Tactics and Vectors 98/99
                           

Home

Table of  Contents

Great Circle Hypotheis  

Magnetoclinic Hypothesis

Magnetic-Latitude Hypothesis

Compass Bearings Hypothesis

Suns' Azimuth Hypothesis

Expansion-Contraction Hypothesis

Always Advance Hypothesis

Never Go Back Hypothesis

* Adapted from Gibo, D. L.,  1986,  1990

Definitions of abbreviations and symbols:  N = number in sample, SW = Southwest, r = length of mean vector, A.D. = Angular deviation, and C.I. = Confidence Intervals.  Three asterisks (***) mean that the significance level for a Rayleigh test was  P<   0.001. 

Comments

1. Mean Bearing of the Population:  This is the mean bearing (direction) for all sightings and includes all flight behaviours (i.e. Category III vanishing bearings).  Because higher flying butterflies are usually travelling in a somewhat different bearing, or even a very different bearing, than lower flying individuals, and almost certainly travelling faster over the ground, the mean bearing of the population is not a good measure of the mean displacement of the population.
   
   
2. Preferred Bearing I:  This is the mean bearing of the subgroup flying within 3 m of the ground (i.e. Category II vanishing bearings).  Butterflies flying within this altitude band are assumed to have more control over their direction of displacement than those flying at higher altitudes above the ground.   Monarch butterflies in southern Ontario engage in a greater proportion of low level flight when the encounter left crosswinds and headwinds (to a SW bearing).
   
   
3. Preferred Bearing II:  This is the mean direction of the subgroup of flying with their bearings and headings aligned (i.e. Category I vanishing bearings).  In other words, these individuals are either flying in: (1)  calm conditions, (2) with direct tailwinds, or (3) with direct headwinds.   The altitude of the butterfly above the ground is not a consideration.  These individuals are the only ones that are not in a crosswind and either experiencing wind drift off course, or are turning into the wind to compensate for the crosswind component.
   
   
4. The purpose of a Rayleigh test is test whether the population from which the sample is drawn differs significantly from randomness (Batschelet, 1981, page 55).    In each case, the Rayleigh test showed that the length of the mean vector differed significantly from zero.   For each of  the three samples, the probability that the population (i.e. all  monarch butterflies in southern Ontario during late summer and fall) from which the sample (e.g. 575 vanishing bearings of monarch butterflies that I observed) was been taken  has no directional bias (i.e. the true vanishing bearings are  randomly distributed in the population) was less than one in a thousand.   In other words, the probability was less than one in a thousand that the directional bias of my sample of monarchs was due to an incredible run of (good? bad?) luck and that the D. plexippus in southern Ontario actually fly in random directions.
   
   
5. It should be noted that the 95% Confidence Intervals show that measurements of vanishing bearings of individuals flying within 3 m of the ground (Sample 2) provide about as good an estimate of the preferred bearing as vanishing bearings of monarch butterflies that had their headings and bearings aligned (sample 3) because the migrants were either flying in calm conditions, flying directly upwind, or flying directly downwind.   Because it is difficult to observe a sufficient sample size of  individuals that have aligned bearings and headings, it is helpful to know that this exercise is probably unnecessary.