Christopher Costello
email: costello@bren.ucsb.edu

Bruno Nkuiya
email: nkuiyamb@ualberta.ca.

Nicolas Quérou
email: nicolas.querou@supagro.fr.

Spatial renewable resource extraction under possible regime shift *

Keywords: Regime shift, spatial management, renewable resources, property rights JEL Classification Codes: C73, H23, H73, Q22

published or not. The documents may come

Introduction

Because renewable resources such as fish, water, game, and infectious diseases are mobile, extraction and productivity in one location affect economic opportunities in other locations.

The resulting spatial externality can be dealt with by a central planner using a number of instruments including spatial taxes, limits of extractive effort, or other means. But in practice, many spatially connected renewable resources are managed via private property rights where autonomous entities (such as countries, villages, cooperatives or individual property right owners) choose their own extraction rates, taking as given the mobility of the resource and the extraction of their competitors. Indeed spatial property rights are implicitly the default approach for managing many renewable resources, 1 despite their potential for inducing spatial externalities driven by resource mobility. This general problem has become a canonical model in spatial resource economics. The main finding from that literature is that non-cooperative extraction will necessarily entail over-extraction (relative to a social optimum) because no single owner is incentivized to account for the effects of her extraction on others [START_REF] Kaffine | Unitization of spatially connected renewable resources[END_REF][START_REF] Fenichel | The control of invasive species on private property with neighbor-to-neighbor spillovers[END_REF]. The aggregate effects of this non-cooperation can range from extremely deleterious (see [START_REF] White | Close the high seas to fishing?[END_REF]) to practically insignificant (see [START_REF] Gisser | Competition versus optimal control in groundwater pumping[END_REF]). 2 This growing literature has evolved in a deterministic setting in which productivity and dispersal functions are common knowledge and fixed over time. Rather, a growing scientific literature suggests that global change may induce regime shifts that affect resource dynamics, and thus may alter future economic incentives and returns. While there are many types of documented (and speculated) regime shifts, they generally share three common features.

First, regime shifts tend to be abrupt -over a relatively short period of time they can shift the state of the world from one state to another. Second, the occurrence date of a regime shift is probabilistic -while scientists might have a sense of the likelihood of a regime shift occurring, we do not know with certainty when it will occur. Third, many regime shifts are thought to be irreversible -once the shift occurs, due to hysteresis, it may be difficult to return to its pre-shift state (Scheffer et al. 2001). 3 Facing this type of regime shift fundamentally alters the constraints and incentives faced by property owners who extract mobile renewable resources. To our knowledge, the earliest contribution along these lines is [START_REF] Mckelvey | Fish-wars revisited: a stochastic incomplete-information harvesting game[END_REF], who use numerical simulations to examine cooperative and non-cooperative behavior in a "fish war" game. Our approach is to tackle this challenge theoretically in a dynamic game, and to obtain analytical results. We are motivated by regime shifts such as those that may be driven by climate change, which could affect the spatial range or dispersal of a migratory fish species such as tuna. For example, under pre-shift parameters, suppose tuna tend to migrate equally between countries A and B. But if a regime shift were to occur, the migratory pattern may shift to favor country A.

If that kind of shift were predictable, it would clearly alter the incentives of countries A and B: Recognizing the improvement in future conditions, country A might be willing to forego harvest today to build resource stocks and capitalize on improved future conditions. 4 And recognizing the deterioration of future conditions, country B would, intuitively, increase its current extraction. These intuitive predictions of countries' behaviors are consistent with results in [START_REF] Hannesson | Global warming and fish migrations[END_REF] and [START_REF] Diekert | International fisheries agreements with a shifting stock[END_REF], but they turn out to be vulnerable to strategic interactions across players. This stylized example illustrates our key inquiry: How will the presence of a possible future regime shift alter strategic interactions of private property owners who extract a mobile natural resource? Will the threat of regime shift always entail a loss of precaution by one agent and an increase in precaution by the other? We address these and related questions in our investigation.

This paper builds on an emerging literature on natural resource management that addresses related questions, but in a context where the random occurrence of a regime shift inflicts a permanent loss to all harvesters. [START_REF] Polasky | Optimal management with potential regime shifts[END_REF] and [START_REF] Ren | The optimal management of renewable resources under the risk of potential regime shift[END_REF] focus on optimal management whereas [START_REF] Fesselmeyer | Strategic exploitation of a common resource under environmental risk[END_REF], [START_REF] Sakamoto | Dynamic resource management under the risk of regime shifts[END_REF], [START_REF] Miller | Coalition formation in fisheries with potential regime shift[END_REF], and [START_REF] Diekert | Threatening thresholds? the effect of disastrous regime shifts on the non-cooperative use of environmental goods and services[END_REF] analyze the strategic management of a common pool resource. These contributions consider only scenarios in which all harvesters are identical and do not explicitly take into account the spatial movement of the resource. But since we are primarily interested in strategic interactions, our analysis investigates cases in which regime shift will alter the distribution of resource stocks so as to create winners and losers. 5 We thus explicitly take into account the spatial movement of the resource and consider heterogenous harvesters subject to different, but connected, economic, environmental, and biological conditions. In this expanded setting we are able to derive tractable, analytical results, and dissect how features such as heterogeneity and movement change incentives and outcomes relative to the previous literature.

Our analysis thus contributes to an increasingly relevant class of economic papers addressing renewable resource management under the prospect of regime shift in the resource distribution, which advantages one patch and disadvantages the other patch. To gain traction on this challenging problem, existing papers rely either on numerical simulations alone, or require assumptions that we will relax. For example, we will allow for common property in each patch and for biological growth conditions to differ across patches. 6 Prominent previous contributions include [START_REF] Diekert | International fisheries agreements with a shifting stock[END_REF] and [START_REF] Liu | Comparing proactive and reactive management: managing a transboundary fish stock under changing environment[END_REF] that concentrate on deterministic regime shift processes with finite transitional periods, and find that the harvester gaining the stock conserves it, whereas the one losing the stock acts aggressively. [START_REF] Hannesson | Global warming and fish migrations[END_REF] considers a random regime shift process and exogenously assumes that escapement is constant within each environmental regime. He finds that the stock cannot go extinct whenever the intrinsic growth parameter is not too low (and smaller than one) and the shares of the stock are sufficiently heterogenous. And while [START_REF] Mckelvey | Fish-wars revisited: a stochastic incomplete-information harvesting game[END_REF] allow for a fairly general stochastic model, they restrict analysis to numerical simulations.

In this paper, we consider analytically the strategic interactions across agents and adopt a feedback approach such that in a given period, escapement endogenously depends on environmental conditions and the resource stock. The setting we analyze builds on, and attempts to generalize, several of the prominent early papers in this area; we argue that these generalizations deliver insights about new harvest incentives that enhance our understanding of strategic interactions over a spatial resource subject to regime shift.

The simplest case is when marginal harvesting costs are constant. In that setting, analytical findings reveal that harvesters in the advantaged patch (the ones gaining the stock) conserve the resource stock in the initial period whereas those in the disadvantaged patch (those losing the stock) increase initial harvest in response to the possible future shift in the resource distribution. In the steady state, we identify two opposing mechanisms through which uncertainty about the shift affects harvest decisions. The first is a direct effect of the shift that tends to reduce harvest incentives in the disadvantaged patch (relative to the no-shift case). The second mechanism is the strategic effect that tends to enhance harvest in the disadvantaged patch, because harvesters in the advantaged patch conserve the resource stock in the short run, which has a positive effect on the disadvantaged patch in the long run through migration. Interestingly, we find that each of these mechanisms can dominate the other depending on environmental, biological, and economic conditions.

To better understand the effects of uncertainty about the shift, we also investigate the case in which harvesting costs decline as the resource stock increases. We find that a stock effect plays a pivotal role in extraction incentives. When the stock effect is sufficiently strong, we find that the conventional wisdom is weakened or even reversed.

The paper unfolds as follows. Section 2 presents the model. Section 3 focuses on behavior following the regime shift. Section 4 analyses harvesters' incentives in anticipation of a future regime shift. Section 5 extends the analysis to the case where each patch is itself common property. Section 6 provides an illustrative example and some further discussion of corner solutions. Section 7 concludes.

The model

A renewable resource stock is distributed heterogeneously across an ecosystem consisting of two areas or "patches," A and B. Patches may differ in shape, size, environmental, and economic characteristics; for example, patches may be countries, private lands, or communal harvesting areas. The time index is denoted by t = 1, 2, 3, .... and h jt represents the extraction in patch j during period t. The resource stock at the beginning of period t in a given patch j is denoted by x jt while the remaining residual stock (or "escapement") e jt is defined as e jt ≡ x jt -h jt , which is the post-harvest stock at the end of period t. As such, when there is no harvest, say, in patch j, the current escapement is equal to the current resource stock: e jt = x jt .

Resource mobility will induce a spatial connection across patches. In period t, a fraction K ijt of patch i's resource stock moves to patch j, i = j while the fraction K iit stays within patch i. Therefore, K ijt + K iit ≤ 1 for i, j ∈ {A, B} with i = j. In the case where this inequality is not binding, a fraction of the resource population living in patch i moves out of the system at date t. The current resource distribution across patches is determined by the 2 × 2 dispersal matrix K t , whose element K ijt takes the value D ij before the shift and D s ij after the shift.

[Figure 1 Here] At the beginning of the initial period, dispersal is in its pre-shift form, so

K ij0 = D ij .
Regime shift occurs at an unknown future date denoted by τ > 0 (that may be infinite), and dispersal irreversibly shifts to regime s, characterized by a dispersal matrix with terms D s ij .

We assume that the shift will give a bio-physical advantage to region A and a disadvantage to region B, so Dispersal is thus characterized as follows:

D s BA > D BA ≥ 0, D BB > D s BB ≥ 0, D AB > D s AB ≥ 0, and D s AA > D AA ≥ 0.
K ijt =        D ij for t < τ, D s ij for t ≥ τ for i, j = A, B.
The regime shift process described above can be represented by the stochastic process t that may either take the values I (for "initial") or S (for "shift") with transition probabilities

P ( t+1 = S| t = S) = 1; P ( t+1 = S| t = I) = λ, (1)
P ( t+1 = I| t = I) = 1 -λ. (2)
At the outset of the initial period, the resource stocks x A0 and x B0 in patches A and B are known. In the absence of harvest, the resource stock grows according to the growth and dispersal equation

x jt+1 = i=A,B g i (x it )K ijt , j = A, B, (3)
where g i (.) represents patch i's growth function that satisfies standard conditions. It is increasing, concave and twice continuously differentiable.

In the presence of harvest, growth depends on escapement, so the law of motion becomes

x jt+1 = i=A,B g i (e it )K ijt , j = A, B. (4)
The evolution of the resource population is stochastically determined by harvest, growth, and environmental conditions K ijt . The timing is thus: the present period stock (x jt ) is observed and then harvested (h jt ) giving escapement (e jt ), which then grows (g j (e jt )), and disperses to itself (K jjt ) and to the other patch (K jit ).

Suppose now that each patch is owned by a single entity. For example, this could be a Territorial User Right Fishery (TURF), a farm on which bees reside, or a country hosting migratory fish, game, or waterfowl. We allow for, but do not require, prices and costs to be patch specific. Denote by p j the unit price associated with patch j; costs may depend on the stock size and are treated later in this section.

The harvester's period-t harvest is (x jt -e jt ) so, in the absence of harvesting costs, her period-t profit is p j (x jt -e jt ). In the presence of harvesting costs, the period-t profit for harvester j is simply the integral of profit flows over the entire extractive period and can be written as follows

π jt = x jt e jt [p j -c j (v)]dv. ( 5
)
Here, the term c j (v) is the marginal harvesting cost when the available stock is v. The integral

x jt e jt c j (v)dv in Equation 5 represents the total cost of harvest during period-t, which may be patch specific. We assume that c j (v) ≤ 0, that is, the marginal cost weakly decreases in the resource stock level. The rationale is that at a given date, a larger resource stock entails a smaller marginal harvest cost. In the case where c j (.) = 0, the profit in patch j is linear in harvest; profit is strictly concave in harvest as long as c j (.) < 0 (i.e. a stock effect is present). To determine whether or not the marginal cost is constant is an empirical issue (see for instance, [START_REF] Atewamba | Testing the assumptions and predictions of the Hotelling model[END_REF], for the case of non-renewable resources). We will show that a sufficiently strong stock effect can fundamentally alter harvest incentives, particularly in the presence of regime shift. We separately examine both cases below.

In our model, regime shift affects the geographic distribution of the resource stock, but does not inherently affect p j or c j (v). In reality, shifts in prices and costs may also occurfor example as range shifts change the location of harvest (which could affect fishing costs) or as the supply chain is altered (which could affect prices). As long as after the regime shift, these changes still adhere to our main assumptions (marginal fishing cost is a function of contemporaneous stock and price is constant in a period and exogenous to harvest), then these kinds of shifts in economic parameters could be accommodated in our modeling framework. But for more complicated kinds of economic shifts (for example, if the regime shift induces a downward-sloping demand curve), a new model would be required, which is unlikely to have analytically tractable results. For all of these reasons, we focus on the case when prices are constant (though may be patch specific), marginal harvest costs depend only on current resource stock in each patch, and regime shift affects only the geographic distribution of the stock.

Prior to the shift, i.e. for any date t < τ , the present value payoff function in patch

j = A, B is given by τ -1 k=t δ (k-t) p j (x jk -e jk ) - x jk e jk c j (v)dv + δ τ -t W j (x(τ )),
where W j (x(τ )) represents the period-τ continuation value of the problem for that harvester and δ is the discount factor. We next solve the post-shift problem and use its result to derive the complete solution for the uncertain regime shift problem presented above.

The post-shift problem

In this section we examine the game between harvesters that will occur following the regime shift. We follow the growing literature, starting with the seminal paper of [START_REF] Reed | Optimal escapement levels in stochastic and deterministic harvesting models[END_REF], that uses escapement as the control variable. 7 In this setting, the patch-j harvester chooses an escapement strategy to maximize her present discounted profits taking as given the escapement strategy of her rival. Thus, immediately following the regime shift, the owner of patch j solves:

W j (x τ ) = max e jt ,t≥τ +∞ k=τ δ (k-τ ) [p j (x jk -e jk ) - x jk e jk c j (v)dv],
subject to (4) with x τ ≡ (x Aτ , x Bτ ) given.

We seek a Markov Perfect Nash Equilibrium (MPNE), which will define the equilibrium harvest decisions (e A (x A , x B ), e B (x A , x B )) following the regime shift. The escapement decision rule (e A (x A , x B ), e B (x A , x B )) is a MPNE if, given the resource stock at the outset of period τ (x τ ≡ (x Aτ , x Bτ )), at any date t ≥ τ , {e j (x As , x Bs ), s ≥ t} is a solution to the optimization problem above. The feedback Nash equilibrium is a MPNE and can be found by specifying and manipulating the Bellman equations for both players. The Bellman equation for the harvester operating in patch j can be written as:

W j (x t ) = max e jt p j (x jt -e jt ) - x jt e jt c j (v)dv + δW j (x t+1 ) ,
which is subject to (4) with the initial resource stock x τ ≡ (x Aτ , x Bτ ) given.

The first-order conditions require

p j = c j (e jt ) + δ i=A,B ∂W j ∂x it+1 (x t+1 )g j (e jt )D s ji , j = A, B. (6)
This equation states that patch j chooses her escapement level to equate the resource price with its augmented marginal cost, which is the marginal cost, augmented by the value forgone by harvesting today rather than keeping the resource for future harvests. The challenge is that the form of the value function W j (x) is unknown. However, its properties can be derived given the structure of this problem. These derivations allow us to characterize an equilibrium over the post regime shift phase, summarized as follows:

Lemma 1. Over the post regime shift phase, the following results hold.

(i) Patch j is harvested down to the escapement level e j . These state-independent (i.e. constant) escapements constitute an equilibrium of the game and are given by the solution to

p j -c j (e j ) = δD s jj [p j -c j g j (e j )D s jj + g i (e i )D s ij ]g j (e j ), i = A, B and i = j. (7)
(ii) Each patch's equilibrium resource stock reaches its steady state in one period, and is thereafter time-independent.

Proof. All proofs reside in the appendix.

Equation 7 implicitly defines harvester j's best response function, e j (e i ), and suggests that her actions depend on the flow of the resource to her own patch (D s ij ), but that e j (e i )

does not depend on D s ji nor D s ii , i = j, which are terms defining the resource flow to the other patch. This is the case because harvester j does not ascribe any value to additional resource stock located out of its boundaries (i.e.

∂W j ∂x it (x t ) = 0 for i = j) because she knows that the best response of her rival would be to harvest any additional stock that arrives.

The equilibrium escapement level corresponds to the intersection of best response functions e A (e B ) and e B (e A ). We can also employ this analysis to identify the equilibrium escapement level of the no-shift case, in which the resource distribution is deterministic and never shifts.

We denote the no-shift variables by a tilde (e.g. ẽjt ). They can be retrieved from Condition 7 by replacing (for i, j = A, B) D s ij by D ij and (x Aτ , x Bτ ) by (x A0 , x B0 ). Because the no-shift case takes the same form as the post shift case (albeit with different parameter values), the equilibrium escapement level of the no-shift case is also time and stock independent (see Lemma 1). The equilibrium resource stock outcome of the no-shift case converges to its steady state in the second period of the game.

To better understand the effects of the shift, we next compare the outcomes of the post regime shift case and the no-shift case.

Proposition 1. Assume that marginal costs are constant (i.e., c j (x) = 0 for all x, j = A, B).

Over the post regime shift phase, the following results hold.

(i) For j = A, B, e j is implicitly defined by: We have so far addressed the cases where the shift has already occurred and where the shift will never occur. We next use these results to completely characterize the equilibrium behavior prior to the shift.

g j (e j ) = 1 δD s jj . ( 8
4 Non-cooperative behavior in advance of a regime shift So far we have analyzed the deterministic spatial game induced either following an irreversible regime shift or in the complete absence of regime shift. But our central research question asks how harvesters interact under the threat of a possible regime shift in the future. In this section, we focus on harvesters' responses to the prospect of a future regime shift. Taking the escapement strategy of the other harvester as given, harvester j = A, B chooses the escapement strategy that maximizes her expected present discounted net profits

V j (x t ) = max e js ,s≥t E +∞ k=t δ (k-t) [p j (x jk -e jk ) - x jk e jk c j (v)dv], (9)
which is subject to (4). We are interested in identifying a MPNE that we next derive using the feedback Nash equilibrium approach. Harvester j's value function given in (9) satisfies:

V j (x t ) = max e jt [p j (x jt -e jt ) - x jt e jt c j (v)dv + δ(1 -λ)V j (x t+1 ) + δλW j (x s t+1 )], (10)
subject to (4), where x s jt+1 = g j (e jt )D s jj + g i (e it )D s ij and x jt+1 = g j (e jt )D jj + g i (e it )D ij . The first two terms on the right hand side of Equation 10 are just contemporaneous revenue and cost from harvesting the resource in patch j. The third term is the discounted expected value in the case where the regime shift does not occur at the end of period t (this occurs with probability (1 -λ)). The final term is the discounted expected value in the case where regime shift does occur at the end of period t, in which case we invoke the value functions from the post regime shift problem derived in Section 3 (this occurs with probability λ).

To interpret Equation 10, it is instructive to rewrite it as follows:

V j (x t ) = max e jt [p j (x jt -e jt ) - x jt e jt c j (v)dv + δj V j (x t+1 )], (11)
where δj = δ + δλ[W j (x s t+1 ) -V j (x t+1 )]/V j (x t+1 ) can be thought of as a risk-adjusted discount factor. Equation 11 can be interpreted as the Bellman equation associated with a deterministic model in which the discount rate endogenously accounts for the possibility of regime shift.

The first-order condition for this maximization problem can be written as

p j = c j (e jt )+δλg j (e jt ) i=A,B ∂W j ∂x it+1 (x s t+1 )D s ji +δ(1-λ)g j (e jt ) i=A,B ∂V j ∂x it+1 (x t+1 )D ji , j = A, B.
Under the assumptions adopted above, since x At+1 , x Bt+1 , x s At+1 and x s Bt+1 depend on e At , e Bt and do not explicitly depend on x At and x Bt , this optimality condition suggests that e At and e Bt are time and stock independent. This intuition is verified in the following lemma.

Lemma 2. Prior to the spatial regime shift, the following results hold:

(i) The pair (e A , e B ) constitutes a MPNE, where e j is implicitly defined as follows:

p j -c j (e j ) = δλD s jj [p j -c j g j (e j )D s jj + g i (e i )D s ij ]g j (e j ) (12)
+ δ(1 -λ)D jj [p j -c j (g j (e j )D jj + g i (e i )D ij )]g j (e j ), i = j.

(ii) e j is stock and time independent.

(iii) A given patch equilibrium resource stock is time dependent and reaches its steady state in the second period.

Lemma 2 suggests that the MPNE in escapement has a simple structure that depends on spatial characteristics, but is state independent. As such, the equilibrium escapement level in patch j is simply e j as defined in (12). In contrast to results obtained in Lemma 1, Equation 12suggests that the escapement level in a patch depends on the probability of regime shift, the patch's self retention rate before and after the shift. Moreover, in (12), terms multiplying λ capture harvester j's strategic responses to the threat of regime shift. Interestingly, for the particular case where λ = 0, (12) characterizes the equilibrium escapement levels for the no-shift case. The outcome of Lemma 2 allows us to derive the following results.

Proposition 2. Assume that marginal costs are constant (i.e., c j (x) = 0 for all x, j = A, B).

Over the pre-regime shift phase, the following results hold:

(i) The equilibrium escapement level in patch j = A, B, satisfies

g j (e j ) = 1 δ(λD s jj + (1 -λ)D jj ) . ( 13
)
(ii) The equilibrium escapement level in Patch A Patch B is increasing decreasing in the likelihood of the shift:

∂e A ∂λ > 0 and ∂e B ∂λ < 0. ( 14
)
Result (i) reveals that harvester j chooses her escapement to equate the financial rate of return with the expected biological return, which is biological growth g j (e j ), discounted by patch j's expected retention rate, λD s jj + (1 -λ)D jj . In other words, the non-cooperative golden rule (13) obtained in the certainty case is modified in response to the threat of future spatial regime shift. As the probability of the shift is raised, anticipating the shift, harvester

A adjusts her harvest decisions to increase escapement (to take advantage of improved future conditions), while harvester B reduces her escapement in response to the threat (to extract the resource before it shifts migration out of her region). Since the no-shift outcome is the special case where λ = 0, result (ii) of Proposition 2 implies that over the pre-regime shift phase, the equilibrium escapement level is lower in patch B and larger in patch A, compared to the no-shift levels.

These results contribute to the literature using escapement strategies as control variable to address the management of a renewable resource. The seminal paper by [START_REF] Reed | Optimal escapement levels in stochastic and deterministic harvesting models[END_REF] established the optimality of constant escapement in a stochastic fishery model. Extending that model, for example, [START_REF] Costello | Partial enclosure of the commons[END_REF] examine the implications of partial enclosure of a renewable common resource in a deterministic setting where the resource distribution regime never shifts; and [START_REF] Costello | Optimal harvesting of stochastic spatial resources[END_REF] focus on the effects of environmental variability on optimal spatial harvest responses. These papers find that the optimal escapement level is time and state independent (and are thus constant). The above results suggest that regime shift creates a discontinuity in the equilibrium escapement levels, so optimal escapements shift in response to the regime shift. This is consistent with previous analyses of optimal resource management of a single, aspatial stock under cyclical population dynamics [START_REF] Carson | Fisheries management under cyclical population dynamics[END_REF] or with environmental predictions [START_REF] Costello | Renewable resource management with environmental prediction[END_REF][START_REF] Kennedy | Renewable resource management with environmental prediction: the importance of structural specification[END_REF].

We have focused on the implications of spatial regime shift on escapement decisions. But we can also analyze the effects of regime shift on harvest and resource stock. We summarize these results as follows:

Proposition 3. Assume that marginal costs are constant (i.e., c j (x) = 0 for all x, j = A, B).

Over the phase prior to the shift, the following results hold.

(i) h B0 > hB0 and h A0 < hA0 .

( At the initial date, anticipating that a shift may occur in the future, harvester B is more aggressive (harvests more than she would in the no-shift case) while harvester A adopts precautionary behavior (reduces her harvest compared to the no-shift case), see Proposition 3i. This seems intuitive because harvester B stands to lose from the regime shift. In the steady state, however, these strategic interactions may be altered to induce a larger or a smaller harvest rate in each patch depending on the values of the spatial characteristics.

To better understand the intuition underpinning this result, it is instructive to decompose the difference between the steady-state harvest rate for harvester B under the threat and no-threat cases as follows:

h Bt -hBt = [g A (e A ) -g A (ẽ A )]D AB Term 1>0 + [(D BB g B (e B ) -e B ) -(D BB g B (ẽ B ) -ẽB )] Term 2<0
.

This condition shows that the effect of the threat of regime shift on harvester B's steady state harvest is driven by two opposite mechanisms, captured by the two underbraced terms in the right hand side. Harvester B will tend to harvest less in the steady state because she harvests aggressively in the initial period. This effect (captured by Term 2) represents the direct effect of the threat and is negative. 8 Moreover, harvests in both patches are linked:

As harvester A reduces her initial harvest, a larger stock will end up in patch B. Term 1 represents the strategic effect on resource growth; it is positive by Proposition 2 and the fact that function g A is increasing. This effect tends to raise harvester B's steady-state harvest rate under the threat.

As D AB is reduced, it is entirely possible that harvester B decreases her own steady state harvest, and that harvester A increases her harvest. Result (iii) of Proposition 3 provides conditions on D AB and D BA under which these statements hold.

For ease of exposition, we have primarily focused on the linear cost case. While this is a common assumption in resource economics, and delivers intuitive results, it fails to capture the stock effect under which the harvest cost increases as the resource stock tends to decline.

Such a stock effect is present whenever c j (x) < 0. It turns out that the presence of a stock effect can fundamentally alter strategic behavior across patches. We find here that these altered strategic incentives can be sufficiently strong to reverse the conventional wisdom about how harvesters will react to the threat of regime shift. The results are summarized in the following proposition.

Proposition 4. In the case where the cost functions are non-linear, it is possible that e A < ẽA and e B > ẽB .

This result suggests that despite the fact that the typical harvester in B will be disadvantaged by the regime shift, she may increase escapement in the pre-shift phase. 9 More precisely, the escapement level in patch j under the threat is larger than under the no-threat case if and only if

λD s jj [p j -c j (x s j )]g j (e j ) + (1 -λ)D jj [p j -c j (x j )]g j (e j ) > D jj [p j -c j (x j )]g j (ẽ j ), (15)
where x s j = g j (e j )D s jj + g i (e i )D s ij , x j = g j (e j )D jj + g i (e i )D ij , and xj = g j (ẽ j )D jj + g i (ẽ i )D ij .

Like the result in Proposition 2, this counterintuitive finding arises as a consequence of strategic interactions, except that here, strategic interactions are being driven by the stock effect. It is illustrative to consider the case of two countries harvesting a mobile fish stock.

When the stock effect is absent (so c = 0), there is no strategic feedback across countries.

In other words, while the countries' harvest choices are inextricably linked, the escapement choices are not; when c = 0, the escapement choice of country B has no impact on the escapement choice of country A. In that setting, a country that stands to lose its stock in the future is incentivized to leave fewer fish in the water and a country that stands to gain its stock is incentivized to leave more fish in the water.

But when a stock effect is present (c < 0), the optimal escapement level of country B does depend on the escapement level in country A. When country A leaves more fish in the water, some of them swim to country B, so the stock of fish in B is larger. This lowers the cost of fishing, and it turns out that this always leads country B to leave more fish in the water. So, under the threat of regime shift such as climate change, there are two important effects on country B. First, country B knows it will be disadvantaged by climate change, so it tends to want to leave fewer fish in the water. But, because country A tends to leave more fish in the water (for precisely the same reason), spatial movement means it is possible that the stock of fish in country B will be larger, which tends to make country B want to leave more fish in the water. Which effect dominates will depend on the relative magnitudes of the two effects. Thus, it is quite possible that the country that will be disadvantaged by climate change may leave more fish in the water than under the no-threat case.

An extension to common property

While patches act non-cooperatively over harvest, we have assumed that each individual patch is harvested by a single owner. But in some settings, it is more realistic for patch j to be common property, and is itself harvested by several non-cooperative agents. For example, if patches are the exclusive economic zones of countries, then harvesters would be individual fishing vessels within those territorial waters. The incentives engendered by this common property feature of non-cooperative spatial patches have not been analyzed previously. In this section we explore the role of common property in determining harvest and conservation incentives in anticipation of a regime shift.

To capture the common property feature, we generalize the previous model such that patch j is now harvested by N j ≥ 1 harvesters acting non-cooperatively in all periods.

Harvesters are identical within each patch, but we maintain the assumption that incentives may differ across patches with respect to economic, environmental, and biological conditions.

As we assumed previously, we denote by p j and c j the unit price and marginal harvesting cost function associated with patch j. Each of the N j harvesters in patch j have access to an equal fraction of the stock,

x jt N j . This available stock will be extracted by harvester k down to escapement level ξ (k) jt . Since harvesters are identical within each patch, the harvester's period-t harvest is (

x jt N j -ξ (k)
jt ) so, in the absence of harvesting costs, her period-t profit is p j (

x jt N j -ξ (k)
jt ). In the presence of harvesting costs, the period-t profit for harvester k = 1, 2, ..., N j associated with patch j can thus be written as

π (k) jt = x jt N j ξ (k) jt [p j -c j (v)]dv. ( 16
)
Recall that we are interested in a Markov Perfect Nash Equilibrium (MPNE), which determines the equilibrium escapement decision rules for all harvesters as functions of (x A , x B ):

(ξ

(1)

A (x A , x B ), ξ (2)
A (x A , x B ), ...., ξ

(N A ) A (x A , x B ), ξ (1)
B (x A , x B ), ..., ξ

(N B ) B (x A , x B )
). Since we are in a two-patch setting and harvesters operating in a given patch are identical, from now on, we concentrate on symmetric equilibria within each patch:

ξ A (.) = ξ (k) A (.) and ξ B (.) = ξ ( ) B (.)
for all k, . Therefore, it suffices to find a pair of escapement strategies (e A (.), e B (.)) with e A (.) = N A ξ A (.) and e B (.) = N B ξ B (.).

Lemma 3. Prior to the spatial regime shift, the following results hold:

The pair (e A , e B ) constitutes a MPNE, where e j is implicitly defined as follows:

p j -c j ( e j N j ) = δλD s jj [p j -c j g j (e j ) D s jj N j + g i (e i ) D s ij N j ] g j (e j ) N j (17) + δ(1 -λ)D jj [p j -c j g j (e j ) D jj N j + g i (e i ) D ij N j ] g j (e j ) N j , i = j.
This proposition provides an interesting characterization of the equilibrium strategic behaviors. In particular, it reveals that the equilibrium escapement levels prior to the shift depend on the biological return, N j , economic and environmental conditions as well as the probability of spatial regime shift. Because the common property case considered in this section generalizes the model from Section 2 to N j harvesters, a natural question is how the number of harvesters affects strategic responses to the possible occurrence of the shift. We formally address this and related questions below. For the sake of tractability, we restrict our attention to the standard logistic growth function case, which can be written as follows g j (e j ) = e j + r j e j (1 -

e j K j ) for j = A, B, (18)
where K j represents the carrying capacity and r j the intrinsic growth rate for patch j.

We first examine the effects on harvest decisions prior to the shift.

Proposition 5. Adopting g j as defined in (18) and provided marginal harvesting costs are constant:

(i) Patch j's equilibrium escapement, e j , is decreasing in N j and satisfies

g j (e j ) = N j δ(λD s jj + (1 -λ)D jj ) . ( 19
)
(ii) Assume that N A = N B = N . In comparison with the no-shift case, the threat of a shift increases (resp. reduces) the steady-state harvest rate in patch A (resp. patch B) whenever

the inequalities δ(1 + r A )D AA > N > N s A > 0 (resp. δ(1 + r B )D s BB > N > N s B > 0) hold. (iii) Assume that N A = N B .
Relative to the no-shift case, the threat of a shift increases (resp. lowers) the steady-state harvest rate in patch A (resp. patch B), but only if N A (resp.

N B ) satisfies δ(1 + r A )D AA > N A > NA (resp. δ(1 + r B )D s BB > N B > NB ),
where N s j and Nj > 0 (j = A, B) are real numbers that depend on spatial characteristics and are given in the appendix.

In the above results, conditions δ(1 + r A )D AA > N A and δ(1 + r B )D s BB > N B respectively ensure that, in each period, resource stock in patch A (resp. B) is not fully depleted.

These results highlight the implications of having more than one harvester in a patch. For instance, in the case where N A = N B = 1, Result (ii) of Proposition 5 implies that the steady-state harvest rate in patch A (resp. patch B) is lower (resp. higher) as long as

min(N s A , δ(1 + r A )D AA ) > 1 (resp. min(N s B , δ(1 + r B )D BB ) > 1) holds.
In the case where

N A = N B = N ≥ 2 this result does not hold for patch A (resp. patch B) when δ(1+r A )D AA > N > N s A > 1 (resp. δ(1 + r B )D BB > N > N s B 1) are satisfied.
The above results also highlight the importance of group size asymmetry (i.e., N A = N B ).

For example, in the case where the number of harvesters across patches is identical (i.e.,

N A = N B = N ≥ 1)
, we show in the appendix that the steady-state harvest rate in patch A (resp. patch B) is lower (resp. greater) irrespective the number of harvesters when

N s A < 0 (resp. N s B < 0) holds.
Irrespective the sign of N s A , Result (iii) of Proposition 5 highlights, however, the non-validity of that finding for patch A when N A = N B and

δ(1 + r A )D AA > N A > NA hold.
Similarly, the finding fails to hold for patch B as long as

N B = N A and δ(1 + r B )D BB > N B > NB are satisfied.
These findings show the importance of group size asymmetry. Whether N A or N B is small or large, our analysis reveals harvest responses to the threat that are qualitatively different than those in models where the number of players in both patches is identical (i.e., standard models in which there is a single agent in each patch).

It is also possible to derive the implications of considering common property vs. sole ownership and heterogenous growth on conservation. Our findings are summarized in the following proposition.

Proposition 6. Provided that c j (.) = 0 and g j (.) is defined in (18):

(i) Patch j is fully depleted in each period of time, but only if

N j ≥ δ(1+r j )(λD s jj +(1-λ)D jj )
is satisfied.

(ii) In the case where r j ≤ 1, patch j is fully depleted at each period of time whenever N j ≥ 2.

(iii) Patch j is not fully depleted when N j = 1, λD s jj + (1 -λ)D jj > 1/δ(1 + r j ) and r j ≤ 1.

Our motivation to tackle these extended results is that the related literature has focused on the cases in which (i) the growth functions are logistic and identical with the intrinsic growth rate smaller than one; (ii) each patch is owned by a single harvester. Relaxing each of those assumptions yields novel equilibria. For example, Proposition 6 suggests that different property rights structures may shift equilibrium states between depletion and no depletion (in each time period). More precisely, patch j is fully depleted in each time period when r j ≤ 1, c j (.) = 0, and N j ≥ 2. In this context, when that patch is owned by a single agent and its expected self-retention rate is large, result (iii) shows that strategic harvest decisions no longer lead to full depletion.

Moreover, we find that when patches have different growth functions, equilibria arise that cannot be captured by models with identical growth functions. For instance, when N j = 1 and c j = 0, depletion is the equilibrium outcome in both patches when r A = r B and

δ(1 + r j ) × min(λD s AA + (1 -λ)D AA ), λD s BB + (1 -λ)D BB ) < 1.
However, only patch B is fully depleted when r A = r B , N j = 1, c j = 0 and the following conditions hold:

δ(1 + r A ) × [λD s AA + (1 -λ)D AA )] > 1 and δ(1 + r B ) × [λD s BB + (1 -λ)D BB ] ≤ 1.
These findings illustrate potential biases that may arise when heterogeneities in property rights structure or biological return exist, but are not taken into account.

Simultaneously accounting for common property and a stock effect on harvesting costs quickly becomes analytically intractable. But one way to get traction in the case where c j < 0 is as follows. Let c i = 0 and other model parameters be such that the optimal aggregate residual stock level e i does not depend on N j and is characterized by

g i (e i ) = N i δ (λD s ii + (1 -λ)D ii )
Assume that the other patch is such that c j < 0 and initial model parameters (in particular the size of the user group, denoted N 0 j ) are such that the optimal aggregate residual stock level e j is characterized by Condition 17. Assume moreover that p j -c j (0) > 0 and g j (.) is bounded from above. Then we have:

Proposition 7. There exists Ñj > N 0 j such that the resource will be fully depleted in patch j at each time period for any size of user group greater than or equal to Ñj .

This result reinforces the general intuition from above: Adding common property to the regime shift game induces a depletion threat that could jeopardize the very existence of the renewable resource. Thus, common property and regime shift spatial interactions are, in some sense, substitutes for one another. In our model, either is sufficient on its own to induce severe resource depletion (or, depending on harvesting costs, possibly full depletion), but the two can also work in tandem, where both effects in combination can be a double whammy for a renewable resource subject to spatial interactions and regime shift.

Further discussion

In this section we provide a brief numerical illustration and some further analysis regarding corner solutions.

Illustrative Example

To visualize and briefly illustrate our theoretical findings, we develop a simple, stylized numerical example. We assume growth in patch j is logistic (as given in Equation 18), and that harvesting cost is linear in harvest, so Propositions 1, 2, 3, 5, and 6 apply. Baseline parameters are given as follows: r A = r B = 1.8, K A = K B = 100, δ = .95, D AA =

.85, D BB = .85, D AB = .1, D BA = .1, λ = .05, and N = 2. Regime shift results in D s AA = .9, D s BB = .8, D s AB = .05, and D s BA = .15. Our approach is to solve for the pre-regime-shift MPNE escapement levels under these baseline parameters, and then to derive the comparative statics of escapement with respect to each of these parameters.

To begin, suppose no regime shift was to occur. In that case, the socially-optimal [Figure 2 Here] Continuing with the regime shift scenario, we ask how sensitive escapements in each patch are to changes in the underlying parameters. To examine this question, we calculate elasticity of pre-regime-shift escapement in patch A with respect to each parameter. For example, we ask: If the growth rate increases by 1%, by what percentage will escapement in patch A increase? And because these elasticities need not be constant, we examine how the escapement changes for larger and larger changes in each parameter.

The top panel of Figure 3 displays these comparative statics results for six parameters.

Consistent with our theory, most parameters have a positive effect on escapement (indicated by an upward-sloping relationship between the parameter and the escapement); the only parameter examined that has a negative effect on escapement is the number of harvesters in each patch, N . The bottom panel of Figure 3 shows the point estimates of the elasticity of escapement with respect to each parameter. For this parameterization, the parameters with the largest elasticities are the discount factor (δ, whose elasticity is +7.6), the number of users in patch A (N A , whose elasticity is -7.5), 10 and the growth rate of the biological population in patch A (r A , whose elasticity is +4.5). Under this parameterization, changes in the regime shift likelihood had a relatively small effect on optimal escapement in patch A (elasticity of < 1, though still positive).

[Figure 3 Here]

Feasibility of interior solutions

At this stage we briefly discuss the feasibility of interior solutions. 11 We focus on the case of constant marginal costs so we can adopt general function forms for growth. First, the feasibility of a strictly interior MPNE is ensured when the following conditions are satisfied, for any j = A, B:

g j (0) > 1 min{δD jj , δD s jj } , g j (x j0 ) < 1 max{δD jj , δD s jj }
, g j (x j ) < 1 δD s jj where x j = D s jj g j (e j ) + D s ij g i (e i ) (i = j) and e j (j = A, B) denotes the pre-shift MPNE outcome in patch j. Under these conditions, following the shift, a corner solution where any agent would harvest nothing during a given time period is infeasible. In other words, for any j = A, B, denoting e P j agent j's post-shift interior equilibrium strategy 12 , the feasibility of interior MPNE requires that D s jj g j (e j ) + D s ij g i (e i ) > e P j be satisfied in the time period where the shift occurs. 13 Due to the concavity of g B (.) and the fact that D BB > D s BB , this condition is satisfied for j = B without further assumptions. Regarding agent A, this condition is satisfied as long as D S BA g B (e B ) is sufficiently large. 14

These conclusions allow us to derive conditions under which corner solutions may emerge.

Specifically, if the following conditions hold:

g A (0) > 1 δD AA , 1 δD s BB ≥ g B (0) > 1 δ [λD s BB + (1 -λ) D BB ] ; and g A (x A0 ) < 1 δD s AA , g A (D s AA g A (e A ) + D s BA g B (e B )) < 1 δD s AA
then an interior post-shift equilibrium outcome is not feasible, but a MPNE exists where the induced pre-shift outcome is interior, and where agent B fully depletes the resource in her own patch at any post-shift time period. Note finally that additional results on corner solutions are provided in Propositions 6 and 7: They rely on a specific form of the growth functions, but they highlight the differences induced by different ownership structures.

One of the most widely anticipated effects of global environmental change is the shift in the spatial distribution and migration of natural resource stocks. While effects will range from moderate to severe, and are occasionally predictable, in most cases, the occurrence of these shifts is uncertain. We have examined the effects of the threat of future spatial regime shift on strategic interactions between spatial property rights owners harvesting a mobile natural resource. Because our model allows for different economic returns, heterogenous growth, stock effects on costs, and spatial migration of the resource, we have been able to extract a number of novel results about how strategic behavior interacts with the threat of spatial regime shift. Our main contribution is to examine how the prospect of a spatial regime shift will affect non-cooperative incentives and equilibrium behavior across heterogeneous property owners. We considered as a baseline the no-shift case in which the resource distribution is deterministic and never shifts; this amounts to a non-cooperative spatial game in a deterministic environment. Then, introducing the possibility of a future regime shift, we examined the non-cooperative behavior of competing spatial property rights holders across a range of shift magnitudes. We modeled spatial regime shift as an abrupt change in the biophysical conditions that govern dispersal of the resource. The shift confers a clear advantage to one harvester and a clear disadvantage to the other. Our focus is on how the harvesters compete prior to the shift (but with common knowledge about the likelihood of the shift).

Our analysis allows for an arbitrary degree of regime shift. In the extreme, the shift could irreversibly drive the entire resource population out of one of the patches and into the other.

In keeping with the literature, we call this the "complete shift" case, where D s BB = D s AB = 0 and D s BA = 1. Using methods similar to those in Propositions 3 and 4, our analysis yields qualitatively similar results as for the partial regime shift case.

Whatever its magnitude, the threat of regime shift always increases initial harvest in the disadvantaged patch when harvest costs are linear. But we also found that strategic interactions can induce that patch to harvest less in steady state under the threat of regime shift. Indeed, this finding can maintain even under the threat of complete shift. The equilibrium escapement level in a patch depends on the resource price, marginal harvesting costs, but only when a patch is common property. We find that strategic interactions can even lead to full depletion, provided the common property feature is sufficiently strong relative to the growth of the resource. In the case where harvest costs are non-linear, new strategic incentives arise; this effect can even reverse the conventional wisdom, so even a patch that will be advantaged by the regime shift may act more aggressively in advance of the shift.

Our results may shed some light on the economic literature examining renewable resource management under the threat of a doomsday event (see for instance, [START_REF] Polasky | Optimal management with potential regime shifts[END_REF].

When the probability of regime shift is exogenous and utility is linear in harvest, the wisdom so far is that aggressive behavior always prevails prior to the shift. In this paper, the probability of regime shift is exogenous and a harvester makes her harvest decisions under the threat of the shift. In contrast to the aforementioned literature, we find conditions under which such harvesters may be cautious in response to disadvantageous regime shifts.

These results also relate to an interesting emerging policy debate. Many resource stocks such as marine fish, waterfowl, and some economically-significant game species migrate across national or other jurisdictions. At the same time, these migratory patterns are expected to change as a consequence of future climate change [START_REF] Molinos | Climate velocity and the future global redistribution of marine biodiversity[END_REF]. The results in this paper help inform our understanding about the behavioral responses of countries or other jurisdictions in advance of shifts, and unveil some counterintuitive predictions arising from strategic interactions to capture the resource. While informative in their own right, these results could be leveraged to inform policy responses for managing transboundary resources subject to possible future regime shift. jj g j (e P j ) + D s ij g i (e P i ) > e P j holds for any agent j. Yet, it is satisfied for j = A, B without further assumptions due to the concavity of g j (.) and the characterization of e P j . 14 A sufficient condition is that D s BA g B (e B ) ≥ D s AA g A (e P A ) -g A (e A ) be satisfied, which is more likely as the growth parameter becomes larger in patch B, or as λD s BB + (1 -λ) D BB becomes smaller. (iii) For t ≥ τ + 1:

x jt ≡ g j (e j )D s jj + g i (e i )D s ij > g j (ẽ j )D jj + g i (ẽ i )D ij ≡ xjt if and only if

D s ij > Ds jx ≡
g j (ẽ j )D jj + g i (ẽ i )D ij -g j (e j )D s jj g i (e i ) .

(iv) For t ≥ τ + 1, the relation

Proof of Lemma 2

Similar to the proof of Lemma 1.

B Details for Section 4

Proof of Proposition 2 (i) In the case where c j (.) = 0, the first order condition p j -c j (e j ) = δD s jj [p j -c j g j (e j )D s jj + g i (e i )D s ij ]g j (e j ), i = A, B and i = j.

simplifies to g j (e j ) = 1 δ(λD s jj + (1 -λ)D jj )

.

(ii) Using the implicit value theorem, e j is a continuously differentiable function of λ.

We then differentiate both sides of ( 22) with respect to λ. Rearranging the outcome yields Using a similar reasoning, we find that x At > xAt if and only if

D BA < Dx A ≡ D AA g A (e A ) -g A (ẽ A ) g B (ẽ B ) -g B (e B ) .
(iii) Using a similar method as for the proof of result (ii) along with the fact that h j = x j -e j , j = A, B, we get Notice that for j = A, B, Dh j and Dx j depend only on λ, δ, D kk , and D s kk , k = A, B.

Proof of Proposition 4

Recall that for j = A, B, e j satisfies the first-order condition (j = i) p j -c j (e j ) = δλD s jj [p j -c j g j (e j )D s jj + g i (e i )D s ij ]g j (e j )+δ(1-λ)D jj [p j -c j (g j (e j )D jj + g i (e i )D ij )]g j (e j ).

Hence, the left-hand side of condition λD s jj [p j -c j (x s j )]g j (e j ) + (1 -λ)D jj [p j -c j (x j )]g j (e j ) > D jj [p j -c j (x j )]g j (ẽ j ) is equal to p j -c j (e j ) while the right-hand side corresponds to p j -c j (ẽ j ). So this condition can be rewritten p j -c j (e j ) > p j -c j (ẽ j ). Since function j (x) = p j -c j (x) is increasing, we conclude that this condition holds if and only if e j > ẽj .

Proof of Lemma 3

Similar to the proof of Lemma 2.

Figure 1

 1 Figure 1 illustrates the pre-shift (top panel) and post-shift (bottom panel) migration patterns where the arrow thickness and size of the patches indicate the strength of connectivity.

(

  [START_REF] Kaffine | Unitization of spatially connected renewable resources[END_REF], where D s jj acts like an additional discount factor. Result (ii) of Proposition 1 is driven by the facts that (a) in each patch, the equilibrium escapement level and the patch retention rate are positively related; (b) patch B's retention rate decreases with the shift whereas this result is reversed for patch A. Results (iii) and (iv) of Proposition 1 lead to an unexpected outcome. Despite the fact that the shift inflicts biophysical damages to patch B (i.e., D s AB < D AB and D s BB < D BB ), the resource stock in patch B may be larger depending on the resource growth and spatial characteristics. In addition, harvester B may have incentives to increase her harvest compared to the no-shift case.

  escapements would have been e A = e B = 47.0 (just under half of the carrying capacity in each patch). Again, with no threat of regime shift, if patches A and B compete with each other, equilibrium escapements in the two areas are both substantially lower (e A = e B = 9.0, about an 81% reduction in escapement in each patch). This reduction in conservation arises from two factors: First, the non-cooperative nature of exploitation leads each owner to over-extract the fishery before her competitor can. The second, larger influence arises because N > 1, which gives rise to a commons problem internal to each patch.Allowing for the regime shift (which, for this parametrization has a very low probability of occurrence each period of just λ =5%) alters the harvest incentives of both parties. Prior to the shift, the MPNE of the regime shift game is given by e A = 9.2 and e B = 8.8. Consistent with Proposition 2, the patch that will be advantaged by regime shift (patch A) increases escapement and the patch that will be disadvantaged decreases escapement, relative to the case without regime shift. After the regime shift occurs, escapement diverges still further to e A = 12.8 and e B = 4.7. For illustrative purposes, suppose the regime shift occurs at time 10. Then the top panel of Figure2shows escapement in each patch (as a fraction of K i ) during the pre-regime-shift phase and the post-regime-shift phase. The bottom panel of Figure2shows the pre-regime-shift steady state harvest as a function of the regime shift risk, λ. Consistent with Proposition 3, the bottom panel of Figure2shows the case in which as the threat of regime shift increases, steady state harvest in patch A increases and steady state harvest in patch B decreases.

  Figures

h

  jt ≡ x jt -e jt = g j (e j )D s jj+ g i (e i )D s ij -e jt > g j (ẽ j )D jj + g i (ẽ i )D ij -ẽjt = xjt -ẽjt ≡hjt holds if and only ifD s ij > Ds jh ≡ g j (ẽ j )D jj + g i (ẽ i )D ij -g j (e j )D s jj + (e j -ẽj ) g i (e i ).Equation 21 suggests that for j = A, B, e j depends only on δ and D s jj whereas for j = A, B,ẽjdepends only on δ and D jj . As such, Ds jx and Ds jh depend only on δ, D s jj , D AA , D BB , and D ij , i = A, B and i = j.

  e j )[λD s jj + (1 -λ)D jj )] 2 , for j = A, B. Since g j (e j ) < 0, D BB > D s BB and D s AA > D AA , the result follows.Proof of Proposition 3(i) Since e A0 ≡ x A0 -h A0 > ẽA ≡ x A0 -hA0 , we necessarily have hA0 > h A0 . Moreover, since e B0 ≡ x B0 -h B0 < ẽB ≡ x B0 -hB0 , we necessarily have hB0 < h B0 .(ii) Using the facts thatx jt ≡ g j (e j )D jj + g i (e i )D ij , xjt = g j (ẽ j )D jj + g i (ẽ i )D ij , g A (e A ) > g A (ẽ A ),and g B (e B ) < g B (ẽ B ), we get x Bt > xBt if and only if D AB > Dx B ≡ D BB g B (ẽ B ) -g B (e B ) g A (e A ) -g A (ẽ A ) .

•

  h Bt > hBt if and only ifD AB > Dh B ≡ D BB (g B (ẽ B ) -g B (e B )) -(ẽ B -e B ) g A (e A ) -g A (ẽ A ) . D BA < Dh A ≡ D AA (g A (e A ) -g A (ẽ A )) -(e A -ẽA ) g B (ẽ B ) -g B (e B ).

)

  (ii) Relative to the no-shift case, the equilibrium escapement level in the post regime shift problem is larger in patch A and smaller in patch B: e Bt ≤ ẽBt and e At ≥ ẽAt for all t ≥ τ .(iii) At any date t ≥ τ + 1, the equilibrium resource stock (x jt ) in patch j is greater relative to the no-shift case (x jt ) if and only if D s ij > Ds jx .

(iv) At any date t ≥ τ + 1, the equilibrium harvest rate (h jt ) in patch j is larger relative to the no-shift case ( hjt ) if and only if D s ij > Ds jh , where Ds jx and Ds jh depend only on δ, D s jj , D AA , D BB and D ij , i = A, B, i = j, and are given in the appendix.

Result (i) of Proposition 1 suggests that harvester j = A, B chooses her escapement level to equate the biological return of the resource discounted by the patch retention rate (D s jj ) and the financial rate of return. This is a non-cooperative "golden rule" for spatial growth models

  ii) At any date t ≥ 1, x Bt > xBt if and only if D AB > Dx

B and x At > xAt if and only if D BA < Dx A . (iii) At any date t ≥ 1, h Bt > hBt if and only if D AB > Dh B and h At > hAt if and only if D BA < Dh A , where Dx j and Dh j depend only on λ, δ, D kk , D s kk , k = A, B and are given in the Appendix.

We are grateful for useful comments from seminar participants at various institutions including University of Alberta and EAERE 2016. Christopher Costello acknowledges the Sustainable Fisheries Group and Environmental Market Solutions Lab at UC Santa Barbara for helpful comments and financial support. Bruno Nkuiya gratefully acknowledges financial support by the Social Sciences and Humanities Research Council of Canada (SSHRC). Nicolas Quérou acknowledges financial support by the ANR project GREEN-Econ (Grant ANR-16-CE03-0005).

Notes

1 For example, migratory waterfowl and fish are managed by the multiple countries whose boundaries they traverse, groundwater is managed by overlying landowners, game is often managed by private wildlife management areas or hunting clubs, and invasive species are controlled by adjacent landowners.

2 See [START_REF] Stavins | The problem of the commons: still unsettled after 100 years[END_REF] for empirical evidence on tragedies of the commons.

3 Issues of uncertainty and irreversibility have been analyzed in the context of the timing of environmental policy adoption (see [START_REF] Ulph | Global warming, irreversibility and learning[END_REF] for an early contribution). They also matter significantly for problems of voluntary contributions to global public goods (see [START_REF] Elsayyad | Technology transfers for climate change[END_REF] for issues raised by technology transfers, or [START_REF] Nordhaus | Climate clubs: overcoming free-riding in international climate policy[END_REF] and [START_REF] Harstad | The dynamics of climate agreements[END_REF] for discussions on the design of a climate treaty).

4 See [START_REF] Costello | Renewable resource management with environmental prediction[END_REF] and [START_REF] Carson | Fisheries management under cyclical population dynamics[END_REF] for aspatial models of resource management with environmental predictions.

5 For example, climate change may irreversibly trigger local scarcity or extinction in the sub-polar regions and invasion in the arctic for many species of fish [START_REF] Cheung | Projecting global marine biodiversity impacts under climate change scenarios[END_REF]).

6 In reality, habitat, salinity, water currents, nutrients, oxygen concentration and extraction costs may vary across patches leading to heterogenous biological, economic, and environmental conditions.

7 This is a benign assumption because harvest and escapement are linked by the identity h t ≡ x t -e t . This approach has subsequently been adopted by numerous authors including [START_REF] Costello | Optimal harvesting of stochastic spatial resources[END_REF], [START_REF] Costello | Partial enclosure of the commons[END_REF], [START_REF] Kapaun | Does the optimal size of a fish stock increase with environmental uncertainties?[END_REF], and many papers cited therein.

8 To formally see why Term 2 is negative, it can be shown by using (13) that D BB g B (e B ) -1 > 0. For the special case where λ = 0, we thus have D BB g B (ẽ B ) -1 > 0. Combining this result with the facts that ẽB > e B and the function D BB g B (x) -x is concave, we conclude that D BB g B (e B ) -e B < D BB g B (ẽ B ) -ẽB . 9 Using the growth functions defined in (18) along with the marginal costs c j (v) = θ j /v, j = A, B and the set of parameters δ = 0.95, x A0 = 0.84, x B0 = 0.8, D AA = 0.7, D BB = 0.77, D AB = 0.26, D BA = 0.22,

.45, we find numerically that e A and e B are both increasing in λ.

10 While N should technically be an integer, we calculate the elasticity of escapement with respect to N by varying it a small amount, to maintain consistency with the approach taken with other parameters.

11 Further technical details are available from the authors upon request.

12 This residual stock level is characterized by e P j = (g j ) -1 ( 1 δD s jj ) as shown in the analysis.

30 (i) The equilibrium value function can be written as

where the law of motion is:

Note that the vector x t+1 depends only on e i and e j . The first-order condition is

In general many possible equilibria can exist in this game. We show that a profile of state-independent strategies (where e i and e j are stock independent) solve the dynamic programming problem and therefore constitute an equilibrium of the model; we will focus on this class of solutions. Under these strategies, the first two terms in the first-order condition are independent of vector x t by inspection. The derivative of the value function in period t+1 depends on the period t+1 state, but is independent of the period t state. For an interior solution we have e it < x it and, using the expression of the law of motion, we conclude that x it+1 depends on e it but not on x it . Therefore, the terms in the sum in the above first-order condition are independent of x t . Under these assumptions, the period t game problem has state independent strategies: e it does not depend on x t for any i = A, B. Using the expression of the law of motion yields ∂x jt+1

∂e it = D ij g i (e it ) for any j = A, B, moreover we obtain ∂W j ∂x it (x t+1 ) = 0, for all i, j ∈ {A, B}, i = j.

Combining this result along with (20), it follows that

Substituting this expression and that of

∂e it into the first-order condition we obtain the required condition. This necessary condition is also sufficient provided the assumption of concavity of returns in the strategy (residual stock level) is satisfied. Finally, since the growth function is independent of time, as is spread, self retention and economic costs, the optimal choice e it is independent of time for any i = A, B.

(ii) This is a simple consequence of the fact that e i and e j are time and state independent.

Proof of Proposition 1

(i) In the case where c j (x) = 0 for all x, j = A, B, using the first-order condition p j -c j (e j ) = δD s jj [p j -c j g j (e j )D s jj + g i (e i )D s ij ]g j (e j ), i = A, B and i = j.

we obtain

(ii) Since D s AA > D AA and D BB > D s BB , it follows that g A (e A ) < g A (ẽ A ) and g B (e B ) > g B (ẽ B ). Hence, ẽA < e A and ẽB > e B . This is the case because functions g j (.), j = A, B are concave such that functions g j (.), j = A, B are decreasing.

Proof of Proposition 5

(i) In the case where c j = 0, the first-order condition

. Using the fact that g j is concave and partially differentiating both sides of this equality and rearranging, it can be shown that e j is decreasing in N j .

Moreover, using this first-order condition in the case of logistic growth functions, we derive

Notice that e A > 0 if and only if δ(1 + r A )D AA > N A . Likewise, we have e B > 0, but only

(iii) Denote by h s A , the steady-state total harvest for patch A. According to the law of motion of the stock in patch j, we have h s A = g A (e A )D AA + g B (e B )D BA -e A . Combining this result with (23) and keeping in mind that the growth function is logistic, we derive

where,

Denote by NA the unique positive root of (24). Clearly, h s A > h s A | λ=0 whenever N A > NA .

(ii) A similar reasoning as for the proof of Result (ii) suggests that

where,

(ii) Assume that N A = N B = N . In this case, condition (24) simplifies to

irrespective of the number of harvesters whenever N s A < 0.

A similar reasoning as for patch A leads to:

of the number of harvesters whenever N s B < 0.

Proof of Proposition 6

Using g j (e j ) = N j δ(λD s jj +(1-λ)D jj ) and keeping in mind that the growth function is logistic, it can be shown that if escapement for patch j is strictly positive, it is necessarily given by ( 23).

(i) Using (23), we derive e j ≤ 0 iff N j ≥ (1 + r j )(λD s jj + (1 -λ)D jj ). The result then follows.

(ii) Assume that N j ≥ 2 and r j ≤ 1. These assumptions imply N j ≥ (1 + r j )(λD s jj +

(1 -λ)D jj ) such that by Result (i), patch j is harvested down to extinction.

(iii) In the case where N j = 1, we can verify that e j defined in ( 23) is strictly positive as long as r j ≤ 1 and [λD s jj + (1 -λ)D jj )] > 1/δ(1 + r j ). The result then follows.

Proof of Proposition 7

It suffices to show that e j cannot be positive for N 0 j arbitrarily large. Using the first-order condition, if e j > 0 we know that the following inequality holds: 25) goes to zero. However, its right hand side term remains positive (as N 0 j gets arbitrarily large) because p j > c j (0) and c j is decreasing. Therefore, as N 0 j becomes arbitrary large, condition (25) (that characterizes interior solutions in escapement) cannot hold. Hence, there exists Ñj such that patch j is fully depleted at each time period as long as N j ≥ Ñj .


Stéphane Bonelli
email: stephane.bonelli@cemagref.fr

-Didier Marot
email: didier.marot@univ-nantes.fr

Micromechanical modeling of internal erosion

Keywords: internal erosion, suffusion, interfacial erosion, homogenization. MOTS-CLÉS : érosion interne, suffusion, érosion interfaciale, homogénéisation

Suffusion is the internal erosion process by which finer soil particles are detached from the solid matrix, and transported through constrictions by seepage flow. At the macroscopic scale, this is a bulk erosion process and corresponds to a source term in the mass balance equations. This paper constitutes a step towards bridging the gap between the counter intuitive bulk erosion model and our intuition that erosion is essentially an interfacial process. We suggest that a framework within suffusion can be viewed as a clay/water interface erosion at the microsopic scale. The coefficient of surface erosion appears to be a relevant parameter for the suffusion bulk erosion law. The comparison between the results of the present modeling study and previously published experimental data supports the validity of our approach.

RÉSUMÉ. La suffusion est le processus d'érosion interne de détachement de particules fines de sol, et de transport de ces particules par écoulement d'eau entre les particules les plus grosses. A l'échelle macroscopique, ce phénomène est une érosion de volume, et il est représenté par un terme source dans les équations de conservation de la masse. Notre intuition est que l'érosion est un phénomène intrinsèquement interfacial. Ce travail vise à comprendre ce paradoxe apparent. Nous proposons un cadre de travail qui permet de considérer la suffusion comme une érosion d'interface argile/eau à l'échelle microscopique.

Le coefficient d'érosion de surface apparaît comme un paramètre pertinent de la loi d'érosion de volume de suffusion. La comparaison entre les résultats de cette modélisation et des résultats expérimentaux publiés confirme la pertinence de notre approche.

Introduction

Erosion is the removal of material caused by the eroding power of the flow and is essentially an interfacial process. Suffusion (or suffosion) is an internal erosion process by which finer soil particles are detached from the solid matrix and transported through pore constrictions by seepage flow. The term suffusion (L. suffossio, from suffodere, to dig under) was introduced by Pavlov in 1898. The internal instability that results from suffusion gives rise to a wide range of hazards including piping and sinkholes [START_REF] Fell | Internal Erosion of Dams and Their Foundations[END_REF][START_REF] Gutiérrez | A genetic classification of sinkholes illustrated from evaporite paleokarst exposures in Spain[END_REF].

Two scales must be considered: a micro-scale defined at the level of the pore constrictions, and a macro-scale corresponding to the Representative Elementary Volume (REV). The REV is typically a laboratory sample, or a spatial integration point in the Finite Element Method (e.g. a Gauss point).

The process of bulk erosion may be considered at the macro-scale as a transition from solid-like to fluid-like behaviour. This transition is smooth, and usually described by using a three-phase model (solid, fluid and fluidized solid). These three phases interact while being constrained by the balance equations. For the sake of clarity, the equations are written here in one-dimensionnal evolution and dilute suspension flow as follows [START_REF] Papamichos | Volumetric sand production model and experiment[END_REF][START_REF] Papamichos | Sand erosion with a porosity diffusion law[END_REF][START_REF] Papamichos | Erosion and multiphase flow in porous media[END_REF][START_REF] Wan | Analysis of sand production in unconsolidated oil sand using a coupled erosional-stress-deformation model[END_REF]: F w q (Darcy constitutive law) [4] In these equations, is the porosity, Clay is the eroded material density, q is the seepage velocity, p is the water pressure, F represents the mechanical interaction between the pore-fluid and the solid matrix, w is the water viscosity and is the geometric permeability.

The term r is introduced into the mass balance Equation [1] to describe the detachment of particles via an erosion law. The description of suffusion at the macroscopic scale therefore corresponds to a bulk erosion process. The erosion constitutive law is introduced as usual by phenomenological analysis, at the macroscale.

Few attempts have been made to relate macroscopic and microscopic quantities in a multi-scale description of the suffusion process. This paper is intended as a further step in this direction, on the basis of the results published by [START_REF] Bouddour | Erosion and deposition of solid particles in porous media: Homogenization analysis of a formation damage[END_REF]. What follows is not a program aimed at reformulating the theory of suffusion as a whole based on homogenization, rather attention focuses explicitly on the goal of bridging the gap between the counter intuitive bulk erosion model and our intuition that erosion is essentially an interfacial process.

Description of the representative elementary volume

Let us consider a representative elementary volume (REV) of a soil partitioned into three disjoint regions

1 Sand Sand s | | , S Sand | Sand | | Sand | [7]
The meso-pore domain contains an incompressible two-phase flow of water and clay particles, assumed to be a dilute suspension (density w , viscosity w ). The clay matrix, which is considered as a continuum, consists of two phases: water, and clays particles Clay V sand

[8]

The porosity V Clay can be divided into macro-porosity corresponding to the meso-pore domain, and micro-porosity Clay in the clay matrix. This description accounts for two observation scales:

(i) local (microscopic) scale, which is associated with the characteristic size of heterogeneities (clay aggregates, sand grains, meso-pores);

(ii) the macroscopic scale, which corresponds to entire soil layers and is most important for practical purposes.

The scale corresponding to clay particles and micro-pores in the clay matrix is not considered. This type of heterogeneous soil is similar to the classical doubleporosity model for modelling flow in fractured porous media, developed in the early 1960s. However, while the meso-pore domain makes up only a small percentage of the total pore volume, it transmits a major portion of the flow through the REV, and the fluid exchange between the clay matrix and the meso-pore domain can be neglected.

We assume that the soil is a clayey sand [START_REF] Revil | Permeability of shaly sands[END_REF][START_REF] Revil | Mechanical compaction of sand/clay mixtures[END_REF], and that the size of the meso-pores is much greater than the size of the micropores. Therefore, V Sand and the sand matrix is connected while the clay matrix may not be.

We assume that erosion occurs only at the interface Clay

Pore of the clay matrix and the meso-pore domain (sand grains are not erodible) (Figure 2). The erosion process is therefore external to the clay matrix, and does not affect its porosity Clay , nor its density Clay .

Sand

Meso-pores

Sand s

Clay s

Clay Pore

1 Sand V (1 Clay ) V Sand V V Clay Sand (1 Clay ) V 1 Sand V (1 Clay ) Clay matrix

Micro-pores

Solids Pores

Figure 1. Relative proportions of each constituent for a sand-clay mixture in the clayey sand domain

As illustrated in Figure 3, suffusion may occur when 0

V sand , whereas no suffusion occurs when V 0 (no clay) or V Sand (the pore constrictions are too small).

Microscopic equations of Stokes flow with interfacial erosion

The mechanical state is such that effective stresses and matrix deformations are negligible. Physico-chemical effects (like dissolution or deposition) are not considered. The porosities Sand and Clay are thus constant. The Reynolds number is small and inertia is neglected.

We take a(x) to denote any quantity a varying at the micro-scale. The Stokes equations for the pore flow within Pore are:

Fluid mass balance equation 0 u in Pore [9] Fluid momentum balance equation 0 T in Pore [10] 1 2 Fluid constitutive law 2 ( ), ( ) ( ) T w p T I D u D u u u in Pore [11]
The jump equations on the sand/water interface are The jump equations on the clay/water interface are (Bonelli and Brivois, 2006;[START_REF] Brivois | Soil erosion in the boundary layer flow along a slope: a theoretical study[END_REF] Total mass jump equation Shear induced interfacial erosion is usually described with threshold laws such as [START_REF] Ariathurai | Erosion rates of cohesive soils[END_REF][START_REF] Knapen | Resistance of soils to concentrated flow erosion: A review[END_REF][START_REF] Partheniades | Erosion and deposition of cohesive soils[END_REF][START_REF] Zhu | Comparison of concentrated-flow detachment equations for low shear stress[END_REF]

Inferring a suffusion law by means of spatial averaging

The complete set of Equations [9]- [17] has been previously used to study different systems by means of periodic homogenization [START_REF] Bouddour | Erosion and deposition of solid particles in porous media: Homogenization analysis of a formation damage[END_REF]. Under the dilute flow assumption, the system obtained is formally similar to Equations [1]-[4]. [START_REF] Bouddour | Erosion and deposition of solid particles in porous media: Homogenization analysis of a formation damage[END_REF] Equation [18] involves the spatial average of the stress c on er . As er depends on , this spatial average is not straightforward. The threshold stress c is not considered, as the average of a local threshold law involving local stresses is known not to be a threshold law involving the average of the local stresses. We propose a first order approximation leading to:

H[ c ]( c )d Pore | er | | Pore | d Pore [20]
In order to model the average shear stress on The local (viscous) shear stress is defined as follows [START_REF] Bouddour | Erosion and deposition of solid particles in porous media: Homogenization analysis of a formation damage[END_REF]:

2 w D(u) n P L , i K ip x q K qp x i e p n q [22]
where K is the microscopic geometric permeability tensor. The volume average of K gives the macroscopic permeability tensor. Equation [21] shows that the spatial average of is equal to e , which gives:

1 | | Pore d [23]
The spatial average of is more complicated:

1 | | d Pore P L [24] 1 | | Pore Pore Pore T T Pore d d d [25] T I n n on Pore [26]
Thorough analysis performed by means of numerical simulations [START_REF] Golay | Interfacial erosion: A three-dimensional numerical model[END_REF][START_REF] Golay | Numerical modelling of interfacial soil erosion with viscous incompressible flows[END_REF], which is not detailed here for conciseness, shows that the normal component of the viscous stress vector on is usually several orders of magnitude lower than the tangential component ( 2 | ( ) | w n D u n

). At the first order we can consider that T , and that 1 .

The orientation of the microscopic velocity is given by u K e as u w 1 K e | P / L | [START_REF] Bouddour | Erosion and deposition of solid particles in porous media: Homogenization analysis of a formation damage[END_REF]. If this microscopic velocity has mostly the same direction as the macroscopic pressure gradient ( u e ), then the use of Equation [22] shows that 1 (considering that T

). This situation can be viewed as an isotropic behaviour with weak inhomogeneities and low tortuosity.

In reality, the porous domain of a soil comprises a mixture of pores of different shapes and orientations. This porous domain is modified by the microscopic erosion process. Suffusion is likely to induce anisotropy, but permeability anisotropy is also likely to affect the suffusion process. The quantity accounts for the relationship between the local permeability gradient and the global pressure gradient. This parameter certainly plays a role in describing the strong coupling between the evolution of the microstructure due to erosion, and the macroscopic suffusion law.

We take i w to denote the hydraulic gradient:

i w 1 w P L [27]
where w w g is the water specific weight. Assuming a constant coefficient of erosion k er , Equation [18] is re-written as follows:

r w k er | er | | Pore | i w [28]
The macroscopic volume erosion appears here to be driven by the global pressure gradient, and not by the seepage velocity, as earlier suggested by [START_REF] Einstein | Der Geschiebetrieb als wahrschein-lichkeits Problem[END_REF] and [START_REF] Sakthivadivel | A review of filtration[END_REF]. In addition, the suffusion process and the evolution of the permeability anisotropy are coupled. Modelling this coupling phenomenon is, however, beyond the scope of this paper. For the sake of simplicity, we assume that is a constant.

A model for | er | / | Pore | is now introduced.

Modeling the evolution of internal surfaces

The dimensionless quantity | er | / | Pore | can be split as follows:

| er | | Pore | | Clay Pore | | Pore | | er | | Clay Pore | [29]
From the identity

| Clay |= | Clay Pore | | Clay Sand | [30]
we obtain

| Clay Pore | | Pore | | Clay | | Pore | | Sand | | Pore | [31]
where

| Clay Sand | | Sand | [32]
Inserting the identity

| Pore |=| Clay | + | Sand | 2 | Clay Sand | [33] into Equation [32] yields 1 2 1 | Clay | | Sand | | Pore | | Sand | [34]
Now inserting this result into Equation [31] gives

| Clay Pore | | Pore | | Clay | | Pore | | Sand | 2 | Pore | [35]
From Equations [5], [6], [7] and [35] we finally obtain

| Clay Pore | | Pore | S Clay V S Pore S Sand (1 Sand ) 2S Pore [36]
We now assume that a domain surface area scales with the corresponding domain volume raised to the power of 2/3. This is a simple surface/volume scaling: for example, the surface of a sphere scales with its volume with this exponent. The expressions S Clay V and S Pore can therefore be modelled as follows: Here, the superscript 0 in any quantity a 0 denotes the initial value of a .

From Equations [5], [6], [7] and [33] we obtain:

S Pore S Clay V S Sand (1 Sand )(1 2 ) [39]
The quantity of Equation [32] must be equal to zero if V 0 ( Clay ). By using Equations [38] and [39], we infer the identity:

S Pore 0 ( 0 ) 1/3 ( Sand ) 2/3 S Sand (1 Sand ) [40]
By using Equation [40], Equations [37] and [38] can now be written as follows:

erosion, as erosion stops if V V stop . It is also a threshold hydraulic gradient for the onset of erosion, as erosion occurs if

i w i c 0 V 0 V 2/3 1 [47]

Parametric analysis

We consider a homogeneous sample, and we assume dilute flow and no deposition or clogging. As a result, the pore-fluid viscosity and the geometric permeability are homogenous. The pressure gradient is therefore homogeneous. The system Equations [1] and [2] (balance equations) and [28]-[46] (erosion constitutive law) leads to the following differential equation:

2/3 2/3 ( ) 1 1 (0) 1 w sand V V V w sand V i d dt i [48]
In Equation [48], V is the dimensionless clay volume fraction, sand the dimensionless sand porosity, and w i the dimensionless hydraulic gradient defined as follows:

0 0 0 , , V sand w V sand w V V c i i i [49]
The dimensionless time t is defined by means of a characteristic time of erosion This characteristic time is the main result of this work: it is established here that the global pressure gradient for suffusion plays the same role as the pressure drop p / L in a pipe of length L , where a similar characteristic time has been defined (Bonelli et al., 2006; Bonelli and Brivois, 2008).

The effects of varying the dimensionless hydraulic gradient w i are shown in Figure 4 for 1/ 0.99 sand , and Figure 5 for 2 sand . Figure 6 and Figure 7 show the effects of varying the initial dimensionless amount of clay sand for 1 w i , and 10 w i respectively.

For a given clayey sand ( i c 0 , k er ), the model captures two important aspects:

-the greater the initial clay volume fraction V 0 , the slower the erosion;

-the greater the hydraulic mechanical state i w , the faster the erosion.

Comparison with experimental results

The suffusion law is now compared with previously published data, giving the mass fraction of eroded material as a function of time for several hydraulic gradients [START_REF] Sterpi | Effects of the erosion and transport of fine particles due to seepage flow[END_REF].

The mass fraction of eroded material is a function of the clay volume fraction as follows:

(t) (1 clay ) V 0 V (t) (1 clay ) V 0 (1 sand ) [51]
Table 1 summarizes the parameters of the suffusion equation. Parameters sand and V 0 were inferred from [START_REF] Sterpi | Effects of the erosion and transport of fine particles due to seepage flow[END_REF], while a characteristic value of clay was chosen as this value was not available.

Equation [46] suggests that it is possible to estimate the parameter i c 0 from the total amount of eroded material in a given test, if at least equilibrium is attained, which was not the case in the available data [START_REF] Sterpi | Effects of the erosion and transport of fine particles due to seepage flow[END_REF]. The results of the identification are mean values and standard deviations for i c 0 and k er .

It is notable that although most of the research on suffusion has been directed towards the threshold stress, the rate of erosion is at least equally as important. This rate of erosion appears to be primarily driven by t er Equation [50], which is essentially affected by the coefficient of surface erosion k er . For piping erosion, k er was found to range from 10 -6 s/m to 10 -3 s/m for fine-grained soils (Bonelli et al., 2006; Bonelli and Brivois, 2008). [START_REF] Sterpi | Effects of the erosion and transport of fine particles due to seepage flow[END_REF]) versus model results (continuous lines)

The coefficient of erosion is about 10 -5 s/m. This is a low value, which corresponds to a slow erodable soil for surface erosion (Bonelli et al., 2006;Bonelli and Brivois, 2008). This finding is consistent with previous experimental results: internal erosion rates have been found to be smaller than surface erosion rates [START_REF] Reddi | Comparison of internal and surface erosion using flow pump tests on a sand-kaolinite mixture[END_REF].

Figure 6 gives the increase in the mass fraction of eroded material as a function of time. The data are well-described by our model over the full range of hydraulic gradients explored.

Conclusion

Erosion, which is the removal of material caused by the eroding power of the flow, is essentially an interfacial process. The objective of this work was to understand the bulk erosion process, which is counter intuitive at the pore scale, as an interfacial erosion process.

By means of homogenization reasoning, the macroscopic volume flux of eroded material is related to the microscopic surface flux of eroded material, at the clay water interface. This surface erosion is considered as shear stress driven. The macroscopic volume erosion is therefore driven by the global pressure gradient. In addition, the coefficient of clay surface erosion appears to be a relevant parameter to describe the kinetics of the suffusion process.

The total clay/water interface and the eroded clay/water interface are related to the clay volume fraction. The amount of erodable clay is related to the hydraulic gradient. Comparisons with published experimental results show that this model gives good results over the full range of hydraulic gradients explored. This paper was not intended to propose a new suffusion law, or to reformulate the theory of suffusion as a whole based on homogenization. Other important phenomena should be considered in a more comprehensive model of internal erosion: permeability anisotropy induced by microstructure erosion, particle transport and filtration, deposition and clogging, two-phase seepage flow, concentrated flow, and dissolution and physico-chemical effects.

  constituent, and not the solid phase. For the clay domain, the solid phase Clay is the sum of the solid constituent Clay s and the micro-pores, saturated by water. All solid constituents are of density s . The sand grains are homogeneous and impervious. The clay matrix density is Clay (1 Clay ) s Clay w . From Figure 1, we infer the important connection:

Figure 2 .Figure 3 .

 23 Figure 2. Schematic of the suffusion process viewed as the surface erosion of the clay matrix

  established that the global volume flux of eroded material r Equation [1] is related to the local surface flux m Equation [16] of eroded material as follows:

  Pore , let | P / L | e the macroscopic pressure gradient denoted by p / X in Equation [3], with magnitude | P / L | and orientation e ( 1 e ). This is the leading term of the microscopic Stokes flow, as the microscopic balance equation is T | P / L | e . As p in Equation [11] now denotes only the pressure fluctuation, this gives the well-known result:

Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 3456 Figure 3. Effects of varying i w on V for sand = 1 / 0.99

  An additional(and usual) constitutive assumption on the interface is that all tangential velocities are assumed to be continuous across Pore , implying the continuity of across Pore :

	where c (Pa) is the threshold stress, k er (s/m) is the coefficient of surface erosion,
	H	[ ]	is the Heaviside step function,	|| || is the tangential shear stress at the
	interface, with			[I n n] T n .
			u	T	0,	u	T		I n n u on Pore	[17]
										:
			er m k H	[		c	](	c	)	on Clay	Pore	[16]

Table 1 .

 1 Numerical values of the parameters

	sand	Sand porosity	0.49
	clay	Clay porosity	0.60
	V 0	Initial clay volume fraction	38%
	w	Water density	1 000	kg.m -3
	Clay	Clay saturated density	1 688	kg.m -3
		Anisotropy and tortuosity	2
	g	Gravitational constant	9.81	m.s -2
	k er	Coefficient of erosion	4.33 10 -5 ± 0.30 10 -5	s.m -1
	i c 0	Threshold hydraulic gradient	2.95 ± 0.64

Figure 6. Suffusion tests with a constant pressure drop, mass fraction of eroded clay is shown as a function of time, test results (symbols,

Acknowledgements

This project is sponsored by the Région Provence Alpes Côte d'Azur, the French National Research Agency under grant 0594C0115 (ANR-ERINOH), and the French Institute for Applied Research and Experimentation in Civil Engineering (IREX).

Inserting the above results into Equation [36] finally yields:

For the sake of simplicity, we retain a first order approximation:

In the erosion process of the enlargment of a pipe (Bonelli et al., 2006;Bonelli and Brivois, 2008), the limitation of the amount of eroded material is a structural piece of information (it may be the earth-dam height for example). For suffusion, the situation is entirely different. This time the amount of eroded material (the clay) is limited, and depends on the hydraulic gradient, as established by experimental results (Bendahmane et al., 2006;[START_REF] Bendahmane | Parametric study of suffusion and backward erosion[END_REF][START_REF] Sterpi | Effects of the erosion and transport of fine particles due to seepage flow[END_REF]. Parameter | er | appears to be the relevant quantity accounting for this material information. Consequently, we assume that the surface fraction of eroded material | er | scales with the clay volume fraction as follows:

The final clay volume fraction V stop is assumed to depend on the hydraulic gradient:

The parameter i c 0 is a phenomenological parameter which is not explicitly connected to the local threshold c . It is a threshold hydraulic gradient to stop