Bocar Ba

Raphael Soubeyran
email: raphael.soubeyran@inra.fr

Exploitation and Recycling of Rare Earths *

Keywords: Rare earths, Phosphorus, recycling, competition, market power, optimal control JEL codes: Q31, Q53

. We then show a new reason why the price of an exhaustible resource may decrease: when the primary sector is monopolistic, the primary producer has incentives to delay its production activities in order to delay recycling. As a consequence, the price path of the recyclable resource may be U-shaped. We also show that a technological improvement in the recycling sector increases the price in the short term but decreases it later.

Introduction

Recyclable exhaustible resources, such as rare earths and Phosphorus, are increasingly important industrial inputs. Indeed, rare earth elements are important inputs for the production of many modern technologies, such as cell phones, light bulbs, automobiles, hybrid car batteries and gearboxes, and wind turbines [START_REF] Chakhmouradian | Rare earth elements: Minerals, mines, magnets (and more)[END_REF]. Phosphorus (derived from Phosphate rocks) is essential for soil fertility and has no substitute in agricultural production processes [START_REF] Cordell | The story of phosphorus: Global food security and food for thought[END_REF]. Due to economic development and an increasing world population, demand for rare earths and phosphorus has been growing rapidly and is expected to grow even more in the future (Alonso et al. 2012[START_REF] Steen | Phosphorus availability in the 21st century: management of a nonrenewable resource[END_REF]). 1Historically, the supply of these resources has been highly concentrated. Until 2010, China controlled 95% of the production of rare earths [START_REF] Chakhmouradian | Rare earth elements: Minerals, mines, magnets (and more)[END_REF], while a handful of countries, including Morocco, China, and the U.S.A, controlled most of the world's Phosphate rock production [START_REF] Ifdc | World phosphate rock: Reserves and resources[END_REF]. However, prospects for the supply of rare earths and Phosphate rocks differ. Since 2010, the supply of rare earths has become less concentrated as China currently possesses less than 40% of rare earth reserves, while the supply of Phosphate rocks has become more concentrated, as 85% of these reserves are currently located in Morocco and Western Sahara. 2 One strategy to increase the supply of these resources is recycling. The recycling of rare earths is in its infancy (UNEP, 2011), currently focused on scrap materials that contain high amounts of rare earth elements and on high value rare earths, including magnets, lamp phosphors, and nickel hybrid batteries [START_REF] Golev | Rare earths supply chains: Current status, constraints and opportunities[END_REF]. European fertilizer firms, such as Ostara and CNP-Technology already produce recycled Phosphorus fertilizers (Hukari and Nätto 2015) and, according to the U.K. Environment Agency [START_REF] Ea | Freshwater eutrophication: A nationally significant water management issue[END_REF], approximatively 70% of Phosphorus from sewage sludge produced in the U.K. is currently recycled as biosolids to be used as fertilizer. 3 In order to assess the effect of recycling, it is necessary to understand how the primary sector may react. Since the main input to the production of recycled materials is the stock of scrap, the emergence of recycling activities may affect the dynamics of both the extraction of the exhaustible resource as well as the price of the final goods.

In this paper, we study the impact of a recycling sector in a stylized economic model of exhaustible resource extraction. We develop a Hotelling model of resource extraction in which the consumption good is produced from virgin or recycled materials. Virgin materials are extracted from a finite stock of a virgin resource and recycled materials are derived from the stock of scrap. The stock of recyclable scrap grows with current consumption of the virgin resource at a given recyclability rate. We assume a competitive recycling sector in which production costs decrease with the stock of recyclable scrap. As a consequence, production in the primary sector generates a positive externality that benefits to the recycling sector. To ensure consistency with the various possible (future and present) market structures in the rare earths and Phosphate rocks extraction sectors, we consider two polar cases: competitive and monopolistic extraction.

Our main results are the following. We show that, if the primary sector is competitive, the optimal level of production for firms in the primary sector is such that the price of the resource grows at the discount rate (this is the so-called Hotelling rule) because these firms assume that their production will not increase the stock of scrap. We then show that, if the extraction sector is monopolistic, the dynamics of the price of the resource depend on the recyclability rate. When the recyclability rate is sufficiently low, the price of the resource increases over, which is similar to the case without a recycling sector. When the recyclability rate is sufficiently high, the monopolistic firm has an incentive to postpone extraction. As a consequence, the price of the resource is U-shaped, i.e. the price first decreases and then increases. We also generate insights concerning the impact of technological progress in the recycling sector when the primary sector is monopolistic. We show that an increase in the recyclability rate has first a positive and then a negative effect on the price of the resource. This suggests that technological progress in the recycling sector can be detrimental to consumers in the short-run and beneficial for them in the long run. We also show that the exhaustion date increases with both the initial stock and the recyclability rate of the resource.

The present paper is related to the literature dealing with durable resources. Durable resources differ from other resources (among which, recyclable resources) in that their demands are for quantities of stock in circulation rather than for flows of production. Producers of durable goods use similar production technologies and consumers typically consume durable goods for a certain period of time. Primary and recycled goods, in contrast, are typically produced using two different production processes. As stated in [START_REF] Levhari | The pricing of durable exhaustible resources[END_REF], demand is a stock relationship for durable resources while it is a flow relationship for recyclable resources. Our results highlight important differences between recyclable and durable resources. We show that the two assumptions lead to quite different results. Indeed, [START_REF] Levhari | The pricing of durable exhaustible resources[END_REF] show that, in the case of a competitive industry that produces a durable good, the price of the resource first decreases and may increase thereafter. In contrast, we find that the price of the resource is always increasing in the context of a competitive extraction sector.

Our model reveals a new reason why the price of a resource may decrease: a firm with market power in the extraction sector will (strategically) choose to delay extraction in order to reduce the opportunities for recycling. This differs from the case of durable resources [START_REF] Levhari | The pricing of durable exhaustible resources[END_REF], in which the initial price decrease is due to the growing amount of the durable good in circulation and the resulting decrease in demand (the price of the resource increases at some point in time only if the durable good depreciates).

There are other explanations for U-shaped price profiles of exhaustible resources. [START_REF] Pindyck | The optimal exploration and production of nonrenewable resources[END_REF] shows that this may occur when exploration and reserve accumulation are taken into account. In a model with exogenous technical change and endogenous change in grades, [START_REF] Slade | Trends in natural-resource commodity prices: An analysis of the time domain[END_REF] also finds that U-shaped price profiles may occur. These studies do not consider the possibility of recycling.

The environmental advantages of recycling have long been recognized in the economic literature [START_REF] Smith | Dynamics of waste accumulation: Disposal versus recycling[END_REF][START_REF] Weinstein | Use patterns for depletable and recycleable resources[END_REF][START_REF] Hoel | Resource extraction and recycling with environmental costs[END_REF]. 4 The present paper is related to literature that focuses on the possible adverse social consequences of market power in the extraction sector. A series of papers consider a monopolistic extraction sector facing a competitive recycling sector [START_REF] Gaskins | Alcoa revisited: The welfare implications of a secondhand market[END_REF][START_REF] Swan | Alcoa: The influence of recycling on monopoly power[END_REF][START_REF] Martin | Monopoly power and the recycling of raw materials[END_REF][START_REF] Suslow | Estimating monopoly behavior with competitive recycling: An application to alcoa[END_REF][START_REF] Hollander | Monopoly and the preemption of competitive recycling[END_REF] and show that, despite the presence of a competitive recycling sector, the extraction firm maintains (at least some of) its monopoly rents.5 However, none of these papers show that market power in the primary sector can result in a U-shaped price profile of the non-renewable resource.

The remainder of the paper is structured as follows. Section 2 introduces the model in which the monopolist of the exhaustible resource faces a competitive recycling sector. Section 3 studies price dynamics in the case of a competitive primary producer sector. Section 4 focuses on the qualitative properties of the optimal path in the case of a monopolistic primary producer. The main conclusions from this analysis are gathered in section 5.

The Model

The economy produces a quantity Q of a consumption good. The consumption good can be produced from a non-renewable resource or from recycled materials. For simplicity we assume that the virgin and recycled materials are perfect substitutes. The primary sector faces a competitive sector of recycling firms.

Non-renewable resource and scrap dynamics

Let X (t) ≥ 0 be the residual stock of virgin resource at time t, X 0 be the initial stock, with X (0) ≡ X 0 > 0, and x (t) ≥ 0 be the extraction rate at time t, so that:

Ẋ (t) = -x (t) .
(1)

The unit cost of extraction of the virgin resource is assumed to be zero. Let S (t) ≥ 0 be the stock of (recyclable) scrap at time t, with a zero initial stock, S (0) = 0. Let r (t) ≥ 0 be the quantity of recycled materials marketed at time t, so that the total quantity consumed at time t is then Q (t) = x (t) + r (t). Let α ∈ [0, 1] be the proportion of the resource that is not recycled and becomes recyclable scrap. The dynamics of the scrap material writes

Ṡ (t) = α (Q (t) -r (t)) or, Ṡ (t) = αx (t) , (2)
where α represents the recyclability rate of the non-renewable resource.

The recycling sector

The recycling sector is assumed to be competitive and the marginal cost of recycling to be a decreasing function of the stock of scrap and an increasing function of the quantity of recycled materials, c (S, r).

In equilibrium in the recycling sector, the price of the consumption good must equal the marginal cost of recycling:

p (Q (t)) = c (S (t) , r (t)) . (3)
The primary sector

The discounted profits in the primary sector, with discount rate δ ≥ 0, are given by:

+∞ 0 e -δt pxdt, (4)
In the following, we will consider two polar cases: the case of a competitive primary sector and the case of a monopolistic primary sector. In the case of a competitive primary sector, resource owners behave as price takers, and they consider the price of the resource to be a function of time, p ≡ P (t). In the case of a monopolistic primary sector, the owner of the resource takes into account how extracted quantities affect the total quantity of material supplied (virgin as well as recycled) and the effect of this supply on the price of the resource, i.e. p ≡ p (Q (t)).

Competitive primary sector

In this section, we consider the case of a competitive primary sector. In this case, producers take the price, P , as well as the total quantity, Q, as given. They consider the following problem:

M ax {x(t),t≥0} +∞ 0 e -δt P (t) x (t) dt, (5)
s.t. Ẋ (t) = -x (t) , (6)
X (t) ≥ 0, x (t) ≥ 0. ( 7
)
The Hamiltonian and the Lagrangian for this optimal control problem are as follows:6

H = P x + λ X (-x) , (8)
L = H + µ X X + µ x x, (9)
where λ X is the co-state variable associated with the stock X, and, µ X , µ x are the multipliers associated with the non-negativity constraints X ≥ 0, and x ≥ 0. The competitive solution is found by solving problem (5) subject to ( 6) and ( 7) and then using (3), (2) and P (t) = p(Q(t)), ∀t, to determine the recycling level and the market clearing price. The Maximum Principle requires that the following conditions hold:

∂L ∂x = P -λ X + µ x = 0, (10) λX = δλ X - ∂L ∂X = δλ X -µ X , (11)
x ≥ 0,

µ x ≥ 0, µ x x = 0, ( 12
) X ≥ 0, µ X ≥ 0, µ X X = 0, (13)
and one transversality constraint is given by:

lim t→+∞ e -δt λ X (t) X (t) = 0, (14)
When both extraction and residual stock levels, x(t) and X(t), are strictly positive, we have µ x = 0 and µ X = 0. Substituting these respective values into (10) and ( 11) yields:

P -λ X = 0, ( 15
) λX = δλ X (16)
Differentiating ( 15) with respect to time gives: λX

λ X = Ṗ P (17)
From ( 16), we have: λX

λ X = δ (18)
The combination of ( 17) and ( 18) yields:

Ṗ P = δ (19)
Equation ( 19) is known as Hotelling's rule: the price of the resource grows at the discount rate. Since δ > 0, the price of the resource increases over time. This result reveals a major difference between recyclable goods and durable goods. The price of a durable exhaustible resource decreases with the amount of the durable good in circulation. It is then either always decreasing or U-shaped when the resource extraction sector is competitive [START_REF] Levhari | The pricing of durable exhaustible resources[END_REF].

Monopolistic primary sector

In this section, we consider the case of a monopolistic primary sector. We derive several qualitative properties regarding the optimal time path of virgin resource extraction, the equilibrium recycling quantity, and the price of the consumption good. We then provide comparative statics in order to highlight the role of recycling technology in the dynamics of the price.

For simplicity, we assume in the rest of the paper that the inverse demand for the consump-tion good and the cost of recycling are linear,

p (Q (t)) = 1 -Q (t) and c (S (t) , r (t)) = 1 -b -β (S (t) -r (t)) , (20)
with β > 0 and b ∈ (0, 1). Parameter b is a measure of the added value of recycled material compared to scrap. Solving the recycling sector equilibrium condition (3), we characterize the equilibrium quantity of recycled material at time t as follows:

r (t) = b + βS (t) -x (t) 1 + β . ( 21
)
Thus, the quantity of recycled material at time t increases with the quantity of scrap and decreases with the quantity of extracted resource. This result is quite intuitive. Since recycling relies on scrap, the higher the stock of scrap, the larger the recycling firms' production. Recycling at time t decreases with the quantity of virgin product sold at time t because recycled and virgin products are strategic substitutes. In the following, we assume that the right hand side of ( 21) is nonnegative.

The current value Hamiltonian H and Lagrangian L are defined as follows:

H = p (Q) x + λ X (-x) + λ S (αx) , (22)
L = H + µ X X + µ S S + µ x x, (23)
where λ X and λ S are the co-state variables associated with the stocks X and S, and µ X , µ S , µ x are the multipliers associated with the non-negativity constraints X ≥ 0, S ≥ 0, and x ≥ 0. The total quantity writes

Q = x + r = [b + β (S + x)] / (1 + β).
The necessary conditions include:

∂L ∂x = β 1 + β p (Q) x + p (Q) -λ X + αλ S + µ x = 0, ( 24
) λX = δλ X - ∂L ∂X = δλ X -µ X , (25) λS
= δλ S - ∂L ∂S = δλ S -µ S - β 1 + β p (Q) x, (26)
x ≥ 0,

µ x ≥ 0, µ x x = 0, ( 27
) X ≥ 0, µ X ≥ 0, µ X X = 0, ( 28
) S ≥ 0, µ S ≥ 0, µ S S = 0, (29)
and two transversality conditions:

lim t→+∞ e -δt λ X (t) X (t) = 0, ( 30
) lim t→+∞ e -δt λ S (t) S (t) = 0, (31)
and S (0) = 0 and X 0 > 0 are given.

In the rest of the paper we focus on solutions in which the exhaustion date of the virgin resource, T * , is finite, that is T * < +∞.7

Optimal extraction path, recycling path and price dynamics

Full resolution of the monopolist's programme yields the extraction and recycling paths as well as the dynamics of the price of the consumption good. We first show that the optimal extraction and the recycling paths are monotonic.

Proposition 1 [Optimal Path]: The optimal extraction path is such that the quantity of extracted material decreases while the quantity of recycled material increases over time: ẋ * (t) ≤ 0 and ṙ * (t) ≥ 0.

All proofs can be found in the appendix.

Proposition 1 states that the optimal level of extraction decreases through time. This result is in line with the standard Hotelling model. Indeed, the extracting firm discounts time, choosing to extract more of the resource today and less tomorrow. The quantity of marketed recycled material, in contrast, increases over time. The intuition of these results is as follows. The stock of scrap increases over time, which reduces the unit cost of recycling. This, in turn, provides incentives for recycling firms to increase their production. At the same time, the quantity of extracted material decreases, causing the market price of the resource and the level of recycling to increase.

The price dynamics of the consumption good is established in the following proposition for the case in which scrap material has a low level of recyclability.

Proposition 2 [Price Dynamics]: The price of the final good is never decreasing if the recyclability rate is sufficiently low. Formally, there exists α > 0 such that ∂p * /∂t ≥ 0 for all t if α ≤ α.

Proposition 2 states that the standard result of an increasing resource price holds if the level of recyclability is low. In this case, the existence of the recycling sector has a limited impact on the optimal choice of the monopolistic firm. We show in the following Proposition that this result no longer holds when the level of recyclability is sufficiently large.

Proposition 3 [Non Monotonic Price]: If the recyclability rate is sufficiently large, then the price of the final good first decreases and then increases. Formally, if α > α there exists

0 < t < T * such that ∂p * /∂t < 0 if t ∈ [0, t) and ∂p * /∂t ≥ 0 if t ∈ [ t, T * ].
Proposition 3 states that, when the recyclability rate is sufficiently large, then the optimal price path is U-shaped. The intuition behind this result lies in the fact that the monopolistic firm has an incentive to postpone extraction of the virgin resource. In delaying extraction, the monopolist delays the increase in the stock of scrap, which in turn delays the reduction of recycling costs. This results in a delayed increase in the quantity of marketed recycled materials.

Propositions 2 and 3 can be illustrated using the following numerical example: Numerical Example A: Let X 0 = 1, β = 1 and δ = 0.02. Figure 1 shows the price path over time for each value of α.

The Role of Recyclability

In this section, we focus on the influence of the recyclability rate on the exhaustion date, the extraction rate, and the price of the consumption good.

We first show that the exhaustion date increases with the availability of the virgin and recycled materials.

Proposition 4 [Exhaustion date and Recyclability]: The optimal exhaustion date increases with both the initial stock and the recyclability rate of the resource. Formally:

∂T * ∂X 0 > 0 and ∂T * ∂α > 0
Proposition 4 states that the date of exhaustion of the resource is delayed with the initial stock, which is intuitive. It also states that the exhaustion date increases with the recyclability rate of the resource. This is due to the fact that a higher recyclability rate provides stronger incentives for the monopolist to delay extraction of the virgin resource.

In the next proposition, we focus on the effect of an increase in the recyclability rate on the optimal extraction and recycling paths.

Proposition 5 [Extraction and Recyclability]: Early extraction decreases while later extraction increases with recyclability. Formally, there exists a date 0 < t < T * such that ∂x * ∂α < 0 for t ∈ [0, t) and ∂x * ∂α ≥ 0 for t ∈ t, T * .

Proposition 5 shows that when the level of recyclability of the resource increases, extraction is delayed. Thus, extraction first decreases and then increases as the recyclability rate increases. The intuition of this result is as follows. When the virgin resource is not exhausted (X > 0), the dynamics of the shadow price of the virgin resource follows Hotelling's rule. This (shadow) price grows at a rate equal to the discount rate, λX /λ X = δ > 0. This means that the extracting firm has incentives to extract the resource early. However, unlike in a standard Hotelling model, the extracting firm in our model also faces the recycling sector. When there is a stock of scrap (S > 0), the dynamics of the shadow price of the scrap stock is driven by the following differential equation: λS = δλ S + β 1+β x. If there is no extraction (x = 0), then the Hotelling rule holds for the stock of scrap, λS /λ S = δ > 0, and the owner of the virgin resource has an incentive to delay extraction in order to maintain a small stock of scrap. If the level of extraction is positive, β 1+β x > 0, there is a tendency for the shadow price of scrap to increase. This reinforces the owner's incentives to delay extraction. Thus, the higher the recyclability rate, the stronger the incentives to postpone extraction. Since the resource is exhausted in finite time and the initial stock is fixed, extraction will increase with the recyclability rate at some point in time. This result can be illustrated in the following numerical example.

Numerical Example B: Let X 0 = 1, β = 1 and δ = 0.02. Figure 2 depicts the optimal extraction path for different values of α. We can now derive comparative static results regarding the effect of an increase in the recyclability rate on the dynamics of the consumption good price.

Proposition 6 [Price and Recyclability]: The price first increases and then decreases with the recyclability rate . Formally, there exists a date 0 < t < T * such that ∂p * ∂α > 0 for t ∈ [0, t ) and ∂p * ∂α ≤ 0 for t ∈ t , T * .

A consequence of Proposition 6 is that a higher recyclability rate is not always beneficial to consumers. In the short-run (t < t ), consumers must pay a higher price for the consumption good. However, in the long run (t < t), consumers pay a lower price for the consumption good (thanks to the increased supply). The intuition underlying this result is as follows. The price of the final good negatively depends on the marketed quantity of virgin and recycled materials. When the recyclability rate increases, the stock of scrap tends to increase, which leads to an increase in recycling. However, because the stock of scrap increases with extraction by a factor α < 1, the increase in recycling cannot compensate for the short-run decrease in extraction, and the marketed quantity of the consumption good decreases. This explains why the price of the final good increases in the short run. Approaching the exhaustion date, an increase in the recyclability rate increases both the stock of scrap and the level of extraction. This explains why the price of the final good decreases in this period. [START_REF] Gaskins | Alcoa revisited: The welfare implications of a secondhand market[END_REF] also finds that the price of the resource increases in the short run in the presence of a recycling sector. However, in contrast with the present analysis, he focuses on the steady state and finds that the steady state price of the resource is close to the competitive price. This result can be illustrated in the following numerical example.

Numerical Example C: Let X 0 = 1, β = 1 and δ = 0.02. Figure 3 depicts the optimal price path for different values of α.

Conclusion

Recycling rare earths and Phosphorus appears to be a promising strategy to increase the supply of these important exhaustible resources. Although the production of the these resources is concentrated in a few countries, reserve estimates suggest that the supply of rare earths will become more competitive while the supply of Phosphorus will become less competitive in the future.

Motivated by these two examples, we built a model of resource extraction in which the primary sector faces a recycling sector. We have shown that, when the primary sector is competitive, the price of the recyclable resource increases through time. This result stands in contrast to durable resources, for which the optimal price path is either decreasing or U-shaped [START_REF] Levhari | The pricing of durable exhaustible resources[END_REF]. We have also shown that, when the primary sector is monopolistic, the price of the recyclable resource may be U-shaped when the recyclability rate is sufficiently large. This occurs because the primary producer has incentives to delay the extraction of the resource in order to delay recycling. We have also shown that the price of the resource first increases and then decreases as the recyclability rate increases. Our results suggest that market power in the primary sector may lead to phases in which the price of the resource decreases. They also suggest that consumers may oppose progress in recycling technologies if they sufficiently weight the negative short term effect at the expense of the positive long term effect. Differentiating (43) with respect to time, we find -2 ẋ -Ṡ -δc 1 e δt + α λS = 0.

(45)

Using ( 43) and ( 45), we find

-2 ẋ -Ṡ -δc 1 e δt -δ a -2x -S -c 1 e δt + α λS -δλ S = 0, (46)
or,

-2 ẋ -Ṡ + δS + (α + 2δ) x -δa = 0, (47)
Differentiating ( 2) with respect to time, we obtain

S = α ẋ. (48)
Substituting ( 48) and ( 2) into (47), and rearranging, we have

S -δ Ṡ - αδ 2 S + αδ 2 a = 0. ( 49
)
Solving for the stock of scrap S we find

S = c 2 e γ + t + c 3 e γ -t + a, (50)
where

γ + = δ+ √ δ(2α+δ) 2 and γ -= δ- √ δ (2α+δ) 2
. Differentiating (50) with respect to time, we obtain

Ṡ = γ + c 2 e γ + t + γ -c 3 e γ -t . (51)
Using (2), we have

x = γ + α c 2 e γ + t + γ - α c 3 e γ -t . (52)
Substituting ( 52) into (44), we obtain

λS -δλ S = γ + α c 2 e γ + t + γ - α c 3 e γ -t . (53)
Solving for the shadow price of the stock of scrap λ S we find

λ S = De δt + γ + α (γ + -δ) c 2 e γ + t - γ - α (δ -γ -) c 3 e γ -t . (54) Using X 0 -X (t) = t 0
xdt and integrating (52) between 0 and t, we find

X 0 -X (t) = 1 α c 2 e γ + t -1 + c 3 e γ -t -1 . (55)
Now consider the second phase in which t ≥ T . We have x (t) = 0 = X (t) and S (t) > 0.

Using (38), we have µ S = 0. Condition (35) writes

λS = δλ S , (56)
and then

λ S = c 5 e δt , (57)
where c 5 is a constant to be determined later.

Notice that Ṡ = αx = 0, and then

S = c 4 , (58)
where c 4 is a constant to be determined later. Using ( 58) and ( 57), transversality constraint (40) becomes

c 4 c 5 = 0. ( 59
)
Assume c 5 = 0. Then, using (58) at t = T , we have S (T ) = c 4 = 0. Combining ( 50) and ( 55) and taking t = T , we have αX 0 = S (T ). Hence, we must have X 0 = 0, which is false. We conclude that c 5 = 0. Thus, for t ≥ T , we have

λ S = 0. (60)
In order to solve for c 1 , c 2 , c 3 , c 4 , D and T , we focus on solutions such that λ S is continuous. Using ( 54) and (60) at t = T , we obtain

De δT + γ + α (γ + -δ) c 2 e γ + T - γ - α (δ -γ -)
c 3 e γ -T = 0. (61)

Using x (T ) = 0 and (52), we have

γ + c 2 e γ + T + γ -c 3 e γ -T = 0. ( 62
)
Using X (T ) = 0 and (55), we have

αX 0 = c 2 e γ + T -1 + c 3 e γ -T -1 . (63)
Using (50) at t = 0, we have

S 0 = c 2 + c 3 + a. (64)
Using ( 43) and (50) at t = T , we obtain c 2 e γ + T + c 3 e γ -T + c 1 e δT = 0. ( 65)

Using ( 50) and ( 58) at t = T , we have

c 4 = c 2 e γ + T + c 3 e γ -T + a. (66)
Solving for c 1 , c 2 , c 3 , c 4 , D and T from conditions ( 61)-( 66), we obtain

c 1 = γ + -γ - γ + e γ + T -γ -e γ -T a -S 0 , c 2 = - γ -e γ -T γ + e γ + T -γ -e γ -T S 0 -a , c 3 = γ + e γ + T γ + e γ + T -γ -e γ -T S 0 -a , c 4 = S 0 -a (γ + -γ -) e δT γ + e γ + T -γ -e γ -T + a, D = a -S 0 α γ + -γ - γ + e γ + T -γ -e γ -T ,
and the exhaustion date T * is implicitly characterized by :

αX 0 = a -S 0 1 - γ + -γ - γ + e γ + T * -γ -e γ -T * e δT * .
(67)

We conclude that the optimal extraction path is, for t ∈ [0, T ] :

x * (t) = a -S 0 δ 2 e γ + T * e γ -t -e γ -T * e γ + t γ + e γ + T * -γ -e γ -T * , (68)
the stock of scrap is, for t ∈ [0, T ],

S * (t) = a -S 0 1 -γ + e γ + T * e γ -t -γ -e γ -T * e γ + t γ + e γ + T * -γ -e γ -T * , (69) and the market price, for t ∈ [0, T ],

p * (t) = S 0 + a -S 0 γ + -γ - 2 e γ + T * e γ -t + e γ -T * e γ + t γ + e γ + T * -γ -e γ -T * . (70)
Since γ + > 0 > γ -, the extraction level x * (t) characterized in (68) decreases through time, while the stock of scrap increases through time, Ṡ * (t) = αx * (t) ≥ 0. Recycling is given by

r * (t) = b β + a   1 - γ + + δ 2β e γ + T * e γ -t -γ -+ δ 2β e γ -T * e γ + t γ + e γ + T * -γ -e γ -T *   , (71)
and increases through time.

Proof of Proposition 2: From (70), we know that the price of the consumption good is

p * (t, α) = a 2 δ (2α + δ) e γ + T * e γ -t + e γ -T * e γ + t γ + e γ + T * -γ -e γ -T * , (72)
The sign of the derivative with respect to time is given by

∂p * ∂t ∝ γ -e γ + T * e γ -t + γ + e γ -T * e γ + t , (73)
which is positive if and only if

t ≥ T * + 1 γ + -γ -ln 1 - δ γ + . (74)
Hence, ∂p * ∂t ≥ 0 for all t ∈ [0, T ] if and only if

T * ≤ 1 γ + -γ -ln γ + γ + -δ . ( 75
)
We know from Proposition 3 that the left hand side of condition ( 75) is increasing with α. The derivative of the right hand side with respect to γ + is -1

(2γ + -δ) 2 2 ln γ + γ + -δ + δ (2γ + -δ) γ + (γ + -δ) < 0. ( 76
)
It is decreasing with α. When α goes to 0, γ + goes to δ and the right hand side in (75) goes to +∞. A first order approximation of (67) at α = 0 leads to X 0 a δ + 1 -δT * e δT * 1. The solution of this equation is T * < +∞ because the left hand side is X 0 a δ + 1 > 1 at T * = 0. The solution of the equation increases up to T * = X 0 a and then decreases and goes to -∞ when T * → +∞. This concludes the proof. Proof of Proposition 3: The result directly follows from the proof of Proposition 2. Proof of Proposition 4: The optimal exhaustion date is implicitly characterized by ( 67), which can be rewritten as:

f γ + , T * , α, X 0 , δ ≡ 1 - 2γ + -δ γ + e γ + T * + (γ + -δ) e (δ-γ + )T * e δT * - αX 0 a = 0, (77)
where γ + = δ + δ (2α + δ) /2. Its derivative with respect to T * is given by

∂f ∂T * = γ + (γ + -δ) (2γ + -δ) e δT * γ + e γ + T * + (γ + -δ) e (δ-γ + )T * 2 e γ + T * -e -(γ + -δ)T * . ( 78
) Since γ + ≥ δ, we have ∂f ∂T * > 0. ( 79
)
The derivative of f with respect to γ + is given by

∂f ∂γ + = - δ e γ + T * -e -(γ + -δ)T * + γ + e γ + T * -(γ + -δ) e -(γ + -δ)T * (γ + -δ) T * (γ + -δ) 2 γ + γ + -δ e γ + T * + e -(γ + -δ)T * 2 e δT * . (80) Since γ + > δ, we have ∂f ∂γ + < 0. ( 81
)
The derivative of f with respect to α is

∂f ∂α = -X 0 /a < 0 (82)
Using ( 77) and the implicit function theorem, we have:

∂T * ∂X 0 = - ∂f /∂X 0 ∂f /∂T = α/a ∂f /∂T > 0. ( 83
)
Using the implicit function theorem again, and ∂γ + /∂α > 0, (81), ( 79) and ( 82), the derivative of the exhaustion date with respect to the recyclability rate is such that:

∂T * ∂α = - (∂f /∂γ + ) (∂γ + /∂α) + ∂f /∂α ∂f /∂T > 0. ( 84
)
Proof of Proposition 5: The proof proceeds in three steps. We first show that the growth rate of extraction is decreasing through time. Second, we show that the growth rate is increasing with the recyclability rate. We then combine these properties in order to prove the result. Differentiating (68), we can write the growth rate of extraction:

τ γ + , T * ≡ ẋ * (t) x * (t) = - (γ + -δ) e (2γ + -δ)(T * -t) + γ + e (2γ + -δ)(T * -t) -1 . ( 85
)
The derivative of the growth rate with respect to T * is

∂τ ∂T * = γ + + δ + (2γ + -δ) γ + e (2γ + -δ)(T * -t) -1 2 δe (2γ + -δ)(T * -t) > 0. ( 86
)
The derivative of the growth rate with respect to γ + is

∂τ ∂γ + = δ G (t)
e (2γ + -δ)(T * -t) -1 2 ,

where G (t) = 2 (T * -t) (2γ + -δ) e (2γ + -δ)(T * -t) + 1 -e 2(2γ + -δ)(T * -t) . Notice that G (t) = -2 (T * -t) (2γ + -δ) 2 e (2γ + -δ)(T * -t) < 0 and G (T * ) = 0. Hence G (t) > 0 and then ∂τ ∂γ + > 0.

(87)

We know from Proposition 4 that T * increases with α and we also know that γ + increases with α. Using ( 86) and (87), we conclude that

dτ dα > 0. ( 88
)
In other words, we have ∂ 2 ln (x) ∂t∂α > 0.

(89)

Hence ln x has the single crossing property with respect to time and the recyclability rate.

According to Proposition 4, the exhaustion date increases with recyclability, ∂T * ∂α > 0. Hence recyclability necessarily increases extraction when time approaches the exhaustion date. Since the initial stock does not depend on the recyclability rate, recyclability necessarily decreases extraction at some point in time. Thanks to the single-crossing property, there exists a date 0 < t < T * such that ∂x * ∂α < 0 ⇐⇒ t < t.

Proof of Proposition 6:

The stock of scrap can be rewritten as follows:

S * (t) = F γ + , T * = a 1 -γ + e γ + (T * -t) + (γ + -δ) e -(γ + -δ)(T * -t) γ + e γ + T * + (γ + -δ) e -(γ + -δ)T * e δt .

The derivative of this function with respect to T * is:

∂F ∂T * = -a
γ + (γ + -δ) (2γ + -δ) e δT -γt -e δT +(γ + -δ)t

γ + e γ + T * + (γ + -δ) e -(γ + -δ)T * 2 e δt ≥ 0, (90)

and its derivative with respect to γ + is ∂F ∂γ + = -a -t (γ + ) 2 e γ + (2T * -t) + (2T * -t) (γ + ) 2 -δ e δT -γ + t -2T * (γ + ) 2 -δ e δT +(γ + -δ)t

γ + e γ + T * + (γ + -δ) e -(γ + -δ)T * 2 e -δt ≥ 0.

(91) Since T * and γ + both increase with α, using ( 90) and ( 91) we conclude that S * increases when α increases. Now consider the growth rate of the price. Using (70), it can be written as follows:

ṗ * p * = γ + -(γ + -δ) e (2γ + -δ)(T * -t) 1 + e (2γ + -δ)(T * -t) ≡ H γ + , T * .

(92)

Its derivative with respect to γ + is given by:

∂H ∂γ + =
1 -e 2(2γ + -δ)(T * -t) -2 (2γ + -δ) (T * -t) e (2γ + -δ)(T * -t)

1 + e (2γ + -δ)(T * -t) 2

≤ 0. ( 93
)
Since H is also decreasing with T * and both T * and γ + increase with α, we conclude that

∂ 2 ln p * ∂t∂α < 0. ( 94
)
This means that ln p * has the single-crossing property with respect to t and α. We know that p * (T * ) = 1 -S * (T ) -x * (T ) = 1 -αX 0 .

Then p * (T * ) decreases with α. Moreover, we have p * (0) = 1 -S * (0) -x * (0)

= 1 -x * (0) .

Using Proposition 5, we know that x * decreases with α at t = 0. Hence p * increases with α at t = 0. Using the single-crossing property (94), we conclude that there exists t ∈ (0, T * ) such that ∂p * /∂α > 0 ⇐⇒ t ∈ [0, t ).

Figure 1 :

 1 Figure 1: The price path is either increasing or U-shaped

Figure 2 :

 2 Figure 2: Optimal extraction path for various levels of the recyclability rate

Figure 3 :

 3 Figure 3: Comparative statics: The price first increases and then decreases with α

Alonso et al. (2012) predict that the demand for rare earths will increase by 5 to 9 percent per annum until

According to EFMA (2000) and[START_REF] Steen | Phosphorus availability in the 21st century: management of a nonrenewable resource[END_REF], the demand for phosphorus may increase by as much as 50 to 100% by 2050 with increased global demand for food.2 After the 2008 peak in demand due to the pro-biofuel policy in the U.S. and to the 150% increase in the export tariff on phosphate in China, Phosphate rock reserve estimates for Morocco and Western Sahara increased from 6 to 50 billion tons, while reserve estimates remained below 6 billion and 2 billion tons for China and other countries (U.S.A, South Africa, Jordan, and Russia), respectively (USGS, 2011).3 A new German ordinance (AbfKlärV) will make phosphorus recovery obligatory for most sewage sludge.It has been notified to the European Commission and may enter into force in 2018. See http://www.phosphorusplatform.eu/scope-in-print/scope-in-press/1327-german-sludge-p-recycling-ordinancenotified-to-europe

[START_REF] Alonso | On the Dynamics of Recycling and Natural Resources[END_REF] provide a model that takes into account the interactions of the material composition of output and waste as potentially recyclable products.

[START_REF] Gaudet | Recycling redux: A Nash-Cournot approach[END_REF] consider the possibility of imperfect competition in the recycling sector. They show that, when primary and secondary production decisions are made simultaneously, the presence of the recycling sector may increase the market power of the primary producer.Weikard and Seyhan (2009), motivated by the case of Phosphorus, consider a model of competitive resource extraction and the possibility of saturated demand (i.e. taking into account the possibility that soil can become saturated with Phosphorus).

We drop the time index when there is no possible confusion.

A sufficient condition is 1 -βαX 0 1-b+β > 0. This condition can be derived from equation (67) in the proof of Proposition 1.

Appendix

Proof of Proposition 1: Using the equilibrium recycling condition (21) and substituting, we have p (Q) = β 1+β (a -x -S), where a = (1 -b + β) /β. The maximization problem then formally writes:

subject to ( 21), (1), (2), X, S, x ≥ 0, X 0 and S 0 given. Thus, for the new problem, the necessary conditions include:

x ≥ 0,

and two transversality constraints:

lim t→+∞ e -δt λ S (t) S (t) = 0. (40)

Let us assume that the solution is such that x (t) > 0 and X (t) > 0 over [0, T ) and x (t) = X (t) = 0 for t ≥ T . We also assume that S (t) > 0 for all t > 0. First consider the first phase in which t ∈ [0, T ). Since x (t) > 0, X (t) > 0 and S (t) > 0, using (36), (37), and (38), we have µ x = µ X = µ S = 0. Then (34) writes λX = δλ X ,

and then

where c 1 is a constant to be determined later. (44)


Summary

In growing animals, feed efficiency is often described from the ratio between growth and corresponding feed consumption during a certain period. Feed efficiency is determined by the efficiency at digestive level and by the efficiency at a metabolic level. The efficiency measured at digestive level is difficult to assess in a large number of animals. In pigs, its measurement requires to house animals in a digestibility cage during a prolonged period to measure feed intake and totally collect the faeces that should be analysed for the nutrient contents. Because of this, this method cannot be used in selection farms of breeders. The variability in digestive ability has always been ignored whereas it has been shown to exist when animals are fed a diet with a high fibre content. The objectives of this deliverable are to identify an alternative methodology to characterize the digestive ability in pigs. For this purpose, an experiment was conducted at UMR Pegase, INRA, France with 63 pigs from Large White, Piétrain, and Duroc breeds that were fed alternatively two diets differing in crude fibre content (3.1 and 8.5% of dry matter) during four periods of three weeks each. The diets were supplemented with three molecules that may be used as indigestible marker: silicone oil, the plastic resin Kynar®, and polyethylene glycol. At each period, digestibility was measured using the gold standard (i.e., measurement of feed intake, total collection of faeces, and dedicated lab analyses to calculate the digestibility coefficients of dry matter, organic matter, energy, N, and crude fibre). At the end of each period, a sample of faeces was also collected directly from the rectum of the pigs. All samples were also analysed by Near InfraRed Spectroscopy (NIRS). The digestive ability of pigs was variable during the growing period, especially when the diet contained a high level of fibre and the digestibility of this type of diet increased when body weight increased. The utilization of indigestible markers did not give satisfying results, because of low recovery in the faeces (plastic resin or polyethylene glycol) or because of large variability in the recovery (silicone oil). The NIRS prediction of digestibility coefficients from a sample collected directly in the faeces was adequate (bias of digestibility coefficients for dry matter, organic matter, energy, and N close to 0%) when pigs are heavier than 60 kg and when they are fed a diet with a high fibre content. This type of diet is susceptible to challenge the digestive ability of pigs and allows identifying specifically the poor digesters in a population using NIRS. The method of collecting a single sample of faeces directly from the rectum of the pig and analysed by NIRS can be used in selection farms of breeders to give information on the digestive ability of individual pigs in a population.

Introduction

In growing animals, feed efficiency is often described from the ratio between growth and corresponding feed consumption during a certain period. Nevertheless, this ratio largely depends on the feed composition and the way animals use nutrients from the diet for growth. Feed efficiency for growth can be further partitioned in a digestive efficiency and a metabolic efficiency. Whereas the variation in metabolic efficiency because of growth characteristics (e.g., composition of bodyweight gain, ratio between viscera and body) has been characterized, the variation in digestive efficiency between animals from different breeds or from different genetic lines within a breed has received little attention. Recently, in pigs, it has been shown that this variation exists and that differences in energy digestibility between animals from different sires can differ up to 2 points of digestibility when pigs are fed a fibrous diet [START_REF] Noblet | Evidence of genetic variability for digestive efficiency in the growing pig fed a fibrous diet[END_REF].

In pigs, the gold standard to measure digestibility of nutrients and energy in animals is rather complex because it requires the adaptation of the animal to the diet during at least 10 days, followed during at least 7 days for accurate measurements of feed intake and simultaneous total collection of faeces, that should not be contaminated by urine. Consequently, experimental animals are most of the time of male gender and housed individually in a digestibility cage (at least during the collection period) to allow for the physical separation between faeces and urine. Apart from the collection of faeces, laboratory analyses involve additional steps, extensive labour, various laboratory equipment, and extensive associated costs. Only a limited number of animals can thus be included in digestibility studies, and this method cannot be used as a large-scale phenotyping tool to be used in selection schemes of breeders.

From a technical, laboratory point of view, predictive analyses, such as those based on Near InfraRed Spectroscopy (NIRS), are promising to reduce the time of analysis because a large variety of sample characteristics can be predicted from the NIRS spectra of a single measurement, once calibration equations have been established. First results in using this technique to predict nutrient and energy contents in faeces have been recently reported in pigs [START_REF] Bastianelli | Predicting feed digestibility from NIRS analysis of pig faeces[END_REF][START_REF] Schiborra | Using faecal near-infrared spectroscopy (FNIRS) to estimate nutrient digestibility and chemical composition of diets and faeces of growing pigs[END_REF] but they still require the measurement of feed intake and total collection of faeces to calculate satisfactorily the digestibility of nutrients and energy. Indigestible markers can be used as an alternative to the necessity to measure feed intake and total collection of faeces to assess digestibility. It involves the analysis of the marker content in diet and in a sample of faeces collected directly from the rectum of the pigs. Nevertheless, the markers currently used (e.g., titanium dioxide, chromium oxide, or ytterbium oxide) require additional laboratory analyses that are expensive and timeconsuming. Analyses of these markers cannot be done by a prediction from NIRS because these markers do not have a spectral trace in NIRS wavelengths (from 400 to 2500 nm). Ideally, the simplest method to estimate the ability of a pig to digest energy and nutrient will be based on a sample of faeces collected directly from the rectum and analysed by NIRS.

One objective in task 2.2 was to develop a method to assess the ability of a pig to digest energy and nutrients from a NIRS prediction based on a sample of faeces collected from the rectum of the animal.

Materials and methods

The methodology was developed during an experiment conducted at UMR Pegase, INRA, France from September 2016 to May 2017 with 63 pigs from Large White, Piétrain, and Duroc breeds (n=21 pigs per breed) that were fed alternatively two diets differing in crude fibre content (3.1 and 8.5% of dry matter; annexe 1) during four periods of 21 days. The diets were supplemented with 0.5% of silicone oil, 0.5% of plastic resin (Kynar®), and 0.5% of polyethylene glycol as possible candidates for indigestible markers that can be analysed by NIRS. The pigs entered in the experiment at the end of the post-weaning phase (i.e., at 70 days of age). During each period, the pigs were adapted to their diet during two weeks. They were then housed individually during one week in a digestibility cage to measure their feed intake and to collect all their faeces. The dry matter (DM) content of the feed offered during this week was measured on a representative sample of diet. Refusals and spillages were totally collected at the end of the week, weighed and their DM content was measured. During the measurement week, the faeces were collected daily in buckets stored at 4°C. At the end of the week, faeces were thoroughly mixed, DM content was measured in two aliquots and a sample was freezedried and ground in a hammer meal for further lab analyses. At the end of each measurement week (i.e., on the morning of the first day after), a sample of faeces was also collected directly from the rectum of the pigs, freeze dried and ground. All samples of faeces and diets were stored at 4°C until laboratory analyses.

At the end of the experiment, samples of diet (one pooled sample per type of diet) and faeces from the total collection (one sample per pig per period) were analysed for their DM, nutrient and energy contents to calculate the digestibility coefficients of DM, nutrients and energy (refer as gold standard method). The DM content was measured from the weight loss during 24 h at 103°C. The ash content was measured from the weight of the residues after burning the sample in an oven at 550°C during 8 h. The N content was measured according to the Dumas method on an Elementar RapidN automatic analyser. The crude fibre was measured according to the Weende method, adapted to an automatic Ankom Fibersac analyser. The gross energy content was measured on an IKA adiabatic calorimeter. Samples of faeces from the total collection period and those collected directly from the rectum of the pigs were also analysed by NIRS using a Bruker MPA equipment (three spectra per sample, each being the average of 64 measurements performed with a 16 cm -1 resolution in a quartz cup at ambient temperature). Calibration equations for indigestible marker, nutrient concentrations and digestibility coefficients were then developed using the manufacturer's software with PLS regression using the data obtained from total collection of faeces and lab analyses as results to be predicted.

Results

Variability of digestibility between breeds, periods and diets

The main results obtained during the experiment are given in Table 2. According to the experimental design, body weight varied during successive periods and Piétrain pigs were heavier than Duroc pigs.

As pigs got older, the weight difference between breeds became larger. The digestibility coefficients of energy and nutrients were strongly affected by diet composition and the increase in fibre content resulted in an 8 to 10-point decrease in digestibility coefficients of dry matter, organic matter, energy, and nitrogen. The digestibility coefficient of crude fibre was 11 points higher with the high-fibre diet compared to the low-fibre diet. There was no difference between breeds for the digestibility coefficients of dry matter, organic matter, energy, ether extract, and crude fibre but digestibility coefficient of N was higher in the Piétrain breed, especially when they got older. Except for N and ether extract, there was a significant interaction between diet and period for the digestibility coefficients of energy and nutrients, because of an improved ability to digest fibres in the large intestine as the pigs got older. Consequently, the experimental design was adequate to provide samples of faeces with a wide range in digestibility coefficients (e.g., digestibility coefficient for energy ranged from 70 to 90%; Figure 1), that can be used to develop predictive equations of nutrient composition in faeces and digestibility coefficients in pigs.

Utilization of silicone oil, plastic resin, and polyethylene glycol as indigestible markers

Apart from the gold standard for digestibility measurements, indigestible markers can be used to assess digestibility from the concentrations of the marker (and nutrients) in the diet and in a faecal sample. The methodology assumes that the marker should be totally indigestible, which means that recovery in faeces should be close to 100%. The marker should also be excreted homogeneously with the faeces during the day. If this is the case, total collection of faeces is not required, and a faecal spot sample can be used. To check whether silicone oil, plastic resin, and polyethylene glycol can be used as indigestible markers, the diets were supplemented with 0.5% of each marker and the faeces collected during the seven days of the collection period were analysed by NIRS for their marker concentration.

For this purpose, NIRS calibration equations for faecal silicone oil, plastic resin, and polyethylene glycol concentrations were developed using the methodology proposed by Casasus and Albanel (2014). Briefly, known amounts of each marker were included in the faeces of non-experimental pigs that were not fed these markers. The faeces were then mixed thoroughly, humidified, freeze-dried, and ground. Equations for each marker were developed with very good accuracies, as reported in Table 2. The R² of prediction were all close to 100%, whereas the intercept and the slope of the relationship between theoretical and predicted concentrations of the validation dataset did not differ significantly from 0 and 1, respectively. Intercept and slope of the regression equation between predicted and theoretical concentrations of the marker.

These equations were then applied to samples of faeces obtained during the total collection period, assuming that it was possible to calculate the faecal concentrations in markers from their concentrations in the diet, feed intake, and faecal dry matter excretion. The best results were obtained for silicone oil. The total recovery of silicone oil did not differ between diets and breeds and averaged 97.4% (Table 3), which is a satisfactorily recovery for an indigestible marker. Nevertheless, the recovery was significantly lower during period 2 and the standard deviation of recovery was high (standard deviation: 20.7%). Additionally, the correlation between predicted and theoretical concentrations in faeces was not sufficient (Figure 2), mainly with the faeces obtained with pigs fed the low-fibre diet (i.e., when the concentration of the marker was the highest). The total recovery of plastic resin and polyethylene glycol differed significantly between diets and periods (Table 3) and the prediction of their faecal concentration (Figure 2) indicated that these markers were not good candidates as indigestible markers to be used to predict the digestive ability in growing pigs.

Utilization of NIRS to predict digestive ability in growing pigs

Prediction of digestibility coefficients in the faeces of growing pigs

Because the equations to predict the faecal concentrations using indigestible markers were not adequate to be used to assess the digestive ability of pigs, a database with samples of faeces from the experiment conducted in the framework of Feed-a-Gene and samples of faeces from previous experiments was constituted. In this database, 831 samples of faeces from growing pigs with values of digestibility coefficients for dry matter, organic matter, energy, and N were included. Among them, 578 samples were also characterized for digestibility coefficients of crude fibre. In the database, 280 samples of faeces were obtained from direct collection from the rectum of the pigs, whereas others originated from experiments where the digestibility coefficients were measured using the gold standard method (i.e., measurement of feed intake and total collection of faeces over 6 to 10 days). , The validation dataset included 82 samples obtained from spot-sampling in the rectum of the pig to develop the NIRS prediction equations for digestibility coefficients. The results of the calibration for prediction equations for digestibility coefficients of dry matter, organic matter, energy, N, and crude fibre are presented in Table 4. The equations to predict the digestibility coefficient of dry matter, organic matter, energy, and N are good, according to the values of the ratio between the standard deviation of laboratory results and the standard error of prediction (RPD), that are close to 3. The value of RPD for the digestibility coefficient of crude fibre is lower (1.74), which can be explained by the inaccuracy of measuring crude fibre in faeces by the Weende method. Consequently, the slope of the relationship between predicted and measured digestibility coefficient is lower for crude fibre than for other nutrients (Table 4).

The NIRS equations developed with the whole database were then applied to the samples obtained during the Feed-a-Gene experiment. When the samples harvested during the total collection period are considered (Table 5), the bias in digestibility coefficient depended on diet composition and period of collection. Additionally, the bias in predicting the digestibility coefficients of dry matter and organic matter did not depend on period of collection when pigs were fed a low-fibre diet whereas it was lower when pigs fed the high-fibre diet got older. The bias in predicting the digestibility coefficient of energy decreased when pigs got older. The bias in predicting the digestibility coefficient of N averaged 1.35% when pigs were fed the high-fibre diet during period 1, but was close to 0 during other periods and with other diets. When the samples directly taken from the rectum of the pigs at the end of the collection period are considered (Table 6), the bias in predicting the digestibility coefficients were on average three times larger than the bias obtained from samples obtained during the total collection period (Figure 3). This illustrates the heterogeneity of the faeces produced during the day, even though the samples were obtained 3 weeks after the beginning of the adaptation period to the diet. Because the bias during periods 1 and 2 was larger than that measured during periods 3 and 4, the homogeneity of the faeces or the ability of the NIRS equations to predict digestibility coefficients may be better as pigs get older. The bias was also close to 0 when the diet contained high level of fibre. This suggests that the NIRS methodology can give satisfying results when pigs are heavier than 60 kg and when they are fed a diet with a high level of fibre.

Utilization of NIRS prediction to classify pigs for their digestive ability

To detect the ability of NIRS to identify groups of pigs with a low or a high digestive ability, predicted digestibility coefficients of dry matter, organic matter, energy, N, and crude fibre from NIRS analysis of samples of faeces directly collected from the rectum of the pigs were analysed in a principal component analysis (package FactomineR in R), followed by a clustering analysis to spread the animals over six groups. The analysis was performed on samples of faeces collected at the end of periods 3 and 4.

The same approach was applied to the measured digestibility coefficients during periods 3 and 4 using the gold standard method as a reference. The clustering analysis made a clear distinction between pigs fed the high-(clusters 1 to 3) or the low-(clusters 4 to 6) fibre diet (Table 7). Cluster 1 grouped pigs fed the high-fibre diet with the highest digestibility coefficients except for the digestibility coefficient of N, whereas cluster 3 grouped the pigs with the lowest digestibility coefficients. When pigs were fed the low-fibre diet, cluster 6 grouped the pigs with the lowest digestibility coefficients whereas the digestibility coefficients of dry matter, energy, and N of pigs groups did not significantly differ between clusters 4 and 5. When the samples directly collected from the rectum and analysed by NIRS are considered, the hierarchical classification resulted in four groups of pigs fed the high-fibre diet and two groups of pigs fed the low-fibre diet (Table 8). Only two pigs were in cluster 4 and their digestibility coefficients did not differ from those of pigs in cluster 3. Cluster 1 grouped the pigs with the highest predicted digestibility coefficients whereas clusters 3 and 4 grouped the pigs with the lowest predicted digestibility coefficient. The pigs in cluster 1 can thus be considered as good digesters of a high-fibre diet because their predicted digestibility coefficients are the highest. In this group, 17 pigs of the 26 were also classified as good digesters from the hierarchical classification based on measured digestibility coefficients but nine pigs belonged to the second group of pigs (cluster 2 in Table 7). This result suggests that the specific character of the method combining spot-sampling of faeces directly from the rectum of the pig and NIRS prediction of digestibility coefficients is not good to identify the good digesters in a population of pigs. The pigs in clusters 3 and 4 exhibited the lowest digestive ability of a high-fibre diet and they were also classified as bad digesters in the first hierarchical classification shown in Figure 4A (i.e., measured digestibility coefficients from measured feed intake, total collection of faeces and laboratory analyses). Only two groups of pigs could be identified when pigs were fed the low-fibre diet and, except for N, the digestibility coefficients of these pigs did not significantly differ between the two groups. This is further illustrated in figure 4B using the digestibility coefficient of energy as an example. This result thus exhibited the inability of the method combining a spot-sampling of faeces directly from the rectum of the pig and a NIRS prediction of digestibility to classify animals for their digestive ability when they are fed a low-fibre diet. This may be explained by the limited variation in diet digestibility between animals when the diet does not challenge the ability of the pig (e.g., low level of dietary fibre).

Figure 1 .

 1 Figure 1. Effect of diet composition on digestibility coefficient of energy as affected by body weight in growing pigs (n= 246).

  .01 0.02 <0.01 0.04 0.63 <0.01 RSD, residual standard deviation; the data were analysed for the effects of diet (D), breed (B), period (P) and their interactions. Within a diet, breed, or period, LS-means with different superscript letters differ (P<0.05).

Figure 2 .

 2 Figure 2. Relationships between predicted and theoretical faecal concentrations in silicone oil, plastic resin, and polyethylene glycol in growing pigs. The intercept, slope, and R² below each chart refer to the correlation line between predicted and theoretical concentrations. The value of the intercept was tested for the difference from 0 and the value of the slope was tested for the difference from 1.

  -2.79 b -2.49 b -3.13 b 7.23 <0.01 0.92 <0.01 0.70 0.02 <0.01 RSD, residual standard deviation; the data were analysed for the effects of diet (D), breed (B), period (P), and their interactions. Within a diet, breed, or period, LS-means with different superscript letters differ (P<0.05). Page 13/18

Figure 3 .

 3 Figure 3. Prediction of dry matter (DM), organic matter (OM), energy, and N digestibility (%) from NIRS analysis of samples of faeces in growing pigs (n=245).

Figure 4 .

 4 Figure 4. Hierarchical classification of animals from principal component analysis performed on measured digestibility coefficients of dry matter, organic matter, N, energy, and crude fibre during total collection (panel A) or from principal component analysis performed on predicted digestibility coefficients of dry matter, organic matter, N, energy, and crude fibre by NIRS on samples of faeces collected in the rectum of pigs (panel B).

Table 1 .

 1 Effect of diet composition, breed, and period on digestibility coefficients of energy and nutrients in growing pigs.

		Diet		Breed			Period				Statis tics
		Low fibre	High fibre	Duroc	Large White	Piétrain	1	2	3	4	RSD	D	B	P	D×B D×P B×P
	n	123	123	76	84	86	64	60	61	61
	Body weight (kg)	54.6	54.9	50.0 a	55.7 b	58.4 c	35.7 a 48.1 b 60.9 c 74.1 d	1.3 0.08 <0.01 <0.01 0.43 0.20 0.04
	Feed intake (g DM/d)	1414 1406	1367 a 1462 c 1400 b	1244 a 1368 b 1458 c 1570 d	38 0.09 <0.01 <0.01 0.74 0.95 <0.01
	Digestibility coefficient (%)
	Dry matter	86.7	78.8	82.8	82.5	82.8	81.2 a 82.2 b 83.5 c 84.0 d	1.3 <0.01 0.71 <0.01 0.34 <0.01 <0.01
	Organic matter	88.4	80.7	84.5	84.3	84.8	83.0 a 84.0 b 85.3 c 85.9 d	1.2 <0.01 0.41 <0.01 0.29 <0.01 <0.01
	Energy	86.6	78.3	82.4	82.2	82.8	80.8 a 81.9 b 83.2 c 83.9 d	1.4 <0.01 0.38 <0.01 0.62 <0.01 0.02
	N	86.1	76.1	80.6 a	80.8 a	81.9 b	78.4 a 80.7 b 82.0 c 83.3 d	1.8 <0.01 0.04 <0.01 0.30 0.15 0.02
	Ether extract	75.0	61.3	68.5	67.6	68.3	70.0	67.6	66.9	68.1	6.5 <0.01 0.59 0.06 0.77 0.16 0.24
	Crude Fibber	43.6	54.6	49.5	47.7	50.2	42.4 a 45.8 b 52.6 c 55.8 d	6.9 <0.01 0.29 <0.01 0.13 <0.01 0.02

RSD, residual standard deviation; the data were analysed for the effects of diet (D), breed (B), period (P), and their interactions. Within a diet, breed or period, LS-means with different superscript letters differ (P<0.05).

Table 2 .

 2 Accuracy of NIRS calibration equations to predict silicone oil, plastic resin, and polyethylene glycol concentrations in the faeces of growing pigs.

			Calibration dataset		Validation dataset
		Range (% DM)	n	Standard deviation	R²	n	Standard deviation	R² Intercept 1 Slope 1
	Silicone oil	0-11.13 771	0.25	99.5	257	0.24	99.5	0.029	0.99
	Plastic resin	0-11.33 771	0.30	99.2	254	0.28	99.2	0.025	0.98
	Polyethylene glycol 0-11.15 771	0.19	99.7	271	0.19	99.7	0.014	1.00
	1

Table 3 .

 3 Effect of diet composition, breed, and period on recovery of silicone oil, plastic resin, and polyethylene when used as markers in the faeces of growing

	pigs
		Diet			Breed			Period					Statistics
		Low fibre	High fibre	Duroc	Large White	Piétrain	1	2	3	4	RSD	D	B	P	D×B D×P B×P
	n	122	122	75	84	85	62	60	61	61
	Silicone oil	98.5	96.2	98.7	98.9	94.3	98.8 a 92.7 b 96.7 ab 101.1 a	15.5 0.26 0.48 0.03 0.06 0.12 0.07
	Plastic resin	95.4	89.5	98.0	93.0	89.4	99.8 a 90.

Table 4 .

 4 Accuracy of NIRS calibration equations to predict digestibility coefficients of energy and nutrients in the faeces of growing pigs.

			Calibration dataset				Validation dataset
		Range (%)	n	Standard deviation	R²	n	Standard deviation	R²	RPD	r	Bias Intercept 1 Slope 1
	Dry matter	72.1-90.4	749	1.61	83.5	82	1.71	85.7	2.67	0.93	0.2	12.3	0.85
	Organic matter 73.9-91.8	749	1.63	82.5	82	1.47	89.3	3.05	0.94	0.1	9.2	0.89
	Energy	70.9-90.4	749	1.84	79.2	82	1.73	87.0	2.78	0.94	0.1	15.8	0.81
	N	67.4-90.9	749	2.32	79.1	82	1.82	89.6	3.11	0.95	0.1	14.5	0.82
	Crude fibre	24.0-73.7	497	6.75	61.4	82	6.64	66.4	1.74	0.82	0.8	18.8	0.59
	RPD, ratio between standard deviation of laboratory results and standard deviation of prediction.

Table 5 .

 5 Effect of diet composition, breed, and period on the bias between measured digestibility coefficient by the total collection method and the digestibility coefficient predicted by NIRS in samples of faeces of growing pigs obtained during the total collection period. standard deviation; the data were analysed for the effects of diet (D), breed (B), period (P), and their interactions. Within diet, breed, or period, LS-means with different superscript letters differ (P<0.05).

		Diet			Breed			Period					Statistics
		Low fibre	High fibre	Duroc	Large White	Piétrain	1	2	3	4	RSD	D	B	P	D×B D×P B×P
	n	122	123	76	84	85	63	60	61	61
	Bias in digestibility
	coefficient
	Dry matter	0.12	0.24	-0.33 b	0.25 a	0.63 a	0.65 a	0.31 ab -0.24 c 0.006 bc	1.23 0.44 <0.01 <0.01 0.37 <0.01 0.02
	Organic matter	-0.22	0.32	-0.12	0.09	0.19	0.94 a	0.13 b	-0.59 c -0.26 bc	1.26 <0.01 0.38 <0.01 0.90 0.08 0.05
	Energy	-0.48	0.28	-0.31	-0.04	0.05	0.51 a	-0.06 b -0.52 b -0.33 b	1.45 <0.01 0.30 <0.01 0.90 0.13 0.08
	N	-0.22	0.31	-0.37 ab	0.50 a	0.01 b	0.65 a	-0.10 b -0.22 b -0.14 b	1.85 0.03 0.01 0.03 0.62 0.09 <0.01
	Crude fibre	0.65	0.64	-0.08 b	0.27 ab	1.75 a	3.99 a	1.85 b	-1.09 c	-2.16 c	5.80 0.99 0.07 <0.01 0.19 <0.01 <0.01
	RSD, residual

Table 6 .

 6 Effect of diet composition, breed, and period on the bias between measured digestibility coefficient by the total collection method and the digestibility coefficient predicted by NIRS in samples of faeces of growing pigs collected from the rectum.

		Diet			Breed			Period					Statistics
		Low fibre	High fibre	Duroc	Large White	Piétrain	1	2	3	4	RSD	D	B	P	D×B D×P B×P
	n	122	122	76	84	84	63	60	60	61
	Bias in digestibility
	coefficient
	Dry matter	-0.39	0.06	-0.58 b	-0.22 b	0.32 a	0.02 b	0.23 a	-0.47 b -0.44 b	1.48 0.02 0.02 <0.01 0.63 <0.01 <0.01
	Organic matter	-0.43	-0.11	-0.27	-0.02	-0.52	0.20	-0.71	-0.28	-0.30	1.84 0.06 0.18 0.17 0.74 <0.01 <0.01
	Energy	-0.78	0.18	-0.29	-0.11	-0.51	0.09 a	-0.87 b -0.16 a -0.26 ab	1.93 <0.01 0.40 0.05 0.49 <0.01 <0.01
	N	-0.99	0.55	-0.03	0.08	-0.70	1.19 a	-0.77 b -0.38 b -0.92 b	2.34 <0.01 0.10 <0.01 0.94 <0.01 <0.01
	Crude fibre	-0.50	-3.05	-1.74	-1.58	-2.00	1.32 a

Table 7 .

 7 Description of clusters obtained from hierarchical classification of animals based on principal component analysis performed on measured digestibility coefficients of dry matter, organic matter, N, energy and crude fibre during total collection (periods 3 and 4).

		Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Cluster 6 RSD	P-value
	n of pigs				17	26	18
	Fed low-fibre diet
	Fed high-fibre diet	28	20	13
	n of pigs
	Duroc	10	5	4	5	6	8
	Large White	7	7	7	2	11	8
	Pietrain	11	8	2	10	9	2
	Digestibility
	coefficients (%)
	Dry matter	81.8 c	80.1 d	78.0 e	88.0 a	87.2 a	86.0 b	0.8	<0.01
	Organic matter	83.7 d	82.0 e	79.9 f	89.8 a	89.0 b	87.8 c	0.8	<0.01
	Energy	81.5 c	79.5 d	77.5 e	88.1 a	87.2 a	85.9 b	1.0	<0.01
	N	79.0 c	78.1 c	75.2 d	88.8 a	87.8 a	85.7 b	1.4	<0.01
	Crude fibre	68.8 a	61.2 b	48.8 d	54.8 c	46.6 d	37.7 e	3.1	<0.01
	RSD, residual standard deviation: the data (only periods 3 and 4) were analysed for the effects of
	cluster. LS-means with different superscript letters differ (P<0.05).

Table 8 .

 8 Description of clusters obtained from hierarchical classification of animals based on principal component analysis performed on predicted digestibility coefficients of dry matter, organic matter, N, energy, and crude fibre by NIRS on samples of faeces collected from the rectum of pigs during periods 3 and 4. standard deviation: the data (only periods 3 and 4) were analysed for the effects of cluster. LS-means with different superscript letters differ (P<0.05).

		Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Cluster 6 RSD	P-value
	n of pigs					42	19
	Fed low-fibre diet
	Fed high-fibre diet	26	25	7	2
	n of pigs
	Duroc	7	10	2		12	7
	Large White	8	7	7	2	13	8
	Piétrain	11	8	1		17	4
	Digestibility coefficients
	(%)
	Dry matter	81.6 b	80.3 c	77.8 d	76.7 d	87.3 a	86.5 a	1.1	<0.01
	Organic matter	83.6 b	82.1 c	79.6 d	78.7 d	89.1 a	88.4 a	1.0	<0.01
	Energy	81.3 b	79.7 c	77.0 d	76.7 d	87.3 a	86.4 a	1.3	<0.01
	N	79.2 c	77.7 d	75.3 e	73.5 e	87.9 a	86.4 b	1.7	<0.01
	Crude fibre	67.5 a	61.6 b	48.1 c	43.4 c	47.8 c	43.0 c	6.1	<0.01
	RSD, residual

Intercept and slope of the regression line between predicted and theoretical concentrations in marker.

Conclusions

The digestive ability of pigs is variable during the growing period, especially when the diet contains a high fibre level because the digestibility of this type of diet increases when body weight increases. The possibility to characterize the digestive ability of pigs in selection farms of breeders is limited because of the difficulty in implementing the gold standard methodology to measure digestibility. Alternatively, spot-sampling of faeces collected directly from the rectum of the pig can be used to characterize digestive ability. In our experiment, the utilization of indigestible markers did not give satisfying results, because of a low recovery of the marker in the faeces (for plastic resin or polyethylene glycol) or because of a large variability in the recovery (for silicone oil). The NIRS prediction of digestibility coefficients from a faecal sample collected directly from the rectum is adequate when pigs are heavier than 60 kg and when they are fed a diet with a high fibre content. This type of diet is susceptible to challenge the digestive ability of pigs and allows identifying specifically the poor digesters in a population using NIRS. The method combining a single sample of faeces collected directly from the rectum of the pig and analysed by NIRS can then be used in selection farms of breeders to give information on the digestive ability of pigs in their population.

Annexes


Barigah Ts

Fernández

Dynamics of xylem cavitation during a simulated drought in Eucalyptus camaldulensis saplings, a species with solitary vessels

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Dynamics of xylem cavitation during a simulated drought in Eucalyptus camaldulensis saplings, a species with solitary vessels

Tete Severien Barigah, J. Gyenge, F. Barreto, Philippe Rozenberg, M.E.

Fernandez

To cite this version:

Tete Severien Barigah, J. Gyenge, F. Barreto, Philippe Rozenberg, M.E.

  Fernandez. Dynamics of xylem cavitation during a simulated drought in Eucalyptus camaldulensis saplings, a species with solitary vessels. International Conference "Adapting forests ecosystems and wood products to biotic and abiotic stress", European Union Marie-Curie project TOPWOOD., Mar 2019rate of native embolism (PLC) was similar for both the control and treated plants during the fortnight after the onset of the experiment. But thereafter, PLC for treated plants rose sharply to more than 90% while PLC levelled off for control plants. Percent loss hydraulic conductance (hydraulic method) was highly correlated with percent stained sapwood area (staining method). y = -0.8305x + 92.7196; r² = 0.80 Left panel -Mean ± SE diameter of stained and unstained vessels in different sections of the xylem (inner and outer sapwood) in differently embolized plants. There was no significant difference in the mean size of stained and unstained vessels (Tukey test, p<0.05). Right panel -Percent of unstained vessels in inner and outer portion of sapwood in plants with different degree of PLC. Proportion of unstained vessels was high in all plants (even control plants). It increased steadily in the inner area along the drying steps while this proportion remained low in the outer part when PLC was lower than 70% but increased sharply for highly stressed plantsfunded by the EU H2020-MSCA-RISE-2014 project TOPWOOD (Wood phenotyping tools: properties, functions and quality -645654). The authors are grateful to the field assistants for their help and for the grant .




  the effects of starvation on alevins nutritional status 2) Test the alevins ability to recover from a delayed first-feeding 3) Analyse the effect of temperature on these phenomena @inra.fr F 5 & F 9 : Fed fish gained weight  No difference between 8°C and 11°C S 5 & S 9 : Starved fish lost weight  Fish leaner at 11°C DF 9 : Delayed first-fed fish: intermediate state  Difference close to significance level p-value = 0Both able to start feeding  Same food intake than fed fish  A trend for a better growth at 8°C


How to properly design bacterial communities?

Animal health

Offer alternatives to antibiotic treatments (probiotics)

Food spoilage

Reduce health risks and food losses and waste

Health & wellbeing

Develop functional and designer food against modern lifestyle diseases

Food quality & safety

Develop safe, healthy, sustainable and hedonic fermented foods

What is for us phenotyping?

Design of bacterial communities

Pre selection tools

Various environments

Bacterial communities

Selected strains 1-collect data from literature using FLORILEGE database

Phenotype

  = expression of the bacteria genotype  morphology or form and structure  biochemical and physiological properties  metabolites produced  To test observable characteristics = phenotypes of a bacteria in an environment • Use of media that mimics at best the in situ environments : composition, structure and abiotic conditions • Assess some risk criteria: biogenic amines, antibiotic resistance… method  Some genes are not accurately annotated: cell-wall proteinases …  Pathways are not complete  No direct Links between genome and phenotype:  Case of intracellular enzymes released after bacterial cell death  Presence of genes cannot predict the observed activity: regulation of gene expression and/or of enzyme production and strains for targeting functionalities within bacterial communities 11 In situ assessment through various environmental conditions  And different kinds of interactions within the communities and with the components of the environment Lactic acid bacteria-Propionibacteria (2) L. lactis / S. aureus (3) Iron privation of S. enteritidis by egg white (1) Adapted from Seth and Taga (2014) What next after the pre-selection of the strains? (1) Baron et al .2017. Front.Microbiol., 8, 829, (2) Thierry et al 2015. Dairy Sci. Technol. , 95(6), 895-918 (3) Cretenet et al. 2011 Environ. Microbiol. Reports, 3



  . (2008) reported that Propionibacterium freudenreichii is the main species used as a ripening starter in Swiss-style cheeses. Text mining for automatic extraction of microbial habitats, phenotypes and data gathering Collect data from FLORILEGE : A public database gathering microbial habitats and phenotypes of food interest Text-mining  Propionibacterium freudenreichii INHABITS hard-cheese USED Tools to select strains with the genes of interest to find strains with high anti-inflammatory potential Select strain with genes of interest (1) 19 Deutsch et al. 2017,Scientific Reports volume 7, Article number: 46409 P freudenreichi has anti inflammatory properties highly variable among strains  There was a multi-gene determinism according to the strains  2 surface protein SlpE et SlpB and hsdM3 genes associated with high anti-inflammatory properties  Mastering the anti-inflammatory potential of Propionibacterium freudenreichii  2 genes encoding cytoplasmic proteins associated with antagonist effects of 12 putative esterase-coding genes identified in P. freudenreichii genome Abeijon et at, 2014, Appl Env Microbiol Select strain with genes of interest (2) • Sequence of all esterases are highly conserved • All esterases were expressed regardless of the presence of milk fat But 1 secreted lipolytic esterase with a major role in cheese lipolysis Production of free fatty acid from milk fat by 24 Pf strains • Lipolytic activity markedly strain-dependent: Knowing the composition of the raw material as a way to select bacteria 22 Ferment sugars from various resources :-lactose from milk, -raffinose, stachyose, verbascose and sucrose from legume resources


published or not. The documents may come

The CIRM -BIA, microbial diversity of food related bacteria in the service of innovation for dairy industry Florence Valence

• In vitro highthroughput screening with cheese substrate

• In situ at pilote scale on 3 dairy models (yogurt, pressed cheese and sor cream)

• In situ at industrial scale by industriel partners on their own dairy products

AF strains

Safety (acquired antibiotic resistance, biogenic amines production)

"•••

  The role of the infinitely small is infinitely great'' Preserve the biodiversity • Give access to biological resources for scientific, industrial, agricultural environmental and medical R&D applications • Be repositories of biological reference material • Be repositories of BR for protection of intelectual property • Facilitate and favorize their use  performance of R&D • Be resources for public information and policy formulation More than 20,000 strains of food, plant and pathogenic bacteria, yeasts and filamentous fungi • Partner of MIRRI, the pan-European Microbial Resource Research Infrastructure 4,000 strains of bacteria of food interest (> 150 species) incl. 2,000 lactic acid bacteria biotope diversity of strains  Genera and species diversity  Intra-species diversity Bacterial diversity: contribution to innovation BAKERY ANR Project Microbial diversity (Bacteria and Yeast) of surdough used for traditional bread • PEAKYEAST ANR Project Microbial diversity (Bacteria and Yeast) of grape must used for wine making • Lemzeit Project Bacterial diversity of traditional fermented wheat (Constantine University, Algeria, PhD R. Merabti) 50 original LAB strains from integrated into our collection. One new yeast species: Kazachstania saulgeensis  Merabti et al. , 2015  Merabti et al. , 2018 -Submitted 69 LAB isolates (6 genera, 16 species), among them 10 LAB strains belonging to 7 species = functional starters for wheat fermentation  Lhomme et al. , 2016 Strains biotope diversification   New starters for fermentation of non dairy matrix or milk-legumes mixes 420 vegetal LAB and ACB strains with high level of species diversity (26 genera, 82 species) integrated into our collection • European project ''Gwell'' Development of a traditional fermented milk product from the milk of local Cows ''bretonne Pie Noire: the Gwell'' (Preparatory action EU plant and animal genetic resources in agriculture) ~ 100 LAB isolates, 18 strains of interest (Lactococci) How to define the ''functional associations'' ? How to predict their functionalities ? No starters, only backslopping technic for production of Gwell Sometimes accidental lost of the ''Gwell''  Gwell production securing strains in associations as ''functionnal gwell starters'' ingredients & adjunct for dairy products: a close research partnership F. Valence -Lanzhou -2018 700 bacteria and fungi screened for their ability to inhibate fungal spoilage (6 specific targets)

  the development of antifungal adjuncts and dairy fermentates Applications Adapted and modified from Crowley et al. 2013 • Choice of fungal targets and screening matrix • Development of an easy (high throughput) and reproducible method Screening and Selection • Scale up: In situ activity (challenge & shelf-life tets) • Sensorial analysis • Safety assesment (acquired antibiotic resistance, biogenic amines production) '', depending of the strains and associations, currently under test by industrial partners  Development of antifungal ingredients for dairy products: from in vitro screening to pilot scale application. L. Garnier et al., 2018.  Antifungal Activity of Lactic Acid Bacteria Combinations in Dairy Mimicking Models and Their Potential as Bioprotective Cultures in Pilot Scale Applications. Leyva-Salas et al., 2018  Biogenic amine and antibiotic resistance profiles determined for lactic acid bacteria and a propionibacterium prior to use as antifungal bioprotective cultures. Coton et al., 2018

fungal targets

Acknowledgements

Pr. Jerôme Mounier (LUBEM) Dr, Anne Thierry (STLO-INRA Rennes)

Functionalities: highly strain dependant !!!

Choice of fungal targets


Lucie Aulus Giacosa
email: lucie.aulus@inra.fr

Spatio-temporal evolution of life history traits related to dispersal, Brown trout colonization

In a context of global change, biological invasions are one of the main causes of biodiversity loss, because non-native species can disperse naturally at the expense of others. As a consequence of ice retreat in the highest latitude, newly opened rivers could become welcoming habitats for the establishment of colonizing fish. Salmonids are thought to be good candidates for colonizing such environments since they display ability to become invasive.

Variance analyses

AGE~rpt age ~1 Fish + 1 Reader + 1 Scale , datatype = Poisson, … ; MEASURE~rpt(rt ~(1|Fish) + (1|Reader) + (1|Scale), datatype = Gaussian, … )

This study aims at understanding the evolution of life history traits related to dispersal, through the quantification of variation in freshwater growth and age at first marine migration at various spatiotemporal scales with respect to the colonization process.

𝑳 𝒊,𝒕 = 𝑳 ∞ 𝒊 +𝒆 -𝑲 𝒊 (𝒕 𝒊 -𝒕 𝟎 )
Main variation in size and age are attributable to Fish. Reader and Scale contribute little to inter-individual variance, suggesting that inference was insensitive to intraorganism biological variation. Using additional scales or readers was an inefficient use of sampling resources.

Model the evolution of traits, ongoing work

Objectives

  𝒑 𝒊 logit(𝒑 𝒊,𝒋,𝒌 ) = β 𝟎 𝒋,𝒌 + β 𝟏 𝒋,𝒌 x 𝑳 𝒊,𝒋,𝒌 Is the probablity to migrate at a certain age or size stable? Carry on the ongoing work and answer the main questions. L., Aymes J-C., Gaudin P., Vignon M., A hierarchical variance decomposition of fish scale growth and age to investigate the relative contributions of readers and scales, Marine and Freshwater Research, Special issue Otolith 2018. (Accepted).