Fieberg and Kochanny (2005) made an extensive review of the indices of
overlap between utilization distributions (UD) of two animals. The
function kerneloverlap implements these indices. The argument
method allows to choose an index.
The choice method="HR" computes the proportion of the home
range of one animal covered by the home range of another one, i.e.:
$$HR_{i,j} = A_{i,j} / A_i$$,
where \(A_{i,j}\) is the area of the intersection between
the two home ranges and \(A_i\) is the area of the home range
of the animal i.
The choice method="PHR" computes the volume under the UD of the
animal j that is inside the home range of the animal i (i.e., the
probability to find the animal j in the home range of i). That is:
$$PHR_{i,j} = \int \int_{A_i} UD_j(x,y) dxdy$$ where
\(UD_j(x,y)\) is the value of the utilization
distribution of the animal j at the point x,y.
The choice method="VI" computes the volume of the intersection
between the two UD, i.e.:
$$VI = \int_x \int_y min(UD_i(x,y),UD_j(x,y)) dxdy$$
Other choices rely on the computation of the joint distribution of the
two animals under the hypothesis of independence UD[i](x,y) *
UD[j](x,y).
The choice method="BA" computes the Bhattacharyya's affinity
$$BA = \int_x \int_y \sqrt{UD_i(x,y)} \times \sqrt{UD_j(x,y)}$$
The choice method="UDOI" computes a measure similar to the
Hurlbert index of niche overlap:
$$UDOI = A_{i,j} \int_x \int_y UD_i(x,y) \times
UD_j(x,y)$$
The choice method="HD" computes the Hellinger's distance:
$$HD = \int_x \int_y ((\sqrt UD_i(x,y) - \sqrt UD_j(x,y))^2
dxdy)^{1/2}$$