Mass extinctions: an alternative to the Allee effect

We introduce a spatial stochastic process on the lattice Z to model mass extinctions. Each site of the lattice may host a flock of up toN individuals. Each individual may give birth to a new individual at the same site at rate φ until the maximum of N individuals has been reached at the site. Once the flock reaches N individuals then, and only then, it starts giving birth on each of the 2d neighboring sites at rate λ(N). Finally, disaster strikes at rate 1, that is, the whole flock disappears. Our model shows that, at least in theory, there is a critical maximum flock size above which a species is certain to disappear and below which it may survive.