58 = { t < B → PMin × Max t ∈ [B, P − F[ → ((1 + cos(π(P − F − t)(P − F − B)))⁄2) (Max − PMin × Max) + PMin × Max t ∈ [P − F, P + F[ → Max t ∈ [P + F, E] → ((1 + cos(π(t − P + F)(E − P + F)))⁄2) (Max − PMin × Max) + PMin × Max t > E → PMin × Max (1) The model requires at most seven parameters, all of which have direct biological interpretations. A graphic explanation of these terms is available in Girondot (2010) B and E are the ordinal dates for the start and end of the nesting season. P is the ordinal date for the peak of the nesting season. F is half of the number of days around P for which the curve flattens out. Max is the mean number of nests at the peak of the nesting season. PMin is the mean nightly nest numbers relative to Max before and after the nesting season. The nesting season is described in segments, and all segments form one continuous function. The nesting season is defined as the interval [B, E]. If F is equal to 0, no flat portion is observed. Rather than fitting B and E, it is more convenient to fit LengthB=P – B and LengthE=E – P with LengthB > 0 and LengthE > 0 to ensure that B < P < E. The parameters B, E, P, F, LengthB, and LengthE are hereafter defined as shape parameters, and PMin and Max as scale parameters. In such a situation when several nesting seasons are analyzed, it is also possible to implement a year effect for Peak (P) and/or for LengthB (length of the nesting season from the beginning to the peak) and LengthE (length of the nesting season from the peak to the end). Four categories of models were then fitted depending on the year effects for Peak and/or for LengthB and LengthE. These are defined as the Peak-Global or Peak-Year and Length-Global or Length-Year. For the seasonality model, parameter fitting was performed using maximum likelihood with negative binomial (NB) daily nest distribution with values produced by Equation (1) (m=nt) as theoretical values and the observed counts (x=Nt) as observations. In ecology, NB distribution is used to describe the distribution of an organism while taking into account the mean number of individuals m and an aggregation parameter k (Taylor et al. 1979). The probability mass function of NB distribution is:
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