## Phases of the Invasion Process

Invasion fitness summarizes the essential features of the initial stages of any invasion process. Two possibilities have to be distinguished. The general mathematical theory of branching processes tells that the prospective invader will die out when p is negative or zero, and has positive establishment probability when p is positive. In the latter case, there will be a short initial phase during which the effects of the initial chance fluctuations on the population structure still are discernible in the form ofthe population trajectory. After that phase the population grows roughly exponentially with rate constant p, although with possibly a good amount of wobble due to the variations in E(t). The exponential phase ends only when the infinite dilution Ansatz starts to fail due to the increase ofinvader numbers.

The clear differentiation in phases only holds good under some technical conditions, which, however, are sufficiently relaxed that from a practical point of view there is little to worry about. However, observationally the phases are not always easily assessable, especially in fluctuating environments. In particular, the numerical increase or decrease of a rare type over a number of generations may not be indicative of its future.

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