I have done an analysis of 13 years' worth of data in my birding database to determine the statistical peak of warblers in spring migration in the southeast Michigan area. I am defining "peak" on the basis of number of warbler species, not number of individuals. For an analysis of this sort all data need to be for a particular, defined birding region or location. I have a large amount of data for Magee Marsh Wildlife Area (25 miles east of Toledo; with, and sometimes called, Crane Creek State Park), so I used that rather than data for, say, Nichols Arboretum (Ann Arbor), for this analysis. It can be expected that migration dates for southeast Michigan will lag dates for Magee Marsh (northwest Ohio) by a day or two.

METHODOLOGY

[I have done a similar analysis for total species count instead of warbler species count. Another analysis has determined first arrival and last departure dates for 36 warbler species found in April and May in the Magee Marsh region (not limited to Magee Marsh Wildlife Area).]

I used only Magee Marsh data for this analysis; i.e., I did not include additional species found at Ottawa National Wildlife Refuge, Metzger Marsh, Maumee Bay State Park, and other places that I regularly visit along with Magee Marsh Wildlife Area on my trips to Ohio. All data were collected in morning birding (7:45am and after) with continuation into the afternoon on most days. On most days I was with one to three other birders. I have included in the data all species that the group found, not just birds my own birds. Needless to say, I was never birding when it was pouring rain, but I never let light rain or cold weather stop me. That is, the data are not particularly constrained by weather conditions except that it is undoubtedly true that fewer of the birds present are found when the weather is bad. Predicted fallouts had little bearing on whether or not I was birding on any particular day. That said, there is a greater density of data for the first two and a half weeks of May than for any other period.

From 1995 through May 2007 I made 98 spring migration birding trips to Magee Marsh. Seventy-seven were in May; twenty-one were in April. After tabulating number of warblers recorded for each date for all trips, I calculated three-day moving averages for all dates in May and the last half of April. Note that these are three-DAY moving averages, not three-POINT moving averages. Multiple data points for particular calendar dates were used if available and all data points were given full and equal weight. There were, for example, as many as five years contributing to some dates (May 8 and May 16--for instance, for May 8, 1995, 2002, 2003, 2005, and 2006). [There is, of course, a SINGLE warblers count for each particular birding day--e.g., May 24, 2004. If at some time in the future I supplement my data with data from other birders, there would still be a single value used for any particular birding day; I would probably use the largest of available values rather than an average.] The results are shown in a table and a graph in the Appendix. Calculation of three-day moving averages is described there in a footnote.

Smoothing, done here by calculating moving averages, was deemed necessary because 98 data points, while representing a lot of birding, are insufficient for simple date-by-date averages over a month and a half (in several years) to have great meaning. I.e., the plotted data show unrealistic amplitudes and numbers of peaks and valleys. If I had 40 years' worth of data, smoothing might not be needed at all--but the long-term decline in some species and effects of changes in climate would complicate the analysis. It is certain, in fact, that these factors affect even my 13 years' worth of data to some unknown degree.

I calculated both three-day and five-day moving averages. I cannot argue on any theoretical basis that one or the other is more reasonable for processing my data. However, while in analyses done prior to this year I used the five-day moving averages, I believe that I now have sufficient data that adequate smoothing is done with the three-day method--and that processing with the five-day method may now oversmooth the data. Both sets of results are shown in Table 1. I will comment on only the results for three-day smoothing. The five-day results are not fundamentally different in nature.

RESULTS

The three-day moving average is characterized by three plateaus, each approximately a week in duration. The center of each plateau is indicated in Table 1 by a "P" notation. The five-day moving average is very similar in nature. A local valley in the three-day moving average is indicated by "(-m/-n)", where m and n are depths of the valley relative to adjacent local peaks.

The overall peak of 19 warblers (three-day average) is from May 10 to May 17. This central plateau has a seemingly real one- to two-warbler dip over the middle four days. Two lesser plateaus lie on either side of the central plateau. These are both about a week in duration (April 30-May 8 and May 20-25), and both are at the 13 warbler level. They are separated from the central plateau by about three days of more-or-less uniform transition. Prior to the first plateau the day-by-day average warbler counts increase sharply and uniformly from April 26 to April 30. Following the third plateau, after May 27, average warbler counts decline more or less uniformly through the end of the month. These characteristics of the data are perhaps seen more easily in the graphical data of Figure 1 than in the tabular data.

SUMMARY

Looked at more broadly, the waveform for warbler migration shows a nearly monotonic rise to and fall from a single central peak (plateau) of one-week duration:   18 warbler species per day from May 10 to 17. In an analysis for total species count, a three-peak waveform stands out. The central peak there is short in duration--66 species for May 9, 10, and 11. Significant 10-species valleys separate the central peak from secondary peaks of about 63 species a week earlier and a week later.

To maximize both warbler count and total species count, the optimal time to bird at Magee Marsh Wildlife Area is May 10-11. May 17 stands out as a date nearly as good.

APPENDIX

Actual warbler peaks from my database are listed at the bottom of Table 1.

TABLE 1
             # of warblers          # of warblers  
           3-day moving avg       5-day moving avg (not used)

April 15            1                      1
      16            1                      1
      17            1                      1
      18            1                      1
      19            1                      2
      20            2                      2
      21            4                      3
      22            4                      4
      23            4                      3
      24            3                      3
      25            2                      3
      26            2                      4
      27            4                      4 
      28            7                      7 
      29           10                     10   
      30           13                     12 
 May   1           13                     13        
       2           13                     12
       3           12                     13 
       4           13  P (center of       13  P (center of 
       5           13   plateau at 13)    13   plateau at 13)
       6           14                     13
       7           13                     14
       8           14                     16        
       9           17                     16
      10           19  P (plateau start)  18  P (plateau start)
      11           19                     18      
      12           17                     18
      13           18                     18 
      14           17  (-2/-2)            18  P (center of
      15           18                     18   plateau at 18)     
      16           19                     18        
      17           19  P (plateau end)    18        
      18           17                     17  P (plateau end)
      19           10                     16
      20           13                     13        
      21           15                     13        
      22           14                     14  P (center of    
      23           13  P (center of       14    13-14 plateau)    
      24           13    plateau at 13)   14        
      25           14                     11        
      26           10                     11        
      27           10                     10       
      28            8                      8        
      29           10                      7
      30            5                      7
      31            4                      4

Actual highs were 

  25 on 5-11-2003 (average=20, 3-day moving average=19) 
  24 on 5-10-2006 (average=17, 3-day moving average=19)
  24 on 5-10-2007 (average=20, 3-day moving average=19)
  23 on 5-10-2005 (average=20, 3-day moving average=19)
  23 on 5-12-2006 (average=20, 3-day moving average=17)
  23 on 5-17-2006 (average=19, 3-day moving average=19)
  23 on 5-14-2005 (average=18, 3-day moving average=17)
  22 on 5-16-2006 (average=19, 3-day moving average=19)
  21 on 5-09-1999 (average=18, 3-day moving average=17)
  20 on 5-16-2002 (average=19, 3-day moving average=19) 
  20 on 5-10-2003 (average=17, 3-day moving average=19)
  20 on 5-17-2003 (average=19, 3-day moving average=19) 
  20 on 5-12-2007 (average=20, 3-day moving average=17)

Eighteen-warbler days came as early as April 30 and as late as 
May 28.


The three-day moving average data are plotted in Figure 1.

FIGURE 1
                April                          May
        5   10   15   20   25   30    5   10   15   20   25   30
   |----|----|----|----|----|----|----|----|----|----|----|----|----|
 19                                        **    **
 18                                           * *
 17                                       *  * *   *
 16 
 15                                                   *
 14                                    * *             *  *
 13                              *** ** *            *  **
 12                                 *
 11 
 10                             *                   *      ** *
  9 
  8                                                          *
  7                            *
  6 
  5                                                            *
  4                     ***   *                                 ***
  3                        *
  2                    *    **
  1              ******
   |----|----|----|----|----|----|----|----|----|----|----|----|----|
        5   10   15   20   25   30    5   10   15   20   25   30
                April                          May

            Three-day moving average warblers count data 
            for birding at Magee Marsh Wildlife Area

Example calculation - three-day moving average for May 4

  Data
       May 3, 2001:   9
       May 4, 2002:  18
       May 4, 2003:   7
       May 4, 2005:   6
       May 4, 2006:  15
       May 4, 2007:  13
       May 5, 2002:  16
       May 5, 2004:  17
 
  Three-day moving average for May 4

    [ 9 + (18+7+6+15+13) + (16+17) ] / 8 = 12.6 = 13
 
    The value 13 (smoothed data) is used instead of the 
    average value for May 4 (which would be (18+7+6+15+13)/5 = 12).