A single function to calculate Augspurgers index of synchrony after Primack 1980. This function very warmly, and humbly, acknowledges Carol Augspurger, or 'Dr. A' as many alumni of UIUC have the privilege to know her. One of the few ecologists, to be a great natural historian, accomplished academic, excellent instructor, devout environmentalist, and unparalled mentor.
This from Augspurger 1983.
A. Synchrony of a given individual with its conspecifics:
$X_{i}$ , the index of synchrony for individual i, is defined
$$ X_{i} = (\frac{1}{n-1})(\frac{1}{f_{i}})\sum_{j = i}^{n} e_{j != i} $$ where,
$e_{j}$ = number of days both individuals i and j are flowering synchronously, j != i
$f_{i}$ = number of days individual i is flowering
$n$ = number of individuals in populationWhen X = 1.0, perfect synchrony occurs, i.e., all flowering days of individual i overlap with all flowering days of each other individual, j # i, in the > population.
When X = 0.0, no synchrony occurs, i.e., no overlap occurs among any of the flowering days of individual i and any other individual, j =# i, in the > population.B. Synchrony of the population:
$Z$ , the index of population synchrony, is defined as:
$$ Z= \frac{1}{n}\sum_{j = 1}^{n}X_{i} $$
$\frac{1}{n}$ where
$X_{i}$ is synchrony of individual i with its conspecifics from part A (above)
Augspurger, Carol K. “Phenology, Flowering Synchrony, and Fruit Set of Six Neotropical Shrubs.” Biotropica, vol. 15, no. 4, 1983, pp. 257–267. JSTOR, https://doi.org/10.2307/2387650. Accessed 19 Mar. 2022.
Augspurger, Carol K. “Reproductive Synchrony of a Tropical Shrub: Experimental Studies on Effects of Pollinators and Seed Predators in Hybanthus Prunifolius (Violaceae).” Ecology, vol. 62, no. 3, 1981, pp. 775–788. JSTOR, https://doi.org/10.2307/1937745. Accessed 19 Mar. 2022.
Primack, Richard B. “Variation in the Phenology of Natural Populations of Montane Shrubs in New Zealand.” Journal of Ecology, vol. 68, no. 3, 1980, pp. 849–862. JSTOR, https://doi.org/10.2307/2259460. Accessed 19 Mar. 2022.