BACKGROUND
many species migrate and hence are present at a particular geographic location during only part of the year  <cit> .
this behavior often produces 'tidal-waves' in the density of individuals along the migration route  <cit>  with a lower frequency signal at wintering or aestivating grounds where the animals tend to stay longer.
accurate detection and quantification of individuals of a population during migration is challenging because the movement of individuals often masks or amplifies the true size of the population, particularly when not all individuals migrate each year  <cit> .
periodic signals of high frequency may also be observed even when animals are resident at a particular location.
for example, the presence of auklets peaks in the morning and in the evening at some colonies  <cit> .
in this latter example, population estimates may vary depending on whether the crepuscular peaks in presence were taken into account  <cit> .
marine turtles exhibit variable seasonal migration patterns, with adult females being present on nesting beaches for only part of the year  <cit> .
when on the beach, nesting sea turtles can be observed and counted as part of estimating population size  <cit> .
calculating the size of a marine turtle population is an essential step in assessing population status and trends .
there are various challenges associated with directly counting the total number of individuals in a marine turtle population, including cryptic life history stages, trans-oceanic dispersal, and nonsequential annual reproduction  <cit> .
as a result, researchers have traditionally relied on enumerating numbers of nests laid by a population as an index of population size  <cit> .
because marine turtles nest often on open sandy tropical and subtropical beaches, and because the adult females leave wide deep tracks on the beach after nesting, it is a relatively easy task to identify a sea turtle nesting crawl  <cit> .
however, not every crawl of a female sea turtle on the beach results in a nest; for instance, when a female is disturbed before oviposition, she may abandon the nesting attempt before successfully laying eggs  <cit> .
if tracks from non-nesting and nesting turtles are confounded during monitoring, this may strongly bias the estimated number of nests laid during a night.
in the case of leatherbacks , this potential bias can be greatly reduced by counting only those tracks that also contain a body-pit made during the nesting process: it is relatively rare  for leatherbacks to abandon the nesting process once they have made a body pit  <cit> .
another challenge in counting sea turtle nests is that nesting seasons usually span several months and turtles can lay their eggs on remote beaches that are difficult to access.
as a result, for many sea turtle monitoring programs there are often temporal and/or spatial gaps in monitoring effort that must be corrected for, particularly when comparing datasets from different years or populations.
several correction factors for gaps in sea turtle nest counts have been previously suggested  <cit> .
one simple method used to correct for an incomplete dataset from a sea turtle nesting beach is n^
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgubgbgaqcaaaa@2e21@ = n/ .
this correction assumes that all beach sections being monitored are homogeneous; a bias is introduced if they are not.
this may be irrelevant if the introduced bias is constant and one is interested only in trends of the abundance of adult females  <cit> .
however, if the sections of the beach surveyed are not similar from year to year, the introduced bias will fluctuate, rendering this simple correction factor unsuitable.
a more complicated situation occurs when there are temporal gaps during monitoring of the nesting season.
as the temporal distribution of nests during a season tends to be bell-shaped, a simple extrapolation based on the proportion of the season that was monitored to correct for temporal gaps in nest monitoring will likely give false results.
furthermore, the distribution of nests during the season is dependent on many factors, such as the total number of nesting females, the timing of initial nesting dates for individual females and the distribution of total number of nests laid during the season by each female.
marine turtles are iteroparous  <cit> , nesting up to  <dig> times in a single season in the case of leatherbacks in french guiana  <cit> .
the internesting interval for females varies according to species: from  <dig> days on average for leatherbacks  <cit>  up to  <dig> days for olive ridley  females in mass nesting events  <cit> .
the internesting interval within some species can vary with fluctuating water temperatures  <cit> .
the temporal pattern of internesting intervals within a single nesting season can produce fluctuations in the total distribution of nests, resulting in successive peaks of nests on certain days during the nesting season.
peaks in nesting activity every ~ <dig> days in apparent synchrony with the lunar phase and tide level have been observed on the leatherback nesting beach of ya:lima:po beach in french guiana  <cit> .
this synchrony may be due to some females adjusting their subsequent nesting date to coincide with a particular lunar phase/tide level; nevertheless, the overall mean internesting interval is  <dig> days for this population  <cit> .
the objective of this study was to develop a method that can transform an incomplete dataset of nesting activity for a sea turtle population into a dataset that accurately describes both the pattern of nesting distribution during the nesting season and any possible short-term fluctuations.
the gaps in the data could be both spatial and temporal.
the objective was achieved by both extrapolation and interpolation, based on a derived realistic model of the nesting season that included parameters that account for variation in the shape of the nesting season.
the resultant model is useful not only for calculating the total number of nests laid  during an entire season, but also for facilitating comparisons across nesting beach datasets that contain dissimilar spatio-temporal gaps.
we tested the developed method using an incomplete leatherback nest count dataset  on ya:lima:po nesting beach in french guiana , and tested it using a complete nest count dataset from  <dig>
RESULTS
ya:lima:po entire beach –  <dig> data
a total of  <dig> nests were counted during  <dig> field days from 1/3/ <dig> until 31/12/ <dig> on the ya:lima:po nesting beach.
the akaike information content  of the  <dig> models describing the shape of the nesting season was lowest when the k <dig> parameter was fixed to  <dig>  however, the various model combinations with k <dig> =  <dig> and/or k <dig> =  <dig> had essentially the same support .
in contrast, the models with the min parameter fixed to  <dig> showed considerable lower support.
as outlined above, our initial model selection criteria were a) low aic value and b) fewer parameters.
from this, the initial model we retained used  <dig> parameters .
the goodness of fit for this model was r <dig> =  <dig>  and it estimated a seasonal total of  <dig> ± 73sd nests.
we added a sinusoid function to this pattern, by generating a map of -ln likelihood with varying φ <dig> and Δ <dig> parameters .
the maximum likelihood was observed for φ <dig> =  <dig>  and Δ <dig> =  <dig> .
these values were used as initial points to search for values of α <dig> β <dig>  φ <dig>  Δ <dig>  and τ <dig> parameters that also maximized likelihood.
we then searched for a simpler model assuming α <dig> =  <dig> and/or β <dig> =  <dig> and/or τ <dig> = l. the model with lowest aic had a fitted value of τ <dig>  however, based on akaike weights, this model exhibited a goodness of fit similar to the model with τ <dig> =  <dig> .
as this latter model was simpler, we retained it for further tests.
when one sinusoid was added, the aic =  <dig> , much lower than aic =  <dig>  prior to the addition of the sinusoid.
the goodness of fit for this model was r <dig> =  <dig>  and the total number of nests is estimated to be  <dig> ± 62sd.
a second sinusoid was added using the same procedure.
the map of -ln likelihood showed a minimum for .
the goodness of fit for this two-sinusoid model was r <dig> =  <dig>  and the total number of nests it estimated was  <dig> ± 58sd .
the addition of a third sinusoid gave the model lower a aic , but based on akaike weight the overall fit was not significantly better than the model with only two sinusoids.
therefore, the best-fit model based on both aic and akaike weight criteria had two sinusoids.
analysis of beach sections –  <dig> data
nest distribution for each beach section  was fitted as previously described.
a homogeneity test showed that the global shape of nesting was significantly different for the four beach sections .
a similar conclusion was obtained when only the beginning of the nesting season was assumed to be identical  or the end of nesting season was assumed to be identical .
the index d+10% that describes the beginning of the nesting season showed a significant spatial tendency, with the nesting season beginning later in the western section of the beach , but no spatial influence was observed for either the peak of the season or the d-10% descriptor of the end of the nesting season .
although the shapes of nest distribution among the different beach sections were significantly different, the use of a single model for the global shape of nest distribution during the nesting season did not affect the estimates of total nest numbers for each zone .
when one sinusoid with a period varying from  <dig> to  <dig> days by steps of  <dig>  days was added to the models, the beach sections also showed strong differences.
indeed, the sinusoidal period appeared to be different according to the position of the beach section analyzed.
the  <dig> -days signal was significantly strong within section i whereas the  <dig> -days signal was significantly strong within section iv .
ya:lima:po entire beach –  <dig> data
a total of  <dig> nests were counted during  <dig> field days from 1/3/ <dig> until 15/9/ <dig> on ya:lima:po nesting beach.
only  <dig> daily counts were missing at the end of the nesting season.
this time-series can therefore be considered as essentially complete as the nesting after july is relatively sparse.
the aic of the  <dig> models describing the overall shape of nesting season was lowest when the k <dig> parameter was fixed to  <dig>  but was not significantly different from several other models based on akaike weight .
we choose the first one in the series with akaike weight >  <dig>  and minimum number of fitted parameters.
we added a sinusoid function to this pattern, by generating a map of -ln likelihood with varying φ <dig> and Δ <dig> parameters.
the maximum likelihood was observed for .
the model with the lowest aic had a nearly identical aic value to the model that was had one less parameter .
using the principle of parsimony, we retained the model with fewer parameters for further exploration.
second and third sinusoidal signals were added to the model; we selected the single model within all groups of models  with the lowest aic values.
of these four, we found that the model with  <dig> sinusoidal signals had the lowest akaike weight .
the two sinusoidal periods fitted were  <dig>  and  <dig>  days.
the gratiot et al.
model  <cit>  when applied to this time-series was strongly rejected compared to the model presented here .
the number of observed nests during the  <dig> night patrols was  <dig>  the number of nests calculated by the models was as follows:  <dig>  for no sinusoidal signal,  <dig>  for  <dig> sinusoidal signal added, and  <dig>  for  <dig> sinusoidal signals added .
all three estimates are close to the observed number of nests .
the periodic signal within number of nests that was detected in the  <dig> database comprised of  <dig> consecutive day counts can be also tested using classical mathematical tools.
the multi-taper method  of spectral analysis provides a means for spectral estimation  <cit>  of a time series that is believed to exhibit a spectrum containing both continuous and singular components.
a mtm test for white noise in this time-series indeed detects a significant period signal at both  <dig>  days  and at  <dig>  days .
however, the mtm method can be used only with continuous time-series, which therefore makes it ineffective when analyzing marine turtle datasets that contain gaps in coverage.
discussion
the total number of nests deposited during the nesting season is often used as an index of abundance for marine turtle species.
to obtain reliable data, monitoring of the nesting beach should be conducted on a regular basis during the nesting season.
however, nesting seasons can span up  <dig> months or more, making it challenging to effectively collect data on a daily basis  <cit> .
it is common for nesting beach datasets to be missing information, either during, at the beginning and/or at end of the nesting season.
one means to account for data gaps during the nesting season is to employ interpolation.
for instance, girondot & fretey  <cit>  were able to fill in data gaps for a leatherback nesting population using interpolation with a small associated error .
however, this procedure is inadequate if the beginning and/or the end of a nesting season is not monitored.
troëng et al.
<cit>  used a moving average, but again, the behavior of such a function can be erratic at the boundaries of the time-series.
yet gaps at the beginning or end of the season are common because research teams tend to concentrate monitoring effort at the peak of the nesting season.
to be able to extrapolate data, one needs an equation that accurately describes the global nesting season.
a recently proposed symmetrical model of a leatherback nesting season  <cit>  is inadequate for modeling the dataset from ya:lima:po because the time series analyzed above was not symmetrical  ≠  <dig> aic  <dig>  or s <dig> = s <dig> and k <dig> = k <dig> =  <dig> aic  <dig>  vs. asymmetrical model – see table 1b – s <dig> ≠ s <dig> and k <dig> = k <dig> =  <dig> aic  <dig> ).
also, nest counts outside the main sinusoid were calculated separately before the fitting procedure used by gratiot et al.
<cit> .
finally, a fitting procedure based on two rounds should be avoided, as they generally do not converge on the best fit of the data.
in contrast, we have shown that the best fit model describing the nesting season of leatherback at ya:lima:po exhibits three components: an overall shape describing the nesting season and two sinusoids signals.
for  <dig>  one signal has a period of  <dig>  ±  <dig>  sd days and the other a period of  <dig>  ±  <dig>  sd days.
for  <dig>  one signal has a period of  <dig>  ±  <dig>  sd days and the other a period of  <dig>  ±  <dig>  sd days.
the first signal likely represents the reproductive cycle of females that nest sequentially during the nesting season, with nests separated by approximately nine to ten days  <cit> .
the second signal could be related to the lunar or tidal phase when full and new moon alternate every  <dig>  days  <cit> .
this signal was particularly strong for the beach section located furthest within the estuary .
in this section, the peak of nesting coincided with the spring high tides observed during each full and new moons.
interestingly, it has been observed that greater numbers of female leatherbacks become trapped in exposed mud flats after nesting during neap tides on the nearby beaches in suriname  <cit> .
however, we do not think that the peak of nesting during spring tides represents a selective avoidance response to this potential mortality factor, for the following two reasons.
first, the actual mortality rate is low for females that are temporarily trapped in the mud-flats , making it unlikely that this would constitute a strong selective pressure.
second, the peak of nesting during spring tide is not observed in all beach sections .
nesting at high tide during spring tide could be also viewed as a way to minimize energy expenditure while selecting a suitable site for nesting on the beach  <cit> .
alternatively, the peaks in nesting distribution at spring high tide could result from currents in the maroni river that may prevent individual leatherbacks from nesting too far inside the estuary at neap tide .
however, an expected complementary peak at neap tide in other beach sections further away from the maroni was not detected .
a third possible explanation is that a shift of female turtles from other nesting beaches in the region  to ya:lima:po beach may be associated with the spring high tide.
current data are insufficient to test this hypothesis.
in french guiana, leatherback females can adjust their return nesting date from  <dig> to  <dig> days between two nesting events in order to be closer to a full or new moon  <cit> .
this effect was not observed in leatherbacks nesting on the pacific coast of costa rica  <cit> .
however, the number of observations available from costa rica was only one tenth of the number in girondot & fretey  <cit> , which is insufficient to detect an effect similar to the one observed in ya:lima:po .
note that even with this significant peak of nesting at the spring high tide, leatherbacks in french guiana have a stronger signature at  <dig>  days that is the characteristic nesting interval for this species  <cit> .
stronger and more significant periodic signals were detected at both ends of the ya:lima:po nesting beach .
the lack of significant signal in sections ii and iii is probably the consequence of reduced environmental constraints from tide or river currents for the central sections of the beach.
the numbers of total nests calculated by the various methods proposed here were not significantly different: the additions of one or two sinusoidal signals in  <dig> changed the estimation of total nest number from  <dig> ±  <dig> sd , to  <dig> ±  <dig> sd  and  <dig> ±  <dig> sd .
given the consistency of total nest estimates, one could question why a more complex model is needed.
it should be noted that the standard deviations of the estimates were lower with the addition of sinusoids because these sinusoids explained some of the variation that was expressed by the standard deviation of the simpler models .
therefore, the beta risk was reduced when the estimate was derived with the more complex model.
CONCLUSIONS
the model proposed here offers a variety of uses beyond its application in accurately estimating total counts from an incomplete dataset.
it also can detect intraseasonal concentrations of animals that may be the result of behavioral interactions, synchronization with external factors and/or biological constraints.
when applied to leatherbacks nesting in french guiana, the model detected  <dig> principal components that contribute to the seasonal nest distribution: a global shape that corresponds to the arrival and departure of leatherback females in the guianas region, a sinusoidal pattern with a period of approximately  <dig> days that is related to physiological constraints of nesting leatherbacks, and a sinusoidal pattern with a longer period that likely reflects the influence of the spring high tide on nesting turtles.
the proposed model is an improvement over previously published methods  <cit>  but we should keep in mind that nest number is only an index of the population size.
indeed, variable remigration interval  <cit>  and variable number of nest per female  <cit>  affect also strongly the observed number of nests for a particular year.

 many species migrate and hence are present at a particular geographic location during only part of the year. this behavior often produces'tidal-waves' in the density of individuals along the migration route with a lower frequency signal at wintering or aestivating grounds where the animals tend to stay longer. accurate detection and quantification of individuals of a population during migration is challenging because the movement of individuals often masks or amplifies the true size of the population, particularly when not all individuals migrate each year. periodic signals of high frequency may also be observed even when animals are resident at a particular location. for example, the presence of auklets peaks in the morning and in the evening at some colonies. in this latter example, population estimates may vary depending on whether the crepuscular peaks in presence were taken into account. marine turtles exhibit variable seasonal migration patterns, with adult females being present on nesting beaches for only part of the year. when on the beach, nesting sea turtles can be observed and counted as part of estimating population size. calculating the size of a marine turtle population is an essential step in assessing population status and trends. there are various challenges associated with directly counting the total number of individuals in a marine turtle population, including cryptic life history stages, trans-oceanic dispersal, and nonsequential annual reproduction. as a result, researchers have traditionally relied on enumerating numbers of nests laid by a population as an index of population size. because marine turtles nest often on open sandy tropical and subtropical beaches, and because the adult females leave wide deep tracks on the beach after nesting, it is a relatively easy task to identify a sea turtle nesting crawl. however, not every crawl of a female sea turtle on the beach results in a nest; for instance, when a female is disturbed before oviposition, she may abandon the nesting attempt before successfully laying eggs. if tracks from non-nesting and nesting turtles are confounded during monitoring, this may strongly bias the estimated number of nests laid during a night. in the case of leatherbacks, this potential bias can be greatly reduced by counting only those tracks that also contain a body-pit made during the nesting process: it is relatively rare for leatherbacks to abandon the nesting process once they have made a body pit. another challenge in counting sea turtle nests is that nesting seasons usually span several months and turtles can lay their eggs on remote beaches that are difficult to access. as a result, for many sea turtle monitoring programs there are often temporal and/or spatial gaps in monitoring effort that must be corrected for, particularly when comparing datasets from different years or populations. several correction factors for gaps in sea turtle nest counts have been previously suggested. one simple method used to correct for an incomplete dataset from a sea turtle nesting beach is n^ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwgubgbgaqcaaaa@2e21@ = n/. this correction assumes that all beach sections being monitored are homogeneous; a bias is introduced if they are not. this may be irrelevant if the introduced bias is constant and one is interested only in trends of the abundance of adult females. however, if the sections of the beach surveyed are not similar from year to year, the introduced bias will fluctuate, rendering this simple correction factor unsuitable. a more complicated situation occurs when there are temporal gaps during monitoring of the nesting season. as the temporal distribution of nests during a season tends to be bell-shaped, a simple extrapolation based on the proportion of the season that was monitored to correct for temporal gaps in nest monitoring will likely give false results. furthermore, the distribution of nests during the season is dependent on many factors, such as the total number of nesting females, the timing of initial nesting dates for individual females and the distribution of total number of nests laid during the season by each female. marine turtles are iteroparous, nesting up to times in a single season in the case of leatherbacks in french guiana. the internesting interval for females varies according to species: from days on average for leatherbacks up to days for olive ridley females in mass nesting events. the internesting interval within some species can vary with fluctuating water temperatures. the temporal pattern of internesting intervals within a single nesting season can produce fluctuations in the total distribution of nests, resulting in successive peaks of nests on certain days during the nesting season. peaks in nesting activity every ~ days in apparent synchrony with the lunar phase and tide level have been observed on the leatherback nesting beach of ya:lima:po beach in french guiana. this synchrony may be due to some females adjusting their subsequent nesting date to coincide with a particular lunar phase/tide level; nevertheless, the overall mean internesting interval is days for this population. the objective of this study was to develop a method that can transform an incomplete dataset of nesting activity for a sea turtle population into a dataset that accurately describes both the pattern of nesting distribution during the nesting season and any possible short-term fluctuations. the gaps in the data could be both spatial and temporal. the objective was achieved by both extrapolation and interpolation, based on a derived realistic model of the nesting season that included parameters that account for variation in the shape of the nesting season. the resultant model is useful not only for calculating the total number of nests laid during an entire season, but also for facilitating comparisons across nesting beach datasets that contain dissimilar spatio-temporal gaps. we tested the developed method using an incomplete leatherback nest count dataset on ya:lima:po nesting beach in french guiana, and tested it using a complete nest count dataset from  ya:lima:po entire beach – data a total of nests were counted during field days from 1/3/ until 31/12/ on the ya:lima:po nesting beach. the akaike information content of the models describing the shape of the nesting season was lowest when the k parameter was fixed to however, the various model combinations with k = and/or k = had essentially the same support. in contrast, the models with the min parameter fixed to showed considerable lower support. as outlined above, our initial model selection criteria were a) low aic value and b) fewer parameters. from this, the initial model we retained used parameters. the goodness of fit for this model was r = and it estimated a seasonal total of ± 73sd nests. we added a sinusoid function to this pattern, by generating a map of -ln likelihood with varying φ and Δ parameters. the maximum likelihood was observed for φ = and Δ =. these values were used as initial points to search for values of α β φ Δ and τ parameters that also maximized likelihood. we then searched for a simpler model assuming α = and/or β = and/or τ = l. the model with lowest aic had a fitted value of τ however, based on akaike weights, this model exhibited a goodness of fit similar to the model with τ =. as this latter model was simpler, we retained it for further tests. when one sinusoid was added, the aic =, much lower than aic = prior to the addition of the sinusoid. the goodness of fit for this model was r = and the total number of nests is estimated to be ± 62sd. a second sinusoid was added using the same procedure. the map of -ln likelihood showed a minimum for. the goodness of fit for this two-sinusoid model was r = and the total number of nests it estimated was ± 58sd. the addition of a third sinusoid gave the model lower a aic, but based on akaike weight the overall fit was not significantly better than the model with only two sinusoids. therefore, the best-fit model based on both aic and akaike weight criteria had two sinusoids. analysis of beach sections – data nest distribution for each beach section was fitted as previously described. a homogeneity test showed that the global shape of nesting was significantly different for the four beach sections. a similar conclusion was obtained when only the beginning of the nesting season was assumed to be identical or the end of nesting season was assumed to be identical. the index d+10% that describes the beginning of the nesting season showed a significant spatial tendency, with the nesting season beginning later in the western section of the beach, but no spatial influence was observed for either the peak of the season or the d-10% descriptor of the end of the nesting season. although the shapes of nest distribution among the different beach sections were significantly different, the use of a single model for the global shape of nest distribution during the nesting season did not affect the estimates of total nest numbers for each zone. when one sinusoid with a period varying from to days by steps of days was added to the models, the beach sections also showed strong differences. indeed, the sinusoidal period appeared to be different according to the position of the beach section analyzed. the -days signal was significantly strong within section i whereas the -days signal was significantly strong within section iv. ya:lima:po entire beach – data a total of nests were counted during field days from 1/3/ until 15/9/ on ya:lima:po nesting beach. only daily counts were missing at the end of the nesting season. this time-series can therefore be considered as essentially complete as the nesting after july is relatively sparse. the aic of the models describing the overall shape of nesting season was lowest when the k parameter was fixed to but was not significantly different from several other models based on akaike weight. we choose the first one in the series with akaike weight > and minimum number of fitted parameters. we added a sinusoid function to this pattern, by generating a map of -ln likelihood with varying φ and Δ parameters. the maximum likelihood was observed for. the model with the lowest aic had a nearly identical aic value to the model that was had one less parameter. using the principle of parsimony, we retained the model with fewer parameters for further exploration. second and third sinusoidal signals were added to the model; we selected the single model within all groups of models with the lowest aic values. of these four, we found that the model with sinusoidal signals had the lowest akaike weight. the two sinusoidal periods fitted were and days. the gratiot et al. model when applied to this time-series was strongly rejected compared to the model presented here. the number of observed nests during the night patrols was the number of nests calculated by the models was as follows: for no sinusoidal signal, for sinusoidal signal added, and for sinusoidal signals added. all three estimates are close to the observed number of nests. the periodic signal within number of nests that was detected in the database comprised of consecutive day counts can be also tested using classical mathematical tools. the multi-taper method of spectral analysis provides a means for spectral estimation of a time series that is believed to exhibit a spectrum containing both continuous and singular components. a mtm test for white noise in this time-series indeed detects a significant period signal at both days and at days. however, the mtm method can be used only with continuous time-series, which therefore makes it ineffective when analyzing marine turtle datasets that contain gaps in coverage. discussion the total number of nests deposited during the nesting season is often used as an index of abundance for marine turtle species. to obtain reliable data, monitoring of the nesting beach should be conducted on a regular basis during the nesting season. however, nesting seasons can span up months or more, making it challenging to effectively collect data on a daily basis. it is common for nesting beach datasets to be missing information, either during, at the beginning and/or at end of the nesting season. one means to account for data gaps during the nesting season is to employ interpolation. for instance, girondot & fretey were able to fill in data gaps for a leatherback nesting population using interpolation with a small associated error. however, this procedure is inadequate if the beginning and/or the end of a nesting season is not monitored. troëng et al. used a moving average, but again, the behavior of such a function can be erratic at the boundaries of the time-series. yet gaps at the beginning or end of the season are common because research teams tend to concentrate monitoring effort at the peak of the nesting season. to be able to extrapolate data, one needs an equation that accurately describes the global nesting season. a recently proposed symmetrical model of a leatherback nesting season is inadequate for modeling the dataset from ya:lima:po because the time series analyzed above was not symmetrical ≠ aic or s = s and k = k = aic vs. asymmetrical model – see table 1b – s ≠ s and k = k = aic ). also, nest counts outside the main sinusoid were calculated separately before the fitting procedure used by gratiot et al.. finally, a fitting procedure based on two rounds should be avoided, as they generally do not converge on the best fit of the data. in contrast, we have shown that the best fit model describing the nesting season of leatherback at ya:lima:po exhibits three components: an overall shape describing the nesting season and two sinusoids signals. for one signal has a period of ± sd days and the other a period of ± sd days. for one signal has a period of ± sd days and the other a period of ± sd days. the first signal likely represents the reproductive cycle of females that nest sequentially during the nesting season, with nests separated by approximately nine to ten days. the second signal could be related to the lunar or tidal phase when full and new moon alternate every days. this signal was particularly strong for the beach section located furthest within the estuary. in this section, the peak of nesting coincided with the spring high tides observed during each full and new moons. interestingly, it has been observed that greater numbers of female leatherbacks become trapped in exposed mud flats after nesting during neap tides on the nearby beaches in suriname. however, we do not think that the peak of nesting during spring tides represents a selective avoidance response to this potential mortality factor, for the following two reasons. first, the actual mortality rate is low for females that are temporarily trapped in the mud-flats, making it unlikely that this would constitute a strong selective pressure. second, the peak of nesting during spring tide is not observed in all beach sections. nesting at high tide during spring tide could be also viewed as a way to minimize energy expenditure while selecting a suitable site for nesting on the beach. alternatively, the peaks in nesting distribution at spring high tide could result from currents in the maroni river that may prevent individual leatherbacks from nesting too far inside the estuary at neap tide. however, an expected complementary peak at neap tide in other beach sections further away from the maroni was not detected. a third possible explanation is that a shift of female turtles from other nesting beaches in the region to ya:lima:po beach may be associated with the spring high tide. current data are insufficient to test this hypothesis. in french guiana, leatherback females can adjust their return nesting date from to days between two nesting events in order to be closer to a full or new moon. this effect was not observed in leatherbacks nesting on the pacific coast of costa rica. however, the number of observations available from costa rica was only one tenth of the number in girondot & fretey, which is insufficient to detect an effect similar to the one observed in ya:lima:po. note that even with this significant peak of nesting at the spring high tide, leatherbacks in french guiana have a stronger signature at days that is the characteristic nesting interval for this species. stronger and more significant periodic signals were detected at both ends of the ya:lima:po nesting beach. the lack of significant signal in sections ii and iii is probably the consequence of reduced environmental constraints from tide or river currents for the central sections of the beach. the numbers of total nests calculated by the various methods proposed here were not significantly different: the additions of one or two sinusoidal signals in changed the estimation of total nest number from ± sd, to ± sd and ± sd. given the consistency of total nest estimates, one could question why a more complex model is needed. it should be noted that the standard deviations of the estimates were lower with the addition of sinusoids because these sinusoids explained some of the variation that was expressed by the standard deviation of the simpler models. therefore, the beta risk was reduced when the estimate was derived with the more complex model.  the model proposed here offers a variety of uses beyond its application in accurately estimating total counts from an incomplete dataset. it also can detect intraseasonal concentrations of animals that may be the result of behavioral interactions, synchronization with external factors and/or biological constraints. when applied to leatherbacks nesting in french guiana, the model detected principal components that contribute to the seasonal nest distribution: a global shape that corresponds to the arrival and departure of leatherback females in the guianas region, a sinusoidal pattern with a period of approximately days that is related to physiological constraints of nesting leatherbacks, and a sinusoidal pattern with a longer period that likely reflects the influence of the spring high tide on nesting turtles. the proposed model is an improvement over previously published methods but we should keep in mind that nest number is only an index of the population size. indeed, variable remigration interval and variable number of nest per female affect also strongly the observed number of nests for a particular year. 
the model proposed here offers a variety of uses for both marine turtles and also other taxa when individuals are observed in a particular location for only part of the year. we detected primary components in the nest distribution of leatherbacks: an overall shape that corresponds to the arrival and departure of leatherback females in the guianas region, a sinusoidal pattern with a period of approximately days that is related to physiological constraints of nesting female leatherbacks, and a sinusoidal pattern with a period of approximately days that likely reflects the influence of spring high tides on nesting female turtles. in the case of leatherback sea turtles nesting in french guiana, we developed a mathematical model to explore the phenology of their nesting season, derived from an incomplete nest count dataset. marine turtles deposit their eggs on tropical or subtropical beaches during discrete nesting seasons that span several months.
 marine turtles deposit their eggs on tropical or subtropical beaches during discrete nesting seasons that span several months. the number and distribution of nests laid during a nesting season provide vital information on various aspects of marine turtle ecology and conservation.  in the case of leatherback sea turtles nesting in french guiana, we developed a mathematical model to explore the phenology of their nesting season, derived from an incomplete nest count dataset. we detected primary components in the nest distribution of leatherbacks: an overall shape that corresponds to the arrival and departure of leatherback females in the guianas region, a sinusoidal pattern with a period of approximately days that is related to physiological constraints of nesting female leatherbacks, and a sinusoidal pattern with a period of approximately days that likely reflects the influence of spring high tides on nesting female turtles.  the model proposed here offers a variety of uses for both marine turtles and also other taxa when individuals are observed in a particular location for only part of the year. 