= Home-range overlap
\b[Home-range overlap] describes how the estimated space used by two animals is shared during specified observation periods. It requires an explicit measure: the common <area>, the fraction of each <home range> shared, and overlap between <utilization distributions> quantify different aspects of space use. The result depends on how the <home ranges> were estimated from locations.<Kernel Methods for Estimating the Utilization Distribution in Home-Range Studies by B. J. Worton>[Abstract: utilization-distribution estimation and smoothing choice]{ref}
Let $A$ and $B$ be planar <home range> regions. Their shared region is the <intersection> $A\cap B$. Two directional fractions are
$$
f_{A}=\frac{\operatorname{area}(A\cap B)}{\operatorname{area}(A)},\qquad f_{B}=\frac{\operatorname{area}(A\cap B)}{\operatorname{area}(B)}.
$$
For an invented example with areas2 and8square kilometres and shared area1square kilometre, half of $A$ is shared but only one eighth of $B$ is shared. A symmetric alternative uses the <union>:
$$
J=\frac{\operatorname{area}(A\cap B)}{\operatorname{area}(A\cup B)}=\frac1{2+8-1}=\frac19.
$$
Reporting only a percentage without the denominator leaves these interpretations ambiguous.
A density-based comparison can retain how intensely different places are used. For normalized <probability density functions> $u_A$ and $u_B$, the shared probability mass
$$
O=\iint\min\{u_A(x,y),u_B(x,y)\}\,dx\,dy
$$
lies between0 and1. It is not generally equal to the area fraction above. In the same example, if use is uniform within each region, this coefficient is $1\times\min(1/2,1/8)=1/8$.
<Radio tracking> can supply locations from which such regions and densities are estimated. A2011 free-roaming cat study combined radio telemetry with collar activity sensors, illustrating the distinction between where an animal ranges and its activity. Spatial overlap without matching time information does not prove simultaneous presence, contact or interaction.<Home Range, Habitat Use, and Activity Patterns of Free-Roaming Domestic Cats by Jeff A. Horn and others>[Abstract and Introduction: telemetry and activity measurements; overlap examples independently composed]{ref}
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