Behavioral Ecology Vol. 15 No. 4: 699-700
Behavioral Ecology vol. 15 no. 4 © International Society for Behavioral Ecology 2004; all rights reserved
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Synchrony, asynchrony, and temporally random mating: a new method for analyzing breeding synchrony
a Project Seahorse, Fisheries Centre, University of British Columbia, 2204 Main Mall, Vancouver, British Columbia V6T 1Z4, Canada, and b PO Box 84, Ecology & Entomology Group, Lincoln University, Canterbury, New Zealand.
Address correspondence to D. Marsden, who is now at the Fisheries Economics Research Unit, Fisheries Centre, University of British Columbia, 2204 Main Mall, Vancouver, B.C. V6T 1Z4, Canada. Email address: d.marsden{at}fisheries.ubc.ca. K. Evans is now at the BIOME Group, Department of Animal and Plant Sciences, Sheffield University, S10 2TN, UK
Received 6 January 2003; revised 3 September 2003; accepted 28 September 2003.
Much research has focused on understanding breeding synchrony in animals and its relationship to such issues as mating systems and extrapair copulations in birds (Birkhead and Biggins, 1987
; Emlen and Oring, 1977; Knowlton, 1979; Stutchbury and Morton, 1995). To empirically examine synchrony, one needs an appropriate measure of the degree of synchrony in a population. Kempenaers (1993) presented an index of breeding synchrony (modified from Björklund and Westman, 1986) that has gained wide use as a simple representation of the degree to which breeding is synchronized among animals. This synchrony index (SI) is calculated as follows:
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). Other investigators have used a variation on this formula:
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; Stutchbury et al., 1998). The first formulation of the index calculates the average percentage of females that are fertile on a given day during the period when females are fertile, whereas the second gives the same value for a specific female p. The value obtained for the SI will range from 0–100%, with 0% representing complete asynchrony (i.e., no overlap in fertile periods) and 100% representing complete synchrony (Kempenaers, 1993). When presenting the results of SI calculations, most investigators present either a point estimate for the SI of the entire population or a mean and error term for the individual SI. One can also compare the SI of individual animals with the population mean to assess which animals are more or less synchronized with other animals. Unfortunately, none of these formulations or presentation formats recognizes an important characteristic of the SI and of synchrony in general. It is misleading to view the SI as a continuum from 0% (asynchrony) to 100% (synchrony); it should instead be considered relative to the value expected if animals were mating randomly in time. If the SI is significantly greater than this value, the population can be said to be mating synchronously; if the SI is significantly less than this value, the population is mating asynchronously (i.e., mating is dispersed in time). This can be thought of by using an analogy in terms of spatial dispersion: a synchronous population is analogous to a spatially aggregated population, whereas an asynchronous population is analogous to a dispersed or uniformly distributed one.
We here present a refinement of Kempenaers' (1993) method for investigating breeding synchrony, which allows a significance value to be assigned to a SI for a population, thus allowing more rigorous hypothesis testing. The population SI value expected when animals are mating randomly in time can be simulated by using a Monte Carlo randomization (Sokal and Rohlf, 1995). The SI is first calculated by using the observed timing of the fertile periods of the population in question. To simulate a randomly mating population, the fertile periods of all animals should then be set to begin on a randomly selected day during the fertile period of the population as a whole. The SI is calculated for this random population, and the first day of each fertile period is then rerandomized. The procedure is repeated 1000 times, yielding a distribution of simulated SI values. This set of values is sorted, and the 25th and 976th values will be the lower and upper ends of the 95% confidence interval, respectively (Sokal and Rohlf, 1995).
All simulations were run in Microsoft Excel 2000, using a simple macro program to repeat the randomization of mating periods and to compile the SI values. Our computer, which has a processor speed of 1000 MHz, took about 1 min to complete 1000 repetitions for the simulation outlined below.
The use of 1000 repetitions is common in Monte Carlo methods, although more or fewer repetitions may be used depending on (1) the precision that is required in the resulting parameters and (2) the available computing power. We recommend that several levels of repetition be used in pilot work to determine the appropriate number of repetitions. For example, we found that 10 simulations of 1000 repetitions each yielded mean SI values ranging from 22.77–22.93%, whereas 10 simulations of 500 repetitions each yielded mean SI values of 22.68–23.15%. The variability of the simulated values will change with the characteristics of the system under study, so the appropriate number of repetitions must be determined on a case-by-case basis.
As an example of the methodology, we simulated a hypothetical population of 20 animals that mate over a period of 25 days, in which each animal has a fertile period tp of 5 days and is restricted to one fertile period during the 25 days. Three hypothetical populations are shown in Figure 1 as histograms of the first day on which animals are fertile. The SI values calculated were 18% for population A (Figure 1A), 24% for population B (Figure 1B), and 32% for population C (Figure 1C). The Monte Carlo randomization, shown in Figure 1D, yielded a mean SI of 23%, with a 95% confidence interval of 19–30%. Thus, a population with SI > 30% should be considered to be mating synchronously, whereas a population with SI < 19% is mating asynchronously. Given these confidence limits, population A can be said to be mating asynchronously, population B is mating randomly, and population C is mating synchronously.
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We can also calculate an exact p value for our populations, which will represent the probability of obtaining the observed SI if the population is mating randomly in time. In the sorted list of 1000 simulated SI values, 18% ranks ninth, so the p value for the asynchronous population (popuation A) is 0.009. The SI value for the synchronized population (32%) ranks 993rd, so the p value for this population (population B) is 0.007. In contrast, the SI value for the randomly mating population (24%) falls near the mean and ranks 617th, so its p value is 0.383.
This technique is important for two reasons. First, previous investigators (see Chu et al., 2002; Dunn et al., 1999) have been restricted to comparing the SI observed in their study to published reviews (see Stutchbury and Morton 1995). In contrast, our technique allows a significance value to be given to a SI for a population, thus allowing more rigorous hypothesis testing. Second, the technique recognizes that synchrony and asynchrony are opposite forms of deviation from temporally random mating. In combination, these two advantages result in an ability to define synchrony and asynchrony more accurately and should prove valuable in theoretical and empirical work that explores the significance of synchrony, for example, in the debate over whether breeding synchrony promotes polygamy or monogamy (cf. Emlen and Oring, 1977; Stutchbury and Morton, 1995).
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ACKNOWLEDGEMENTS |
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This is a contribution from Project Seahorse. The work was conducted while investigating seahorse mating patterns with Amanda Vincent, who we thank for the collaboration. We thank Brian Giles for help with programming the Monte Carlo procedure, and Brian Giles, Janelle Curtis, and three anonymous reviewers for comments on the manuscript. D.M. thanks Guylian Chocolates Belgium for financial support.
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