Hi There,
In theory, you should get the same results from ArcGIS, QGIS and GME because if they are running the same mathematical function in the same way, you should get the same results. However, in practice this is not the case. This is because different programs use different default settings and make different assumptions about the data. One of the biggest differences you have to look out for here is the type of kernel generated. In ArcGIS, by default this is Gaussian. GME, because it uses the ArcGIS functions to generate its kernels, also uses this by default, but from what I can work out, the Heatmap Plugin for QGIS (which is what most people use to generate kernels in QGIS) uses a quadratic function by default. This is also the same for the Adehabitat library in R. However, in all cases you should be able to change the default to make sure that they are using the same function.
You also have to look out for differences in things like the default smoothing parameters which are selected. When different software uses different ways to generate the default smoothing parameter, you can get very different results. An example of this is between ArcGIS 10.2 and ArcGIS 10.3. They changed the way the default smoothing parameter was changed (from the shortest of width or height of the extent divided by 30 to Silverman's rule of thumb), and this resulted in very different kernels being generated from the same data.
So, to summarise, different software packages should give the same results when doing the same calculation with the same settings on the same dataset, but differences in the way that different software packages are programmed to carry out the calculations (especially with default values chosen) can result in quite different results. This means that you really need to dig down into the help and document files for individual GIS packages to find out exactly what these are (and this is not always very clear), and then make sure you use exactly the same settings if you are wanting to compare the results of different software packages.
This is all fine if you are wishing to compare the differences within your own data, but if you are trying to compare your results to previously published studies, this can be a bit of a nightmare, as often the methods provided in papers are not sufficient for you to work out exactly what settings were used in the GIS software package being used. As a result, if you are wanting to make a comparison to a previously published study, the easiest thing is to try to use the exact same software set up. If you can't do this, then you may find that you have to contact the authors to find out exactly what settings they used, but don't be too surprised if they don't actually know all the details.
I hope this helps, and it strikes me that if anyone out there has the time, this might make an interesting little project to see exactly how different the results are with different software packages, and indeed different versions of the same software package.
However, all this having been said, the differences between software packages is going to be relatively small in comparison to differences in the methods used to select the optimal bandwidth/search radius/h-value, as the value used for this will have the greatest impact on the kernel density values for any given grid cell based on the input data set, and on the resulting isopleths/home range estimates.
All the best,
Colin