General Harris regularity criterion for non-linear Markov branching processes

We extend the Harris regularity condition for ordinary Markov branching process to a more general case of non-linear Markov branching process. A regularity criterion which is very easy to check is obtained. In particular, we prove that a super-linear Markov branching process is regular if and only if the per capita offspring mean is less than or equal to 1 while a sub-linear Markov branching process is regular if the per capita offspring mean is finite. The Harris regularity condition then becomes a special case of our criterion.