END_REF] to obtain that, in C 1 (Ω), we have

z δ,m = K 0 m 0 + o δ→0 (1)
, whence the conclusion.

From Proposition 16 we obtain that, for any η > 0 there exists δ 1 > 0 such that, for any 0 < δ < δ 1 , for any m ∈ N (Ω),

1 -q δ,m 1 - K 0 2m 0 δ - 1 η .
However, we may proceed as in the proof of Theorem I: for any m 1 m 2 , m 1 = m 2 , J δ,µ (m 2 ) < J δ,µ (m 1 ), so that the functional is no longer monotonic.

Proofs of Theorems III-IV

Reformulation of the second-order Gateaux-derivatives The proofs of the concavity of J µ and the characterisation of maximisers of J δ,µ as extreme points of the admissible sets rely on the type of computations carried out in [START_REF] Mazari | The bang-bang property in some parabolic bilinear optimal control problems via two-scale asymptotic expansions[END_REF][START_REF] Mazari | Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate[END_REF][START_REF] Mazari | Qualitative analysis of optimisation problems with respect to nonconstant Robin coefficients[END_REF] and in particular on a certain reformulation of the second-order Gateaux-derivatives of the functionals under consideration.

Reformulation of Jµ We start with Jµ (α)[h, h],

which for notational convenience we write here J(α)[h, h] (in other words we have dropped the subscript µ). Throughout the computations that follow we work with a fixed α ∈ M = (κ, V 0 ) and a fixed admissible perturbation h at α. We recall that from Lemma 11 we have the expression

1 2 J(α)[h, h] = Ω (1 -p α )h θα - Ω p α θ2 α .
Now observe that we may rewrite

h := µ∆ θα + θα (K -α -2θ α ) θ α ,
whence, defining

ψ α := 1 -p α θ α , we derive Ω (1 -p α )h θα = Ω ψ α µ θα ∆ θα + θ2 α (K -α -2θ α ) = Ω ψ α µ 2 ∆ θ2 α -µ ∇ θα 2 + θ2 α (K -α -2θ α ) = -µ Ω ψ α ∇ θα 2 + Ω θ2 α µ 2 ∆ψ α + ψ α (K -α -2θ α ) .
We have thus established the following lemma:

Lemma 17. For any α ∈ M = (κ, V 0 ), for any admissible perturbation h at α, there holds

1 2 J(α)[h, h] = -µ Ω 1 -p α θ α |∇ θα | 2 + Ω θ2 α µ 2 ∆ 1 -p α θ α + 1 -p α θ α (K -α -2θ α ) -p α .
Reformulation of J δ,µ We can carry the same type of computations for the second-order Gateaux derivative of J δ,µ : let m ∈ N (Ω) be fixed and h be an admissible perturbation at h. We know from Lemma 12 that

1 2 Jδ,µ (m)[h, h] = δ Ω (1 -q δ,m )h θK+δm - Ω q δ,m θ2
K+δm .

However, we may rewrite

h = µ∆ θK+δm + θK+δm (-δm -2θ K+δm ) δθ K+δm
and thus obtain, defining

ϕ δ,m := q δ,m -1 θ K+δm δ Ω (1 -q δ,m )h θK+δm = µ Ω ϕ δ,m ∇ θK+δm 2 + Ω θ2 K+δm µ 2 ∆ϕ δ,m -(-δm -2θ K+δm ) .
Hence the following lemma holds:

Lemma 18. For any m ∈ N = (Ω), for any admissible perturbation h at m, there holds

1 2 Jδ,µ (m)[h, h] = µ Ω q δ,m -1 θ K+δm ∇ θK+δm 2 + Ω θ2 K+δm µ 2 ∆ 1 -q δ,m θ K+δm -(-δm -2θ K+δm ) - Ω q δ,m θ2 K+δm .
Proofs of Theorems III-IV We now get to the core of the proofs.

Proof of Theorem III. Theorem III contains two statements, one dealing with the one-dimensional case, the other one dealing with the multi-dimensional case. Both rely on the same estimate of the expression of the second order gateaux derivative give by Lemma 17.

From the proof Lemma 14 we lift estimate (2.20), which ascertains that lim V0→0 sup α∈M (κ,V0)

p α C 1 (Ω) = 0.
Let us introduce, for any α ∈ M (κ, V 0 ), the potential

W α := µ 2 ∆ 1 -p α θ α + 1 -p α θ α (K -α -2θ α ) -p α
as well as, for any α ∈ M (κ, V 0 ), with V 0 small enough to ensure that for any α ∈ M (κ, V 0 ) we have 1 -p α 1 2 , the operator

L α := -µ∇ • 1 -p α θ α ∇ -W α .
Let ξ(α) be the first eigenvalue of L α . ξ(α) is defined, by its Rayleigh quotient, as

ξ(α) := inf u∈W 1,2 (Ω) , ffl Ω u 2 =1 µ Ω 1 -p α θ α |∇u| 2 - Ω W α u 2 . . (2.26)
From Lemma 17, there holds, for any α ∈ M (κ, V 0 ) and any admissible perturbation h at α,

1 2 J(α)[h, h] -ξ(α) Ω θ2 α . (2.27)
The goal is now to get the asymptotic behaviour of ξ(α) as V 0 → 0 and, more precisely, to obtain that lim

V0→0 + inf α∈M (κ,V0) ξ(α) > 0, (2.28)
which would suffice to prove the concavity of the functional. To do so, a first step is to understand the behaviour of the potential W α as V 0 → 0 . As

µ∆ 1 -p α θ α = -µ ∆p α θ α + 2µ ∇p α , ∇θ α θ 2 α -(1 -p α )µ ∆θ α θ 2 α + 2(1 -p α ) |∇θ α | 2 θ 3 α and as -µ∆p α = α + p α (K -α -2θ α )
we deduce that, if we define the limit potential

W 0 := µ 2 ∆ 1 θ + 1 θ (K -2θ), then it follows from (2.18)-(2.20) that ∀p ∈ [1, +∞) , lim V0→0 sup α∈M (κ,V0) W α -W L p = 0. (2.29)
Let ξ be the first eigenvalue of the operator

L := -µ∇ • 1 θ -W .
By a standard method that we detail in Appendix A.2 this implies lim

V0→0 sup α∈M (κ,V0) ξ(α) -ξ = 0. (2.30)
In particular, the proof of the Theorem is complete, provided we can prove that ξ > 0.

First analysis of ξ Let us first observe that we can expand W as follows:

W = µ 2 ∆ 1 θ + 1 θ (K -2θ) = µ 2 - ∆θ θ 2 + 2 ∇θ 2 θ 3 + 1 θ (K -2θ) = 1 2θ (K -θ) since θ solves (2.17) + µ ∇θ 2 θ 3 + 1 θ (K -2θ) F := -µ∇ • 1 θ ∇ -Z (2.32)
is equal to 0. We will then use a monotonicity principle for eigenvalues. We start with the fact we just claimed:

Lemma 19. Z being defined by (2.31) and F being defined by (2.32), the first eigenvalue A of F is zero, and its associated eigenfunction is ϕ = θ Proof. Let ϕ := θ 3 2 . We have

∇ϕ = 3 2 • θ∇θ
and so

∇ϕ θ = 3 2 • ∇θ √ θ . Thus -µ∇ • ∇ϕ θ = 3 2 • -µ∆θ √ θ -µ 3 4 ∇θ 2 √ θ = 3 2 • θ(K -θ) √ θ -µ 3 4 ∇θ 2 √ θ = 3 2 • θ(K -θ) -µ 3 4 ∇θ 2 √ θ = ϕ 3 2 • K -θ θ -µ 3 4 • ∇θ 2 θ 3 .
Thus ϕ is an eigenfunction of F associated with the eigenvalue 0. As ϕ = θ 3 2 > 0 and as the first eigenvalue of F is the only eigenvalue whose associated eigenfunctions have constant signs, we deduce that ϕ is a principal eigenfunction and that the first eigenvalue of F is 0. Now, if we can ensure that W Z , W = Z then by virtue of the monotonicity of the first eigenvalue [24, Lemma 2.1] we have ξ > 0 so that (2.28). Thus the proof would be complete. We now notice that

W -Z = µ 4 • ∇θ 2 θ 3 -1 .
Proving that ξ > 0 boils down to investigating whether or not µ 4 • |∇θ| 2 θ 3 -1 < 0. We do that in the one-dimensional case and, in the higher dimensional case, for resources distributions that are close to a constant.

1. In the one-dimensional case: here we use an estimate of Bai In particular,

1 - µ 4 • θ (x) 2 θ 3 1 - 1 6 = 5 6
so that the proof is concluded.

2. In the higher-dimensional case: in that second case, since we work with variable K, let us add the subscript K to the notation θ and denote by θ K the solution of (2.17). In this case the only thing that should be noted is that, when K is constant,

θ K ≡ 1 Vol(Ω) K 0 = K. In that case, W -Z = -1 < 0.
However, a simple adaptation of the arguments of (2.18) proves that for any δ > 0 there exists a constant ε 3 > 0 such that, for any

K ∈ K(Ω), if K -K L 1 (Ω) ε 3 then θ K -θ K C 1 (Ω) δ .
If δ is small enough, this implies that for any

K ∈ K(Ω) such that K -K L 1 (Ω) ε 3 we have -1 + 1 4 |∇θ K | 2 θ 3 K < - 1 2 .
The conclusion follow in exactly the same way.

Proof of Theorem IV. For the proof of Theorem IV we follow the same type of strategy as the one used for the proof of Theorem III. We start with the expression of the second-order Gateaux derivative given in Lemma 18: for any m ∈ N = (Ω) and any admissible perturbation h at m we have

1 2 Jδ,µ (m)[h, h] = µ Ω q δ,m -1 θ K+δm ∇ θK+δm 2 + Ω θ2 K+δm µ 2 ∆ 1 -q δ,m θ K+δm -(-δm -2θ K+δm ) - Ω q δ,m θ2 K+δm .
Recall that, from Proposition 16, there exists δ 2 > 0 small enough such that, for any δ δ 2 and for any m ∈ N = (Ω) there holds

q δ,m -1 sup m∈N=(Ω) θ K+δm L ∞ (Ω) 2 .
For δ δ 2 we can thus bound the second-order derivative as

1 2 Jδ,µ (m)[h, h] µ 2 Ω ∇ θK+δm 2 + Ω Y δ,m θ2 K+δm ,
where the potential Y δ,m is defined as

Y δ,m := µ 2 ∆ q δ,m -δ θ K+δm -(-δm -2θ K+δm ) -q δ,m .
However, expanding Y δ,m as was done in the proof of Theorem III for ∆ 1-pα θα we obtain the existence of a constant β = β(δ) such that

∀m ∈ N = (Ω) , Y m L ∞ (Ω) β.
Defining γ := µ 2 we thus have, for the second-order Gateaux-derivative, the lower estimate

1 2 Jδ,µ (m)[h, h] γ Ω ∇ θK+δm 2 -β Ω θ2
K+δm .

However, we are now exactly in the proper situation to mimic the proof of [59, Theorem 1]: argue by contradiction and assume that there exists a non-bang-bang solution m * of (Q single =,δ ). In particular the set ω * := {0 < m * < 1} has a positive measure. Let M > 0 be arbitrarily large. Following [59, Proof of Theorem 1, Eq. (2.20) and below] there exists an admissible perturbation h M at m * supported in ω * such that

Ω ∇ θK+δm * 2 > M Ω θ2 K+δm * .
Taking M := β γ + 1 we obtain the required contradiction: for the perturbation h β γ there holds

1 2 Jµ,δ (m * ) h β γ , h β γ γ Ω θ2 K+δm * > 0,
in contradiction with the optimality of m * .

3 Proofs of Theorem V

Proof of Proposition 4

Before we prove Theorem V we prove Proposition 4.

Proof of Proposition 4. We recall that

J 0 : α → Ω α K 0 - Ω α .
Clearly, J 0 is twice Gateaux-differentiable at every α and, for any α ∈ M (κ, V 0 ) and any admissible perturbation h at α there holds

J0 (α)[h, h] = Ω h K 0 -2 Ω α .
In particular, if ffl Ω α V 0 < K0 2 the functional J 0 is increasing on M (κ, V 0 ), so that any solution α * of (P ,single,µ→∞,0 ) satisfies ffl Ω α * = V 0 . Thus, α * is also a solution of (P =,single,µ→∞,0 ). If, on the contrary, we assume that V 0 > K0 2 , consider a solution α * of (P ,single,µ→∞,0 ). Let us prove that we necessarily have ffl Ω α * < V 0 . If, by contradiction, we had

Ω α * = V 0
then, for any non-positive, non-zero perturbation h, we have

J0 (α * )[h] = Ω h K 0 -2 Ω α * > 0,
in contradiction with the optimality of α * . In particular, ffl Ω α * < V 0 and so the two problem (P =,single,µ→∞,0 ) and (P ,single,µ→∞,0 ) do not coincide.

Proof of Theorem V

Proof of Theorem V. Reformulation of J 1 : To prove Theorem V, we need a tractable rewriting of the function J 1 . Let us recall that we defined the constant

M α := K 0 - Ω α.
As we are working with an equality constraint we may drop the subscript α and simply define

M 0 := K 0 -V 0 .
The functional J 1 is defined as

J 1 (α) = Ω αv α where      -∆v α -M 0 (K -α -M 0 ) = 0 in Ω , ∂vα ∂ν = 0 on ∂Ω , ffl Ω v α = 1 M 2 0 ffl Ω |∇v α | 2 .
Let us introduce, for any α ∈ M (κ, V 0 ), the solution vα of

     -∆v α -M 0 (K -α -M 0 ) = 0 in Ω, ∂ vα ∂ν = 0 on ∂Ω , ffl Ω vα = 0. (3.1)
Clearly we have

v α = vα + 1 M 2 0 Ω |∇v α | 2 , so that J 1 (α) = Ω αv α = Ω αv α + 1 M 2 0 Ω |∇v α | 2 Ω α = Ω (α + M 0 -K)v α - Ω (M 0 -K)v α + V 0 M 2 0 Ω |∇v α | 2 = - 1 M 0 Ω |∇v α | 2 + Ω K vα as Ω vα = 0 + V 0 M 2 0 Ω |∇v α | 2 = 2V 0 -K 0 M 2 0 Ω |∇v α | 2 + Ω K vα .
Analysis of the second order derivative of J 1 : But now observe that, if we define

j 1 : α → 2V 0 -K 0 M 2 0 Ω |∇v α | 2 , j 2 : α → Ω K vα ,
then j 2 is linear in α as the map α → vα is linear. As

J 1 = j 1 + j 2
the second order derivative of J 1 is determined by the second-order derivative of j 1 . However, it is straightforward to see, mimicking the computations of [58, Proof of Theorem 1, Step 1], that, for any α ∈ M = (κ, V 0 ) and any admissible perturbation h at α, we have

j1 (α)[h, h] = 2V 0 -K 0 M 2 0 Ω ∇ vα 2 where      -∆ vα + M 0 h = 0 in Ω , ∂ vα ∂ν = 0 on ∂Ω , ffl Ω vα = 0.
In particular, if 2V 0 > K 0 the functional is (strictly) convex. Thus, any solution of (P =,single,µ→∞,1 ) is an extreme point of M = (κ, V 0 ) that is, any solution is a bang-bang function. Conversely, if 2V 0 < K 0 , the functional is (strictly) concave. This ends the proof of the two first-points of the theorem. Now let us move to the characterisation of optimisers in the convex regime (point 3 of the theorem). Assume Ω = (0; 1), assume that 2V 0 > K 0 and that K is a non-increasing, non constant function. To give an explicit description of the maximiser α we need to use the notion of nonincreasing rearrangement. Let us recall the following definition: Definition 21. For any non-negative function f ∈ L 1 (0; 1) there exists a unique non-increasing, non-negative function f # ∈ L ∞ (0; 1) such that ∀t 0 , Vol ({f t}) = Vol({f # t}).

Similarly, there exists a unique non-decreasing, non-negative function f # ∈ L ∞ (0; 1) such that ∀t 0 , Vol ({f t}) = Vol({f # t}).

Two inequalities are of paramount importance when dealing with rearrangements:

1. The celebrated Pólya-Szegö inequality: it states that, if f ∈ W 1,2 (0; 1), then

f # ∈ W 1,2 (Ω)
and, furthermore, that we have

1 0 f # 2 1 0 f 2 . (3.2)
2. The Hardy-Littlewood inequality: it states that, if f , g ∈ L 1 (Ω) are bounded functions then

1 0 f # g # 1 0 f g 1 0 f # g # . (3.3)
While rearrangements are central in the calculus of variations (we refer for instance to [START_REF] Baernstein | Symmetrization in Analysis[END_REF][START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model : I -species persistence[END_REF][START_REF] Kawohl | Rearrangements and Convexity of Level Sets in PDE[END_REF][START_REF] Kesavan | Symmetrization and Applications[END_REF][START_REF] Rakotoson | Réarrangement Relatif[END_REF]) and has wide ranging applications, we focus here on Talenti inequalities. Originating in the seminal [START_REF] Talenti | Elliptic equations and rearrangements[END_REF], in the case of the Schwarz rearrangement for Dirichlet boundary conditions, these inequalities aim at comparing the solution u of a Poisson equation of the form -∆u = f with Dirichlet boundary conditions with the solution v of a symmetrised equation. Among the many results related to possible extensions and to the qualitative analysis of these inequalities to other operators [START_REF] Alvino | Comparison results for elliptic and parabolic equations via Schwarz symmetrization[END_REF][START_REF] Alvino | Comparison result for solutions to elliptic problems with robin boundary conditions[END_REF][START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF][START_REF] Mazari | Quantitative estimates for parabolic optimal control problems under l∞ and l1 constraints in the ball: Quantifying parabolic isoperimetric inequalities[END_REF][START_REF] Mossino | Isoperimetric inequalities in parabolic equations[END_REF][START_REF] Sannipoli | Comparison results for solutions to the anisotropic laplacian with robin boundary conditions[END_REF] let us focus on the results of [START_REF] Langford | Comparison Theorems in Elliptic Partial Differential Equations with Neumann Boundary Conditions[END_REF]. To use them, we need to recall the rearrangement order on L 1 (0; 1): for any two non-negative functions f, g ∈ L 1 (0; 1), we say that f dominates g in the sense of rearrangements and we write f ≺ g if, and only if,

∀r ∈ [0; R] , r 0 f # r 0 g # .
Our goal is to show that minor adaptation of [45, Chapter 5] yields the following result: defining, for any f ∈ L 1 (Ω) such that

ffl 1 0 f = 0, the solution u f of      -(u f ) = f in (0; 1) , u f (0) = u f (1) = 0 , ffl 1 0 u f = 0. (3.4)
we claim that, for any g such that f ≺ g, there holds

u f ≺ u g # . (3.5)
Before we prove (3.5), let us investigate why this yields the required result.

Lemma 22. If estimate (3.5) holds for any non-negative f ∈ L 1 (Ω), if K = K # is not constant and if V 0 > K0 2
then the unique solution of (P =,single,µ→∞,1 ) is given by

α * = κ1 [1-;1]
where κ = 1.

Proof of Lemma 22. The proof of this Lemma rests upon a rewriting of J 1 in terms of natural energy functional associated with vα .

Rewriting of J 1 in terms of an energy functional We start from the fact that for any α we have

J 1 (α) = 2V 0 -K 0 M 2 0 1 0 |v α | 2 + 1 0 K vα
where vα satisfies (3.1) To alleviate notations, define

C 0 := 2 2V 0 -K 0 M 2 0 > 0, so that ∀α , J 1 (α) = C 0 1 0 |v α | 2 + 1 0 K vα .
However, (3.1) admits a natural variational formulation: introduce the space

X := u ∈ W 1,2 (0; 1) , 1 0 u = 0
and define the energy functional

E α : X u → 1 2 1 0 |u | 2 -M 0 1 0 (K -α -M 0 )u.
Then vα is the unique solution of min u∈X E α (u). Now observe that from the weak formulation of (3.1) we have

1 0 |v α | 2 = M 0 1 0 (K -α -M 0 )v α = -E α (v α ) + 1 2 1 0 |v α | 2 (3.6)
so that in the end

1 0 |v α | 2 = -2E α (v α ).
This allows us to rewrite J 1 as

J 1 (α) = -2C 0 E α (v α ) + 1 0 K vα . (3.7)
We will prove that rearranging the coefficients of the equation increases each term appearing in the right-hand-side of (3.7).

Rearranging α increases -E α (v α ) Let us start with the energy functional. From the Pólya-Szegö inequality (3.2) we know that

1 0 |v α | 2 1 0 v# α 2 .
Furthermore, from equimeasurability of the rearrangement, we have

1 0 M 0 vα = 1 0 M 0 (v # α ).
Finally, from the Hardy-Littlewood inequality (3.3) there holds

1 0 K vα 1 0 K # v# α = 1 0 K v# α and 1 0 α # v# α 1 0 αv α .
This gives

E α # vα # E α # v# α E α (v α ).
Rearranging increases ffl 1 0 K vα Let us now observe the effect of rearrangement on the equation satisfied by vα . Assume that the Talenti inequality (3.5) holds. Then we know (taking f = g in (3.5)) that vα ≺ z

where z solves      -z = M 0 (K -α -m 0 ) # in (0; 1) , z (0) = z (1) = 0 , ffl 1 0 z = 0.
In general it is not true that (K -α) # = K # -α # . However, we always have the inequality

(K -α) # ≺ K # -α # = K -α # .
See, for instance, [2, Proposition 3]. Thus, applying (3.5)

with f = M 0 (K -α -M 0 ) and g = M 0 (K # -α # -M 0 ) yields z ≺ vα # , whence vα ≺ vα # .
From the Hardy-Littlewood inequality (3.3) and the definition of the order relation ≺ this gives

1 0 K vα 1 0 K # vα # = 1 0 K vα # .

Conclusion

In conclusion, we have established that

J 1 (α) = -2C 0 E α (v α ) + 1 0 K vα -2C 0 E α # (v α # ) + 1 0 K vα # = J 1 (α # ),
whence the conclusion. To guarantee uniqueness, it suffices to check that equality holds in the Pólya-Szegö inequality if and only if vα = v# α or vα = (v α ) # . This, however, follows from [START_REF] Ferone | Minimal rearrangements of sobolev functions : a new proof[END_REF]. We then conclude that either α = α # or α = α # . However, as K = K # is not constant, the only possibility to also achieve equality in the Hardy-Littlewood inequality is to have α = α # .

It remains to prove the Talenti inequality (3.5). As we said, the proof can be quickly derived from the considerations of Langford in [START_REF] Langford | Comparison Theorems in Elliptic Partial Differential Equations with Neumann Boundary Conditions[END_REF]. For the sake of completeness, we sketch the details of the proof of [START_REF] Langford | Comparison Theorems in Elliptic Partial Differential Equations with Neumann Boundary Conditions[END_REF] here.

Proof of the Talenti inequality (3.5). We may extend f by parity to an even function (still denoted f in the following) on (-1; 1). Similarly, since u f satisfies Neumann boundary conditions at 0, it may be extended by parity to (-1; 1). Thus extended, the function u f satisfies

     -(u f ) = f in (-1, 1) , u f (-1) = u f (1) = 0 , ffl 1 -1 u f = 0. (3.8)
Furthermore, by parity of f , we have We extend it to IR by 2-periodicity. Consequently (see [START_REF] Langford | Comparison Theorems in Elliptic Partial Differential Equations with Neumann Boundary Conditions[END_REF]Proposition 5.2]), we have an explicit formula for u f :

u f = G f : x → 1 -1 K(x -y)f (y)dy.
Furthermore, G is, on (0, 1), a decreasing function, and so it is equal to its decreasing rearrangement. Now, by the Riesz convolution inequality [4, Theorem 1]), for any E ⊂ (-1, 1) of volume 2r < 2,

E u f = 1 -1 1 E (G f ) 1 -1 1 (-r,r) (K f * ) 1 -1 1 (-r;r) (G g # ) = r -r u g # ,
which gives the required result. However, u g # which is, a priori, only defined on (-1, 1), is necessarily symmetric with respect to 0, as g # is. Thus, its restriction to (0, 1) is the solution of the Neumann problem with datum g # . This concludes the proof.

We note that the computations we carried out in the course of proving this theorem also provides an efficient way to reach Proposition 5.

Proof of Proposition 5. We use the optimality conditions for the problem (P =,single,µ→∞,1 ). Note that α is an interior point of M = (κ, V 0 ), so that α is a critical point if, and only if, for any admissible perturbation h at α, J1 (α)[h] = 0.

Adapting the proof of Lemma 11 we obtain the following expression for the first order Gateaux-Derivative of J 1 at any α in any admissible (at α) direction h:

J1 (α)[h] = 2 2V 0 -K 0 M 2 0 =:C1 Ω ∇v α , ∇ vα + Ω K vα with      -∆ vα + M 0 h = 0 in Ω , ∂ vα ∂ν = 0 on ∂Ω , ffl Ω vα = 0.
Introduce the adjoint state q as the solution of

     -∆q = K -K 0 in Ω , ∂q ∂ν = 0 on ∂Ω, ffl Ω q = 0. (3.9)
This allows to rewrite J1 (α)[h] as

J1 (α)[h] = C 1 Ω ∇v α , ∇ vα + Ω K vα = C 1 Ω vα h -M 0 Ω qh.
Thus, if α is a critical point of J 1 , we must have

C 1 vα -M 0 q = µ
where µ is a real constant. Taking the Laplacian on both sides of this equality, this implies that

C 1 M 0 (K -α -M 0 ) = K -K 0 .
However,

C 1 M 0 (K -α -M 0 ) = K -K 0 ⇔ K -α -M 0 = K -K 0 C 1 M 0 ⇔ (K -K 0 ) 1 - 1 C 1 M 0 = 0.
We develop

1 - 1 C 1 M 0 = 1 - M 0 2K 0 -4α = 2K 0 -4α -K 0 + α 2K 0 -4α = K 0 -3α 2K 0 -4α .
we thus derive the conclusion: for α to be a critical point, we must either have K constant, or

K 0 -3V 0 = 0.

Proofs of Theorems VI and VII

Proof of Theorem VI. The proofs of Theorems VI follow in an almost straightforward manner from the previous considerations on single player games. Indeed, observe the following fact: from Theorem III, µ > 0 being fixed, in the one-dimensional case, there exists δ 1 , δ 2 > 0 such that

V 1 < δ 1 , V 2 < δ 2 ⇒ ∀α 2 ∈ M = (κ 2 , V 2 ) , I 1,µ (•, α 2 ) is concave in α 1 , and V 1 < δ 1 , V 2 < δ 2 ⇒ ∀α 1 ∈ M = (κ 1 , V 1 ) , I 2,µ (α 1 , •) is concave in α 2 .
Indeed, it suffices, for the concavity of I 1,µ , to apply Theorem III with K -α 2 as a resources distribution, and similarly for the concavity of I 2,µ . Similarly, in any dimension d, we obtain δ 1 , δ 2 > 0 such that, if

V 1 + V 2 δ 1 , K -K L 1 (Ω) δ 2
then the maps I 1,µ (•, α 2 ) and I 2,µ (α 1 , •) are concave in their respective variables.

So what matters about the assumptions of smallness of V 1 , V 2 (and K -K L 1 (Ω) ) is that the functionals for which we are seeking a Nash equilibrium are concave. The rest of the proof does not depend in any way on the dimension. This concavity property is the natural one in the context of existence of Nash equilibria. Indeed, let us recall [START_REF] Glicksberg | A further generalization of the kakutani fixed point theorem, with application to nash equilibrium points[END_REF][START_REF] Nash | Non-cooperative games[END_REF]

: if ∆ i ⊂ IR d (i = 1, 2) is a convex, compact set, and if L i = ∆ 1 × ∆ 2 → IR is a concave, continuous function (i = 1, 2) then the game find x * i ∈ ∆ i (i = 1, 2) such that L 1 (x * 1 , x * 2 ) = max x1∈∆1 L 1 (x 1 , x * 2 ) , L 2 (x * 1 , x * 2 ) = max x2∈∆2 L 2 (x * 1 , x 2 )
has a Nash equilibrium (x * 1 , x * 2 ). To apply this result to the situation under investigation in the present paper, we need to approximate our infinite dimensional problem by a finite dimensional one.

Reduction to the finite-dimensional setting Let us explain how this reduction is carried out: consider, a fixed integer N being fixed, a measurable partition of Ω as

Ω = n(N ) k=0 ω k,N (4.1)
where, for any k ∈ {0, . . . , n(N )} we have

Vol(ω k,N ) 2 -N .
We consider the auxiliary admissible sets

M N = (κ i , V i ) :=    N k=0 a k 1 ω k,N , 0 a k κ i , n(N ) k=0 a k Vol(ω k,N ) = V i    (i = 1, 2),
and we define ∆ i := M N = (κ i , V i ) (i = 1, 2). Of course, for any α 2 ∈ ∆ 2 , the map I 1,µ (•, α 2 ) is concave on ∆ 1 and, similarly, for any α 1 ∈ ∆ 1 , the map I 2,µ (α 1 , •) is concave on ∆ 2 . The continuity of I i,µ (i = 1, 2) on ∆ 1 × ∆ 2 is obvious. Thus, by the existence theorem for pure Nash equilibria, we conclude that there exists a Nash equilibrium (α * 1,N , α * 2,N ) for the problem

find α * i ∈ ∆ i (i = 1, 2) such that I 1,µ (α * 1 , α * 2 ) = max α1∈∆1 I 1,µ (α 1 , α * 2 ) , I 2,µ (α * 1 , α * 2 ) = max α2∈∆2 I 2,µ (α * 1 , α 2 ).

Conclusion of the proof

We fix, for any N ∈ IN, a Nash equilibrium (α * 1,N , α * 2,N ). Up to a (non-relabelled) subsequence, there exists a couple (α

* 1 , α * 2 ) ∈ M = (κ 1 , V 1 ) × M = (κ 2 , V 2 ) such that α * i,N N →∞ α * i (i = 1, 2)
where the convergence holds weakly in L ∞ -*. However, this weak convergence implies that, weakly in W 2,2 (Ω) (in particular, strongly in L 2 (Ω)), there holds

θ α * 1,N ,α * 2,N ,µ → N →∞ θ α * 1 ,α * 2 . Let us check that (α * 1 , α * 2
) is a Nash equilibrium for our initial problem. To this end, let α 1 ∈ M = (κ 1 , V 1 ) and let us prove that

I 1,µ (α * 1 , α * 2 ) I 1,µ (α 1 , α * 2 ).
By (4.1), there exists a sequence {α 1,N } N ∈IN such that, for any

N ∈ IN, α 1,N ∈ M N = (κ 1 , V 1 ) and such that, strongly in L 1 (Ω), α 1,N → N →∞ α 1 .
By definition of α * 1,N we have, for any N ∈ IN,

I 1,µ (α * 1,N , α * 2,N ) I 1,µ (α 1,N , α * 2,N ).
Passing to the limit as N → ∞ we obtain

I 1,µ (α * 1 , α * 2 ) I 1,µ (α 1 , α * 2 ).
As the symmetric property for I 2 (i.e that α * 2 is a maximiser of

I 2,µ (α * 1 , •) over M = (κ 2 , V 2
)) is proved in the very same fashion we omit it here. The conclusion follows: (α * 1 , α * 2 ) is indeed a Nash equilibrium. This can be seen also with the same philosophy of the bathtub principle, observing that the optimal strategies have to be supported in a subset of a level set of the switch function. Indeed, if the integral constraint V 0 satisfied that V 0 /κ = |{x ∈ R 2 : θ(x)(1 -p(x)) = max x θ(x)(1 -p(x))}| we would nonetheless observe a bang-bang strategy. However, for this simulation, the above property is not satisfied and one has the V 0 /κ is smaller than the volume of the level set corresponding to the maximum of the switch function. Hence, what one observes is supp

(α) ⊂ {x ∈ R 2 : θ(x)(1 -p(x)) = max x θ(x)(1 -p(x))}.

Simulations of the Nash equilibria

In this section, we provide several numerical simulations that illustrate some of the phenomena described in the article. Moreover, it allows us to formulate open problems that may lead to further research on the topic. For all the simulations we have employed a fixed-point algorithm to find Nash equilibria. The algorithm used is the following Algorithm 1. . 1. Initialization: Take a pair of strategies, α

2 ∈ M (κ, V 0 ).

Recursion:

For every k ∈ N, solve sequentially the optimization problems

max α (k) 2 ∈M (κ,V0) ˆΩ α (k) 2 θdx restricted to -µ∆θ = θ(K(x) -θ) -α (k-1) 1 (x)θ -α (k) 2 (x)θ, + Boundary conditions. and then max α (k) 1 ∈M (κ,V0) ˆΩ α (k) 1 θdx restricted to -µ∆θ = θ(K(x) -θ) -α (k) 1 (x)θ -α (k-1) 2 (x)θ, + Boundary conditions.
If the algorithm (1) converges, i.e. if there holds

α (k) 1 → α * 1 and α (k) 2 → α * 2 then the pair (α * 1 , α * 2
) is a Nash equilibrium by definition. We do not have a proof of convergence of the above algorithm. The proof itself would imply the existence of Nash equilibria (but not the Figure 2: In this figure, several optimal policies α (first row) are depicted along with the corresponding switch functions (second row). All the simulations have been done using the same capacity K = K(x) of Figure 1 and same diffusivity µ = 1 although with different control limitations. In the first and second column the integral constraint is V 0 = 0.3, but κ = 7 in the first column while κ = 0.1 in the second column. In the third column, V 0 = 0.05 and κ = 7. other way around). In the case of potential games, Algorithm 1 always converges. However, it can be seen, using a contradiction argument, that our game is not a potential game.

We will employ this algorithm numerically to try to discover if a Nash equilibrium exists and we will use the stopping condition α

(k+1) i -α (k) i L 2
tol for i = 1, 2. Since the algorithm above is forced to stop given a tolerance, one cannot guarantee that the convergence is at a Nash equilibria, but rather at an -Nash equilibria.

Definition 23 ( -Nash equilibria). Fix 0. A pair of strategies (α 1 , α 2 ) ∈ M (κ, V 0 ) 2 is an -Nash equilibria if ∀α ∈ M (κ, V 0 ) I 1 (α 1 , α 2 ) I 1 (α, α 2 ) -, ∀α ∈ M (κ, V 0 ) I 2 (α 1 , α 2 ) I 2 (α 1 , α) -
Note that if = 0 one has the definition of a Nash equilibria. Furthermore, it is important to observe that an -Nash equilibria (with > 0) does not need to be close to a Nash equilibria. Moreover, it is worth noting that an -Nash equilibria can exist without a Nash equilibria existing. If it converges, Algorithm 1 converges to an -Nash equilibria (see Proposition 24 in the Appendix).

Symmetric bounds for both players

In this subsection we perform simulations and discuss the case in which both players have the same fishing capacity, i.e. the game is symmetric. After looking at Figures 3 and4, one should observe five things: 1. All simulations have converged to an -Nash equilibria in which the strategies of both players are the same. Therefore, they suggest that the search of such Nash equilibria can be phrased as finding fixed points for the map Λ :

L ∞ (Ω) -→ L ∞ (Ω) defined as Λ(K) = 1 -α * K
where by α * K is an element of the minimizers of the single player game. Of course, for such map to be well defined, we would need to ensure uniqueness for (P single ,V0 ). 2. When the integral bound V 0 is small, the Nash equilibria are not exhibiting a bang-bang structure and instead are constants (for K = 1, Figure 3 left column). This is in the line with the concavity properties observed in Theorems III and V in this paper.

3. When V 0 is big, the algorithm converges, for both the one and two dimensional problems to a Nash equilibrium that is bang-bang. We have observed that, in the asymptotic regime there are two Nash equilibria for V 0 = 1 3 , (V 0 , V 0 ) and (1 (0,V0) , 1 (0,V0) ) (as a consequence of Theorem VII). For every µ, (V 0 , V 0 ) is a Nash equilibria for the non-linear problem. An interesting question is to determine whether or not bang-bang symmetric Nash equilibria exist for general diffusivities.

4. Simulations in Figure 4, also point that in the two dimensional case, there is no uniqueness. For the same diffusivity, and for the same integral bound, two different -Nash equilibria were found (left and middle columns of Figure 4).

5. An apparent fragmentation phenomena as observed in the simulations. The TV semi-norm of the strategies increases as µ → 0 + . This is a phenomenon observed in the maximisation of the total population size [START_REF] Heo | On the ratio of biomass to total carrying capacity in high dimensions[END_REF][START_REF] Mazari | Optimisation of the total population size for logistic diffusive equations: bang-bang property and fragmentation rate[END_REF][START_REF] Mazari | A fragmentation phenomenon for a non-energetic optimal control problem: optimisation of the total population size in logistic diffusive models[END_REF]. However, this phenomenon is quite surprising with respect to the previous studies since, in this problem, we are dealing with Nash-equilibria for a game whose pay-offs are different from maximizing the total population.

Non-symmetric bounds

In this subsection we introduce some asymmetry in the problem by considering different capacities for the players Figures 5 and6. We remark the following 1. As observed in the previous case, when the integral bound is low, the observed -Nash equilibria consists of a pair of constants (left column in Figure 5). As before, this is a manifestation of Theorem III and Theorem VI.

2. In contrast with the symmetric case, we no longer observe a full bang-bang strategy. Both in the one dimensional case in Figure 5 and in the two dimensional one in Figure 6, we observe that the player with higher capacity adopts a bang-bang strategy while the player with less capacity is not showing this feature.

3. Figure 5 also shows that for high integral bound, the players do not necessarly share the supports of their strategies. In contrast, the simulation done in the two dimensional case, Figure 6 is not showing this particularity. There, it is observed that the player with less capacity fishes in the same area than the player with higher capacity but at a "lesser" intensity in some areas.

4. Note that all the comments made for (P single ,V0 ) in the previous subsection regarding Figure 2 apply in the context of Nash equilibria for understanding a posteriori its geometrical properties. . The strategy of the first (and second) player, with higher (lower) capacity, has been depicted as a blue (orange) subgraph. The second player has a lower integral bound than the first player. In all these simulations, K ≡ 1.

5. Furthermore, we also observe an apparent fragmentation of the Nash equilibria shown for high capacity.

Open problems

Concavity for low fishing abilities in higher dimensions One of the main drawbacks of Theorem III is the fact that it holds only in one-dimension or, in higher dimensions, if the resources distribution K is close to a constant in the L 1 norm. As was seen during the proof, the main possibility to derive a result is so far to establish that

1 - µ 4 • |∇θ| 2 θ 3 > 0 (6.1)
for any K ∈ K(Ω), θ being the solution of (2.17). In the on-dimensional case this was obtained through an estimate of [START_REF] Bai | An optimization problem and its application in population dynamics[END_REF]. In the higher-dimensional setting, however, it is quite likely that there is some serious difficulty in obtaining such an estimate for the following reason: in [START_REF] Inoue | On the unboundedness of the ratio of species and resources for the diffusive logistic equation[END_REF], it is proved that, if we simply assume that K 0 , K = 0, the quantity sup µ>0 ,K ,K =0 ,K 0 ffl Ω θ/ ffl Ω K is infinite. Should an estimate of the form (6.1), such a result could not be true (as one could then apply the technique of [START_REF] Bai | An optimization problem and its application in population dynamics[END_REF] and obtain sup µ,K ,K =0 ,K 0 ffl Ω θ/ ffl Ω K 4, an obvious contradiction. Of course, in constructing a sequence such that the biomass to resources ratio diverges, the authors of [START_REF] Inoue | On the unboundedness of the ratio of species and resources for the diffusive logistic equation[END_REF] blow the L ∞ bound up, but the fact that such phenomena occur in higher dimensions indicates the potentially very intricate nature of the problem. The question of fragmentation for Nash equilibria We have also observed a clear fragmentation phenomenon of Nash equilibria in the low diffusivity limits. Building on [START_REF] Heo | On the ratio of biomass to total carrying capacity in high dimensions[END_REF][START_REF] Mazari | A fragmentation phenomenon for a non-energetic optimal control problem: optimisation of the total population size in logistic diffusive models[END_REF], is it possible to prove a theorem of the form lim

µ→0 + inf (α * 1 ,α * 2 ) Nash equilibria min ( α * 1 BV , α * 2 BV ) = ∞?
At this stage, it seems thoroughly unclear how one could approach that question, as this would require a very fine knowledge of the set of all Nash equilibria of the problem. We plan on tackling this question in future works.

Optimal Game Regulation Problem

In this article we have studied several regimes for which Nash equilibria exist. Furthermore, we also illustrated how Nash equilibria lead to an underperformance of resources, in the sense that there are Nash equilibria for which the sum of the pay-offs of the players is strictly lower than what is optimal to fish. This also has been illustrated in the numerical simulations. Behind this lines, there is a relevant problem to be addressed. What is the optimal regulation so that we avoid overfishing as much as possible? In Figure 7 the total fish harvested is depicted with respect to the volume constraint. One can observe that, for the Nash equilibria found, there is an optimal volume constraint for maximising the total amount harvested. This allows us to propose an optimal regulation problem for the harvesting problem. Let us first define the set of all Nash equilibria given a volume constraint V 0

N (V 0 ) := {(α * 1 , α * 2 ) ∈ M (1, V 0 ) such that (α * 1 , α * 2
) is a Nash equilibria} . Now, the optimal game regulation problem for the harvesting game is the maximisation of the worst Nash equilibria with respect to V 0 , mathematically max V0 min (α1,α2)∈N (V0) ˆΩ(α 1 (x) + α 2 (x))θ α1,α2 (x)dx where θ follows (1.2). To address this problem, it is necessary to characterise all Nash equilibria given a volume constraint V 0 . In Figure 7, we only used the Nash equilibria found with Algorithm 1, but we do not know if there are other Nash equilibria. It is worth noting that, in the case of Figure 7 (K ≡ 1), it would be sufficient to prove that the unique Nash equilibria for V 0 = 0.25 is α 1 (x) = V 0 , α 2 (x) = V 0 . In particular, by the weak formulation of (A.1) there exists M 2 ∈ IR such that

sup k∈IN ϕ k W 1,2 (Ω) M 2 .
Let λ ∞ be a closure point of {λ k } k∈IN and ϕ ∞ be a (weak W 1,2 , strong L 2 ) closure point of {ϕ k } k∈IN . As

V k L 2 (Ω) -→ k→∞ K -2θ.
Passing to the limit in the weak formulation of (A.1), as well as in the normalisation conditions, we obtain, on ϕ ∞ , the equation

     -µ∆ = (K -2θ)ϕ ∞ + λ ∞ ϕ ∞ in Ω , ∂ϕ∞ ∂ν = 0 on ∂Ω , ϕ ∞ 0 in Ω , ´Ω ϕ 2 ∞ = 1.
It thus appears that ϕ ∞ is a constant-sign eigenfunction of the operator -µ∆ -(K -2θ). As the first eigenfunction of an operator is the only one having constant sign we deduce that ϕ ∞ is a first eigenfunction of -µ∆ -(K -2θ), so that λ ∞ = λ(K -2θ). As λ ∞ = lim k→∞ λ(K -α k -2θ α k ), this is a contradiction.

A.2 Proof of (2.30)

Proof of (2.30). We argue by contradiction and assume that (2.30) does not hold. In particular there exists η > 0, a sequence {V 0,k } k∈IN converging to zero and, for any

k ∈ IN, α k ∈ M (κ, V 0,k ) such that ∀k ∈ IN , ξ(α k ) -ξ η > 0.
As W α is uniformly bounded in L ∞ (Ω) for V 0 small enough from (2.18)-(2.20), the sequence {ξ(α k )} k∈IN is uniformly bounded, say by a constant M 0 > 0:

∀k ∈ IN , |ξ(α k )| M 0
and thus, up to extracting a subsequence, it converges to some ξ * .

In turn this implies that, if we define, for any k ∈ IN, the normalised eigenfunction ϕ k as the solution of We can thus extract a converging (weakly in W 1,2 (Ω), strongly in L 2 (Ω)) subsequence of {ϕ k } k∈IN , and, without relabelling we assume that the entire sequence thus converges to a ϕ * ∈ W 1,2 (Ω).

       -µ∇ • |∇ϕ k | θα k -W α k ϕ k = ξ(α k )ϕ k in Ω , ϕ k 0 , ffl Ω ϕ 2 k = 1 ,
Passing to the limit in the normalisation conditions provides us with

ϕ * 0 in Ω , Ω (ϕ * ) 2 = 1.
Since W α k → k→∞ W strongly (in particular, weakly) in L 2 (Ω) we finally obtain, passing to the limit in the eigenequation, that ϕ * solves

       -µ∇ • ∇ϕ * θ -W ϕ * = ξ * ϕ * in Ω , ∂ϕ * ∂ν = 0 on ∂Ω , ϕ * 0 , ffl Ω (ϕ * ) 2 = 1.
In particular, ϕ * is a positive eigenfunction of L. As an eigenfunction of L has a constant sign if, and only if, it corresponds to the first eigenvalue, we deduce that ξ * = ξ, a contradiction.

A.3 Proof of convergence to an -Nash equilibria

Proposition 24. Algorithm 1 with tol= > 0, in case of convergence it converges to an 1/3 -Nash equilibria.

Proof. Assume that one has set the tolerance of the algorithm up to > 0 and that the algorithm has converged. Then one has that

  = θ(K(x) -θ) -α(x)θ

. 2 ) 9 .

 29 Remark Existence and uniqueness of solutions of (2.1)-(2.2) follow from the following crucial observation [49, Comment after eq. (2.6)]: from (1.4), the first eigenvalue of the operator -µ∆ -(K -α -θ α ) is zero. From the monotonicity of the eigenvalue, the first eigenvalue of -µ∆ -(Kα -2θ α ) is positive. The existence and uniqueness of solutions to (2.1)-(2.2) then follow from a standard variational argument.

1 - 1 ξf

 11 (ξ)dξ = 0.We will establish a comparison inequality on this new problem. To this end, let us introduce the fundamental solution of the Neumann Laplacian on (-1, 1). It is the function K defined by G(x)

Figure 1 :

 1 Figure 1: Capacity K(x) used for the simulations shown in Figure 2.

Figure 3 :

 3 Figure 3: The blue line is the state θ α1,α2 . The grey area indicates the are the subgraphs of the strategy of the players (both players play the same strategy). Both players have the same capacity and K(x) = 1.

Figure 4 :

 4 Figure 4: In this figure, several simulations following the fixed point Algorithm 1 have been performed. In all the simulations, both players play the same strategy, and hence, only one strategy is depicted. K(x) = 1 was choosen for all simulations.

Figure 5 :

 5 Figure 5: The blue line represents the state θ α1,α2. The strategy of the first (and second) player, with higher (lower) capacity, has been depicted as a blue (orange) subgraph. The second player has a lower integral bound than the first player. In all these simulations, K ≡ 1.

Figure 6 :

 6 Figure 6: At the left the strategy of the first player, at the right the strategy of the second player. The second player has a lower integral bound than the first player. K(x) = 1.

Figure 7 :-

 7 Figure 7: In the vertical axis, the sum of the total fish harvested for both players at the Nash equilibria found, in the horizontal axis the volume constraint for both players. K(x) = 1.

  the estimate(2.18) and the fact that inf Ω θ > 0 yield the existence of a constant M 1 such that∀k ∈ IN , ϕ k W 1,2 (Ω) M 1 .

  It should be noted that in[START_REF] Bai | An optimization problem and its application in population dynamics[END_REF] Estimate (2.2)] this estimate is proved when θ is monotonic, and that they then integrate this identity on such an interval to obtain an integral estimate. Then, they present, in [6, Steps 2 and 3, proof of Theorem 2.2], a way to glue these integral estimates. The very same strategy works to prove that [6, Estimate (2.2)] is valid on the entire interval.

					, He and Li [6, Estimate (2.2)],
	namely, that, in the one-dimensional case, provided K is bounded (which is the case here by
	assumption) there holds
	µ 2	θ (x)	2	θ(x) 3 3	.
	Remark 20.

For certain parameters, the switch function does not have any flat region and it is uniformly positive. In this case, we have a bang-bang strategies due to the well known bathtub principle: the optimal policy is a characteristic function κ1 ω , where ω is the level set of the switch function with volume V0 κ .

The switch function can combine both aspects, it can have flat region and a nonflat one. This is the case of the third column in Figure2, where a qualitative mixture of the phenomenology described in the previous two points is observed. In this case, one observes that the flat region is at the maximum of the switch function. This is the reason why the optimal strategy does not saturate the upper bound α κ and shows a non-bang-bang structure. In addition, as in the previous case, one can observe that the support of α is not the whole square [0, 1] 2 .[START_REF] Alvino | Comparison results for elliptic and parabolic equations via Schwarz symmetrization[END_REF] In the sense that we pick random Fourier coefficients

Acknowledgment. This work was started during a visit of D. Ruiz-Balet at CEREMADE. I. Mazari was partially supported by the French ANR Project ANR-18-CE40-0013 -SHAPO on Shape Optimization and by the Project "Analysis and simulation of optimal shapes -application to life sciences" of the Paris City Hall.

Proof of Theorem VII. The proof of this theorem follows from Theorem V. Indeed, observe that, if

Consequently, for any α 2 ∈ M = (κ 2 , V 2 ), it follows from Theorem V that, for any α 2 ∈ M = (κ 2 , V 2 ) fixed, the map

is strictly convex on M = (κ 2 , V 2 ). Now let us take α * 1 , α * 2 as defined in the statement of the theorem. We apply Theorem V: taking as a resources distribution K = K 0 -α 2 , which is a non-constant, non-decreasing function, we deduce (from the convexity of the functional) that any solution of max M=(κ1,V1)

is a bang-bang function and (from Theorem V) that the solution is a non-increasing function. Thus, the solution is exactly α * 1 . Similarly, α * 2 is the solution of max

Thus, (α * 1 , α * 2 ) is indeed a Nash equilibrium in the sense of Definition 8.

5 Numerical simulations and comments

Simulations of the optimal harvesting problem

In this section we consider a random 1 positive continuous function K : [0, 1] 2 → R represented in Figure 1 and we consider the optimal harvesting problem (P single ,V0 ). Figure 2 exemplifies the richness of different qualitative behaviours this simple problem can exhibit under modifications of the constraints. More specifically we notice the following facts:

1. The switch function θ(1 -p), can be constant and hence the optimal controls are not necessarily bang-bang. This is a case already emphasised in the particular case in which K(x) = 1.