**Establishment risk index **

The establishment risk index (ERI) is defined as the number of time intervals with a net reproduction rate, R_{o}, above 1 (*I _{i=1}*) divided by the total number of time intervals within a year (

*I*). By default, the maps, as presented in the Atlas, are generated by using a 1-month time scale; however, the calculation can also be based on other time scales (e.g., 1-day intervals). The formula for using monthly intervals is as follows:

_{i}in which *I _{i}* is the interval of the month

*i*(with

*i*= 1, 2, 3, …, 12). Its value is 1 if the population is expected to increase within this interval (

*I*= 1 if

_{i}*R*≥ 1), and the value is 0 if the population is expected to decrease (

_{o}*I*= 0 if

_{i}*R*< 1) according to the established temperature-driven phenology model, and the total number of intervals,

_{o}*I*, is 12.

_{I}**Generation index**

The generation index (GI) is computed by averaging the sum of estimated generation lengths,* T _{i}*, calculated for each Julian day,

*x*, as shown in the following expression:

where 365 is the number of days per year and *T _{i}* is the predicted average generation lengths in days during the month,

*i*(

*i*= 1, 2, 3, …, 12).

** **

**Activity index**

The activity index (AI) is computed by employing the following equation:

where is the finite rate of increase at Julian day*, x *(*x *= 1, 2, 3, …, 365).

where is the finite rate of increase per day during month* i *(*i *= 1, 2, 3, …, 12).