This method uses the area of the intersection between the territorial zones or home bases between each pair of individuals. For a spatial point pattern X, the association of individual j on individual i, Aij, is calculated as the percentage of the overlap of the discs centered at points X_i and X_j of the total area of the territorial area for individual i. The radius for each disc is the inputted interaction radius. The interaction radius for a given population can be identical for each individual, or different. The interaction radius represents the area within which an individual extracts nutrients or exerts its influence, or communicates an action.

This function is similar to the calculate.areas function. The difference, however, between these two functions is that for this function we assume that the strength of interaction for any given individual gradually decreases with distance. As shown in the cartoon below for two individuals, the discs of points denote two individuals distinguished by colour. As the distance from the center of each disc (the inidividual is located at the center of the disc) decreases, the density of the points (representing the strength of the individual's influence or strength of interaction) decreases also. This is an illustration of the concept of the interaction function that we adopt for this function. The associations calcuated using this method can be asymmetric. In this case, the interaction radii for two given individuals would be different, implying that the proportion of the overlap between the zones for the individuals is different for each individual. As as example, Figure 1 illustrates the effect of different interaction radii per individual. Individual i is represented by the filled square and individual j is represented by the filled circle. The percentage of the overlap between the two territorial zones in the total area of territorial zone i is larger than that in territorial j, suggesting that the effect of individual j on i is greater than that of i on j.

The calculations are done based on a Monte Carlo method.

1 | ```
calculate.gradedareas(arg1, arg2, arg3, numpts)
``` |

`arg1` |
x coordinates for individuals |

`arg2` |
y coordinates for individuals |

`arg3` |
interaction radii for each individual (they can all be equal) |

`numpts` |
number of Monte Carlo simulations #' |

1 2 3 4 5 | ```
a = c(0.4, 0.5, 0.5,0.6)
b = c(0.1, 0.2, 0.3, 0.4)
d = c(0.1, 0.1, 0.1, 0.1)
e = 1000000
calculate.gradedareas(a,b,d,e)
``` |

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