I think the best and accurate way to do this job, is by using radio-callering tools, or transmeters fixed around the birds themselfs then tracking them using radar related tools. of course it is a huge work, with huge amount of money expenditure and time consuming

I'm going to second Imadeddin's answer. Likely you'd have to map out where a simple random sample of radio-collared birds and then map that over GIS data that has the protected areas you're interested in overlaid? That's the best answer I can give, unfortunately. Best of luck with this, it seems a very interesting research question!

Dear friend. Goor morning.

 make  use ecological statistics on transec that you determinet. Sample has been proven effective. Thank you.

Do you have just the kernels summarized by month, or do you actually have the individual occurrence data?

Summarized data may be a bit coarse. Polygon overlap extent, month by month, could be used, but would be subject to a lot of uncertainly, and more so if the kernels do not have an associated, spatially-explicit z-value associated to the probability of occurrence of the bird.

If he latter (e.g. time-stamped positions or tracks for each tagged animal, from direct observation, radio-tracking fixes, or satellite tag data), the problem should be relatively simple to tack either by frequency analysis or through Markov matrix.

A large-enough set of primary occurrence records, i.e. a set of time-stamped coordinates should be enough as long as the sample of records do cover randomly or uniformly both activity (day) and roosting (night) times. If the distribution of samples is not random (e.g. time-constrained satellite readings), corrections could be applied but still the problem boils down to no. of occurrences within protected area polygons vs. points occurring outside (frequency analysis) or recording the changes in state, e.g. how often two consecutive readings are within the polygon (or outside it) vs. changes in state, from polygon to outside or vice-versa (Markov process).

One problem you should solve first is when you consider a fix to be inside or outside a polygon, especially if fixes are averaged over time. Although protected areas do have clean limits on a map, (i) they are not so clean in the field (border uncertainty) and (ii) fixes have an implicit precision which might be low, depending on the recording system. So an individual record has a probability associated to it of belonging to the subset of interest (e.g. being recorded inside the area) that decreases with the proximity to the border and the increase in time integration.

We recently had to tackle a rather similar problem, trying to compute biodiversity occurrences within protected areas from disparate sources. In the end we defined a buffer zone around the protected area, and counted records falling within the buffer area. The width of the buffer was calculated from the average precision in the coordinates of the datasets. Here are the papers (Links below.)

Pino-del-Carpio A., Ariño A.H., Miranda R., 2014: Data exchange gaps in knowledge of biodiversity: Implications for the management and conservation of Biosphere Reserves. Biodiversity and Conservation, 23(9): 2239-2258

Pino-del-Carpio A., Ariño A.H., Villarroya A., Puig J., Miranda R., 2013: The biodiversity data knowledge gap: Assessing information loss in the management of Biosphere Reserves. Biological Conservation, 173: 74-79.

Regards

Article Data exchange gaps in knowledge of biodiversity: Implication...

Article The biodiversity data knowledge gap: Assessing information l...

I agree with many of the suggestions and comments above.  If you do have kernel density estimates (KDEs) as shapefiles already, you can clip these by your protected area shapefile to determine the proportion of the volume or area within the protected area.  Just be sure you're clear on the ramifications of choosing either volume or area.  (There is plenty of literature on home range estimates and using polygons vs. 3-dimensional KDEs.)  Volume of intersection for KDE may also be a useful measure for comparing 2 KDEs, such as diurnal vs nocturnal points.  I did some similar analyses in this paper, which also has several citations for KDE home range estimates and volume of intersection:

https://www.researchgate.net/publication/269580922_Effects_of_Habitat_Disturbance_on_Texas_Horned_Lizards_An_Urban_Case_Study

-Alex

Article Effects of Habitat Disturbance on Texas Horned Lizards: An U...

As suggested above, you can calculate the proportion of KDE falling within protected areas, you would then have a proportion of space used by month. But if you have regular locations, you can calculate the proportion of locations overlapping protected areas for daylight and night, it would be more informative. What kind of data do you have exactly? Tracking data? Count data? 

I just published a paper using this kind of analysis, if it can help!