Resource-dependent branching process
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A branching process (BP) (see e.g. Jagers (1975)) is a mathematical model to describe the development of a population. Here population is meant in a general sense, including a human population, animal populations, bacteria and others which reproduce in a biological sense, cascade process, or particles which split in a physical sense, and others. Members of a BP-population are called individuals, or particles.
If the times of reproductions are discrete (usually denoted by 1,2, …) then the totality of individuals present at time n and living to time n+1 excluded are thought of as forming the nth generation. 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. The population decides on a society form which determines the rules how available resources are distributed among the individuals. For this purpose a RDBP should incorporate at least four additional model components, namely the individual demands for resources, the creation of new resources for the next generation, the notion of a policy to distribute resources, and a control option for individuals for interactions with the society.
Definition
A (discrete-time) resource-dependent branching process is a stochastic process Γ defined on the non-negative integers which is a BP defined by
- an initial state Γ0;
- a law of reproduction of individuals;
- a law of individual creation of resources;
- a law of individual resource demands (claims);
- a policy to distribute available resources to individuals which are present in the population
- a tool of interaction between individuals and the society.
History and objectives of RDBPs
RDBPs were introduced by Franz Thomas Bruss (1983) with the objective to model different society structures and to compare the advantages and disadvantages of different forms of human societies.
Tractable RDBPs
Realistic models for human societies ask for a bisexual mode of reproduction whereas in the definition of an RDBP one simply speaks of a law of reproduction. However the notion of a average reproduction rate per individual (Bruss (1984)) shows that for all relevant questions (long-term behavior) it is justified for simplicity to assume asexual reproduction.
Secondly, models for the development of a human society in time must allow for interdependencies between the different components. Such models are in general very complicated. This is why at the beginning the results were modest. The situation changed (around 1990) with the idea not to try to model the development of a society with a (single) realistic RDBP but rather by a sequence of control actions defining a sequence of relevant short-horizon RDBPs. The second was the observation that two special policies stand out as guidelines for the development of any society. The two policies are the so-called weakest-first policy (wf-policy) and the so-called strongest-first policy (sf-policy).
Definition
The wf-policy is the rule to serve in each generation, as long as the accumulated resource space allows for it, with priority always the individuals with the smallest individual claims. The sf-policy is the rule to serve in each generation always with priority the largest individual resource claims, again as long as the accumulated resource space suffices. The societies adapting these policies strictly are called the wf-society, respectively the sf-society.
Survival criteria
In the theory of BPs it is of interest to know whether survival of a process is possible in the long run. For RDBPs this question is even more interesting since it depends also strongly on a feature on which individuals have a great influence, namely the policy to distribute resources.
Let:
- m = mean reproduction (descendants) per individual
- r = mean production (resource creation) per individual
- F = the individual probability distribution of claims (resources)
Further suppose that all individuals which will not obtain their resource claim will either die or emigrate before reproduction. Then using Bruss and Robertson (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 strongest result for RDBPs is the Bruss-Duerinckx Theorem of the envelopment of societies proved in 2013 which shows that, in the long run, any society is bound to live between the wf-society and the sf-society. Interestingly, intuition why this should be true, is wrong. The mathematical proof heavily depends on fine-tuned balancing acts between model assumptions and different notions of Convergence of random variables.
See also
References
- Jagers, Peter (1975). Branching processes with biological applications. London: Wiley-Interscience [John Wiley & Sons].
- Bruss, F. Thomas (1983). "Resource-dependent branching processes". Stochastic Processes and Theoretical Applications. 16: 36.
- Bruss, F. Thomas (1984). "A note on extinction criteria for bisexual Galton–Watson processes". Journal of Applied Probability. 21: 915–919.
- Bruss, F. Thomas; Robertson, James B. (1991). "Wald's Lemma for the sum of order statistics of i.i.d. random variables". Advances in Applied Probablity. 23: 612–623.
- Bruss, F. Thomas; Duerinckx, Mitia (2014). "Resource-dependent branching processes and the envelope of societies". Annals of Applied Probability.