Kam-Rigne Laossi
email: kam-rigne.laossi@bondy.ird.fr

Amandine Ginot

Cristina Diana

Manuel Noguera

Sébastien Blouin

Barot

Wim K Van Der Putten

R Laossi

D C Noguera

: S Barot

M Blouin

S Barot

Earthworm effects on plant growth do not necessarily decrease with soil fertility

Keywords: Earthworms, L, terrestris, A, caliginosa, Plant growth, Soil type, Nutrient availability, Shoot/root ratio

Earthworms are known to generally increase plant growth. However, because plant-earthworm interactions are potentially mediated by soil characteristics the response of plants to earthworms should depend on the soil type. In a greenhouse microcosm experiment, the responsiveness of plants (Veronica persica, Trifolium dubium and Poa annua) to two earthworm species (in combination or not) belonging to different functional groups (Aporrectodea. caliginosa an endogeic species, Lumbricus terrestris an anecic species) was measured in term of biomass accumulation. This responsiveness was compared in two soils (nutrient rich and nutrient poor) and two mineral fertilization treatments (with and without). The main significant effects on plant growth were due to the anecic earthworm species. L. terrestris increased the shoot biomass and the total biomass of T. dubium only in the rich soil. It increased also the total biomass of P. annua without mineral fertilization but had the opposite effect with fertilization. Mineral fertilization, in the presence of L. terrestris, also reduced the total biomass of V. persica. L. terrestris did not only affect plant growth. In P. annua and V. persica A. caliginosa and L. terrestris also affected the shoot/root ratio and this effect depended on soil type. Finally, few significant interactions were found between the anecic and the endogeic earthworms and these interactions did not depend on the soil type. A general idea would be that earthworms mostly increase plant growth through the enhancement of mineralization and that earthworm effects should decrease in nutrient-rich soils or with mineral fertilization. However, our results show that this view does not hold and that other mechanisms are influential.

Introduction

Soil organisms are known to affect plant growth by enhancing mineralisation of soil organic matter and modifying physical and chemical properties of soil [START_REF] Bardgett | A temporal approach to linking aboveground and belowground ecology[END_REF][START_REF] Lavelle | Soil ecology[END_REF]. Within soil organisms, earthworms are in term of biomass and activity among the most important detritivores in terrestrial ecosystems [START_REF] Edwards | Earthworm ecology[END_REF]. They are also known to affect plant growth, generally positively, via five main mechanisms [START_REF] Brown | How earthworms effect plant growth: burrowing into the mechanisms[END_REF][START_REF] Scheu | Effects of earthworms on plant growth: patterns and perspectives[END_REF]: (1) an increased mineralization of soil organic matter (2) the production of plant growth substances via the stimulation of microbial activity;

(3) the control of pests and parasites; (4) the stimulation of symbionts and (5) modifications of soil porosity and aggregation, which induces changes in water and oxygen availability to plant roots. Although these mechanisms are well identified it is difficult to determine their relative influence [START_REF] Blouin | Earthworms (Millsonia anomala, Megascolecidae) do not increase rice growth through enhanced nitrogen mineralization[END_REF]) either in precise cases or in broad classes of cases as defined by plant functional type, geographic area or soil type. [START_REF] Brown | How earthworms effect plant growth: burrowing into the mechanisms[END_REF] has remarked that the response of plants to earthworms should depend on the type of soil and especially its texture and its richness in mineral nutrients and organic matter. Indeed, mechanisms through which earthworms influence plant growth might be either down or up-regulated by soil characteristics. For example, if the positive effect of an earthworm species on plant growth is mainly due to an increase in mineralization, the species might no longer increase plant growth in a soil where nutrients are not limiting. However, few studies [START_REF] Doube | The influence of two species of earthworm (Aporrectodea trapezoides and Aporrectodea rosea) on the growth of wheat, barley and faba beans in three soil types in the greenhouse[END_REF][START_REF] Wurst | Indirect effects of earthworms (Aporrectodea caliginosa) on an above-ground tritrophic interaction[END_REF] have tested the effect of earthworms on plant in different soils in the same laboratory experiment. [START_REF] Doube | The influence of two species of earthworm (Aporrectodea trapezoides and Aporrectodea rosea) on the growth of wheat, barley and faba beans in three soil types in the greenhouse[END_REF] have shown that Aporrectodea trapezoides increased the growth of wheat in a sandy soil but not in a clayey one. They also showed that the growth and the grain yield of barley were increased by Aporrectodea trapezoides and Aporrectodea rosea (both are endogeic earthworms) in the sandy soil but reduced in the clayey. On the contrary, [START_REF] Wurst | Indirect effects of earthworms (Aporrectodea caliginosa) on an above-ground tritrophic interaction[END_REF] have shown that Aporrectodea caliginosa increased the root biomass of Cardamine hirsute in two different soils. Up to our knowledge, no laboratory experiment has so far compared the effect of two earthworm species belonging to different functional groups on the same plant species, in soils differing by their texture and nutrient content. Our experiment aims at meeting this need.

Besides, plant species of different functional groups should respond differently to earthworms [START_REF] Brown | How earthworms effect plant growth: burrowing into the mechanisms[END_REF]) because they are not limited by the same resources and do not have the same resource allocation strategies [START_REF] Laossi | Effects of endogeic and anecic earthworms on the competition between four annual plants and their relative reproduction potential[END_REF]). Legumes, for example, are thought to be less responsive to earthworms than grasses since they are not limited by nitrogen [START_REF] Brown | How earthworms effect plant growth: burrowing into the mechanisms[END_REF][START_REF] Wurst | Do endogeic earthworms change plant competition? A microcosm study[END_REF]. Finally, plant responses may also depend on earthworm functional group since earthworm belonging to different functional groups differ in their behaviors [START_REF] Lavelle | Soil ecology[END_REF]. For example, endogeic earthworms keep moving inside the soil to feed on soil organic matter while anecic feed on plant litter at the soil surface and tend to stay in the same burrow [START_REF] Lavelle | Earthworms as a resource in tropical agroesosystems[END_REF]. Anecic earthworms fragment plant litter and incorporate it into the soil where it can subsequently be ingested by endogeic earthworms. Such an interaction could increase further mineralization and plant growth [START_REF] Brown | Regulation of soil organic matter dynamics and microbial activity in the drilosphere and the role of interactions with other edaphic functional domains[END_REF][START_REF] Jégou | Effects of four ecological categories of earthworms on carbon transfer in soil[END_REF].

We tested how the effects of earthworms belonging to different functional groups on plant growth covary with plant functional group and soil type. Hence, we investigated in a microcosm greenhouse experiment the responsiveness of three annual plant species of different functional groups (Poa annua, a grass; Trifolium dubium, a legume; Veronica persica, a forb) to an endogeic (Aporrectodea caliginosa) and anecic (Lumbricus terrestris) earthworm species as well as to the combination of the two species. The responsiveness of plant was measured in term of biomass accumulation and was compared in two soils (a clayey nutrient rich soil with higher organic matter content and a sandy nutrient poor soil with lower organic matter content) and two mineral fertilization treatments (with and without). Mineral fertilization can be considered either to mimic richer soils or agricultural practices.

We hypothesized that earthworms affected plant growth mainly through an enhancement of mineralization, which is the more often cited mechanism [START_REF] Kreuzer | Decomposer animals (Lumbricidae, Collembola) and organic matter distribution affect the performance of Lolium perenne (Poaceae) and Trifolium dubium (Fabaceae)[END_REF][START_REF] Partsch | Decomposers (Lumbricidae, collembola) affect plant performance in model grasslands of different diversity[END_REF]Wurst et al. 2003). Therefore (see above), both earthworm species should affect (1) plant growth and (2) plant resource allocation only in the soil that is poor in organic matter and mineral nutrient. Similarly, (3) significant interactive effects of the two earthworm species on plant growth and resource allocation should only be found in the poor soil. Finally, according to the same assumption that earthworm mostly influence plants by increasing the availability of mineral nutrients, (4) the impact of A. caliginosa and L. terrestris on plants should decrease with mineral fertilization.

Materials and methods

Experiment set up

The experiment was set up in microcosms consisting of PVC pots (inner diameter 14 cm, height 12.5 cm) that were closed at the bottom with 1 mm plastic mesh to prevent earthworms from escaping. A total of 320 microcosms were filled with 950 g (±20 g) of sieved (2 mm) dry soil in a greenhouse. Before starting our experiment, the microcosms were watered regularly for 2 weeks and germinating weeds from the seedbank were removed. Eight grams of dried litter (72 h at 60°C) of grass leaves were placed at the soil surface and 1 g was mixed with the first cm of soil, prior to the addition of earthworms and seeds. This constituted the essential food resource for the anecic earthworm species.

Soils

We used two different soils: A sandy cambisol (called hereafter in the text the "nutrient poor soil") supporting a meadow (OM=2.55%; C/N ratio=12.4; total carbon content=1.47%; Ntotal=0.12%; pH=5.22) collected at the ecology station of the Ecole Normale Supérieure at Foljuif (France) and a clayey leptosol (OM=9.81%; C/N ratio=12.2; total carbon content= 5,67; Ntotal=0.465; pH=7.45) collected at the ecology station of Brunoy (France) (called hereafter the "nutrient rich soil").

Earthworms

We used an anecic earthworm, Lumbricus terrestris (L.) -LT-, and an endogeic earthworm, Aporrectodea caliginosa (Savigny) -AC-. LT were purchased in a store and AC were collected in the park of the IRD centre in Bondy (France). Our experiment consisted in four treatments: AC, LT, AC + LT and a control without any earthworm species (C). One adult of LT (4.2±0.5 g) and three adults of AC (2.4±0.4 g) were introduced in each treatment including these species. This represents respectively 273 g m -² and 156 g m -², which is comparable to the biomasses found in temperate grassland ecosystems [START_REF] Edwards | Biology and ecology of earthworms[END_REF]. In the treatment with both earthworm species (AC + LT) we have maintained for each earthworm species the biomass used in AC and LT treatment. This was done to maintain the same activity level of each earthworm species, which was the only way to allow testing for a possible interactive effect of the two earthworm species on plant growth. 96% of the earthworms were recovered at the end of the experiment (471 A. caliginosa and 143 L. terrestris individuals among the 480 and 160 that were originally introduced respectively).

Plants

One week after introducing earthworms, 15 seeds of Veronica persica, Trifolium dubium and Poa annua were sown in monocultures. Three weeks later, a single plant per microcosm was kept (the other seedlings were removed and cut down in the original microcosm). Microcosms were weeded every week during the experiment. Microcosms were watered during 7 weeks with 6.5 ml every day and from 8th week to the end (week 16) with 13 ml every day. This allowed us to maintain the soil near its field capacity (this was checked through regular weighing of some pots). Microcosm position within the greenhouse was randomized every 2 weeks.

Fertilizer

For each combination of treatments (soil type × LT × AC × plant species), two fertilization treatments were implemented: without or with mineral fertilization. This treatment consists in an application of fertilizer containing N, P, K, S and Mg. 0.6 g of fertilizer was placed at the soil surface at the beginning of the experiment, 0.6 g 3 weeks after sowing and 0.6 g on week 6, when the first flowers were produced. From week 6 to week 12, 1 g of fertilizer was added every 2 weeks before watering. A total of 5.8 g of fertilizer was then added per pot. This corresponds to 48 kg of N and K; 32 kg of P; 6.7 kg of S and 97 kg of Mg per hectare. Five replicates of each treatment combination, i.e. fertilization × soil type × LT × AC × plant species, were implemented.

Sampling

Plants were harvested on week 16. Shoot (leaves and stems) were cut at the soil surface and roots were separated from the soil by washing on a 600 µm mesh.

Root and shoot biomasses were dried at 60°C for 72 h. Dried shoot and root biomasses were weighted. Because of differences in the timing of seed maturation between plant species, seeds were not harvested.

Statistical analyses

Data were analysed with ANOVAs using SAS GLM procedure (Sum of squares type III, SS3) (SAS 1990). A full model was first used to test all factors ("AC", "LT", "plant species", "fertilizer" and "soil") and all interactions between them (Table 1). When significant interactions between plant species and other factors (AC, LT, soil and fertilization) were detected, data were reanalysed separately for each plant species (Table 2) to describe in a more detailed way the effects of these treatments on each plant species. This allowed for example determining which plant species responded to which earthworm species and in which conditions (soil type and fertilization). Effects of treatments and interactions between treatments were tested on shoot biomass, root biomass, total biomass, and shoot/root ratio.

The residuals of each model were analysed to test for normality and homogeneity of variances. To determine the direction of significant effects, we used multiple comparison tests based of least square means taking into account Bonferroni's correction (LSmeans, LSmeans SAS statement). All tests were achieved with a significance level α=0.05.

Results

In the present experiment, the main effects are due to the anecic earthworm species, L. terrestris. No significant effect of A. caliginosa was found. The full statistical model ( effect on shoot biomass and total biomasses varied with plant species (significant LT × plants species interaction, p<0.01), soil type (significant LT × soil interaction, p<0.05) and fertilizer (significant LT × fertilizer interaction, p<0.001). Its effects on the shoot/root ratio also varied with fertilizer addition (significant LT × fertilizer interaction, p <0.05).

Further the presence of both earthworm species affected the shoot/root ratio and this effect varied with plant species (significant AC × LT × plant species interaction, p<0.001). Finally we found that the shoot/root and total biomasses were affected by the soil type and fertilizer but plant species differed in their responses (significant soil × plant species and fertilizer × plant species interactions, p<0.001).

These general results and the significant interactions involving the plant species justify analysing separately for each plant species the effects of the presence of L. terrestris, soil type and fertilization (Table 2).

Shoot biomass

L. terrestris increased the shoot biomass of T. dubium (+26%) but decreased the shoot biomass of V. persica (-24%) (Table 2). Mineral fertilization increased the shoot biomasses of T. dubium, V. persica and P. annua (Table 2) by 24%, 46% and 48% respectively. The shoot biomass of T. dubium was 140% higher in the nutrient poor soil than in the nutrient rich one. The significant LT × fertilizer interaction for the shoot biomasses of V. persica and P. annua (Table 2) showed that effect of LT on these plants species varied with mineral fertilization. LT reduced the shoot biomasses of V. persica (-37%) and P. annua (-10%) when fertilizer was added (Fig. 1). The significant LT × Soil interaction, (Table 1 and LSmeans comparisons) indicated that the effect of LT on shoot biomass of T. dubium varied with soil type. It increased the shoot biomass of the legume (+130%) only in the nutrient rich soil.

Root biomass

Mineral fertilization decreased the root biomasses of P. annua (-50%) and T. dubium (-25%). The root biomasses of the three plant species were affected by soil type with the highest root biomass in the nutrient rich soil for P. annua and V. persica while T. dubium showed higher root biomass in the nutrient poor soil (Table 2).

Total biomass

LT increased by 26% the total biomass of T. dubium but decreased the total biomass of V. persica by 24%. Mineral fertilization increased the total biomass of T. dubium (16%), P. annua (31%) and V. persica (41%) (Table 2). Significant interactions between LT and fertilizer were found for the total biomasses of P. annua and V. persica suggesting that effects of LT on these plant species varied with mineral fertilization (Table 2). Without mineral fertilization, LT increased the total biomass of P. annua (+18%) while no significant effect of this earthworm was found when the fertilizer was added. For V. persica LT reduced (-48%) the total biomass when the fertilizer was added but without mineral fertilization it did not show significant effect (significant LSmeans comparisons and Fig. 1). For T. dubium the significant LT × Soil interaction (Table 2) showed that the effect of LT on the total biomass of this plant species varied with soil type. In the nutrient rich soil the presence of LT increased significantly the total biomass of T. dubium (+121%) while, no significant effect of LT was found in the nutrient poor soil (Fig. 1). Moreover, in the nutrient rich soil without earthworms, the mineral fertilization reduced (-55%) the growth of the legume (Fig. 1). The effect of soil type on the total biomass of T. dubium varied with fertilizer addition (Table 2, significant Soil × Fertilizer interaction). In the nutrient poor soil, the addition of fertilizer increased (+22%) the total biomass of the legume while no significant effect of fertilizer was found in the nutrient rich soil (Fig. 1).

Shoo/root ratio

The presence of LT increased by 21% the shoot/root ratio of P. annua. Shoot/root ratio of V. persica was increased (+32%) when both earthworm species were present (Fig. 2 and LSmeans comparisons). Moreover the shoot/root ratios of the three plant species were significantly affected by the soil type and mineral fertilization. Their shoot/root ratios were higher in the nutrient poor soil than in the nutrient rich one (Fig. 2). Mineral fertilization increased the shoot/root ratio of T. dubium (193%), V. persica (194%) and P. annua (335%). Effects of LT on the shoot/root ratio of P. annua varied with the soil type (significant LT × soil interaction, Table 2). LT increased (+27%) the shoot/ root ratio of P. annua only in the nutrient poor soil (Fig. 2 and LSmeans comparisons).

Discussion

We hypothesized that: both earthworm species should affect (1) plant growth and (2) plant resource allocation only in the nutrient and organic matter poor soil; (3) significant interactive effects of the two earthworm species on plant growth and resource allocation should only be found in the nutrient and organic matter poor soil; (4) the impact of A. caliginosa and L. terrestris on plants should decrease with mineral fertilization. It is generally thought that positive effects of earthworms on plant growth are more likely in nutrient poor soils than in nutrient rich soils [START_REF] Brown | How earthworms effect plant growth: burrowing into the mechanisms[END_REF]). Although only L. terrestris showed significant effects, our results contradict this hypothesis. Our first hypothesis was not confirmed since LT, increased the shoot biomass and the total biomass of T. dubium only in the nutrient rich soil. Our second hypothesis was confirmed since there was no case where an earthworm species changed the shoot/root ratio in the rich soil but not in the poor soil. For example, for P. annua the presence of LT increased the shoot/root ratio but only in the poor soil as predicted. Few significant interactions between the anecic and the endogeic earthworms were found in this study, and these interactions did not depend on the soil type. Consequently, our third hypothesis was not verified. Similarly, our fourth hypothesis was not supported by our results since there was no case where mineral fertilization decreased or suppressed a positive effect of an earthworm species on the growth of a plant species.

Plant growth

Effects of LT on the growth of T. dubium varied with the soil type but contrary to our expectation, this earthworm species did not have any significant effects on this plant in the nutrient poor soil. Indeed, LT increased the shoot and total biomasses of T. dubium only in the nutrient rich soil and mineral fertilization did not modify these effects (see Fig. 1). These results suggest that LT effect on T. dubium was probably not due to an enhancement of mineralization. This hypothesis is supported by the fact that the fertilizer addition had the opposite effect: it increased the total biomass of this plant only in the poor soil. These results should thus be explained by one of the four other mechanisms through which earthworms influence plant growth [START_REF] Brown | How earthworms effect plant growth: burrowing into the mechanisms[END_REF][START_REF] Scheu | Effects of earthworms on plant growth: patterns and perspectives[END_REF]. A possible mechanism could be the production of growth regulators in the presence of LT [START_REF] Blouin | Earthworms (Millsonia anomala, Megascolecidae) do not increase rice growth through enhanced nitrogen mineralization[END_REF]. Indeed, as microbial biomass is generally higher in clayey soils than in sandy soils (Hendrix . 1998), the activity of LT could have led to a greater production of phytohormones, through the stimulation of bacteria in the nutrient rich soil than in the nutrient poor soil and this resulted in an increase in the growth of T. dubium. Moreover, T. dubium, as a legume, is less sensitive to an increase in mineralization [START_REF] Brown | How earthworms effect plant growth: burrowing into the mechanisms[END_REF][START_REF] Jenerette | Interactions of ecosystem processes with spatial heterogeneity in the puzzle of nitrogen limitation[END_REF], which further supports our rationale. An alternative explanation for the positive effect of LT on the legume is that this earthworm species could have increased microbial biomass burying organic matter in the soil which could in turn increase the immobilization of mineral nutrient [START_REF] Van Der Heijden | The unseen majority: soil microbes as drivers of plant diversity and productivity in terrestrial ecosystem[END_REF]) and allowed T. dubium to grow better because it is better adapted to nitrogen poor soils. This hypothesis is supported by the fact that the legume had a lower total biomass in the nutrient rich soil than in the nutrient poor one. Finally, the soil type itself did not affect the growth of P. annua and V. persica suggesting that these plant species are less sensitive to the soil quality than the legume.

Although LT increased the total biomass of P. annua without mineral fertilization, it decreased the total biomasses of P. annua and V. persica when mineral fertilizer was added. This suggests that, in our experiment, without mineral fertilization, LT affected these plants mostly through an increase in mineralization. An explanation for the negative effect of LT on the total biomass of P. annua and V. persica when mineral fertilizer was added is that this earthworm species could have enhanced the loss of added mineral nutrients through the galleries it produced. This suggests that LT could influence plant growth not only through mineralization of organic matter, at least when the soil contains enough organic matter, but also through its effects on nutrient losses [START_REF] Barot | Nutrient cycling efficiency explains the long-term effect of ecosystem engineers on primary production[END_REF][START_REF] Dominguez | Earthworms increase nitrogen leaching to greater soil depths in row crop agroecosystems[END_REF].

Shoot/root ratio

The effect of earthworms on plant resource allocation is poorly documented and understood [START_REF] Scheu | Effects of earthworms on plant growth: patterns and perspectives[END_REF], but increase of the shoot/root ratio in the presence of earthworms has been documented before [START_REF] Kreuzer | Decomposer animals (Lumbricidae, Collembola) and organic matter distribution affect the performance of Lolium perenne (Poaceae) and Trifolium dubium (Fabaceae)[END_REF][START_REF] Scheu | Links between the detritivore and the herbivore system: effects of earthworms and Collembola on plant growth and aphid development[END_REF]). In our study, LT effect on P. annua shoot/root ratio varied with the soil type, this earthworm species increased the shoot/root ratio only in the nutrient poor soil. The significant effect of the interaction between LT and the soil type on the shoot/root ratio is probably due to the strategies plants have evolved to optimize their resource allocation to their root system to efficiently take up nutrients. Indeed, when mineral nutrients are poorly available it is efficient for a plant to increase locally its root biomass in nutrient rich patches, while, if mineral nutrient availability is high enough in the whole soil, it is more efficient for a plant to decrease its total root biomass [START_REF] Wilson | A review of evidence on the control of shoot: root ratio, in relation to models[END_REF]. However, plants have evolved different thresholds, first, in the local nutrient availabilities triggering local root proliferation, second, in the general nutrient availabilities triggering a change in the shoot/root ratio [START_REF] Hutchings | Differential foraging for resources and structural plasticity in plants[END_REF]. In this way, the observed variations of the responsiveness to earthworms of the shoot/root ratio with the soil type would reflect the scale and the precision at which root systems exploit the soil [START_REF] Campbell | A trade-off between scale and precision in resource foraging[END_REF]. Moreover, the possible local release of plant growth factors [START_REF] Muscolo | Earthworm humic matter produces auxin-like effects on Daucus carota cell growth and nitrate metabolism[END_REF] in earthworm casts and local changes in soil structure due to earthworm activities are also likely to influence local root densities as well as whole shoot/root ratios. This suggests that studying thoroughly changes in the allocation to the root system and in the root system architecture in the presence of earthworms and in different soils would provide useful information on root foraging strategies.

Interactive effect of both earthworm species on plant growth Few significant effects of the interaction between LT and AC were found in our study; moreover the observed effects did not change with the soil type. This suggests that mechanisms through which the two earthworm species might influence plant growth do not interact. In our experiment this is also probably due to the fact that few significant effects of AC were found. Contrarily to other experiments using similar biomass of AC [START_REF] Kreuzer | Decomposer animals (Lumbricidae, Collembola) and organic matter distribution affect the performance of Lolium perenne (Poaceae) and Trifolium dubium (Fabaceae)[END_REF]Wurst et al. 2003[START_REF] Wurst | Do endogeic earthworms change plant competition? A microcosm study[END_REF], no significant effect of this earthworm species was found on plant growth in our study. AC individuals were probably less active than those of LT that produced a lot of casts on the soil surface (personal observations). Another complementary explanation is that AC might have enhanced mineralization or triggered the release of phytohormones in the soil without affecting significantly plant growth.

In this case, AC might have influenced the physiology of these plants, for example the allocation of N to the root and aerial system, without affecting their biomass. This hypothesis is supported by the results of another experiment where AC affected the N content of plant without affecting their growth [START_REF] Laossi | Effects of endogeic and anecic earthworms on the competition between four annual plants and their relative reproduction potential[END_REF]. Other studies are needed to test for this kind of effect.

Conclusion

We have confirmed that earthworm effects (especially L. terrestris) on plants change with soil type [START_REF] Doube | The influence of two species of earthworm (Aporrectodea trapezoides and Aporrectodea rosea) on the growth of wheat, barley and faba beans in three soil types in the greenhouse[END_REF]) and nutrient supply. However, our results suggest that earthworm effects do not necessarily decrease with soil content in mineral nutrients and that earthworms do not affect plant via mineralization only. Consequently, other mechanisms are influential and new experiments are required to predict when and where the different mechanisms are influential. This is required to be able to predict how earthworm effects on plant growth vary with soil characteristics such as the availability of mineral nutrients, organic matter content and texture. The issue should be tackled by systematically comparing earthworm effect on plant, everything else being equal, in many different soils representing various combinations of these characteristics.

Fig. 1

 1 Fig.1Effects of earthworms, soil type and fertilizer on the total biomasses of P. annua (PA), T. dubium (TD) and V. persica (VP). Abbreviations: C, control treatment; AC, A.

Fig. 2

 2 Fig.2Effects of earthworms, soil type and fertilizer on the shoot/root ratio of P. annua (PA), T. dubium (TD) and V. persica (VP). Abbreviations: C, control treatment; AC, A.

Table 1 )

 1 showed that L. terrestris

	Table 1 ANOVA table of F-values for the effects of			Root biomass Shoot biomass Total biomass Shoot/root
	earthworms (AC and LT),
	soils, fertilizer and plant		df F	F	F	F
	species on root, shoot and
	total biomass and shoot/root	AC	1	0.63	2.72	1.83	2.90
	ratio (Total df=209)	LT	1	0.30	0.03	0.07	6.63*
		Soil	1	4.30*	18.14***	12.17***	44.31***
		Fertilizer	1	42.92***	92.65***	55.68***	139.48***
		Plant species	2 199.22***	536.48***	609.26***	6.07**
		AC*LT	1	0.01	1.20	1.08	0.00
		AC*soil	1	3.14	0.11	0.03	0.03
		AC*Fertilizer	1	1.48	0.00	0.06	0.10
		AC*plant species	2	2.53	0.10	0.05	1.95
		LT*soil	1	0.11	4.02*	3.47*	0.00
		LT*Fertilizer	1	3.19	10.57***	11.46***	4.09*
		LT*plant species	2	0.90	4.56**	4.45**	0.01
		Soil*plant species	2	8.65***	17.34***	20.46***	1.10
		Soil*Fertilizer	1	10.70***	4.00*	1.26	42.48***
		Fertilizer*plant species	2	31.48***	27.74***	14.01***	0.94
		AC*LT*soil	1	0.11	0.08	0.11	0.57
		AC*LT*plant species	2	0.59	0.32	0.25	6.07***
		AC*LT*fertilizer	1	0.10	0.99	1.00	0.43
		LT*soil*fertilizer	1	0.07	0.54	0.39	0.02
		AC*soil*fertilizer	1	0.40	1.00	0.62	0.98
		plant species*soil*fertilizer	2	1.07	0.78	1.07	1.26
		AC*LT*soil*fertilizer	1	0.15	0.56	0.61	0.42
		AC*LT*plant species*fertilizer 2	3.12	68.18***	72.46***	5.08*
	*p<0.05; **p<0.01;	r²		0.79	0.90	0.89	0.62
	***p<0.001

Table 2

 2 ANOVA table of F-values for effects of earthworms (AC and LT), soil and fertilizer on root, shoot and total biomass and shoot-to-root ratio of Trifolium dubium (total df= 78), Poa annua (total df=74) and Veronica persica (total df=55)

	Plant Soil (2010) 328:109-118
		Shoot/	root ratio	F	2.17	22.86***	36.70***	5.68**	0.03	0.72	16.20***	0.32	0.71
		Total	biomass	F	6.20**	3.04	12.03**	.	0.01	6.65**	0.20	0.10	0.47
		Shoot	biomass	F	6.51*	1.53	15.41***	.	0.14	6.22**	0.03	0.07	0.46
	V. persica	Root	biomass	F	.	5.46*	0.26	.	.	.	4.69	.	0.15
		Shoot/	root ratio	F	4.66*	24.05***	153.33***	0.31	4.91*	1.81	19.94***	.	0.76
		Total	biomass	F	0.31	1.04	48.25***	.	0.42	14.74***	0.40	0.00	0.51
		Shoot	biomass	F	0.4	2.43	87.66***	.	0.38	13.66**	1.27	0.02	0.63
	P. annua	Root	biomass	F	.	5.93*	99.80***	.	.	.	4.96*	.	0.61
		Shoot/	root ratio	F	1.34	11.56**	29.63***	3.18**	1.79	2.23	17.82***	2.19	0.52
		Total	biomass	F	10.95**	135.03***	4.79*	.	10.11**	2.00	8.31**	3.09*	0.73
		Shoot	biomass	F	9.10**	117.69***	7.99**	.	9.54**	1.64	9.97**	3.07*	0.68
	T. dubium	Root	biomass	d f F	LT 1 .	Soil 1 35.71***	Fertilizer 1 29.40***	AC*LT 1 .	LT*soil 1 .	LT*Fertilizer 1 .	Soil*Fertilizer 1 6.26*	AC*LT*fertilizer 1 .	r² 0.49	*p<0.05; **p<0.01; ***p<0.001

Plant Soil (2010) 328:109-118

PlantSoil (2010) 328:109-118

Acknowledgments We thank Abraham Bartolomé-Lasa, Anaïs Aubert and Ignace Kouassi Kouadio for help during the experiment. We thank Michael Bonkowski for helpful comments on the manuscript. This study was supported by the ANR program "jeune chercheur 2005" through the SolEcoEvo project, (JC05_52230).


Roman G Novikov

Matteo Santacesaria

GLOBAL UNIQUENESS AND RECONSTRUCTION FOR THE MULTI-CHANNEL GEL'FAND-CALDER ÓN INVERSE PROBLEM IN TWO DIMENSIONS

Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions

Roman Novikov, Matteo Santacesaria

Equation (1.2) arises, in particular, in quantum mechanics, acoustics, electrodynamics; formally, it looks like the Schrödinger equation with potential v at zero energy. In addition, (1.2) comes up as a 2D-approximation for the 3D equation (see section 2).

The following inverse boundary value problem arises from this construction.

Problem 1. Given Φ, find v. This problem can be considered as the Gel'fand inverse boundary value problem for the multi-channel 2D Schrödinger equation at zero energy (see [START_REF] Gel'fand | Some problems of functional analysis and algebra[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF]) and can also be seen as a generalization of the 2D Calderón problem for the electrical impedance tomography (see [START_REF] Calderón | On an inverse boundary problem[END_REF], [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF]). In addition, the history of inverse problems for the two-dimensional Schrödinger equation at fixed energy goes back to [START_REF] Dubrovin | The Schrödinger equation in a periodic field and Riemann surfaces[END_REF] (see also [START_REF] Novikov | The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator[END_REF], [START_REF] Grinevich | The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy, (Russian)[END_REF] and references therein). Note also that Problem 1 can be considered as a model problem for the monochromatic ocean tomography (e.g. see [START_REF] Baykov | Mode Tomography of Moving Ocean[END_REF] for similar problems arising in this tomography).

In the case of complex-valued potentials the global injectivity of the map v → Φ was firstly proved in [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF] for D ⊂ R d with d ≥ 3 and in [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF] for d = 2 with v ∈ L p : in particular, these results were obtained by the use of global reconstructions developed in the same papers. This is the first paper which gives global (uniqueness and reconstruction) results for Problem 1 with M n (C)-valued potentials with n ≥ 2. Results in this direction were only known for potentials with many restrictions (e.g. see [START_REF] Xiaosheng | Inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials in two dimensions at fixed energy[END_REF]).

We emphasize that Problem 1 is not overdetermined, in the sense that we consider the reconstruction of a M n (C)-valued function v(x) of two variables, x ∈ D ⊂ R 2 , from a M n (C)-valued function Φ(θ, θ ′ ) of two variables, (θ, θ ′ ) ∈ ∂D × ∂D, where Φ(θ, θ ′ ) is the Schwartz kernel of the Dirichletto-Neumann operator Φ: this is one of the principal differences between Problem 1 and its analogue for D ⊂ R d with d ≥ 3. At present, very few global results are proved for non-overdetermined inverse problems for the Schrödinger equation on D ⊂ R d with d ≥ 2. Concerning these results, our paper develops the two-dimensional works [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF], [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF] and indicates 3D applications of the method. The non-overdetermined inverse problems, including multi-channel ones, are much more developed for the Schrödinger equation in dimension d = 1 (e.g. see [START_REF] Agranovich | The inverse problem of scattering theory, Translated from the Russian by[END_REF], [START_REF] Zakhariev | Direct and inverse problems[END_REF]).

We recall that in global results one does not assume that the potential v is small in some sense or is (piecewise) real analytic or is subject to some other serious restrictions.

Our global reconstruction procedure for Problem 1 follows the same scheme as in the scalar case given in [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF], with some fundamental modifications inspired by [START_REF] Bukhgeim | Recovering a potential from Cauchy data in the two-dimensional case[END_REF].

Let us identify R 2 with C and use the coordinates z = x 1 + ix 2 , z = x 1ix 2 , where (x 1 , x 2 ) ∈ R 2 . We define a special family of solutions of equation (1.2), which we call the Buckhgeim analogues of the Faddeev solutions: ψ z 0 (z, λ), for z, z 0 ∈ D, λ ∈ C, such that -∆ψ + v(x)ψ = 0 over D, where in particular ψ z 0 (z, λ) → e λ(z-z 0 ) 2 I for λ → ∞ (i.e. for |λ| → +∞) and I is the identity matrix.

More precisely, for a matrix valued potential v of size n, we define ψ z 0 (z, λ) as

(1.4) ψ z 0 (z, λ) = e λ(z-z 0 ) 2 µ z 0 (z, λ),
where µ z 0 (•, λ) solves the integral equation

(1.5) µ z 0 (z, λ) = I + D g z 0 (z, ζ, λ)v(ζ)µ z 0 (ζ, λ)dReζ dImζ, I is the identity matrix of size n ∈ N, z, z 0 ∈ D, λ ∈ C and (1.6) g z 0 (z, ζ, λ) = e λ(ζ-z 0 ) 2 -λ( ζ-z 0 ) 2 4π 2 D e -λ(η-z 0 ) 2 + λ(η-z 0 ) 2 (z -η)(η -ζ) dReη dImη
is a Green function of the operator 4 ∂ ∂z + 2λ(zz 0 ) ∂ ∂ z in D, for z 0 ∈ D. We consider equation (1.5), at fixed z 0 and λ, as a linear integral equation for µ z 0 (•, λ) ∈ C 1 z ( D): we will see that it is uniquely solvable for |λ| > ρ 1 (D, N 1 , n), where v C 1 z ( D,Mn(C)) < N 1 (see Proposition 1.3). In order to state the reconstruction method we also define the Bukhgeim analogue of the Faddeev generalized scattering amplitude (1.7) and v| ∂D = 0. Consider, for z 0 ∈ D, the functions h z 0 , ψ z 0 , g z 0 defined above and Φ, Φ 0 the Dirichlet-to-Neumann maps associated to the potentials v and 0, respectively. Then the following reconstruction formulas and equation hold:

h z 0 (λ) = D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 v(z)µ z 0 (z, λ)dRez dImz, for z 0 ∈ D, λ ∈ C.
v(z 0 ) = lim λ→∞ 2 π |λ|h z 0 (λ), (1.8) h z 0 (λ) = ∂D e -λ(z-z 0 ) (Φ -Φ 0 )ψ z 0 (z, λ)|dz|, (1.9) ψ z 0 (z, λ)| ∂D = e λ(z-z 0 ) 2 I + ∂D G z 0 (z, ζ, λ)(Φ -Φ 0 )ψ z 0 (ζ, λ)|dζ|, (1.10)
where

(1.11) G z 0 (z, ζ, λ) = e λ(z-z 0 ) 2 g z 0 (z, ζ, λ)e -λ(ζ-z 0 ) 2 , z 0 ∈ D, z, ζ ∈ ∂D, λ ∈ C, |λ| > ρ 1 (D, N 1 , n), where v C 1 z ( D,Mn(C)) < N 1 .
In addition, if v ∈ C 2 ( D, M n (C)) with v C 2 ( D,Mn(C)) < N 2 and ∂v ∂ν | ∂D = v| ∂D = 0 then the following estimates hold:

v(z 0 ) - 2 π |λ|h z 0 (λ) ≤ a(D, n) log(3|λ|) |λ| 1/2 N 2 (N 2 + 1), (1.12a) v(z 0 ) - 2 π |λ|h z 0 (λ) ≤ b(D, n) (log(3|λ|)) 2 |λ| 3/4 N 2 (N 2 2 + 1), (1.12b) for |λ| > ρ 2 (D, N 1 , n), z 0 ∈ D. Remark 1. Note that in Theorem 1.1, ρ j = ρ j (D, N 1 , n), j = 1, 2 (where v C 1 z ( D,Mn(C)) < N 1 ), are arbitrary fixed positive constants such that 2n c 2 (D) |λ| 1 2 v C 1 z ( D) < 1, |λ| ≥ 1, if |λ| > ρ 1 , 2n c 2 (D) |λ| 1 2 v C 1 z ( D) ≤ 1 2 , |λ| ≥ 1, if |λ| > ρ 2 , (1.13)
where c 2 is the constant in Lemma 3.1.

Remark 2. Note that estimate (1.12b) is not strictly stronger than (1.12a) because of the presence of the N 3 2 factor.

In order to make use of the reconstruction given by Theorem 1.1, the following two propositions are necessary: Proposition 1.2. Under the assumptions of Theorem 1.1 (without the additional assumptions used for (1.12)), equation (1.10) is a Fredholm linear integral equation of the second kind for ψ z 0 ∈ C(∂D).

Proposition 1.3. Under the assumptions of Theorem 1.1 (without the additional assumptions used for (1.12)), for |λ| > ρ 1 (D, N 1 , n), where v C 1 z ( D,Mn(C)) < N 1 , equations (1.5) and (1.10) are uniquely solvable in the spaces of continuous functions on D and ∂D, respectively. Remark 3. Note that the assumption that v| ∂D = 0 is unnecessary for formula (1.9), equation (1.10) and Propositions 1.2, 1.3. In addition, formula (1.8) also holds without this assumption if

(1.14) ∂D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 w(z)|dz| → 0 as |λ| → ∞,
for fixed z 0 ∈ D and each w ∈ C 1 (∂D). The class of domains D for which (1.14) holds for each z 0 ∈ D is large and includes, for example, all ellipses. Note also that if

v| ∂D = 0 but v ≡ Λ ∈ M n (C)
on some open neighborhood of ∂D in D, then estimates (1.12) hold with h z 0 (λ) replaced by The global reconstruction of Theorem 1.1 is fine in the sense that is consists in solving Fredholm linear integral equations of the second type and using explicit formulas; nevertheless this reconstruction is not optimal with respect to its stability properties: see [START_REF] Burov | Practical application possibilities of the functional approach to solving inverse scattering problems[END_REF], [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF], [START_REF] Bikowski | Direct numerical reconstruction of conductivities in three dimensions using scattering transforms[END_REF] for discussions and numerical implementations of the aforementioned similar (but overdetermined) reconstruction of [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF] for d = 3 and n = 1. An approximate but more stable reconstruction method for Problem 1 will be published in another paper.

(1.15) h + z 0 (λ) = h z 0 (λ) + R 2 \D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 Λχ(z)dRez dImz, where χ ∈ C 2 (R 2 , R), χ ≡ 1 on D,
The present paper is focused on global uniqueness and reconstruction for Problem 1 for n ≥ 2. In addition, using the techniques developed in the present work and following the scheme of [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF] it is also possible to obtain a global logarithmic stability estimate for Problem 1 in the multi-channel case. Following inverse problem traditions (e.g. see [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF], [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF], [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]) this result will be published in another paper.

Approximation of the 3D equation

In this section we recall how the multi-channel two-dimensional Schrödinger equation can be seen as an approximation of the scalar 3D equation in a cylindrical domain; in this framework, three-dimensional inverse problems can be approximated by two-dimensional ones.

Let L = [a, b] for some a, b ∈ R and consider the complex-valued potential v(x, z) defined on the set D × L, where x = (x 1 , x 2 ) ∈ D ⊂ R 2 , z ∈ L. We consider the equation

(2.1) -∆ψ(x, z) + v(x, z)ψ(x, z) = 0 in D × L.
Now, for every x ∈ D we can write ψ(x, z) = ∞ j=1 ψ j (x)φ j (z), where {φ j } is the orthonormal basis of L 2 (L) given by the eigenfunctions of -d 2 dz 2 : more precisely

- d 2 dz 2 φ j (z) = λ j φ j (z) for z ∈ L, (2.2)
φ j | ∂L = 0 (for example) (2.3) L φi (z)φ j (z)dz = δ ij and ψ j (x) = L ψ(x, z) φj (z)dz. Now equation (2.1) reads ∞ j=1 (-∆ x ψ j (x)φ j (z) -ψ j (x)∆ z φ j (z)) + v(x, z) ∞ j=1 ψ j (x)φ j (z) = 0. (2.4)
Using (2.2)-(2.4) and the properties of {φ j (z)}, we obtain that equation (2.1) is equivalent to the following infinite-dimensional system

-∆ x ψ i (x) + λ i ψ i (x) + ∞ j=1 V ij (x)ψ j (x) = 0, for i = 1, . . . , (2.5)
where

V ij (x) = L φi (z)v(x, z)φ j (z)dz.
Notice that if v = v then V * = V . Now, if we impose 1 ≤ i, j ≤ n for some n ∈ N, we find equation (1.2).

We also give here the relation between the Dirichlet-to-Neumann (D-t-N) operators of the 3D equation and that of the 2D multi-channel equation. If Φ(θ, z, θ ′ , z ′ ) is the Schwartz kernel of the D-t-N operator of the 3D problem, and (Φ ij (θ, θ ′ )) i,j≥1 that of the 2D infinity-channel problem, we have

(2.6) Φ ij (θ, θ ′ ) = L×L Φ(θ, z, θ ′ , z ′ ) φi (z)φ j (z ′ )dz dz ′ , where θ, θ ′ ∈ ∂D, z, z ′ ∈ L. This follows from ∂D×L Φ(θ, z, θ ′ , z ′ )f (θ ′ , z ′ )dθ ′ dz ′ = ∞ i=1   ∞ j=1 ∂D Φ ij (θ, θ ′ )f j (θ ′ )dθ ′   φ i (z), (2.7
)

for every f ∈ C 1 (∂(D×L)) such that f | D×∂L = 0 and f (θ, z) = ∞ j=1 f j (θ)φ j (z).
Let us remark that reductions of 3D direct and inverse problems to multichannel 2D problems are well known in the physical literature for a long time (e.g. see [START_REF] Baykov | Mode Tomography of Moving Ocean[END_REF]). Nevertheless, we do not know a reference containing formula (2.6) in its precise form.

Preliminaries

In this section we introduce and give details about the above-mentioned family of solutions of equation (1.2), which will be used throughout all the paper.

Let us define the function spaces The functions

C 1 z ( D) = {u : u, ∂u ∂ z ∈ C( D, M n (C))} with the norm u C 1 z ( D) = max( u C( D) , ∂u ∂ z C( D) ), u C( D) =
G z 0 (z, ζ, λ), g z 0 (z, ζ, λ), ψ z 0 (z, λ), µ z 0 (z, λ) defined in Sec- tion 1, satisfy 4 ∂ 2 ∂z∂ z G z 0 (z, ζ, λ) = δ(z -ζ), (3.1) 4 ∂ 2 ∂ζ∂ ζ G z 0 (z, ζ, λ) = δ(ζ -z), (3.2) 4 ∂ ∂z + 2λ(z -z 0 ) ∂ ∂ z g z 0 (z, ζ, λ) = δ(z -ζ), (3.3) 4 ∂ ∂ ζ ∂ ∂ζ -2λ(ζ -z 0 ) g z 0 (z, ζ, λ) = δ(ζ -z), (3.4) -4 ∂ 2 ∂z∂ z ψ z 0 (z, λ) + v(z)ψ z 0 (z, λ) = 0, (3.5) -4 ∂ ∂z + 2λ(z -z 0 ) ∂ ∂ z µ z 0 (z, λ) + v(z)µ z 0 (z, λ) = 0, (3.6)
where z, z 0 , ζ ∈ D, λ ∈ C, δ is the Dirac's delta. (In addition, it is assumed that (1.5) is uniquely solvable for µ z 0 (•, λ) ∈ C 1 z ( D) at fixed z 0 and λ.) Formulas (3.1)-(3.6) follow from (1.5), (1.6), (1.11) and from

∂ ∂ z 1 π(z -ζ) = δ(z -ζ), ∂ ∂z + 2λ(z -z 0 ) e -λ(z-z 0 ) 2 + λ(z-z 0 ) 2 π(z -ζ) e λ(ζ-z 0 ) 2 -λ( ζ-z 0 ) 2 = δ(z -ζ),
where z, ζ, z 0 , λ ∈ C. We say that the functions G z 0 , g z 0 , ψ z 0 , µ z 0 , h z 0 are the Bukhgeim-type analogues of the Faddeev functions (see [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]). We recall that the history of these functions goes back to [START_REF] Faddeev | Growing solutions of the Schrödinger equation[END_REF] and [START_REF] Beals | Multidimensional inverse scatterings and nonlinear partial differential equations, Pseudodifferential operators and applications[END_REF]. Now we state some fundamental lemmata. Let (3.7)

g z 0 ,λ u(z) = D g z 0 (z, ζ, λ)u(ζ)dReζ dImζ, z ∈ D, z 0 , λ ∈ C,
where g z 0 (z, ζ, λ) is defined by (1.6) and u is a test function.

Lemma 3.1 ([17]

). Let g z 0 ,λ u be defined by (3.7). Then, for z 0 , λ ∈ C, the following estimates hold:

g z 0 ,λ u ∈ C 1 z ( D), for u ∈ C( D), (3.8) g z 0 ,λ u C 1 ( D) ≤ c 1 (D, λ) u C( D) , for u ∈ C( D), (3.9) g z 0 ,λ u C 1 z ( D) ≤ c 2 (D) |λ| 1 2 u C 1 z ( D) , for u ∈ C 1 z ( D), |λ| ≥ 1. (3.10) Given a potential v ∈ C 1 z ( D) we define the operator g z 0 ,λ v simply as (g z 0 ,λ v)u(z) = g z 0 ,λ w(z), w = vu, for a test function u. If u ∈ C 1 z ( D), by Lemma 3.1 we have that g z 0 ,λ v : C 1 z ( D) → C 1 z ( D), (3.11) g z 0 ,λ v op C 1 z ( D) ≤ 2n g z 0 ,λ op C 1 z ( D) v C 1 z ( D) ,
where (3.11) and Lemma 3.1 implies existence and uniqueness of µ z 0 (z, λ) (and thus also

• op C 1 z ( D) denotes the operator norm in C 1 z ( D), z 0 , λ ∈ C. In addition, g z 0 ,λ op C 1 z ( D) is estimated in Lemma 3.1. Inequality
ψ z 0 (z, λ)) for |λ| > ρ 1 (D, N 1 , n). Let µ (k) z 0 (z, λ) = k j=0 (g z 0 ,λ v) j I, h (k) z 0 (λ) = D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 v(z)µ (k) z 0 (z, λ)dRez dImz,
where z, z 0 ∈ D, λ ∈ C, k ∈ N ∪ {0}.

Lemma 3.2 ([17]

). For v ∈ C 1 z ( D) such that v| ∂D = 0 the following formula holds:

(3.12) v(z 0 ) = 2 π lim λ→∞ |λ|h (0) z 0 (λ), z 0 ∈ D. In addition, if v ∈ C 2 ( D), v| ∂D = 0 and ∂v ∂ν | ∂D = 0 then (3.13) v(z 0 ) - 2 π |λ|h (0) z 0 (λ) ≤ c 3 (D, n) log(3|λ|) |λ| v C 2 ( D) , for z 0 ∈ D, λ ∈ C, |λ| ≥ 1.
Following the proof of [17, Lemma 6.2] and assuming (1.14), we have that limit (3.12) is valid without the assumption that v| ∂D = 0. In addition, if

v| ∂D = 0 but v ≡ Λ ∈ M n (C)
h (0),+ z 0 (λ) = h (0) z 0 (λ) + R 2 \D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 Λχ(z)dRez dImz,
where χ ∈ C 2 (R 2 , R), χ ≡ 1 on D, suppχ is compact, and the constant c 3 depending also on χ.

Let (3.15) W z 0 (λ) = D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 w(z)dRe zdIm z,
where z 0 ∈ D, λ ∈ C and w is some M n (C)-valued function on D. (One can see that [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]). For w ∈ C 1 z ( D) the following estimate holds:

W z 0 = h (0) z 0 for w = v.) Lemma 3.3 ([
(3.16) |W z 0 (λ)| ≤ c 4 (D) log (3|λ|) |λ| w C 1 z ( D) , z 0 ∈ D, |λ| ≥ 1. Lemma 3.4. For v ∈ C 1 z ( D) and for g z 0 ,λ v op C 1 z ( D) ≤ δ < 1 we have that µ z 0 (•, λ) -µ (k) z 0 (•, λ) C 1 z ( D) ≤ δ k+1 1 -δ , (3.17)
|h z 0 (λ) -h (k) z 0 (λ)| ≤ c 5 (D, n) log(3|λ|) |λ| δ k+1 1 -δ v C 1 z ( D) , (3.18)
where z 0 ∈ D, λ ∈ C, |λ| ≥ 1, k ∈ N ∪ {0}.
The proof of Lemma 3.4 in the scalar case can be found in [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]: the generalization to the matrix-valued case is straightforward.

Lemma 3.5. The function g z 0 (z, ζ, λ) satisfies the following properties:

g z 0 (z, ζ, λ) is continuous for z, ζ ∈ D, z = ζ, z 0 ∈ D, (3.19)
|g z 0 (z, ζ, λ)| ≤ c 6 (D)| log |z -ζ||, z, ζ ∈ D, z 0 ∈ D, (3.20)
where λ ∈ C.

These properties follow from the definition (1.6) and from classical estimates (see [START_REF] Vekua | Generalized Analytic Functions[END_REF]). Lemma 3.6. Under the assumptions of Proposition 1.2, the Schwartz kernel (Φ -Φ 0 )(z, ζ) of the operator Φ -Φ 0 satisfies the following properties:

(Φ -Φ 0 )(z, ζ) is continuous for z, ζ ∈ ∂D, z = ζ, (3.21) |(Φ -Φ 0 )(z, ζ)| ≤ c 7 (D, v, n)| log |z -ζ||, z, ζ ∈ ∂D. (3.22)
For a proof of this Lemma in the scalar case we refer to [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0, Funkt[END_REF][START_REF] Novikov | Formulae and equations for finding scattering data from the Dirichletto-Neumann map with nonzero background potential[END_REF]: the generalization to the matrix-valued case is straightforward. We begin with a matrix version of Alessandrini's identity (see [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] for the scalar case):

(4.1) ∂D u 0 (z)(Φ -Φ 0 )u(z)|dz| = D u 0 (z)v(z)u(z)dRez dImz
for any sufficiently regular M n (C)-valued function u (resp. u 0 ) such that ∆u 0 = 0 (resp. (-∆ + v)u = 0) in D. This follows from Stokes's theorem, exactly as in the scalar case.

The general matrix version of Alessandrini's identity (that will not be used) (4.2)

∂D u 1 (z)(Φ 2 -Φ 1 )u 2 (z)|dz| = D u 1 (z)(v 2 (z) -v 1 (z))u 2 (z)dRez dImz for u 1 , u 2 ∈ C 2 ( D, M n (C)) such that (-∆ + v j )u j = 0 in D,
v(z 0 ) - 2 π |λ|h z 0 (λ) ≤ c 3 (D, n) log(3|λ|) |λ| v C 2 ( D) + c 5 (D, n) log(3|λ|) |λ| 1/2 v 2 C 1 z ( D) ≤ c 8 (D, n) log(3|λ|) |λ| 1/2 ( v C 2 ( D) + v 2 C 1 z ( D) ), for λ such that 2n c 2 (D) |λ| 1 2 v C 1 z ( D) ≤ 1 2 , |λ| ≥ 1,
which implies (1.12a). In order to prove (1.12b) we will need the following lemma:

Lemma 4.1. Let g z 0 ,λ u be defined by (3.7), where u ∈ C 1 z ( D), z 0 , λ ∈ C. Then the following estimate holds:

g z 0 ,λ u C( D) ≤ η(D) log(3|λ|) |λ| 3 4 u C 1 z ( D) , |λ| ≥ 1. (4.4)
Proof of Lemma 4.1. As in the proof of [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]Lemma 3.1], we can write g z 0 ,λ = 1 4 T Tz 0 ,λ , for z 0 , λ ∈ C, where

T u(z) = - 1 π D u(ζ) ζ -z dReζ dImζ, Tz 0 ,λ u(z) = - e -λ(z-z 0 ) 2 + λ(z-z 0 ) 2 π D e λ(ζ-z 0 ) 2 -λ( ζ-z 0 ) 2 ζ - z u(ζ)dReζ dImζ,
for z ∈ D and u a test function. We have that (see [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF]):

T w ∈ C 1 z ( D), (4.5) T w C 1 z ( D) ≤ η 1 (D) w C( D) , where w ∈ C(D), (4.6) Tz 0 ,λ u ∈ C( D), (4.7) Tz 0 ,λ u C( D) ≤ η 2 (D) |λ| 1 2 u C 1 z ( D) , |λ| ≥ 1, (4.8) Tz 0 ,λ u C( D) ≤ log(3|λ|)(1 + |z -z 0 |)η 3 (D) |λ||z -z 0 | 2 u C 1 z ( D) , |λ| ≥ 1, (4.9)
where u ∈ C 1 z ( D), z 0 , λ ∈ C. Let z 0 ∈ D, 0 < δ < 1 2 and B z 0 ,δ = {z ∈ C : |zz 0 | < δ}. We have

|4πg z 0 ,λ u(z)| = D Tz 0 ,λ u(ζ) ζ -z dReζ dImζ (4.10) ≤ B z 0 ,δ ∩D | Tz 0 ,λ u(ζ)| |ζ -z| dReζ dImζ + D\B z 0 ,δ | Tz 0 ,λ u(ζ)| |ζ -z| dReζ dImζ ≤ 2πδ η 2 (D) |λ| 1 2 u C 1 z ( D) + log(3|λ|)η 4 (D) |λ|δ u C 1 z ( D) ,
where we used the following estimate:

D\B z 0 ,δ 1 |ζ -z||ζ -z 0 | 2 dReζ dImζ = B z,δ ∩(D\B z 0 ,δ ) 1 |ζ -z||ζ -z 0 | 2 dReζ dImζ + D\(B z,δ ∪B z 0 ,δ ) 1 |ζ -z||ζ -z 0 | 2 dReζ dImζ ≤ 2π δ + D\(B z,δ ∪B z 0 ,δ ) 1 |ζ -z| 3 + 1 |ζ -z 0 | 3 dReζ dImζ ≤ η 5 (D) δ .
Putting δ = 1 2 |λ| -1 4 in (4.10) we obtain the result. Thus Lemma 4.1 is proved.

We now come back to the proof of (1.12b). Proceeding from (4.3) and Lemma 3.2 we obtain:

v(z 0 ) - 2 π |λ|h z 0 (λ) ≤ c 3 (D, n) log(3|λ|) |λ| v C 2 ( D) + 2 π |λ||h z 0 (λ) -h (0) z 0 (λ)|, (4.11)
for |λ| ≥ 1. In addition, from the definitions of h (k) , µ (k) , Lemmata 3.1 and 3.4, we have

|h z 0 (λ) -h (0) z 0 (λ)| ≤ D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 v(z)g z 0 ,λ v(z)dRez dImz + O log(3|λ|) |λ| 2 n 2 v 3 C 1 z ( D) , for λ such that 2n c 2 (D) |λ| 1/2 v C 1 z ( D) ≤ 1 2 , |λ| ≥ 1.
Repeating the proof of [17, Lemma 3.3] and using also Lemma 4.1, we have, for 0 < ε ≤ 1, D e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 v(z)g z 0 ,λ v(z)dRez dImz (4.12)

≤ D∩Bz 0 ,ε v(z)g z 0 ,λ v(z) C( D) dRez dImz + 1 4|λ| ∂(D\Bz 0 ,ε) v(z)g z 0 ,λ v(z) C( D) |z -z0 | |dz| + 1 2|λ| D\Bz 0 ,ε ∂ ∂ z v(z)g z 0 ,λ v(z) z -z0 dRez dImz ≤ σ 1 (D, n) v C( D) v C 1 z ( D) ε 2 log(3|λ|) |λ| 3/4 + σ 2 (D, n) v C( D) v C 1 z ( D) log(3ε -1 ) log(3|λ|) |λ| 1+3/4 + 1 8|λ| D\Bz 0 ,ε e λ(z-z 0 ) 2 -λ(z-z 0 ) 2 v(z) Tz 0 ,λ v(z) z -z0 dRez dImz , |λ| ≥ 1,
where we also used integration by parts and the fact that ∂ ∂ z g λ,z 0 u(z) = The proofs of the other formulas of Theorem 1.1 are based on identity (4.1). As µ z 0 (z, λ) = e -λ(z-z 0 ) 2 ψ z 0 (z, λ), we can write the generalized scattering amplitude as h z 0 (λ) = D e -λ(z-z 0 ) 2 v(z)ψ z 0 (z, λ)dRez dImz. Now identity (4.1) with u 0 (z) = e -λ(z-z 0 ) 2 I and u(z) = ψ z 0 (z, λ) reads ∂D e -λ(z-z 0 ) 2 (Φ -Φ 0 )ψ z 0 (z, λ)|dz| = D e -λ(z-z 0 ) 2 v(z)ψ z 0 (z, λ)dRez dImz which gives formula (1.9).

Since µ z 0 is a solution of equation (1.5), ψ z 0 (z, λ) satisfies the equation

1 .

 1 Introduction Let D be an open bounded domain in R 2 with with C 2 boundary and let v ∈ C 1 ( D, M n (C)), where M n (C) is the set of the n × n complex-valued matrices. The Dirichlet-to-Neumann map associated to v is the operator Φ : C 1 (∂D, M n (C)) → L p (∂D, M n (C)), p < ∞ defined by: (1.1) Φ(f ) = ∂ψ ∂ν ∂D where f ∈ C 1 (∂D, M n (C)), ν is the outer normal of ∂D and ψ is the H 1 ( D, M n (C))-solution of the Dirichlet problem (1.2) -∆ψ + v(x)ψ = 0 on D, ψ| ∂D = f ; here we assume that (1.3) 0 is not a Dirichlet eigenvalue for the operator -∆ + v in D.

Theorem 1 . 1 .

 11 Let D ⊂ R 2 be an open bounded domain with C 2 boundary and let v ∈ C 1 ( D, M n (C)) be a matrix-valued potential which satisfies(1.3)

  sup z∈ D |u| and |u| = max 1≤i,j≤n |u i,j |; we define also C 1 z ( D) = {u : u, ∂u ∂z ∈ C( D, M n (C))} with an analogous norm.

  on some open neighborhood of ∂D in D, then estimate (3.13) holds with h (0) z 0 (λ) replaced by(3.14)

4 .

 4 Proofs of Theorem 1.1, Propositions 1.2, 1.3 and Corollary 1.4

works if u 1 and v 1 3 )

 13 commute each other (but does not work in general). Proof of Theorem 1.1. Let us begin with the proof of formulas (1.8) and (1.12): we have indeed v(z 0 ) -2 π |λ|h z 0 (λ) ≤ v(z 0 ) -The first term in the right side goes to zero as |λ| → ∞ by Lemma 3.2, while the other by Lemmata 3.1 and 3.4. In addition, for v ∈ C 2 ( D, M n (C)) with v C 2 ( D) < N 2 and ∂v ∂ν | ∂D = 0, using (3.10), (3.11), (3.13) and (3.18) we obtain, from (4.3):

1 4/4 v 2 C 1 z

 121 Tz 0 ,λ u(z). The last term in (4.12) can be estimated independently on ε byσ 3 (D, n) log(3|λ|) |λ| 1+3/4 v C( D) v C 1 z ( D) (4.13)using the same argument as in the proof of Lemma 4.1 (see estimate (4.10)). Now putting ε = |λ| -1/2 in (4.12) we obtain|λ||h z 0 (λ)h (0) z 0 (λ)| ≤ σ 4 (D, n) (log(3|λ|)) 2 |λ| 3( D) ( v C 1 z ( D) + 1),for |λ| > ρ 2 (D, N 1 , n), which, together with (4.11), gives us (1.12b).

(4. 14 ) 1 .

 141 ψ z 0 (z, λ) = e λ(z-z 0 ) 2 I + D G z 0 (z, ζ, λ)v(ζ)ψ z 0 (ζ, λ)dReζ dImζ, for z 0 , z ∈ D, λ ∈ C, |λ| > ρ 1 (D, N 1 , n). Thus again by identity (4.1), with u 0 = G z 0 (z, ζ, λ)I and u(z) = ψ z 0 (ζ, λ), by (3.2) and (4.14) we obtain, for z ∈ ∂D,∂D G z 0 (z, ζ, λ)(Φ -Φ 0 )ψ z 0 (ζ, λ)|dζ| = D G z 0 (z, ζ, λ)v(ζ)ψ z 0 (ζ, λ)dReζ dImζ = ψ z 0 (z, λ)e λ(z-z 0 ) 2 I.This finish the proof of Theorem 1.Proof of Proposition 1.2. By (1.11) we have that G z 0 (z, ζ, λ) satisfies the same properties as g z 0 (z, ζ, λ) in Lemma 3.5, with the difference that the constant in(3.20) depends also on λ. This observation, along with Lemma 3.6, implies that the operator A(λ) defined asA(λ)u(z) = ∂D G z 0 (z, ζ, λ)(Φ -Φ 0 )u(ζ)|dζ|, z ∈ ∂D,for a test function u, is compact on the space of continuous functions on ∂D. Thus equation (1.10) is a Fredholm linear integral equation of the second kind in the space of continuous functions on ∂D.

  suppχ is compact, and with the constants a, b depending also on χ. The aforementioned matrix Λ, for example, can be related with a diagonal matrix composed by the eigenvalues {λ i } 1≤i≤n arising in section 2. Corollary 1.4. Let D ⊂ R 2 be an open bounded domain with C 2 boundary, let v 1 , v 2 ∈ C 1 ( D, M n (C)) be two matrix-valued potentials which satisfy (1.3) and Φ 1 , Φ 2 the corresponding Dirichlet-to-Neumann operators. If Φ 1 = Φ 2 then v 1 = v 2 .

	Theorem 1.1 and Propositions 1.2, 1.3 yield the following corollary:
	Theorem 1.1, Propositions 1.2, 1.3 and Corollary 1.4 are proved in section
	4.

Acknowledgements. We thank V. A. Burov, O. D. Rumyantseva, S. N. Sergeev for very useful discussions.

Proof of Proposition 1.3. First we have that equations (1.5) and (1.10) are well defined (i.e. Fredholm linear integral equations of the second type) on the spaces of continuous functions on D and ∂D respectively. This follows from (3.9) for the first equation and from Proposition 1.2 for the second one. Now if (1.5) admits a solution µ z 0 (z, λ) ∈ C( D), then by (3.8) and (1.5) one readily obtains µ z 0 (z, λ) ∈ C 1 z ( D). This solution is unique by Lemma 3.1 for |λ| > ρ 1 (D, N 1 , n) and by the same arguments as in the proof of Theorem 1.1 one has that ψ z 0 (z, λ)| z∈∂D satisfies equation (1.10).

Conversely, suppose that ψ z 0 (z, λ) ∈ C(∂D) satisfies equation (1.10): we have to show that ψ z 0 (z, λ), defined on D as the solution of the Dirichlet problem (-∆ + v)ψ z 0 (z, λ) = 0 with boundary values given by a solution of equation (1.10), satisfies (4.14).

By identity (4.1), ψ z 0 (z, λ) satisfies already equation (4.14) with z ∈ ∂D. Now, the function

where we called ψ j z 0 (z, λ) the Bukhgeim analogues of the Faddeev solutions corresponding to v j , for j = 1, 2). Thus we also have equality between the corresponding generalized scattering amplitudes,

If v j | ∂D = 0, for j = 1, 2, and D is such that (1.14) holds, then by Remark 3 we can apply Theorem 1.1 and argue as above.

The general case follows from stability estimates which will be published in another paper, following the scheme of [START_REF] Novikov | A global stability estimate for the Gel'fand-Calderón inverse problem in two dimensions[END_REF].


Nicolas Barre

Jo ¨rn Theuerkauf

Ludovic Verfaille
email: verfaille@loyalty.nc

Pierre Primot
email: p.primot@oie.int

Maurice Saoumoe Ŕeceived

T N Friedl

Barre

J Theuerkauf

Á P Primot

Á M Saoumoe
email: m-saoumoe@loyalty.nc

Exponential population increase in the endangered Ouve ´a Parakeet (Eunymphicus uvaeensis) after community-based protection from nest poaching

Keywords: Nest poaching, Public awareness, Recovery plan, Threats

The Ouve ´a Parakeet (Eunymphicus uvaeensis), endemic to Ouve ´a Island (Loyalty Islands, New Caledonia, south-west Pacific), is a rainforest bird that is dependent on tree cavities for nesting. It is threatened by deforestation, but also by competition for nest sites with introduced bees, harvesting for pets, and potentially predation by introduced species. Despite these threats, we show that the Ouve ´a Parakeet population increased exponentially from an estimated 617 (274-996) birds in 1993 to 2,090 (1,413) birds in 2009 (95% confidence interval). We explain this population increase by community-based protection measures that eliminated nest poaching. We recommend that these measures are maintained, remnant forest is protected, and the introduction of rats is prevented.

Introduction

The Ouve ´a Parakeet (Eunymphicus uvaeensis) is endemic to the small island of Ouve ´a (Loyalty Islands, New Caledonia, south-west Pacific). The species was previously considered a subspecies of the Horned Parakeet (Eunymphicus cornutus), which is endemic to mainland New Caledonia (Grande Terre). The Ouve ´a Parakeet is now recognized as a full species [START_REF] Juniper | Parrots-a guide to parrots of the world[END_REF][START_REF] Boon | Morphological, behavioural and genetic differentiation within the horned parakeet (Eunymphicus cornutus) and its affinities to Cyanoramphus and Prosopeia[END_REF]. The two species are the only taxa of the genus Eunymphicus.

The Ouve ´a Parakeet is a rainforest species that nests in cavities in mature trees (Robinet et al. 1996;Robinet and Salas 1996), and is consequently highly susceptible to deforestation and selective harvesting of large trees. It mainly feeds in forest, but also in fruit and vegetable gardens, and young secondary forests. Population estimates in the 1980s and early 1990s varied from less than 200 individuals [START_REF] King | Endangered birds of the world. The ICBP red data book[END_REF][START_REF] Lambert | Impact of protection on nest take and nesting success of parrots in Africa, Asia and Australasia[END_REF] to between 70 and 90 individuals [START_REF] Hahn | Anmerkungen zur Situation des Hornsittichs Eunymphicus cornutus auf Neukaledonien und Ouve ´a[END_REF]. The entire population was presumed to be restricted to 20 km 2 of forest in the northern part of the island [START_REF] Hannecart | Les oiseaux menace ´s de la Nouvelle-Cale ´donie et des ı ˆles proches[END_REF]. In 1963, a translocation of 15 parakeets to the adjacent island of Lifou failed to establish a second population [START_REF] Bruce | L'avifaune de Lifou (Archipel des Loyaute ´)[END_REF][START_REF] Hannecart | Les oiseaux menace ´s de la Nouvelle-Cale ´donie et des ı ˆles proches[END_REF]. In 1993, the first comprehensive census estimated the population at about 620 birds, most occurring in the north of Ouve ´a, but also a few in the south (Robinet et al. 1996).

Based on its small population, its limited range on just one island, the decrease of suitable habitat as a result of coconut plantations, and the impact of human harvest, the Ouve ´a Parakeet was categorised as Vulnerable in 1994 (Collar et al. 1994) Despite protection by local law since 1935 [START_REF] Hannecart | Les oiseaux menace ´s de la Nouvelle-Cale ´donie et des ı ˆles proches[END_REF], the species was a favoured pet and has been captured (by snaring or nest poaching without destroying the nest cavity) and exported from the island for over a century [START_REF] Sarasin | Nova Caledonia: Forschungen in Neu-Caledonien und auf den Loyalty-Inseln-Recherche scientifiques en Nouvelle Cale ´donie et aux Iles Loyalty[END_REF][START_REF] Berlioz | Les Psittacide ´s de la Nouvelle-Cale ´donie et des e ´tablissements franc ¸ais d'Oce ´anie[END_REF]. The impact of this harvesting was dependent on external demands and is thought to have increased in the second half of the twentieth century, with 30-50 young taken each year [START_REF] Robinet | The Ouvea parakeet-state of knowledge and conservation status[END_REF]. This harvest rate might have been close to the species' total annual reproductive output, and was probably unsustainable (Robinet et al. 1996).

Besides collection for pets and trade, there are several other potential threats to the survival and increase of the population: (1) the Little Fire Ant (Wasmannia auropunctata), which can severely impact biodiversity [START_REF] Jourdan | Threats on Pacific Islands: the spread of the tramp ant Wasmannia auropunctata (Hymenoptera: Formicidae)[END_REF], was introduced to Ouve ´a in the early 1990s; (2) the Honey Bee (Apis mellifera ligustica), which competes with the parakeets for tree cavities, was introduced to Ouve ´a in 1994, and by 2005, there were at least 250 colonies throughout the island [START_REF] Taylor | A report on the prevalence and population growth of Apis mellifera feral colonies on Ouvea and management strategies to reduce the impact on the Ouvea parakeet[END_REF]; (3) the New Caledonian Rainbow Lorikeet (Trichoglossus haematodus deplanchii), a potential nest-cavity competitor, was introduced from Grande Terre in the 1970s; and (4) a range of species prey upon the Ouve ´a Parakeet (Robinet et al. 1996), both natural (Brown Goshawk Accipiter fasciatus and Pacific Boa Candoia bibroni) and introduced (Feral Cat Felis catus). Another potential predator, the Black Rat (Rattus rattus), was introduced to Grande Terre and the other Loyalty Islands, but not to Ouve ´a (Robinet and Salas 1996).

In an effort to save the Ouve ´a Parakeet, the ASPO (Association for the Protection of the Ouve ´a Parakeet) was created in 1993. The ASPO was established as a nongovernmental organisation of mostly local members to ensure that its proposed conservation measures were accepted by the Ouve ´an community. The ASPO initiated a long-term study of the biology and ecology of the species [START_REF] Robinet | Ecology and conservation of the Ouvea parakeet Eunymphicus cornutus uvaeensis (Loyalty Islands, New Caledonia[END_REF] as a basis for two recovery plans (1997-2002 and 2003-2008) that were prepared with Ouve ´an inhabitants, and accepted by local authorities and the Loyalty Islands administration [START_REF] Aspo | Second se ´minaire pour la conservation de la Perruche d'Ouve ´a[END_REF][START_REF] Anonymous | Un exemple de gestion concerte ´e avec les coutumiers: la Perruche d'Ouve ´a. Bilan du second se ´minaire. 11-14. Le nouveau plan de sauvegarde de la Perruche d'Ouve ´a[END_REF]). The recovery plans recommended increasing law enforcement to tackle illegal trade, mitigating habitat degradation and restoring suitable habitat, monitoring nesting sites, testing artificial nests, preventing the introduction of Black Rats, locating and destroying Honey Bee nests in the wild, and improving public awareness of the value of the Ouve ´a Parakeet.

In order to meet the recommendations formulated by the Species Survival Commission of the IUCN [START_REF] Lambert | Impact of protection on nest take and nesting success of parrots in Africa, Asia and Australasia[END_REF], the South Pacific Regional Bird Conservation Programme Plan for Action [START_REF] Sherley | South Pacific regional bird conservation programme plan for action[END_REF], and the priorities defined in the two recovery plans, a population monitoring programme was established in 1993 to evaluate the efficacy of the conservation management programme. In this paper, we present the results of eight censuses of the Ouve ´a Parakeet conducted over 16 years, and discuss the reasons for the observed population trend.

Methods

Ouve ´a lies 100 km north-east of Grande Terre (20°23-44 0 S, 166°25-40 0 E), and forms part of the Loyalty Island archipelago. The island is relatively small (130 km 2 ), and flat (max. elevation 41 m), and is inhabited by 5,100 people, mostly of Melanesian and Polynesian origin. Ouve ´a receives an average annual rainfall of 1,250 mm, and is mainly covered by tropical rainforest on coral substrates (Fig. 1). Ouve ´a is described in detail by [START_REF] Robinet | The Ouvea parakeet-state of knowledge and conservation status[END_REF]. The entire island has recently been classified as an Important Bird Area [START_REF] Spaggiari | Zones importantes pour la conservation des oiseaux de Nouvelle-Cale ´donie[END_REF]. Besides recovery plan meetings, the ASPO organises community work through regular discussions with inhabitants of Ouve ´a, to explain the aims and activities of the association by various means of communication (including fact sheets, stickers, leaflets, posters, a newsletter, T-shirts, radio and TV spots, and films). In 1993, the ASPO employed three local guides from villages all over the island to spread conservation messages and to protect parakeet nests in the field. Since 2006, the number of employed guides has increased to nine. Employing local guides from as many villages as possible was essential for the ASPO conservation strategy, because nest poaching was formerly organised by each village. We believe that local guides, as accepted members of the local community, had a much higher impact on the attitudes of people than laws established by the government of Grande Terre. As nest takes were illegal, it is difficult to obtain detailed information on poaching rates before local guides were employed. However, we have no evidence of nest poaching occurring since at least 2000.

Ouve ´a Parakeets use a limited number of habitats (Robinet et al. 1996): medium to high rainforest (canopy height [6 m), early stages of secondary forests, and active or abandoned gardens with papaya (Carica papaya) and other plants consumed by the bird. About 41.5 km 2 of south Ouve ´a, and 20 km 2 of north Ouve ´a are covered by suitable habitats (analyses based on digital versions of 1:10,000 scale maps of IGN 1992, Robinet et al. 1996).

From 1993 to 2009, we surveyed parakeet numbers on eight occasions, all in November-December (Table 1).

Transects were walked slowly (1-2 km/h), usually by three observers, who followed either a forest track or a compass bearing through forest. Transects were typically carried out from 0600 to 1100 and from 1500 to 1800 hours. We did not survey in strong wind or rain. Whenever possible, the same itineraries were used on each census. However, small orientation errors, caused by the absence of permanent tracks or marks, meant that some portions of the transects did not correspond exactly during each survey; therefore, the length was not always the same. There were eight transects in the north and five in the south repeated during each survey (Fig. 1).

From 1993 to 2000, we used the line transect method of [START_REF] Bibby | Species factsheet: Eunymphicus uvaeensis[END_REF], as described in detail by Robinet et al. (1996). We used two distance belts to estimate population density: up to 15 m perpendicular from the transect line, and beyond 15 m. We chose the distance of 15 m to separate belts as we were confident that we were able to find all parakeets within this distance from the transect line. We divided transects into units of 500 m, which allowed us to calculate the mean number of birds per 500 m (based on all birds seen), a density (using only encounters within the inner 15 m belt), and an estimation of the population, as described in detail in Robinet et al. (1996).

Since 2003, we have estimated parakeet numbers by distance sampling [START_REF] Buckland | Distance sampling: estimating abundance of biological populations[END_REF], and have therefore recorded the exact distance of each observation to the transect line. When parakeets were in a flock, we estimated the distance to the flock based on the average distance to each individual. We calculated parakeet numbers and confidence intervals with Distance 6.0 Release 2 [START_REF] Thomas | Distance 6.0 Release 2. Research Unit for wildlife population assessment[END_REF]) and truncated any distances beyond 60 m from the transect line, which is proposed as standard for estimating parrot density in New Caledonia (A. Legault, J. Theuerkauf, S. Rouys et al., unpublished manuscript). We estimated parakeet numbers by multiplying the estimated densities with the area of suitable habitats on Ouve ´a. In order to allow for statistical comparison among years and methods, we compared the number of parakeets per 500 m (up to a maximum distance of 60 m from the transect line). To check if there was an increase in parakeet numbers, we tested whether the slope variable of the regression model was significant. We tested five regression models (linear, quadratic, growth, exponential, logistic) with PASW Statistics 18 (SPSS) to find out which model best fitted the data (ranked by F value). We aged parakeets in the field. Young parakeets (up to a few months old) are recognisable by a yellow bill and the absence of a crest. During surveys, we noted whenever we encountered Rainbow Lorikeets, Brown Goshawks or Little Fire Ants.

Results

The mean Ouve ´a Parakeet density increased threefold from 10 (4-16) birds per km 2 in 1993 to 34 (21-56) birds per km 2 (95% confidence interval) in 2009 (Fig. 2). Growth, logistic and exponential models best fitted the increase in the number of parakeets encountered per 500 m of transect per year (data in Table 1). F and P values of these three models were identical (entire island: F = 10.8, P = 0.017; south: F = 30.5, P = 0.001; north: F = 2.8, P = 0.153). The linear model was also significant for the entire island (P = 0.037) and the south (P = 0.050), whereas the quadratic model was not significant in any region.

Out of the 349 encounters with parakeets over the entire survey period, 40% were single birds, 43% were pairs and 17% were groups of three or more birds. In total, we aged 188 birds, including 146 in the north and 42 in the south. Of these, 19% (21% in the north, 12% in the south) were juveniles.

We did not detect any change in the relative abundance of Brown Goshawks during the 16-year monitoring period (1-2 birds observed per day throughout the study). The Rainbow Lorikeet was rare and was never observed in the habitat used by the Ouve ´a Parakeets. The Little Fire Ant was recorded on most of our transects in 1998-2003, but was a little less widespread from 2004 to 2009.

The financial investment for the conservation of the Ouve ´a Parakeet (ASPO budget) was insignificant before 1998, but increased afterwards continuously due to the support of the Loyalty Islands administration (Fig. 3). Initially, the money was mainly spent on public awareness (organisation of meetings and publications), but later most of the money was spent for salaries of guides (recruitment of local personnel all over the island for field surveys) and stationary costs (office, phone, etc.).

Discussion

In 2009, we estimated a mean population density of 34 parakeets per km 2 across Ouve ´a. Parakeet density was highest in the north, where it reached 57 birds per km 2 (see Fig. 2). These densities are about twice as high as the mean and maximum densities of Horned Parakeets observed on mainland New Caledonia (A. Legault, J. Theuerkauf, S. Rouys et al., unpublished manuscript). In the north, the population was relatively stable and high throughout the monitoring period. Most of the population increase has occurred in the south where the bird numbers were very low in 1993. As the percentage of young in the north was twice that in the south, it is likely that at least a part of the population increase in the south was caused by dispersal of birds from the productive northern subpopulation through the central isthmus of the island. Robinet and Salas (1996) observed that Ouve ´a Parakeets had a relatively high reproductive output, which we can confirm, as 19% of birds observed were fledglings. Ouve ´a Parakeets had an annual fledging rate of 63%, a clutch size of 2.9, and 1-2 broods per year (Robinet and Salas 1996), which is comparable to that of Horned Parakeets from the mainland of New Caledonia [START_REF] Theuerkauf | Cooperative breeding, mate guarding, and nest sharing in two parrot species of New Caledonia[END_REF]). Robinet et al. (1996) therefore suspected that the scarcity of tree cavities in mature trees was the limiting factor for Ouve ´a Parakeet density. This could explain why the bird is absent or scarce in some forested parts of the island: the transects on Ohnyot in the north and Hulup in the south are in rainforest, but the trees are of medium size, and are probably unsuitable for nests. Furthermore, two trials to install artificial nests (50 wooden nests of two different sizes in [START_REF] Buckland | Distance sampling: estimating abundance of biological populations[END_REF]35 PVC nests in 2005) in rainforests in the north of Ouve ´a (Teuta, Weneki and Goossana) were unsuccessful, as birds did not use the nests. Parrots only use artificial nest boxes if they are suitable, even if they have to compete for nests [START_REF] Brightsmith | Ecology and management of nesting blue-and-yellow macaws (Ara ararauna) in Mauritia palm swamps[END_REF]. Thus, we are unsure if the parakeet population in the high density areas was limited by the availability of suitable nest cavities. To contribute to the population increase, we suggest that additional artificial nest-cavities be provided in forested areas that lack mature trees and have low parakeet numbers.

The Ouve ´a Parakeet population increased exponentially during the monitoring period, despite little or no change in the state of potential threats caused by Brown Goshawks, Rainbow Lorikeets or Little Fire Ants. We have no reason to believe that other potential threats, such as Pacific Boas, Pacific Rats (Rattus exulans) or Feral Cats decreased or increased in numbers. Honey Bees occupied 10-16% of the known Ouve ´a Parakeet nest-sites [START_REF] Aspo | Second se ´minaire pour la conservation de la Perruche d'Ouve ´a[END_REF]. The ASPO staff destroyed or removed 187 bee colonies during 2002-2008, although this did not result in a noticeable reduction of bee colonies. We therefore believe that the increase was mainly due to the prevention of poaching since 1993.

Nest poaching is a serious problem for parrot species worldwide [START_REF] Wright | Nest poaching in neotropical parrots[END_REF], Pain et al. 2006). We think that the successful prevention of nest poaching on Ouve ´a was not primarily facilitated by legal protection, which was established since 1956, but by community-based measures taken after the creation of the ASPO. The continuous presence of local guides effectively prevented nest poaching, and the increased financial support allowed awareness and conservation actions. Today, many people on Ouve ´a appreciate the value of the bird, which has become the island emblem: the Ouve ´a council has included a parakeet in its logo, and a monumental sculpture representing the bird was erected at the Ouve ´a airport in 1999 as a reminder of the richness of the local biodiversity. This demonstrates that by remaining in the wild, the parakeet can be a natural resource providing income for those inhabiting an island where unemployment is high. Smuggling seems to have disappeared since the implementation of the awareness campaign and other ASPO actions.

We recommend continuing actions to raise awareness, implementing measures to prevent future introductions of potential predators (e.g. the Black Rat) and preserving all of the remnant rainforest, as proposed in the recovery plans [START_REF] Aspo | Second se ´minaire pour la conservation de la Perruche d'Ouve ´a[END_REF][START_REF] Anonymous | Un exemple de gestion concerte ´e avec les coutumiers: la Perruche d'Ouve ´a. Bilan du second se ´minaire. 11-14. Le nouveau plan de sauvegarde de la Perruche d'Ouve ´a[END_REF]. We believe that it is also important to improve incomes and job opportunities for the inhabitants of Ouve ´a, in order to prevent nest poaching and smuggling of the Ouve ´a Parakeet in the long term. We conclude that Ouve ´a Parakeet nest poaching has been eliminated because of the protection provided by local people, which is potentially due to the more reliable incomes offered through tourism than nest poaching.

Fig. 1

 1 Fig. 1 Distribution of rainforest (dark grey), other vegetation types (light grey), transects (lines) and Ouve ´a Parakeet (Eunymphicus uvaeensis) encounters during censuses and occasional observations from 1993 to 2009 (dots) on Ouve ´a, Loyalty Islands, New Caledonia (locality names from IGN 1992)

Fig. 2

 2 Fig. 2 Estimated parakeet numbers on Ouve ´a and parakeet density (with 95% confidence intervals) in the north and south with exponential regression lines (x = years after 1993)

  and upgraded to Endangered in 2000 (BirdLife International 2009). It is legally protected from trade by New Caledonian law, and in 2000, the genus Eunymphicus was transferred from Appendix II of CITES (since 1975) to Appendix I (UNEP-WCMC 2009).

Table 1

 1 Number of Ouve ´a Parakeets (Eunymphicus uvaeensis) recorded per 500 m, dates and total length of transects in the north (N), south (S) and the entire island (weighted by surface of suitable habitat) of Ouve ´a

	Year	Dates	Zone	Length of	Birds (0-15 m	Birds (16-60 m	Birds/500 m
				transects (km)	from transect)	from transects)
	1993	28/11-06/12	N	35.9	29	19	0.67
			S	29.1	2	1	0.05
			Ouve ´a				0.26
	1998	15-18/12	N	15.9	46	18	2.00
			S	9.9	3	0	0.15
			Ouve ´a				0.77
	2000	28/11-01/12	N	22.8	30	6	0.80
			S	21.1	4	3	0.17
			Ouve ´a				0.38
	2003	06-09/12	N	24.9	7	40	0.94
			S	27.8	11	18	0.52
			Ouve ´a				0.66
	2004	29/11-03/12	N	25.7	18	30	0.93
			S	27.8	10	9	0.34
			Ouve ´a				0.54
	2006	02-04/12	N	26.7	25	49	1.39
			S	27.0	9	28	1.37
			Ouve ´a				1.38
	2008	06-10/12	N	25.6	37	62	1.93
			S	25.7	16	24	0.78
			Ouve ´a				1.16
	2009	01-03/12	N	27.8	40	49	1.60
			S	26.3	18	12	0.57
			Ouve ´a				0.91

  Budget of the ASPO for public awareness, salaries and stationary costs in 1993-2009 Zusammenfassung Exponentieller Populationszuwachs des Ouve ´a-Sittichs (Eunymphicus uvaeensis) nach selbst durchgefu ¨hrten Schutzmaßnahmen der Inselbevo ¨lkerung gegen Nestwilderei Der Ouve ´a-Sittich (Eunymphicus uvaeensis), eine endemische Art der Loyalita ¨tsinsel Ouve ´a (Neukaledonien, Su ¨dwestpazifik), ist eine Feuchtwaldart, die Ba ¨umho ¨hlen zum Nisten beno ¨tigt. Die Art ist bedroht durch Waldzersto ¨rung, durch eingefu ¨hrte Bienen, die mit den Sittichen um Nistho ¨hlen konkurrieren, durch die Entnahme von Jungvo ¨geln aus den Nestern und potentiell durch eingefu ¨hrte Pra ¨datoren. Wir zeigen, dass trotz dieser Bedrohungen die Population der Ouve ´a-Sittiche exponentiell von gescha ¨tzten 617 (274-996) Vo ¨geln im Jahre 1993 auf 2.090 (1.280-3.413) Vo ¨gel im Jahre 2009 gewachsen ist (95% Konfidenzintervall). Diesen Populationszuwachs erkla ¨ren wir damit, dass die einheimische Bevo ¨lkerung die Schutzmaßnahmen selbst umsetzte, so dass die Entnahme der Jungvo ¨gel unterbunden werden konnte. Wir empfehlen diese Maßnahmen weiterzufu ¨hren, den verbliebenen Feuchtwald zu schu ¨tzen und die Einfu ¨hrung von Ratten zu verhindern.

	Annual budget in Euro	20,000 40,000 60,000	stationary salaries awareness
		0
		1992	1996	2000	2004	2008
				Year
	Fig. 3

J Ornithol (2010) 151:695-701

Acknowledgments This study was financed by the Province des Iles Loyaute ´(New Caledonia), the ASPO (New Caledonia), the Socie ´te ´Cale ´donienne d'Ornithologie (New Caledonia) and the British Birdwatching Fair (U.K.). J.T. was supported by the Loro Parque Fundacio ´n (Spain) and Fonds fu ¨r bedrohte Papageien-Zoologische Gesellschaft fu ¨r Arten-und Populationsschutz (Germany). We thank the members of the ASPO-S. Baoutuau, J.F. Chaouri, J.B. Dao, P. Dialla, V. Mindia, B. Tangopi, W. Wamo, L. We ´a, and A. We ´te

´we ´a-for guiding us on their lands; T.C. Greene and R.G. Powlesland for advice; M.F. Barre ´, J.C. Birot, M. Boissenin, V. Chartendrault, F. Desmoulins, J. Spaggiari, and C. Vanhoye for their help during field work; and D. Brightsmith, G. Dutson, T.W.P. Friedl, A. Legault, T. Mu ¨ller, J.M. Wunderle and anonymous reviewers for useful comments and corrections.