A branching process (BP) (see e.g. Jagers (1975)) is a mathematical model to describe the development of a population.
Simple BPs are defined by an initial state (number of individuals at time 0) and a law of reproduction, usually denoted by pk, k = 1,2,.... A resource-dependent branching process (RDBP) is a discrete-time BP which models the development of a population in which individuals are supposed to have to work in order to be able to live and to reproduce.
In these processes individuals have a means of interaction with the society which determines the rules how the current available resources should be distributed among them.
For RDBPs this question depends also strongly on a feature on which individuals have a great influence, namely the policy to distribute resources.
Then using results on expected stopping times for sums of order statistics (1991) the survival criteria can be explicitly computed for both the wf-society and the sf-society as a function of m, r and F. The arguably strongest result known for RDBPs is the theorem of the envelopment of societies (Bruss and Duerinckx 2015).
The mathematical proof depends on the mentioned results on expected stopping times for sums of order statistics (1991) and fine-tuned balancing acts between model assumptions and different notions of Convergence of random variables.