Say we have $N$ birds, $r$ is the probability that one bird sings. What is the probability $p$ that any of $N$ birds sings?
If we assume independence, there is a simple model describing the situation:
$$p = 1 - (1 - r)^N$$
But in reality, the independence is invalid - if one bird sings, the others which are nearby are more likely to sing as well. How can this be modelled? We would for sure need at least one another parameter in the model representing the autocorrelation of the events. But I have no clue how to do that... thanks for your help.
P.S.: There are certainly many ways how to build such a model. I would be grateful for very simple one (very simple formula like above, adding perhaps one extra parameter) and then perhaps gradually adding complexity. Thanks.
EDIT: I am actually interested in birds. The data I have is basically $N_i$, which is number of singing birds within the radius of 100 meters from point $i$. I do not have particular coordinates of the singing individuals, therefore I want the model as simple as possible.