Etienne Baras

Jacques Slembrouck

Agus Priyadi

Darti Satyani

Laurent Pouyaud

Marc Legendre

Biology and culture of the clown loach Chromobotia macracanthus (Cypriniformes, Cobitidae): 3-Ontogeny, ecological and aquacultural implications

Keywords: Tropical freshwater fish, Ornamental fish, Clown loach, Aquaculture, Ontogeny, Pigmentation, Growth, Food intake

Hatchlings of clown loach (3.8 mm in total length, TL) are pelagic. When reared at 26-28 • C, they become benthic 3 days after hatching (dah), when their swim bladder is inflated. The fin development sequence (pectorals < caudal < dorsal = anal < pelvic) is typical of the clade Ostariophysi. All fins and finrays are fully elongated at 20 mm TL, but the finfold persists until 25-26 mm TL (start of juvenile stage). Melanophores appear at 3 dah, they form a 5-bar pattern at 5 dah, then two bars (III and V) vanish progressively, producing at 26 mm TL the 3-bar pattern that is typical of adults. Throughout the ontogeny, the pigment pattern exhibits a structural regularity (bars spaced at regular intervals), which is interpreted in a functional perspective by reference to the maintaining of crypsis and signalling throughout. Exogenous feeding commences at 4 dah (5.5 mm TL). Food intake (FI) increases rapidly, from 6% wet body mass (WM) at 5.5 mm TL to >20% WM in fish >7 mm TL. Gut evacuation rate (R g ) increases with increasing meal size and fish size, as a result of gut coiling (from 8 to 15 mm TL), and is highest at 11 mm TL (about 10% WM h -1 in fish feeding maximally). The allometric increase of FI and R g during the early larval stages is accompanied by increasing capacities for growth, so early sizes differences amplify rapidly during the ontogeny. Nevertheless, growth remains slow (mean of 0.4 mm TL day -1 from 4 to 29 dah; 0.9 mm TL day -1 for top growers). By contrast, unfed fish display long resistance to starvation (until 14-15 dah). The combination of slow growth and long resistance to starvation is discussed in respect to the reproductive phenology of the species, as the capacity of making metabolic economies prevails over fast growth for seasonal strategists spawning mainly at the start of the rainy season.

Introduction

The understanding of how a species can adapt to environmental conditions requires the capacity of identifying them at any life stage. This can be straightforward in adults, but more difficult at younger life stages, whereas these stages are generally more sensitive than others because of their small size, limited movement capacities, low energy reserves and restricted food spectrum [START_REF] Balon | Reproductive guilds of fishes: a proposal and definition[END_REF][START_REF] Kamler | Early Life History of Fish: An Energetics Approach[END_REF]. Genetic tools such as DNA barcoding nowadays bridge this gap [START_REF] Ward | The campaign to DNA barcode all fishes, FISH-BOL[END_REF]), but nevertheless remain of limited use in many instances, because of their cost for large sample size or destructive nature for very small organisms. Morphological descriptors (including pigmentation) thus remain tremendously a Corresponding author: etienne.baras@ird.fr useful and central to all identification keys of fish larvae (e.g. [START_REF] Koblitskaya | Identification keys for young of freshwater fishes[END_REF][START_REF] Pinder | Keys to Larval and Juvenile Stages of Coarse Fishes from Fresh Waters in the British Isles, Ambleside, The Ferry House, Freshwater Biological Association[END_REF]. Likewise, the study of fish behaviour during the early stages can provide essential insights into their ecology, for example as regards their propensity to drift, which can be inferred from their use of the water column (e.g. [START_REF] Araujo-Lima | Transport of larval fish in the Amazon[END_REF][START_REF] Lucas | Migration of Freshwater Fishes[END_REF]. Similarly, the study of their physiology, in particular the dynamics of yolk absorption, resistance to starvation, food intake and growth, can be extremely useful, as these traits intimately condition their survival in the wild [START_REF] Kamler | Ontogeny of yolk-feeding fish: an ecological perspective[END_REF][START_REF] Yúfera | The onset of exogenous feeding in marine fish larvae[END_REF]. It is difficult to obtain accurate data on these traits in natural environments, whereas this can be achieved among fish born in captivity and raised for aquacultural purposes. All aforementioned variables, of physiological, behavioural or morphological nature, are also central to the concerns of aquaculturists to maximise survival, growth, size homogeneity and food conversion efficiency.

Article published by EDP Sciences

The present study investigates the ontogeny of the clown loach Chromobotia macracanthus (Bleeker 1852), with the objectives of providing information for its propagation in captivity, its univocal identification and better understanding of its ecology in natural environments. Clown loach is a key species for ornamental aquaculture [START_REF] Ng | Freshwater fishes of Southeast Asia: potential for the aquarium fish trade and conservation issues[END_REF], but for which all marketed specimens exclusively come from capturebased fisheries. The current degree of endangerment of this species is largely unknown, but is presumably worth considering in view of its endemic distribution (rivers of Sumatra and Borneo Islands) and intensity of fisheries, which collect 20-50 million juveniles annually. Recent research efforts have enabled the maturation of clown loach broodfish in captivity, the production of functional gametes following hormonally induced reproduction, and the birth of viable embryos and larvae [START_REF] Legendre | Biology and culture of the clown loach Chromobotia macracanthus (Cypriniformes, Cobitidae): 1-Hormonal induced breeding, unusual latency response and egg production in two populations from Sumatra and Borneo Islands[END_REF][START_REF] Slembrouck | Biology and culture of the clown loach Chromobotia macracanthus (Cypriniformes, Cobitidae): 2-Importance of water movement and temperature during egg incubation[END_REF]). Here, the ontogeny of clown loach is analysed, focusing on embryos, larvae and young juveniles, which are the main critical life stages in the wild and those of interest for aquaculturists, as the market size for clown loach does not exceed 4-5 cm. In addition to standard morphological descriptors, special attention is dedicated to the pigment pattern of clown loach, which consists in a typical alternation of yellow-orange and black bars in adults, but is also found at much younger life stages. The emphasis is also laid on other traits that could contribute to making the culture of clown loach sustainable, in particular the ontogenetic variations of meal size and gut evacuation rate, which are essential to the optimisation of feeding schedules and costs [START_REF] Jobling | Fish Bioenergetics[END_REF]Houlihan et al. 2001).

Materials and methods

Fish and rearing conditions

Experimental fish were obtained from the hormonally induced ovulation and spermiation of broodfish originating from the River Musi (Sumatra) and held captive in the experimental facilities of the BP2BIH research station (Depok, West Java, Indonesia). All observations were done in indoor recirculating systems, under a day length of 12L:12D, with light intensities of 50-100 and <0.01 lux during the hours of light and darkness, respectively. Oxygen was maintained near saturation (>90%) throughout in all rearing structures. Throughout the experiments in the present study, water quality stood as follows (ranges): pH: 7.75-7.86; hardness: 12.2-18.2 mg L -1 ; conductivity: 123-254 μS cm -1 ; total ammonia nitrogen, TAN: 0.002-006 mg L -1 ; nitrite, N-NO - 2 : 0.001-0.002 mg L -1 . Hormonal-induced breeding and ova fertilization were performed after [START_REF] Legendre | Biology and culture of the clown loach Chromobotia macracanthus (Cypriniformes, Cobitidae): 1-Hormonal induced breeding, unusual latency response and egg production in two populations from Sumatra and Borneo Islands[END_REF]. Incubation was carried out at 26-28 • C, which is near optimal for this species, in zugger jars (30 cm in diameter, 35 cm high; flow of 1-2 L min -1 ), as developmental deformities occur in absence of egg agitation [START_REF] Slembrouck | Biology and culture of the clown loach Chromobotia macracanthus (Cypriniformes, Cobitidae): 2-Importance of water movement and temperature during egg incubation[END_REF]. Following hatching, embryos ascended the water column in the zugger jars and spontaneously entered hapas (i.e. fine mesh cages) placed in the main tank of the hatchery water recirculation system.

At the age of 36 h after hatching (hereafter hah), 30 fish were randomly collected for the study of resistance to food deprivation. The fish were housed individually in small plastic containers (0.125 L; 6.5 h × 5.0 cm in diameter) that were filled with aerated well water and partly immersed in the main tank of the hatchery (next to the hapas) for maintenance of temperature (range of 25.8-27.5 • C). Housing in isolation was desirable to avoid possible biases originating from necrophagy or cannibalism. Water was changed every two days. Survival was verified every morning and evening by moving gently the fish with a water flow produced by a pipette.

The other fish were transferred into 90-L aquaria (50 × 50 × 36 (h) cm) in an indoor water recirculation system at ambient temperature (mean of 27.5 • C, range of 25.7-29.3 • C). The stocking density was 1.33 fish L -1 (120 fish per tank). From the age of 72 hah onwards, freshly hatched Artemia nauplii were distributed six times a day (02:30, 06:30, 10:30, 14:30, 18:30 and 22:30 h). At each meal, feeding was supposedly in slight excess, as verified by the presence of live nauplii in the water column 30 min after the distribution of food. Excess food, faeces and dead fish were removed once a day with a siphon.

Sixty fish were randomly sampled in the aforementioned tanks for a more accurate study of fish growth and size dispersal involving the measurement of all fish. Except for stocking density (two 90-L aquaria, with 30 fish each), the environmental conditions were identical as above. The study was not commenced before fish were aged 8 dah (days after hatching), as it was uncertain whether younger and smaller fish could tolerate handling. To minimize operational stress to the larvae, measurements were done on a weekly basis (8, 15, 22 and 29 dah). On days of measurement, all fish were captured, anaesthetised (2-phenoxy-ethanol, 0.35 ml L -1 ) and their body lengths were measured with a graduated eyepiece under the stereomicroscope (magnification: ×6-25, depending on fish size).

Morphological observations

Samples for morphological analyses were collected twice a day until the age of 5 dah, at daily intervals until 12 dah and less frequently thereafter, until the transition between the larval and juvenile stages (i.e. when all fins and finrays were developed, and the finfold was fully absorbed). Fish were anaesthetised (2-phenoxy-ethanol, 0.35 ml L -1 ) and photographed in profile, dorsal and ventral views under the stereomicroscope (magnification: ×6-50). Additional photographs were taken under the light microscope for a more detailed examination of mouth dimensions (magnification: ×50-200). Dimensions were measured to the nearest pixel from digital photographs on the computer, by reference to a finely graduated scale that had been photographed at the same magnification(s). The morphomeristic variables under scrutiny were those that are essential to feeding and locomotion: i.e. fish size (total, fork and standard lengths), yolk dimensions (length, depth and width), body depth, fin length and finray extension, head width and length, mouth width and jaw length. Gape height was estimated from jaw length, on the assumption of a 90 • opening capacity [START_REF] Shirota | Studies on the mouth size of fish larvae[END_REF]. The ontogeny and regularity of the pigment pattern was analysed as follows. The centre of each vertical bar was marked on photographs at the level of the lateral line, then the distance between this mark and the fish snout was measured and expressed as a proportion of the fish total body length (hereafter TL).

Fish wet body mass (hereafter WM) was measured (nearest 0.1 mg) under anaesthesia in fish with empty guts (morning samples), either individually for fish >10 mg or in groups of 5-10 individuals of homogeneous size (nearest 0.2 mm) for smaller larvae, so as to document the relationship between WM and TL. Following the measure of WM, anaesthetised fish were euthanized with an excess dose of anaesthetics (2-phenoxyethanol, 2.0 ml L -1 ), placed at 105 • C for 12 h then weighed again for the measurement of dry body mass (hereafter DM, AOAC 1995). Measurements were done on groups of fish producing at least 30 mg of DM (tens of fish for small larvae, at least three individuals for larger fish).

Food intake and gut evacuation rate

All experiments took place in a thermostated room (26.5-27.0 • C) and used the protocols developed by Baras et al. (2012, in press). The experimental fish had been removed from the rearing tanks during the preceding evening and offered no food overnight in order to ascertain that they would have empty guts on the next morning. Fish were allowed to feed on Artemia nauplii during 20 min and then they were transferred in water devoid of food.

The measure of food intake took place about 40 min after the start of feeding, well below the start of defecation in larvae of this species (see results). The fish were anaesthetised (2-phenoxy-ethanol, 0.35 ml L -1 ), placed in a small (50 mm in diameter) Petri dish filled with the anaesthetic solution under the stereomicroscope and measured with a graduated eyepiece. Their abdominal region was photographed in profile view under magnification ×12-50 (depending on fish size) by reference to a finely graduated scale (0.1 mm). Prior to this study, it had been tested whether accurate estimates of gut content could be obtained from profile views only, by comparing the width and depth of the gut content from photographs of clown loach in ventral and profile views. The ratio between gut content depth (measured from profile views) and width (measured from ventral views) was close to 1 (mean ± SD of 0.98 ± 0.02) and did not vary between fish of different sizes.

The protocol for measuring gut evacuation rates (R g ) was slightly different as it involved sequential observations of the same individual fish, and no anaesthetics was used as it could have interfered with the fish metabolism (and thus with R g ). About 20 minutes after feeding, five or six fish were sampled and housed in isolation in 300-ml plastic containers (12.5 × 8.0 × 3.0 (h) cm) filled with water devoid of food. At the time of observation, an individual fish was gently captured with a pipette, placed in a Petri dish filled with water under the dissection microscope, then about 80% of the water in the Petri dish was pumped with a pipette until gravity gently forced the fish in a lateral recumbent position, The gut region was photographed rapidly, water was poured again and the fish was returned to its enclosure. The operation, which took no longer than 30 s, was applied to the other fish, so all individuals were photographed within less than 3 min. The same sequence was repeated at regular intervals (about 40 min) until gut evacuation had (almost) ended. Thereafter, all fish were anaesthetised and photographed in full for measuring their body length (same protocol as above).

The measure of R g in the same individual fish was preferred to the measures of gut volumes of different fish at different times after feeding, because the value of R g can depend strongly on meal size and degree of gut fullness [START_REF] Jobling | Fish Bioenergetics[END_REF]. Nevertheless, it was uncertain whether repeated observations of the same fish interfered with R g . To test for this possible bias, a post hoc comparison was made between the R g values obtained with the sequential observations of the same individuals in absence of anaesthesia, and those from different anaesthetised individuals at different times after feeding (same protocol as in Baras et al. 2012, in press). The scope of this particular comparison was restricted to fish with very high food intake. The R g curves that were produced with the two methods were highly consistent (data not shown), thereby suggesting that the individual protocol was not excessively stressful.

Calculations and statistics

Morphological variables were expressed as a proportion of the fish total body length (TL), to facilitate comparisons between fish of different sizes. TL was preferred to the standard body length SL, which could not be measured before the notochord was bent. Fish growth was expressed in terms of linear growth (GL, mm day -1 ), which was calculated as GL = (TL 2 -TL 1 ) (t 2t 1 ) -1 , where TL 2 and TL 1 are the total body lengths (mm) of fish at times 2 and 1, respectively. Size heterogeneity was expressed by the coefficient of variation (CV, %), which was calculated as CV = 100 SD × TL -1 m , where SD is the standard deviation of the mean body length (TL m ). The water content of fish (WC, % WM) was calculated as WC = 100 (WM -DM) WM -1 , where WM and DM are the wet and dry body mass of fish (mg).

The volume (V) of yolk sac was calculated from its length (L), depth (D) and width (W), which were measured on profile and ventral views of the fish. The yolk of clown loach was pear-shaped or strongly conical in its caudal region, so its volume was systematically overestimated with an ellipsoidal model using L, W and D as diameters (i.e. V = 0.167 π L W D). To correct for this, the perimeter of the yolk was contoured with a hand drawn closed polygon, and the surface area (S ) was calculated with the freeware Image J [START_REF] Abramoff | Image Processing with Image[END_REF]. Thereafter, the diameters (D 1 and D 2 ) of a planar ellipse with a surface area equal to S were calculated on the assumption that the ratio between the two diameters had to be identical to the ratio between the actual dimensions of the yolk (D 1 /D 2 = L/D for profile views and D 1 /D 3 = L/W for ventral views). The values of D 1 , D 2 and D 3 substituted those of L, H and W in the ellipsoidal model for calculating the yolk volume.

For the measures of gut content and gut evacuation rate (i.e. decline of gut content over time), the stomach and intestine were analysed separately, because of their contrasting shapes (details in Baras et al. 2012, in press). The stomach region has an ellipsoidal shape, so its volume (V s ) was calculated in the same way as for the yolk sac. For the intestinal content, which has a cylindrical shape, the surface of the gut content (S i ) was contoured and its length (L i ) was measured by tracing an open polygon passing in its centre. The S i /L i ratio gave the mean diameter (D i ) of the intestinal content, the volume of which (V i ) was calculated as V i = 0.25 π L i D 2 i . The total gut content volume V g was calculated as V g = V s + V i , and expressed as a proportion of the fish WM, assuming that a gut content volume of 1 mm 3 weighed 1 mg.

Simple (linear, logarithmic or power) and polynomial regression analyses were used to describe the relationships between body dimensions, WM and TL, DM and WM, and between TL and fish age. A log-logistic regression model was used for modelling the resistance (P 50 ) to starvation. A stepwise multiple-regression analysis was used to identify the variables (fish size, food intake and their interaction) that influenced significantly the gut evacuation rate. Null hypotheses were rejected at p < 0.05.

Results

Morphology

General

The ova used in this experiment averaged 1.18 mm in diameter and 0.88 mg WM, with a water content of 66.6%. Hatching took place 19-20 h after fertilization (27 ± 1 • C). Hatchlings averaged 3.8 mm TL and their yolk sac (0.49 mm 3 ) exhibited a marked posterior axial protrusion (yolk extension, Fig. 1a). Until 78-84 hah, before the inflation of the swim bladder, embryos were strongly pelagic, swimming up and down in the water column. Exogenous feeding started at 96 hah. Larvae were strongly thigmotactic and essentially benthic by then. First-feeding larvae averaged 5.5 mm TL and 1.2 mg WM, with a water content of 87%.

The ontogeny of clown loach is illustrated in Figure 1. The following sections lay the emphasis on the development of the fins, cephalic region, swim bladder, gut, and pigmentation. Morphometric relationships are given in Table 1. The model between WM and TL in young clown loach larvae has a polynomial nature, as it depicts the steps of allometric growth of body depth and width that take place during the early larval stage of this species (see below).

Fins

At hatching, only a continuous, non-structured finfold was present. Two days later, it was structured in the caudal, dorsal and abdominal regions. The fin development sequence, apparition and completion of finrays are illustrated in Figure 2. Pectoral fins grew first (20 hah, 4.6 mm TL), but the first finrays to become differentiated were those of the caudal fin, slightly before the flexure of the notochord, which was conspicuous at about 6.7 mm TL and attained its maximal bend (45-50 • ) at 8.8 mm TL. The anal and dorsal fins followed similar developmental patterns, and pelvic fin grew last. All fins had complete finrays and were fully elongated at 20 mm TL, but the finfold in the caudal and abdominal regions did not vanish before 26 mm TL, which can be regarded as the cut-off size between larvae and juveniles in this species.

Cephalic region

Head length and width attained their maximal dimensions relative to body length (20 and 17% TL, respectively) at the start of exogenous feeding (96 hah). The eyes became pigmented at about 12 hah and were fully pigmented at 60 hah (4.4 and 5.2 mm TL, respectively). The erectile subocular Table 1. Morphometric relationships in clown loach larvae and small juveniles. V is the volume of the yolk sac (mm 3 ), A is the age (days after hatching), FL, SL and TL are the fork, standard and total body lengths (mm), WM is the wet body mass (mg), WC is the water content (%), and TL M is the total body length of the fastest growing fish in this study (data in Fig. 8), respectively; p < 0.0001 for every coefficient in every model. spines that are found in adults were not observed in fish 26 mm TL. The gill arches appeared at 30 hah (4.9 mm TL) and the primary gill lamellae about 24 h later. Secondary gill lamellae were not observed before larvae attained 6.7 mm TL. The mouth opened at 30 hah (4.9 mm TL) and one day later the upper and lower jaws had already attained their longest dimensions relative to fish size (i.e. 4.0 and 3.2% TL, respectively), producing a gape of 5.5-6.0% TL for a 90 • opening (Fig. 3). By contrast, mouth width continued growing until fish were 7.5-8.0 mm TL and attained 11% TL by then. At the start of exogenous feeding (96 hah, 5.5 mm TL) gape height and mouth width thus do not exceed 0.25 and 0.33 mm, respectively, which is small by reference to the size of Artemia nauplii (on average 0.60 mm in length and 0.15 mm in diameter, excluding appendages) and can complicate feeding (see Sect. 3.4 on food intake).

Clown loach grew four pairs of circumoral barbels. The pair of lateral maxillary barbels started growing in between 60 and 72 hah (5.2-5.4 mm TL), whereas the two pairs of central maxillary barbels and the pair of central mandibular barbels were not observed before 6.0 and 6.5 mm TL, respectively.

Thoracic and abdominal regions

Fish body depth at hatching was 26-27% TL. It decreased rapidly as embryos absorbed their yolk and averaged 15-16% TL at the start of exogenous feeding (Fig. 3). Thereafter, it increased in a curvilinear way, up to a plateau at 22-23% TL in fish greater than 12.5 mm TL. The swim bladder started developing and became pigmented between 48 and 54 hah (5.1-5.2 mm TL) and one day later it was inflated (Fig. 1b). Until larvae were 8.0 mm TL, the swim bladder exhibited no specialisation. Thereafter, the anterior (hearing-specialised) chamber started differentiating. Its development could not be traced accurately in absence of micro-dissection, because of the development of pigmentation in this part of the thoracic region.

At the start of exogenous feeding, the gut of clown loach was straight (Fig. 4a). Gut coiling started at 7.8-8.0 mm TL (Fig. 4b). Thereafter, the stomach progressively acquired the shape of a pouch, while the pyloric bulb was progressively bent upwards and migrated slightly cranial, as far as under the centre of the swim bladder at 15 mm TL (Fig. 4d).

Pigmentation

No pigmentation was observed before 72 hah (5.3 mm TL). Then, six patches of melanophores appeared almost simultaneously in three body regions on each side of the body (at the level of the swim bladder, slightly cranial and slightly caudal to the anus), with a dorsal and a ventral patch in each region (Fig. 1b). A seventh patch of melanophores appeared ventrally close to the caudal end of the notochord at 100 hah (5.5 mm TL), and an eighth patch on the dorsal part of the head, slightly behind the eyes, at 120 hah (5.6-5.7 mm TL). In the meanwhile, the six other patches had extended ventrally and dorsally, including over the finfold, and merged to form three vertical bars. At 6.2 mm TL, the vertical bars on the left and right sides of the fish had merged in their dorsal region to form saddles. At this size, larvae exhibited five saddles, which lied over the eyes (I), the swim bladder (II), the dorsal (III) and anal fins (IV), and over the hippural complex of the caudal fin (V). Saddle V was always less pigmented than the four others. It vanished progressively and was no longer conspicuous in fish >11 mm TL. Saddle III was still complete by then (Fig. 1d). Thereafter, it vanished progressively in its ventral region and only a small patch of melanophores persisted on the basis of the dorsal fin and underlying epaxial muscles in fish >15-16 mm TL. This patch had almost completely disappeared at 26 mm TL (Fig. 1g). It is worth noticing that although melanophores vanished from two body regions during the ontogeny of clown loach, the saddles were spread at almost regular intervals throughout (Fig. 5).

Yolk absorption and resistance to food deprivation

Before the start of exogenous feeding (96 hah), embryos absorbed their yolk at a steady rate (on average 0.103 mm 3 day -1 ; Fig. 6, Table 1). By contrast, their growth in length was strongly curvilinear during this period. Until 30 hah growth averaged 1.0 mm TL day -1 , whereas it amounted to 0.25 mm TL day -1 during the next 24 h, and less than 0.15 mm T L day -1 until the start of exogenous feeding at 96 hah. Thereafter, the rate of yolk absorption decreased rapidly, and remnants of yolk were occasionally observed until 178 hah. Yolk volume did not vary much between hatchlings (CV of 3.3%), but became strongly heterogeneous afterwards (CV of 13-14% from 30 to 66 hah; Fig. 6), thereby suggesting marked individual variations in yolk absorption rates. The sizes of embryos and the remaining yolk volumes at different ages were negatively correlated (r 2 = 0.660), but a better correlation was found between fish age and yolk volume with the same data set (r 2 = 0.907; Table 1). This comparison further supports the view that clown loach embryos absorb their yolk at variable rates, but also that they do not transform yolk equally well. These findings account for why size heterogeneity at the start of exogenous feeding was twice as high as at hatching (CV TL of 3.9 versus 2.0%).

The P 50 survival of larvae housed individually and deprived of food stood at 14.7 dah (Fig. 7). No fish died before 13 dah but none survived beyond 16.5 dah This 3-day range compares with the variation between the ages at which embryos had fully exhausted their yolk (from <120 to 178 hah, Fig. 6).

Survival, growth and size heterogeneity of larvae and small juveniles

Survival at 29 dah was high in both tanks (26 and 28 out of 30 fish stocked at 8 dah) and no cannibalism occurred. Almost identical growth trajectories were observed in the two replicate groups, so the data were pooled for the description below. During the first days of exogenous feeding, the mean growth rate of clown loach larvae increased rapidly, from 0.15 to 0.5 mm day -1 at 5 and 8 dah, respectively (Fig. 8). The average growth rate remained stable around 0.5 mm day -1 until 15 dah and slowed down slightly afterwards (0.4 mm day -1 ). At 29 dah fish averaged no more than 15.4 mm TL and 59 mg WM. However, in each replicate group, at least one fish was about 10 mm larger than the average fish size, with a steady growth of 0.9 mm day -1 thereafter.

Size heterogeneity (CV TL) increased with increasing age, from 3% at 4 dah, to 10% at 8 dah (corresponding values of 15 and 36% for CV WM), i.e. an average increase of 1.7% day -1 . Thereafter CV TL continued increasing, but at a much slower pace (0.75% day -1 ) and attained 26% at 29 dah (corresponding CV WM of 84%).

Food intake and gut evacuation rate

The development of xanthophores in the thoracic and abdominal regions prevented the observation of gut content in fish >15.4 mm TL (Fig. 1f), except in an individual with an exceptionally low degree of pigmentation. Food intake (FI) varied substantially during the first stages of exogenous feeding (Fig. 9). In fish <5.5 mm TL (< 1.3 mg), FI never exceeded E. Baras et al.: Aquat. Living Resour. 25, 119-130 (2012) ) and in all survivors from 8 to 29 dah (2 groups of 30 fish at 0.33 fish L -1 ) (see methods). Curves produced by interpolation.

6.5% WM, whereas it attained 13.5% WM at 6.0 mm TL (1.8 mg) and 20.0% WM at 6.5 mm TL (2.3 mg). Thereafter, FI continued increasing, but at a much slower pace, and attained 23% at 10 mm TL (12 mg). No decrease in FI was observed in larger fish, at least until 15 mm TL. The examination of gut contents a few minutes after feeding revealed that in fish <5.5-5.6 mm TL, the outline and eyes of Artemia nauplii in the Total body length (TL, mm) Fig. 9. Size-dependent variations of food intake in clown loach larvae of different sizes. Food intake is expressed as a proportion of the fish wet mass (WM). Symbols refer to measurements in individual fish (n = 124). The plain curves were modelled from the highest values of gut content (closed circles). Two separate models were used for fish smaller and larger than 6.5 mm TL (vertical dashed line; see Table 2 for equations and statistics).

foregut were never conspicuous, as if they had been chewed or sucked in. By contrast, in fish >5.7-5.8 mm TL, the contours of nauplii were sharp and their eyes were visible (Fig. 4a), as if they had been swallowed straight.

The gut evacuation rate (R g ) was examined in 22 fish ranging from 5.9 to 14.2 mm TL. All fish had been selected on the basis of their high FI by reference to the curve shown in Figure 9. The decline in gut content over time was curvilinear in fish of all ages and sizes, but was best described by a linear relationship during the first two hours following the start of defecation (Fig. 10). Defecation started earlier and R g was faster in large than in small fish. Proportionally, size-dependent differences in R g ) were highest around the size of fish at the time of gut coiling (7.8-8.0 mm TL; Fig. 4b). A stepwise multiple-regression analysis revealed that R g was significantly dependent on TL, FI and their interaction (Table 2). Based on this model, R g would increase rapidly until about 11 mm TL and decrease in larger fish. This steep variation in the capacity of young clown loach to evacuate -and thus consumefood compares well to the curvilinear increase of growth rates observed during this study (Fig. 8).

Discussion

Morphology

The fin developmental sequence in Chromobotia macracanthus (Fig. 2) is found in other loaches (Krizanovsky 1949;[START_REF] Kim | Illustrated Encyclopedia of Fauna and Flora of Korea[END_REF][START_REF] Shimizu | Embryonic and larval development of a Japanese loach Cobitis takatsuensis[END_REF][START_REF] Pinder | Keys to Larval and Juvenile Stages of Coarse Fishes from Fresh Waters in the British Isles, Ambleside, The Ferry House, Freshwater Biological Association[END_REF][START_REF] Kottelat | Botia kubotai, a new species of loach (Teleostei: Cobitidae) from the Ataran River basin (Myanmar), with comments on botiine nomenclature and diagnosis of a new genus[END_REF] Table 2. Models depicting the variations of maximal food intake (FI M ) in clown loach larvae of different sizes (total body length, TL), and the effects of fish size and food intake on the gut evacuation rate (R g ). The probabilities in the outer right column refer to the intercept and variables, given in the same sequence as in the equations. Fig. 10. Decline of gut content volume (% wet body mass, WM) in clown loach larvae of different size (total body length, T L) fed a single meal. On each graph, symbols and whiskers are the means and standard deviations of the same (four or five) individual fish, having fed near maximally and examined at regular intervals (see Fig. 9 for size-dependent maximal food intake). Only fish with similar sizes are shown here (14 of the 22 fish under examination). The maximal rate of gut evacuation (% WM h -1 ) was calculated on the linear part of the curve, during the first two hours following the start of defecation.

and is shared by all Ostariophysi (E. Baras, unpubl. data). The presence of a yolk extension in clown loach embryos is no diagnostic criterion either, as it is a shared trait among the clade Cypriniformes [START_REF] Virta | Ontogeny and phylogeny of the yolk extension in embryonic cypriniform fishes[END_REF]. By contrast, the early development of the barred pigment pattern in clown loach should make their identification almost univocal from a very young age. It is frequent that skin pigment cells appear during the early ontogeny of fishes (e.g. 2 dah in Cobitis bilineata, 3-4 dah in C. taenia ;[START_REF] Bohlen | Differences in egg size, clutch size and larval pigmentation between Cobitis taenia and C. bilineata (Cobitidae)[END_REF]) and sometimes before hatching (review in [START_REF] Price | Pigments, patterns and fish behavior[END_REF]). However, with the exception of species with very large eggs (e.g. ariid catfishes), it is rare that a pigment pattern be structured at the start of exogenous feeding, and exceptional that it be conserved throughout the ontogeny. The pigment pattern of clown loach passed from five bars in young larvae to three bars in small juveniles (the latter pattern being maintained throughout adulthood). However, at all stages it consisted in an alternation of clear and dark bars spaced at regular intervals (Fig. 5).

Barred or striped pigment patterns serve a broad series of functions, including crypsis, predator evasion or warning, signalling conspecifics or competitors, and it is frequent that a single pattern serves several purposes [START_REF] Kenward | Spots and stripes: the evolution of repetition in visual signal form[END_REF][START_REF] Ruxton | Avoiding Attack: the Evolutionary Ecology of Crypsis, Warning Signals and Mimicry[END_REF][START_REF] Price | Pigments, patterns and fish behavior[END_REF]. Whatever the function, pattern repetition and regularity can enhance the effect [START_REF] Endler | Natural selection on color patterns in Poecilia reticulata[END_REF][START_REF] Guilford | Receiver psychology and the evolution of animal signals[END_REF][START_REF] Armbruster | Convergence of a cryptic saddle pattern in benthic freshwater fishes[END_REF][START_REF] Mcrobert | The influence of body coloration on shoaling preferences in fish[END_REF], both in mobile and immobile animals (mechanisms in [START_REF] Coren | Sensation and Perception[END_REF][START_REF] Kenward | Spots and stripes: the evolution of repetition in visual signal form[END_REF]). The benefits of structural regularity for a pigment pattern might be one of the reasons why clown loach have evolved developmental mechanisms that enable the (almost) maintenance of a regular distance between bars throughout their ontogeny.

Growth depensation

It is generally assumed that siblings possess slightly different capacities for growth, and genuine differences are amplified in the course of dominance hierarchies, resulting in increasing dispersal in fish of increasing size, unless cannibalism emerges, removes the smallest fish and stabilizes size dispersal [START_REF] Huston | Size bimodality in monospecific populations: a critical review of potential mechanisms[END_REF][START_REF] Hecht | A review of cannibalism and its implications in fish larviculture[END_REF][START_REF] Kubitza | Formulated diets, feeding strategies and cannibalism during intensive culture of juvenile carnivorous fishes[END_REF][START_REF] Baras | Dynamics of intracohort cannibalism in cultured fishes[END_REF][START_REF] Kestemont | Size heterogeneity, cannibalism and competition in cultured predatory fish larvae: biotic and abiotic influences[END_REF]. The present study comprised no detailed study of clown loach behaviour or individual growth trajectories, so it is uncertain whether the marked size dispersal of clown loach under culture conditions was a matter of dominance hierarchies or intrinsic differences in metabolism and capacities for growth. There is some indirect evidence that not all clown loach embryos transformed their yolk with the same efficiency (Fig. 6), which suggests between-individual differences in metabolic rates. However, similar differences can originate from variable yolk composition or resource allocation (e.g. swimming versus growth, [START_REF] Bagatto | Growth and metabolism of larval zebrafish: effects of swim training[END_REF]; for a review, see [START_REF] Kamler | Resource allocation in yolk-feeding fish[END_REF].

Nevertheless, the present study provided information that can account for why size dispersal in clown loach is higher than in most cultured fishes, in particular why early size differences can amplify genuinely, even if all siblings possess very similar capacities for growth. Clown loach larvae >5.6 mm TL swallow Artemia nauplii whole, whereas smaller larvae chew them or suck them in, which is certainly longer and presumably more energy demanding. The transition between the two ingestion modes corresponds to a period of rapid allometric growth of mouth dimensions (Fig. 3). If clown loach larvae are fed Artemia nauplii, early size differences are likely to amplify rapidly if the size distribution of larvae at the start of exogenous feeding encompasses the aforementioned cutoff size. This was the case here, and in almost all progenies of clown loach studied since then (authors' unpublished data). Growth depensation further continues because of the allometric increase of food intake (until 7 mm TL; Fig. 9) and gut evacuation rate (until 10 mm TL; Fig. 10) in parallel with gut coiling (Fig. 4). These genuine differences might account for why size heterogeneity in the growth study increased regularly until all individuals had grown beyond this size (Fig. 8). Clown loach larvae possess no oral teeth, their mouth is small and their body is deep (Fig. 3), so sibling cannibalism is almost impossible, i.e. it could only occur if the cannibal were three times as long as its victim (corresponding WM ratio of 25-30), which is much higher than in most cultured species (review in [START_REF] Baras | Cannibalism in fish larvae: what have we learned?[END_REF]. In absence of cannibalism, size dispersal can continue increasing under culture conditions, especially if rearing protocols are not tailored yet [START_REF] Kubitza | Formulated diets, feeding strategies and cannibalism during intensive culture of juvenile carnivorous fishes[END_REF][START_REF] Kestemont | Size heterogeneity, cannibalism and competition in cultured predatory fish larvae: biotic and abiotic influences[END_REF]. Further studies are needed to determine the proximal factors (fish density, food density, temperature, etc.) that can limit growth dispersal of clown loach under culture conditions.

Slow and heterogeneous growth of clown loach: an ecological perspective

Although larvae and juveniles of clown loach were fed abundantly with high-energy feed (Artemia nauplii) distributed all day round, their growth averaged 0.4 mm day -1 and did not exceed 0.9 mm day -1 in top growers. This is a slow growth in comparison to most freshwater tropical fishes in culture conditions, but not in comparison to other cobitids, as this group seemingly comprises no fast growing species. The growth of young clown loach was strongly heterogeneous, for reasons that were debated above. The present study also provided evidence that clown loach larvae could survive protracted periods of food deprivation, as the P 50 mortality at 26.7 • C did not occur before 14-15 dah (Fig. 7). This is not exceptionally long in comparison to species producing large eggs or living in cold climates [START_REF] Kamler | Ontogeny of yolk-feeding fish: an ecological perspective[END_REF][START_REF] Kamler | Parent egg-progeny relationships in fish: an energetics perspective[END_REF], but substantially longer than in most warmwater fishes with similar egg size (Araujo-Lima 1994; [START_REF] Yúfera | The onset of exogenous feeding in marine fish larvae[END_REF].

The combination of these three traits (resistance to starvation, slow and heterogeneous growth) can be interpreted in an ecological perspective. In the River Musi (Sumatra), adult clown loach live in the upper and medium reaches and there are strong indications that they spawn during flood pulses following heavy rainfalls [START_REF] Legendre | Biology and culture of the clown loach Chromobotia macracanthus (Cypriniformes, Cobitidae): 1-Hormonal induced breeding, unusual latency response and egg production in two populations from Sumatra and Borneo Islands[END_REF]. Eggs of clown loach are not adhesive, they undergo substantial swelling and become semi-buoyant [START_REF] Slembrouck | Biology and culture of the clown loach Chromobotia macracanthus (Cypriniformes, Cobitidae): 2-Importance of water movement and temperature during egg incubation[END_REF], so they probably enter the drift, as those of many other seasonal strategists with similar characteristics [START_REF] Lowe-Mcconnell | Ecological Studies in Tropical Fish Communities[END_REF][START_REF] Araujo-Lima | Transport of larval fish in the Amazon[END_REF][START_REF] Lucas | Migration of Freshwater Fishes[END_REF]. Free embryos of clown loach in the present study were observed swimming actively in the water column during the first 3 days after hatching. Hence, it is likely that their wild counterparts continue drifting at this age, until they fill their swim bladder, acquire their pigment pattern, shift to benthic behaviour and settle in the recently inundated floodplain. These interpretations are supported by field observations. Because of an increasing shortage of market size specimens in the wild, Indonesian fishermen nowadays harvest clown loach larvae (<10 mm TL). Yet, they continue fishing in the same places where they used to collect juveniles in the floodplain, but they focus their fishing effort when water starts receding after a flood pulse, thereby supporting the view that young larvae drift onto the recently inundated floodplain.

Survival in the wild is generally size-dependent and it is frequently assumed that young-of-the-year fishes must have attained a minimal size at the onset of the harsh season [START_REF] Sibly | Are patterns of growth adaptive?[END_REF][START_REF] Arendt | Adaptive intrinsic growth rates: an integration across taxa[END_REF]). In the tropics, the larvae of seasonal strategists born at the start of the rainy season can spend months in the floodplain before water recedes and forces them in the main river channel, where predation hazards are high [START_REF] Winemiller | Effects of seasonality and fish movement on tropical river food webs[END_REF][START_REF] Lucas | Migration of Freshwater Fishes[END_REF]. In view of the length of the growing season, there has probably been little selection pressure on fast growth in these species (Bailey 1988). By contrast, the productivity of the floodplain is generally low at the start of the rainy season, and the first seasonal rainfalls are frequently followed by more or less protracted dry periods, which result in decreasing water levels, habitat fragmentation and temporary food shortage (e.g. [START_REF] Loubens | Les milieux aquatiques de la région de Trinidad (Béni, Amazonie bolivienne)[END_REF]. This context might have favoured the selection of a longer resistance to starvation. In view of the unpredictability of rainfall patterns and associated levels of food availability in the early rainy season, it might be advantageous for early seasonal spawners to produce heterogeneous progenies as regards metabolic rates, as at least a part of the progeny would be adapted to the environment. An analogous way of adapting to unpredictability through intrinsic heterogeneity has been found in annual fishes of genus Notobranchius (Cyprinodontidae). Their eggs may enter one or more diapauses of variable durations, postponing hatching at different times of the year, and almost suppressing the risk that an entire progeny dies from hatching in inadequate conditions [START_REF] Wourms | The developmental biology of annual fishes. III. Pre-embryonic and embryonic diapause of variable duration in the egg of annual fishes[END_REF].

Conclusion, perspectives for the larviculture of clown loach

This study highlighted that the ontogeny of clown loach is particular as regards its pigment pattern, which is well structured before the start of exogenous feeding. It is frequent that the expression of marked pigment patterns varies between social contexts, and this makes clown loach an excellent biological model for the study of social interactions among fish larvae.

In respect to the propagation of clown loach in captivity, survival is expectedly not a major issue, in view of the high fecundity of the species (around 100 000 eggs kg -1 ; [START_REF] Legendre | Biology and culture of the clown loach Chromobotia macracanthus (Cypriniformes, Cobitidae): 1-Hormonal induced breeding, unusual latency response and egg production in two populations from Sumatra and Borneo Islands[END_REF]) and of the low mortality during the embryonic and larval stages in the present study, although the rearing protocols had not been refined yet. Clown loach larvae do not grow rapidly, but this is not a major issue, as they are marketed at a small size (4-5 cm). Nevertheless, improvements in growth rates and size homogeneity are desirable to reduce production costs, especially if broodfish can be reproduced all year round in captivity. Likewise, clown loach in this study consumed large food rations but grew at a slow rate, thereby indicating a low feed efficiency (estimated here as 9-10 mg of Artemia nauplii for producing 1 mg of fish). This trait is particularly important because Artemia are expensive, so a low feed efficiency could make aquacultural products less competitive than those from capture-based fisheries. Further research in this field is thus needed, essentially by testing the effects of food density, fish density and water temperature, which have been shown to impact on food intake, growth and size dispersal in a broad series of fish species. To this respect, the results on the ontogenetic variations of food intake and gut evacuation rate in the present study provide the bases for tailoring feeding schedules more adequately in respect to the actual feeding capacities of clown loach of different sizes and ages.

Fig. 1 .

 1 Fig. 1. Ontogeny of clown loach with emphasis on the pigment pattern. I-V: dark pigment bars; B: swim bladder; Ba and Bp: anterior and posterior chambers of the swim bladder after specialization; E: eye; F: finfold; O: otic capsule; X: xanthophores; Y: yolk; YE: yolk extension. (a) 3.7 mm TL, 4 h after hatching, no pigment; (b) 5.8 mm TL, 5 days after hatching (dah); (c) 7.8 mm TL, 10 dah; (d) 10.0 mm TL, 14 dah; (e) 13.5 mm TL, 21 dah; (f) 17.0 mm TL, 28 dah; (g) 26.0 mm TL, 37 dah.

Fig. 2 .

 2 Fig. 2. Fin development in clown loach. The upper part of the figure indicates when finrays start developing and are fully elongated. The lower part shows fins' elongation as a function of the fish total body length. Dimensions are equated to fish size rather than age, owing to the marked growth heterogeneity in clown loach. Symbols refer to individual fish. Closed triangles: caudal fin (n = 88); open circles: anal fin (n = 80); grey circles: dorsal fin (n = 80); open diamonds: pectoral fins (n = 76); closed diamonds: pelvic fins (n = 76).

Fig. 3 .

 3 Fig. 3. Ontogenetic variations of gape height (closed diamonds, n = 76), mouth width (open diamonds, n = 54), body depth (closed circles, n = 80) and head width (open triangles, n = 54), which also corresponds to maximal body width over the size interval under study. Gape height is calculated from jaw lengths, for a gape opening of 90 • .

Fig. 4 .

 4 Fig. 4. Gut coiling in clown loach. (a): 6.5 mm TL; (b): 8.0 mm TL; (c): 9.0 mm TL; (d):12.4 mm TL. On every photograph, the horizontal bold bar in the lower right corner stands for 1 mm.

Fig. 5 .

 5 Fig. 5. Ontogenetic variation of the pigment pattern in clown loach. The five dark bars (I-V) are numbered in cranio-caudal sequence. Symbols refer to the position of the centre of dark bars on the lateral line, by reference to the fish total length. Curves were produced by interpolation. The dashed curves illustrate where the centres of bars III and V would have lied if they had not vanished during the ontogeny. Measurements in 30 fish (5.3-26.0 mm TL).

Fig. 6 .

 6 Fig. 6. Dynamics of yolk absorption and growth in clown loach. At each age, the closed circle stands for the mean, the box for the standard deviation, and the whiskers indicate the minimal and maximal values over a sample of 5-10 fish. The dashed vertical line indicates the start of exogenous feeding.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Resistance to starvation in clown loach embryos and larvae housed in isolation (26.7 ± 0.8 • C, mean ± SD). Curve produced with a log-logistic regression model.

  Equation -FI M (% WM), TL (mm) Statistics Fish <6.5 mm TL R 2 = 0.976, df = 10 FI M = -1063 + 2606 log TL -1568(log TL) 2 p = 0.0031, 0.0041, 0.0060 Fish 6.5-15 mm TL R 2 = 0.934, df = 12 FI M = -34.3 + 107.2 log TL -49.8(log TL) 2 p = 0.0035, 0.0002, 0.0003 Fish 5.9-15 mm TL, R g (% WM h -1 ) R 2 = 0.956, df = 21 log R g = -4.480 + 0.818 log FI log TL + 9.469 log TL -5.083 (log TL) 2 p < 0.0001, <0.0001 <0.0001,

Acknowledgements. The authors wish to express their most sincere thanks to Slamet Sugito and Mochamad Hasan (BP2BIH, Depok) for their technical assistance. Mrs. Dominique Caseau-Baras contributed to improve the English style of the manuscript. Etienne Baras is an honorary research associate of the Belgian FNRS.


Liang Zhou

Benoît Geller

Baoyu Zheng

Anne Wei

Jingwu Cui

System Scheduling for Multi-Description Video Streaming over Wireless Multi-Hop Networks

Keywords: multiple description coding, video transmission, rate control, multipath routing

Providing real-time multimedia applications over wireless multi-hop networks is a challenging problem because the wireless channels are highly sensitive to delay, interference and topology change.

Multiple description coding (MDC), as a new emerging error-resilient technique, has been widely used recently in wireless video transmission. Its fundamental principle is to generate multiple correlated descriptions such that each description approximates the source information with a certain level of fidelity. Inevitably, MDC introduces many description streams which may influence each other and thus, reasonable system scheduling is needed to provide a satisfied video quality. The novelty of this work is to investigate the optimal distributed scheduling for multiple competing MDC streams in a resourcelimited wireless multi-hop network. This is achieved by joint optimization of MDC, rate control and multipath routing. Two joint optimal algorithms, namely a distributed rate control and routing (DRCR) and a simplified DRCR algorithm, are proposed to solve this problem with constraints that arise from the multiple description streams among multiple users via multiple paths. Both algorithms are designed in a distributed manner that is amenable to on-line implementation for wireless networks. Theoretical analysis and simulation results are provided which demonstrate the effectiveness of our proposed joint schemes.

I. INTRODUCTION

Rapid growth in wireless networks is fueling the demand that services traditionally available only in wired networks, such as video, be available to mobile users. However, the characteristics of wireless systems provide a major challenge for reliable transport of video since the wireless multimedia transmission is highly sensitive to delay, interference and topology change, which can cause both packet losses and bit-errors. Current and future wireless systems will have to cope with this lack of QoS (Quality of Service) guarantees [START_REF] Zhou | Joint QoS Control for Video Streaming over Wireless Multihop Networks: A Cross-Layer Approach[END_REF].

The issue of supporting error-resilient video transport over error-prone wireless networks has received considerable attention recently. Along one thread, some works presented some source coding-based error-resilient approaches that divide the original bit-stream into multiple streams, called multiple description coding (MDC), to tradeoff the error-resilience and the coding complexity (see [START_REF] Akyol | A Flexible Multiple Description Coding Framework for Adaptive Peer-to-Peer Video Streaming[END_REF], [START_REF] Radulovic | Multiple Description Coding with Redundant Expansions and Application to Image Communication[END_REF] and the references therein). The fundamental principle of MDC is to generate multiple correlated descriptions of the source such that each description approximates the source with a certain level of fidelity [START_REF] Zhou | A roubust resolution-enhancement scheme for video transmission over mobile ad-hoc networks[END_REF]. [START_REF] Kim | Distributed Video Streaming Using Multiple Description Coding and Unequal Error Protection[END_REF] and [START_REF] Tian | Robust and efficient path diversity in application-layer multicast for video streaming[END_REF] amplified the benefits of using MDC by combining it with path diversity; in this context, each stream is explicitly transmitted over an independent path to the receiver in order to achieve higher tolerance to packet loss and delay due to network congestion. Along another thread, some researchers studied the network congestion control and optimal routing for wireless video transmission (see [START_REF] Key | Combining multipath routing and congestion control for roubustness[END_REF], [START_REF] Zhu | Joint Routing and Rate Allocation for Multiple Video Streams in Ad Hoc Wireless Networks[END_REF] and the references therein) so that the network can be stable, robust and the users can have better QoS for the applications. In [START_REF] Key | Combining multipath routing and congestion control for roubustness[END_REF], congestion control and multipath routing were studied and it demonstrated that there are significant advantages when each source randomly selects multiple paths from all its available choices. In [START_REF] Zhu | Joint Routing and Rate Allocation for Multiple Video Streams in Ad Hoc Wireless Networks[END_REF], it showed that the optimal allocated rate strikes a balance between the selfish motivation of minimizing video distortion and the global goodness of minimizing network congestions, while the routes are chosen over least-congestion links in the network. [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF] researched MDC video streaming over wireless network, however, it only considered the multi-routing problem and did not present a systematic scheduling algorithm. Typically, for real-time video communications over resource-limited wireless networks, the key point is how to allocate the resource to different users to minimize the total video distortion.

In this paper, we will employ MDC as our error-resilient target. For such a case, there are many and different description streams in multiple paths which may influence each other and thus the sources should choose optimal rate-distortion points and provide reasonable transmission rate and routing such that the video sources can be both error-resilient and network-adaptive.

However, to the best of our knowledge, the current literature considered MDC, congestion control and multipath routing separately and independently. In order to achieve improved video quality supported by wireless networks, and to provide a more robust video delivery system, these factors are jointly considered in this paper.

In this work, we propose a framework to combine multipath routing and rate allocation with the asymmetric MDC in a general wireless multi-hop network in order to minimize the total distortion of all users. Our framework reflects the intrinsic tradeoff between the rates and distortions at source coding, and the tradeoff between the allocated rates and packet loss due to congestion and unreliable links. The main contributions and novelties of this paper are: (1) providing a general framework to combine multipath routing and rate allocation with multiple asymmetric MDC streams in a wireless multi-hop network; (2) proposing a joint routing and rate control algorithm for system scheduling to maximize the end-to-end quality of all the users; (3) extending and simplifying this joint algorithm to improve its convergence rate. Note that both algorithms are designed in a distributed manner that is amenable to on-line implementation for wireless networks.

The rest of the paper is organized as follows. In section II, we formulate the problem for the network with asymmetric MDC. Based on the problem analysis, a distributed joint rate control and routing scheme is proposed in section III. In section IV, an extended algorithm is presented to improve its convergence rate, and related works are given in section V, followed by the concluding remarks in section VI.

II. PROBLEM FORMULATION

A. Video Distortion

Consider a wireless network with L links, and each link l ∈ L with a capacity of C l . In this system, there are S users, and each user s ∈ S uses asymmetric MDC. In this work, we focus on two-description since it is the most widely used for MDC video. Each description of user s is denoted as H i s for i = 1, 2, each using a set of links L(H i s ) from source to destination. Each link l is shared by a set of descriptions ∪ 2 i=1 H i s (l) of each user s. For each user s, denoting d 0 s as the central distortion when both descriptions are received, d i s for i = 1, 2 as the side distortion if description H i s is received, and d 3 s as the distortion if none of the description is received. Besides, denoting R i s and P i s for i = 1, 2 as the rate and the packet loss probability of the path for description H i s , respectively. Our system scheduling aims at finding the optimal rate-distortion operating points to minimize the total distortion

min s∈S D s , ( 1
)
where D s is the distortion of each user s, the expected average video distortion at the receiver can be approximated as:

D s = d 0 s 1 -P 1 s 1 -P 2 s + d 1 s 1 -P 1 s P 2 s + d 2 s 1 -P 2 s P 1 s + d 3 s P 1 s P 2 s . ( 2
)
In this paper, assuming Gaussian sources with zero and variance σ 2 , we employ the following distortion-rate regions [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF]:

               d 0 s ≥ 2 -2(R 1 s +R 2 s ) 2 -2R 1 s +2 -2R 2 s -2 -2(R 1 s +R 2 s ) • σ 2 d 1 s ≥ 2 -2R 1 s • σ 2 d 2 s ≥ 2 -2R 2 s • σ 2 d 3 s = σ 2 .
(3)

B. Packet Loss Approximation

In wireless video communications, the reconstructed video quality is affected by quality degradation due to packet losses either caused by late arrival or transmission error. Late arrival is induced by the packet end-to-end delay exceeding delay constraints T and transmission error is caused by unreliable wireless links. In this paper, we don't employ any retransmission policy when transmission errors occur, so we can view the two previous kinds of distortion as independent and additive. Thus, we can calculate P i s as:

P i s = P i,late s + P i,error s , (4)
where P i,late s and P i,error s refer to the packet losses due to late arrival and transmission error, respectively. Assume that on each link l, in addition to the MDC traffic r l we are interested in, there is some background traffic f l . In a bandwidth-limited network, this combined loss rate can be further modeled based on the M/G/1 queuing model. In this case, the delay distribution of packets over a single link is exponential [START_REF] Zhu | Congestion-distortion optimized video transmission over ad hoc networks[END_REF]. Note that, since the end-to-end delay of packet delivery in wireless network is dominated by the queuing delay at the bottleneck link, the empirical delay distribution for realistic traffic patterns can still be modeled by an exponential formulation:

P i,late s ≈ max l∈L(H i s ) e -µ l (1-θ l )T (5)
where T reflects the maximum tolerate delay, µ l is the service rate C l /K (K is the average packet length) and θ l is the link utilization (r l + f l )/C l for link l. If each link is independent with each other, the end-to-end packet loss probability induced by transmission error for description H i s is

P i,error s = 1 - l∈L(H i s ) (1 -p l ), (6)
where p l is the packet error probability of link l. According to [START_REF] Lee | Price-based distributed algorithm for optimal rate-reliability tradeoff in network utility maximization[END_REF], p l is an increasing function of r l , and a bound on this function is

p l ≥ 1 2 2 -KC l (1-θ l ) . ( 7
)
Assuming that the error probability of each link l is small, we can approximate P i,error s as

P i,error s ≈ l∈L(H i s ) p l ≈ l∈L(H i s ) 1 2 2 -KC l (1-θ l ) . (8)
Therefore, the overall end-to-end packet loss P i s can be expressed as

P i s ≈ max l∈L(H i s ) e -µ l (1-θ l )T + l∈L(H i s ) 1 2 2 -KC l (1-θ l ) = F (r l ), (9)
where F is just a function chosen for notation convenience.

C. Optimization Problem

We investigate the optimal system scheduling by jointly optimizing the rate-distortion adaptation and congestion control. Based on the previous discussion, the optimal problem is:

min s∈S D s ( 10
)
subject to P i s ≥ F (r l ), l ∈ L(H i s ), ∀s, i = 1, 2 r l = 2 i=1 H i s ∈H i s (l) R i s ≤ θ l C l -f l , ∀l d 0 s , d i s , R i s , P i s , r l , θ l , ∀s, i = 1, 2, ∀l
where D s is defined by [START_REF] Akyol | A Flexible Multiple Description Coding Framework for Adaptive Peer-to-Peer Video Streaming[END_REF]. The first constraint is for the packet loss and the second is the flow constraint. This problem is a non-convex optimization problem. But if the ordering of R i s (i = 1, 2) is known for every user s 1 , the optimization problem is a convex optimization if we apply a logarithmic change of variable [START_REF] Chiang | Balancing Transport and Physical Layers in Wireless Multihop Networks: Jointly Optimal Congestion Control and Power Control[END_REF]. With the additional log change of variable to P i s , P i s = exp( P i s ), P i s ≤ 0, the optimization problem [START_REF] Altman | Performance analysis of AIMD mechanisms over a multi-state Markovian path[END_REF] becomes: 1 In practical, R i s is known by setting its original value for all users. It should be noted that when one stream updates in current time slot, the other steams which share links with this stream unchange in this time slot. One iteration contains all of the streams completing one update. To make [START_REF] Altman | Performance analysis of AIMD mechanisms over a multi-state Markovian path[END_REF] strictly convex in R i s , we add -ξr 2 l to the right side of the constraints in [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF], where ξ is a small number such that ξr 2 l is small compared with r l . Since there is a fixed closed interval for R i s , the optimization problem ( 11) is a convex optimization given the ordering of possible R i s for every user s [START_REF] Boyd | Convex Optimization[END_REF]. In the next section, we propose a distributed algorithm where each source and each link solve their own problem with only local information through the standard dual decomposition.

min s∈S D s ( 11
)
subject to P i s ≥ log F (r l ) = F (r l ), l ∈ L(H i s ), ∀s, i = 1, 2 2 i=1 H i s ∈H i s (l) R i s ≤ θ l C l -f l -ξr 2 l , ∀l d 0 s , d i s , R i s , P i s , r l , θ l , ∀s, i = 1, 2, ∀l

III. DISTRIBUTED RATE CONTROL AND ROUTING

In this section, we first describe the Distributed Rate Control and Routing (DRCR) algorithm.

Then, we evaluate DRCR 's convergence and robustness.

A. DRCR Algorithm

DRCR is a joint multipath routing and rate allocation scheme where the sources split traffic for each source-destination pair over multiple paths. The key challenges in designing DRCR are how to select optimal paths L(H i s ) as well as allocated rates R i s to ensure that the resulting system is both stable and optimal. We illustrate the interplay between the sources (that determine L(H i s ) and compute the rates R i s ) and the network links (that feedback link price φ l and congestion price κ l ) in Fig. 1.

DRCR proceeds first by determining the available paths between the source and destination, and then by deciding paths according to the minimum distortion. It proceeds in two phases, the path discovery and path reservation phases, respectively. To this aim, control messages are exchanged between the source and the destination via forwarding by the intermediate nodes. In order to derive exact bounds on the performance of DRCR, we assume that the control channel is reliable, and the nodes are synchronized2 (i.e., there is a bounded time interval in which all nodes receive all dedicated control packets).

TABLE I THE DRCR ALGORITHM

Source s: determine the optimal path for each source-destination pair min P s∈S Ds -

P i P l∈L(H i s ) φ l (t) e P i s - P i R i s κ i s (t)
where φ l (t) denotes the price for link l and κ i s (t) refers to the end-to-end congestion price for H i s at iteration t. Assuming the congestion price for link l is κ l (t), κ i s (t) can be expressed as:

κ i s (t) = P l∈L(H i s ) κ l (t)
• Congestion Price Update:

κ l (t + 1) = [κ l (t) + λκ(t) `r l (t) -θ l C l + f l + ξr 2 l
´]+

where [x] + = max(x, 0), λκ(t) is the step size and

r l (t) = P 2 i=1 P H i s ∈H i s (l) R i s (t)
is the aggregate rate of all the sources on link l at iteration t.

Link l: determine the optimal traffic in each link

min φ l (t) e F (r l ) + κ i s (t) `θl C l -f l -ξr 2 l ẃhere φ l (t) = P 2 i=1 P H l s ∈H i s (l) φ l (t)
is the aggregate traffic load reduction price paid by sources using link l.

• Link Price Update:

φ l (t + 1) = [φ l (t) + λ φ (t) e F (r l )] + , l ∈ L(H i s ) where λ φ (t) is the step size.
The source sends all outgoing links with path discovery messages, which are forwarded by the intermediate nodes on the control channel. At each intermediate node, the path discovery messages contain the information of congestion price and link price related to every possible flow between the source and intermediate nodes. Each intermediate node then extends the path as the source does. Upon reception of path discovery messages from the destination, the source determines the possible paths L(H i s ) between the source and destination based on explicit feedback from the links, in the form of link price φ l and congestion price κ l . In particular, the source minimizes the total distortion while balancing the price of using path L(H i s ). The path price is the product of the source rate with the price per load for path L(H i s ) (computed by summing φ l over all the links in the path). In fact, it is similar to the standard TCP dual algorithm except that the maximization problem is conducted over a vector not a scalar, to reflect the multi-path nature of DRCR.

The assignment of R i s at the source is determined by the total traffic that traverses each link in L(H i s ). The resulting aggregate traffic load price on link

l ∈ L(H i s ) is i H i s ∈H i s (l) φ l F (r l )
, which serves as an implicit feedback that the link uses to compute the congestion price κ l . By the standard dual decomposition approach [START_REF] Lee | Price-based distributed algorithm for optimal rate-reliability tradeoff in network utility maximization[END_REF], DRCR is realized by each source and each link solving their own problem with only local information to get an optimal solution for Eq. [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF]. In this case, each source adjusts its offered congestion price per unit traffic load for each description and each link in its path determines its total traffic that maximize the "net income" of the network based on its link price. The whole DRCR algorithm is shown in Table I.

Since the fixed descriptions for each source, the computations at the sources and the links are linear with the number of the sources. In addition to computation overhead, there are three new functionalities required by DRCR. First, DRCR requires MPLS (Multi-Protocol Label Switching) for splitting traffic over multiple paths. Second, DRCR requires frequent link-load measurements which is possible using the SNMP (Simple Network Management Protocol). Finally, DRCR requires an explicit rate limit for the incoming traffic, and this can be done by dropping packets sent above the allowed rate.

B. Optimality and Stability Characteristics

In this subsection and Appendix I, we provide the analytical derivation and theoretical foundation of the DRCR algorithm.

Theorem 1: The algorithm DRCR converges to the joint global optimum (D s , R i s ) of Eq. ( 11) for sufficiently small step sizes λ κ and λ φ .

Outline of the Proof: The idea of DRCR algorithm is to decouple the coupled objective function in Eq. ( 11) by introducing auxiliary variables and additional constraints, and then use Lagrange dual decomposition to decouple all of the constraints. There are two exact steps: [START_REF] Zhou | Joint QoS Control for Video Streaming over Wireless Multihop Networks: A Cross-Layer Approach[END_REF] Introducing new variables to enable decoupling; (2) Employing dual decomposition and gradient descent method to derive the DRCR algorithm. See Appendix I for the detailed proof.

So far, we have taken a deterministic model with a static population of sources, and stability here means global asymptotic convergence. Given the dynamic nature of DRCR, it is natural to wonder whether it would also behave well with stochastic variations in traffic. Consider sessions arriving according to a Poisson process with exponentially-distributed file sizes. A session leaves the network after it finishes transmitting a file. The service rates are determined by the solution of the DRCR algorithm. Note that sessions may arrive and depart even before the DRCR algorithm converges, i.e., we do not assume time-scale separation between the algorithm convergence and the stochastic stability of DRCR (whether the number of active sessions and the sizes of the queues in the network remain finite for DRCR in such dynamic environment). The answer is positive, as summarized in the following theorem.

Theorem 2: The DRCR algorithm is stochastically stable if the average arrival load in each link is smaller than its capacity, i.e., the stochastic region of DRCR is the largest possible one: the interior of the feasible region of problem [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF].

Outline of the Proof:

The key idea is to show that dual variables are scaled versions of queue lengths, and then to find that the DRCR algorithm is a special case of dual-based algorithms for generalized minimal distortion whose stochastic stability has already been established [START_REF] Liu | Distributed Queue Length Based Algorithms for Optimal End-To-End Throughput Allocation and Stability in Multi-Hop Random Access Networks[END_REF].

C. Simulations

To study the proposed DRCR algorithm, we display some experiment with two wireless multihop networks shown in Fig. 2. Fig. 2(a) is a geometrical-mesh topology, which is representative of a specific network structure. Of the many possible source-destination pairs, we choose 1-6.

If the number of user S = 2 and S = 3, the source-destination pair is replaced by 2 and 3 uses, respectively; Fig. 2(b) is a general network, and we choose 10 source-destination pairs:

1-4, 2-4, 1-3, 2-6, 3-8, 4-5, 7-5, 9-5, 4-6, and 6-9. Similarly, if the number of user S = 20 and S = 30, the source-destination pair is replaced by 2 and 3 uses, respectively. It should be noted that, in order to avoid negative values, the link capacity of each network is assumed to follow a truncated distribution, with an average varying from 200Kbps to 800Kbps.

In all experiments, we start with an initial routing configuration (i.e., the earliest path known by the source) that splits the traffic evenly among the paths for each source-destination pair.

For background flow f l , it is generated according to an on/off source model with exponential distribution of staying time, and average rates spanning between 0 and 100Kbps. To simulate the video application, one HD (High-Definition) sequence (City) is used. In terms of HD video, the sequence has spatial resolution of 1280 × 720 pixels, and the frame rate of 60 frames per second. In addition, we employ the 3-D SPIHT coder, an asymmetric MDC coder introduced in [START_REF] Zhou | A roubust resolution-enhancement scheme for video transmission over mobile ad-hoc networks[END_REF]. In the following, we set K = 40Kbits, T = 300ms unless otherwise specified. In order to evaluate the performance of the joint scheme, the proposed MDC and DRCR scheme (noted as MDC+DRCR) is benchmarked against joint Layered Coding (LC) with Additive-Increase-Multiplicative-Decrease (AIMD) rate allocation scheme (noted as LC+AIMD). LC, which is also called hierarchical coding, can be considered as an extreme case of MDC, where streams are useful only after successful reception of higher priority layers; AIMD-based rate allocation method is used by TCP congestion control [START_REF] Altman | Performance analysis of AIMD mechanisms over a multi-state Markovian path[END_REF]. Fig. 3 shows the distortion comparison for different users under different networks. The proposed MDC+DRCR scheme can be seen to achieve a higher performance in terms of end-to-end distortion (10log s D s ) compared to the competing scheme. For the 8-link network, when S = 1 and C l = 800Kbps, the average distortion using the proposed scheme is -62 while it is -49 for the case of LC+AIMD, thus, 13 performance gain can be achieved on average using the proposed scheme. Similarly, for the 16-link network, when S = 1 and C l = 800Kbps, around 11 performance gain can be achieved on the average. It should be noted that as the user number increases or link capacity decreases, the gap between the proposed MDC+DRCR and LC+AIMD reduces. That is because when users are numerous or link capacities are limited, the total congestion price of each possible path is large and relatively approaches to each other. In this case, the performance difference between the DRCR and AIMD is small.

Then, we test the efficiency of the proposed scheduling scheme when the link capacity is time varying from 200Kbps to 800Kbps. To demonstrate the advantages of using MDC, we compare MDC with single description coding (SDC) and the method in [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF]. Similarly, our MDC+DRSR is benchmarked against SDC+DRSR, LC+AIMD, and Ref. [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF]. The simulation results are presented in Table II. From Table II, we can see that the proposed MDC+DRSR scheme has some performance improvement compared to other competing schemes. It should be noted that although SDC has high coding efficiency, however, it is very sensitive to network conditions, especially to the case of limited network resources. For example, for the 8-link network, when S = 1, the performance gap between MDC+DRSR and SDC+DRSR is 4, while it increases to 20 when S = 30. This in turn illustrates the benefits of using MDC in wireless networks. Moreover, we can also observe that MDC+DRSR also outperforms Ref. [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF]. That is because Ref. [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF] only considers the routing problem, while our paper joint considers the video distortion, rate allocation and multi-path routing.

Then, to present a clear picture of how DRCR works, the proposed DRCR algorithm is operated over the 8-link network when f l = 0, S = 1 and all of the links are with the same capacity. The graphs in Fig. 4 illustrate both the rates and the total distortion of each iteration for 800Kbps and 200Kbps link capacity, respectively. It can be observed that the possible paths for 1-6 pair have changed over the iterations, and re-dispensing its own traffic over the network to avoid already congested links. Changes in the paths also affect the congestion-increment information, which in turn leads to changes in the rate allocation decisions. Note that, DRCR needs numerous iterations to get the optimal value even when S = 1. Therefore, to make DRCR suitable for a more general condition, it is necessary to improve its convergence rate. In addition, a number of interesting observations can be made from these graphs. For example, the final selected paths are both disjoint and the optimal rates in these selected paths are very close to the link capacity.

In the next section, we will study some possible rules to simplify DRCR.

IV. EXTENSION: SIMPLIFIED DRCR FOR IMPROVING CONVERGENCE RATE

So far, we have discussed the equilibrium behavior of the DRCR algorithm. Since the convergence rate for any distributed algorithm on wireless networks is particularly important because the network resource and system traffic are dynamic and source traffic may exhibit low degree of stability, this section simplifies the DRCR algorithm to improve its convergence rate.

A. Design Guide

In this subsection, we derive several theorems to guide the design of a simplified DRCR strategy. This subsection shows that, in the optimal rate allocation, a flow is either used at full bandwidth or not used at all. Furthermore, the optimal rate allocation always chooses the lowest congestion price paths, i.e., a path is selected because there are no other paths with a lower congestion price. We start from an ideal streaming scenario with adequate disjoint paths, and then add the bottleneck links and streaming constraints.

First, assuming that there are adequate disjoint paths for all of the descriptions H i s , ∀s ∈ S, i = 1, 2. From the minimum distortion point of view, any of the two different descriptions H i s streaming over different paths with the same packet loss rate P i s , then, they can be viewed as a single flow with an aggregated rate with packet loss rate P i s . In this case, we first claim that the optimal rate either uses a full bandwidth path, or does not use it at all.

Lemma 1: Given any description

H i s having rate R i s ∈ [0, R max ] (R max : maximum potential rate for R i s
) and a distortion metric s∈S D s (D s is in Eq. ( 2)), the optimal solution of the rate allocation problem when all the paths are disjoint and the optimal value of R i s is either R max or 0, ∀s, i = 1, 2.

Outline of the Proof:

The proof consists of two main steps. First, we show that the optimal rate allocation problem is achieved when paths are disjoint. In this case, we need to prove

that R i s = r l , l ∈ L(H i s ), ∀s ∈ S, i = 1, 2.
We derive the distortion function with respect to rate R i s and r l , respectively. Then, we observe and analyze the conditions of an extremum ∂ s∈S D s /∂R i s = 0 for any R i s and r l . See Appendix II for the detailed proof. Next, we consider the case when there exists joint paths (bottleneck links) in the network. In order to get the minimum distortion, we choose the paths with minimum total network congestion.

Proposition 1: If there are some joint paths in the network, the optimal path selection strategy is to choose the "least" joint paths which leads to the minimum total network congestion, and the rate allocation in this case is the same as Lemma 1.

Outline of the Proof: From the point view of distortion, the less congestion, the less distortion.

Therefore, the basic proof of Proposition 1 is identical to Lemma 1, and the only difference is that the flow constraints have changed. Then we just show that the optimal solution is achieved when R i s gets to the largest possible value approaching to r l . Note that the previous theorem deals with how to choose optimal paths when "joint paths" occurs. The following we address how to allocate the rate for the bottleneck link. Assuming l b is a bottleneck link, and L l b = {L(H i s )}, ∀s, i : l b ∈ H i s , be set of paths (at least two distinct descriptions) sharing the bottleneck link l b . Note that l b may, or may not be a bottleneck link for any of the paths L(H i s ), treated independently. The following theorem regulates the sharing the bandwidth of l b among these joint paths: Theorem 3: Let l b be a bottleneck link for the set of paths L l b = {L(H i s )}, ∀s, i : l b ∈ H i s , the bottleneck link bandwidth shall be shared among paths L(H i s ) in a greedy way, starting with the path with the lowest congestion price.

Outline of the Proof: Let the path L(H i s ) ∈ L l b be arranged in an increasing order of their congestion price

κ i s = i l∈L(H i s ) κ l . Let R k = {R k } k∈L l b
denote a valid rate allocation among these joint paths. Recall that a valid rate allocation has to satisfy the multiple flow constraints.

Let L(H i s ) be the path with lowest congestion price in L l b . If the rate in L(H i s ) is not the largest one, one can always find a better solution by transferring rate from other flows sharing the same bottleneck link. Since the total rate stays constant at that moment, the rate transfer does not violate the multiple flow constraints. It however changes the total source distortion, resulting in a decreased overall performance. By induction, the proof can be extended to all the joint paths. This shows that, for any valid rate allocation R k = {R k } k∈L l b , there exists a best solution that fills up in priority the lowest congestion price path.

B. Simplified DRCR

The previous theorems represent the keys for designing a simplified DRCR algorithm to improve its convergence rate. There are two main changes in the simplified DRCR: (1) cancel the link price. r l value just depends on whether it serves for source-destination pairs or not and on each link's capacity; (2) decouple the joint paths. For the joint paths, each R i s is re-allocated in a greedy way starting with the path with the lowest congestion price, and recalculate the corresponding congestion price of each path again. Therefore, the convergence rate of the DRCR can be improved dramatically by simplifying the rate control procedure. Table III proposes a sketch of the simplified DRCR.

In the following, we show some properties of the simplified DRCR: (1) algorithm converges in one round if paths are disjoint; (2) algorithm terminates in a finite number of rounds in any condition; (3) algorithm converges to the joint global optimum value of Eq. [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF].

Property 1: If the paths requested by the sources do not share any bottleneck joint link l b , the simplified DRCR converges in one round.

Proof: Let L * (H i s ) = {L(H i s )} be the optimal set of paths chosen by the destination for transmission. According to Lemma 1 and Proposition 1, for any available path rate R i s is less than the available bandwidth of each link l ∈ L(H i s ). Since, by hypothesis, the chosen paths L * (H i s ) do not contain any joint bottleneck link, this means that r l ≥ s i R i s , l ∈ L(H i s ). This means that any node, upon the reception of reservation packets, can allocate the request rate on the outgoing links for all requested flows. Therefore, the source can compute the optimal allocation after one round of the protocol. Proof: This result can be seen as an extension of Property 1. For the simplified DRCR, since using the greedy way based on the known congestion price of each possible path, the algorithm can deal with at least one joint link, and the available rate of the links and the corresponding

r l = θ l C l -f l , l ∈ L(H i s ); 13:
R i s = min l∈L(H i s ) r l ; 14:

κ l (t + 1) = [κ l (t) + λκ(t) `P2 i=1 P H i s ∈H i s (l) R i s (t) -θ l C l + f l + ξr 2 l
´]+ ;

15: rate allocation for each H i s can be adjusted at each round. Hence, on subsequent rounds of the algorithm, the sources will be able to deal with a finite number of flows and the algorithm terminates in N l b + 1 rounds.

κ i s (t) = P H i s ∈H i s (l) κ l (

Property 3:

The simplified DRCR converges to the joint global optimum value of Eq. [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF].

Proof: In fact, as for the path selection, the simplified DRCR scheme is identical to the DRCR.

The only difference between them is how to allocate the rate to the given possible paths. From Lemma 1, Proposition 1, and Theorem 3, we can know that the final allocated rates by simplified DRCR is the same with the DRCR. From Theorem 1, we can find that simplified DRCR also converges to the joint global optimum value of Eq. [START_REF] Kompella | Cross-layer optimized multipath routing for video communications in wireless networks[END_REF].

C. Simplified DRCR Versus DRCR

Following the experimental set-up in subsection.III-C, we compare the simplified DRCR with DRCR. At first, we target DRCR as an optimal solution, and test the simplified DRCR in both networks. In Fig. 5, we plot the distortion gap using simplified DRCR with C l = 200Kbps. We can observe that the aggregate distortion gap follows an increasing concave trajectory, converging close to the optimum in a finite iterations. While the graphs in Fig. 5 are for one particular initial conditions, we have done simulations for a variety of initial conditions to verify that the convergence and distortion gap are independent of the initial conditions. Fig. 6 illustrates the number of iteration times corresponding to Fig. 5. Clearly, the simulation results are consistent with the analytical properties in subsection.IV-B. In particular, the simplified method can reduce the number iteration times dramatically compared to DRCR.

V. RELATED WORKS

In this section, we present some related works on the multi-path routing and multi-streaming rate control that have not been discussed earlier in the paper. In addition, we also indicate the difference between our proposed method with the previous works.

A. Multi-Path Routing

Multi-path routing has been an active research topic over the years. For example, various polynomial time algorithms have been proposed to compute multiple shortest paths. Other important works include, node-or link-disjoint path routing, and braided multiple path routing.

However, most of these algorithms do not explicitly consider optimizing performance for the video streams. The problem of path selection for multiple description video streams has recently been explored in [START_REF] Begen | Multi-path selection for multiple description encoded video streaming[END_REF]. [START_REF] Begen | Multi-path selection for multiple description encoded video streaming[END_REF] studies the problem of path selection for double-description video in the context of overlay networks, where path selection is formulated as an optimization problem that minimizes video distortion. The problem is solved by an exhaustive search over the exponential solution space. In a recent work, [START_REF] Wei | Multiple Tree Video Multicast Over Wireless Ad Hoc Networks[END_REF] presents a distributed heuristic for finding two maximally disjoint source trees for double description video streaming in ad hoc networks.

Unlike the aforementioned works that just consider multi-path routing for data traffic over wireless networks, we take into account the specific video characteristics in the routing and rate control scheme. Network congestion is considered in the route selection metric, to meet the stringent delay requirement for video transmission. In addition, each source's rate-distortion characteristic is also incorporated in the joint routing and rate control procedure to provide multiple streams with various contents and complexities.

B. Multi-Streaming Rate Control

The issue of multi-streaming rate control is still an open problem and has received considerable attention recently. A mathematical framework of multi-user rate allocation is presented in [START_REF] Kelly | Rate control for communication networks: Shadow prices, proportional fairness and stability[END_REF],

where the authors also analyzed two classes of distributed solutions, corresponding to the primal and dual decomposition of the optimization objective. In wireless networks, adaptive transmission techniques are typically used to protect the video stream against the time-varying channel [START_REF] Shen | Error resilient video communications over CDMA networks with a bandwidth constraint[END_REF].

When multiple streams are involved, centralized channel time allocation among multiple wireless stations has been investigated in [START_REF] Van Der Schaar | Cross-layer wireless multimedia transmission: challenges, principles, and new paradigms[END_REF]. Distributed algorithms have also been proposed, using rate-distortion optimized packet scheduling in [START_REF] Chakareski | Rate-distortion optimized distributed packet scheduling of multiple video streams over shared communication resources[END_REF] for rate allocation among streams sharing a bottleneck link. What's more, rate allocation algorithm combined with a packet partitioning algorithm has been proposed to support video streaming from multiple sources to a receiver over the Internet [START_REF] Nguyen | Multiple sender distributed video streaming[END_REF]. The rates are chosen to adapt the available network bandwidth for each stream, and the packet partitioning is designed to minimize the start up delay. For video streaming over a wireless multihop networks, a rate control scheme has been proved to efficiently utilize the available wireless link capacity [START_REF] Chen | Rate Contcrol for Streaming Video over Wireless[END_REF].

Our proposed scheme jointly considers the rate control and the routing, and the optimization function contains both network congestion and video distortion. This differs from previous works where routing and rate allocation are considered separately.

VI. CONCLUDING REMARKS

In this paper, we have studied a distributed scheduling for multiple competing MDC streams interacting in a resource-limited wireless multi-hop network. The framework is based on the availability of asymmetric MDC, multipath routing and rate control to jointly optimize the endto-end video distortion of all the users. As detailed in the paper, our proposed distributed rate control and routing scheme as well as its simplified version can be adapted to dynamic wireless networks by adjusting the routing and the allocated rate for each video stream. The theoretical analysis and simulation results demonstrate the effectiveness of our proposed joint scheme for multi-description video streaming transmission over wireless multi-hop networks.

APPENDIX I PROOF OF THEOREM 1

Since ( 11) is a convex optimization problem satisfying Slater's condition, the duality gap is zero. Therefore, a distributed algorithm for (11) can be derived through the Lagrange dual problem. First we form the following Lagrangian: κ l (t)R i s [START_REF] Chiang | Balancing Transport and Physical Layers in Wireless Multihop Networks: Jointly Optimal Congestion Control and Power Control[END_REF] • Each Link l:

L(D s , R i s , φ l , κ l ) = s∈S D s - i l∈L(H i s ) φ l (t) P i s -F (r l ) (12)
+ i H i s ∈H i s (l) κ l (t) R i s -θ l C l + f l +
min φ l (t) F (r l ) -κ i s (t) θ l C l -f l -ξr 2 l ( 14
)
Recall that κ i s (t) = l∈L(H i s ) κ l (t) and φ l (t) =

where (φ l , κ l ) are the dual variables. Note that ( 15) is a convex minimization. Since L d (φ l , κ l ) may be non-differentiable, an iterative subgradient method can be used to update the dual variables to solve (15):

• Link Price Update: φ l (t + 1) = [φ l (t) + λ φ (t) F (r l )] + , [START_REF] Wei | Multiple Tree Video Multicast Over Wireless Ad Hoc Networks[END_REF] where λ φ (t) represents the link price step size.

• Congestion Price Update:

κ l (t + 1) = [κ l (t) + λ κ (t) 2 i=1 H i s ∈H i s (l) R i s (t) -θ l C l + f l + ξr 2 l ] + , ( 17
)
where λ κ (t) represents the congestion price step size. This is exactly the DRCR algorithm described in Table I. Certain choices of step sizes , such as λ κ (t) = λ 1 /t, λ φ (t) = λ 2 /t where λ 1 > 0, λ 2 > 0, guarantee that this algorithm will converge to the joint optimum. As to the relationship between the step size and iteration bounds, please refer to [START_REF] Dantzig | Linear Programming and Extensions[END_REF]. In this case, the convergent point is a globally optimal (D s , R i s ) to the problem (11) since we have shown that the problem can be written as convex optimization.

APPENDIX II PROOF OF LEMMA 1 First, we view R i s and r l , l ∈ L(H i s ) as two different variables and r l ≥ R i s . Using the Lagrange dual function Eq. ( 12), we derive the L(D s , R i s , φ l , κ l ) with respect to R i s and r l respectively, we obtain: Then, in order to get the optimal value of the L(D s , R i s , φ l , κ l ), it is necessary to get the following equations at the same time:

∂L(D s , R i s , φ l , κ l ) ∂R i s = (1 -P 1 s )(
∂L(D s , R i s , φ l , κ l ) ∂R i s = 0, ∂L(D s , R i s , φ, κ) ∂R l = 0. (20)
However, it is easy to find that Eq. ( 20) can't be realized when r l = R i s . Likewise, if r l = R i s , taking into account the previous results in ( 16) and ( 17), when ∂L(Ds,R i s ,φ l ,κ l ) ∂R l = 0 we can find that the optimal solution for R l is:

r * l = θ l C l -f l -2 H i s ∈H i s (l) ξr * l . ( 21
)
Since ξr l are small values for r l , so r * l is:

r * l = θ l C l -f l . ( 22
)
Likewise, the optimal solution for R i s is:

R * i s = min l∈L(H i s ) (θ l C l -f l ). (23)
Therefore, the optimal solution of rate the (11) when R i s = r l = min(θ l C l -f l ), for l ∈ L(H i s ), ∀s, i = 1, 2. If l does not belong to any of the description paths, obviously, r l = 0.

Fig. 1 .

 1 Fig. 1. The diagram for DRCR algorithm.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Two network topologies.

Fig. 4 .

 4 Fig. 4. Plots of rate and distortion versus time for source pair 1-6 in 8-link network for C l = 800Kbps and C l = 200Kbps, respectively. Step sizes: λκ = 4 × 10 -5 , λ φ = 2 × 10 -5 .

Property 2 :

 2 If the paths requested by the sources have shared bottleneck joint links and their number are N l b (N l b ≥ 1), simplified DRCR terminates in N l b + 1 rounds.

Fig. 5 .

 5 Fig. 5. Plots of aggregate distortion gap using simplified DRCR (C l = 200Kbps, f l = 0).

Fig. 6 .

 6 Fig. 6. Iteration times comparison used by Simplified DRCR and DRCR (C l = 200Kbps, f l = 0).

  ) φ l (t) refer to the end-to-end congestion price for H i s and the aggregate traffic load reduction price paid by sources using link l at iteration t, respectively. The Lagrangian dual function L d (φ, κ) is defined as the maximized L(D s , R i s , φ, κ) over D s and R i s for given φ and κ. Each source can compute an optimizer D * s and each link l can compute an optimizer r * l (φ, κ). The Lagrange dual problem of (11) is:min L d (φ l , κ l ) = L(D * s , r * l (φ l , κ l ), φ l , κ l ),

TABLE II PERFORMANCE

 II COMPARISON FOR DIFFERENT METHODS IN DIFFERENT NETWORKS WITH VARIABLE LINK CAPACITIES

	User	Network		Average Distortion of Different Methods	Average
	Number	Type	MDC+DRSR SDC+DRSR LC+AIMD	Ref. [11]	Improvement
	1	8-link 16-link	-48 -77	-44 -70	-38 -64	-38 -60	4∼10 7∼17
	2	8-link 16-link	-47 -72	-41 -65	-33 -55	-33 -53	6∼14 7∼19
	3	8-link 16-link	-45 -66	-36 -58	-30 -50	-27 -53	9∼18 8∼16
	10	8-link 16-link	-33 -47	-15 -27	-24 -37	-20 -33	9∼18 10∼20
	20	8-link 16-link	-31 -44	-11 -22	-19 -30	-21 -32	10∼20 12∼22
	30	8-link 16-link	-27 -41	-7 -16	-15 -26	-5 -23	12∼22 15∼25

  ξr2

	φ l (t) P i s +
	i l∈L(H i s )	i H i s ∈H i s (l)

l • Each Source s: min s∈S D s -

We use the method proposed in[START_REF] Liu | Distributed Queue Length Based Algorithms for Optimal End-To-End Throughput Allocation and Stability in Multi-Hop Random Access Networks[END_REF] to guarantee the node synchronization in one practical experiment.

This work is supported by the International Project PRA-SI (financed by France and China government) under Grant No. SI04-03, and the Key Project of Nature Science Foundation of Jiangsu (China) under Grant BK2007729.