59 ( ; ; ) = ( = ) = ( + ) ! ( ) ( + ) ( + ) , > 0, > 0 (2) When the mean number of nests is low during all the season, a Poissonian distribution rather than a NB distribution can be used (Girondot 2017). Poissonian distribution is a special case of NB distribution when k is +Inf. When Nt is equal to 0, it is replaced with 10-9 as the negative binomial and the Poissonian models are not defined for m=0. When the count (Nt) represents the exact sum of activity during several nights, the probability mass function of the sum of NB is used (Girondot and Barry 2023). When the count represents the minimum number of activities because of the track loss in time, the likelihood is the integral of the probability mass function of NB for all the values between Nt and the maximal number of nests Nmax. Nmax is determined taking into account the probability of detection of a body pit after a known period on the beach. Nesting seasonality was modeled following the Girondot phenological model (Girondot 2010, 2017), using the R package ‘Phenology’ available in the Comprehensive R Archive Network (https://cran.rproject.org) (Girondot 2023b). This model can be applied to any proxy of nesting such as nest or track counts. Scripts are provided in the following GitHub repository https://github.com/Marc-Girondot/NC2024/. 3.3.3 Interannual spatial and temporal trends A model is used to estimate the number of nest tracks for any of the considered islets that has not been surveyed on a given season, based on the number of tracks recorded on this islet during other monitored seasons, the relative frequency of tracks on the different sites and the total number of tracks for each nesting season. Let the total theoretical number of nest tracks be Ti for season i in the entire region where K islets were monitored during a range of Y years. The distribution of the nests across the different sites is defined by the proportion pj of Ti nests in the j beach. For a total of K islets, K–1 parameters p are necessary due to the relation: ∑ =1 = 1 (3) The values pj can be modeled as constant (K–1 parameters) or first (2.K–2 parameters) or second (3.K–3 parameters) order as a function of time to represent situations with changes in the relative use of the different nesting sites. The expected number of nests for year i in the beach j is then: , = ∗ (4)
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