Polygons statistics

Hi all,

I have a polygon shapefile of animal spices with > 15000 overlapping polygons. I would like to end up with a raster of a certain grid size containing for each cell the NUMBER of species present.

At the moment we work as follows:
- Split by attribute from Fox tool to create single shapefiles for each species.
- Rasterize each shapefile through a batch process
- Sum up all the rasters and calculate the count.

This way is extrremely slow and each procedure crashes often being the data very large.
Any suggestion from the community?

Thanks in advance

Marco