Some of the simpler theoretical models which have been proposed for phenomena (for example, the competition between species or the occurrence of epidemics) which involve stochastic interactions between several populations have the common feature that they are Markov processes, homogeneous in time, with a countable set of states (m, n) where m and n represent the sizes of two populations. These processes are specified by prescribing the rates at which transitions occur, only transitions to "neighboring" states being allowed: a formal definition of such "competition processes" will be given in section 2. It is usually difficult to find explicit formulas for the transition probabilities pij(t) of a Markov process, or even for their limiting values 7rij as t -oo, when the process is defined in terms of the transition rates. However, there are simpler questions worth an answer, relating to recurrence and mean recurrence times and, if there are absorbing states, to absorption probabilities and mean absorption times. We shall discuss such problems for competition processes.
[1]
F. G. Foster.
Markoff chains with an enumerable number of states and a class of cascade processes
,
1951,
Mathematical Proceedings of the Cambridge Philosophical Society.
[2]
Mark Bartlett,et al.
Deterministic and Stochastic Models for Recurrent Epidemics
,
1956
.
[3]
F. G. Foster.
On the Stochastic Matrices Associated with Certain Queuing Processes
,
1953
.
[4]
David G. Kendall,et al.
On non-dissipative Markoff chains with an enumerable infinity of states
,
1951,
Mathematical Proceedings of the Cambridge Philosophical Society.
[5]
David G. Kendall,et al.
The calculation of the ergodic projection for Markov chains and processes with a countable infinity of states
,
1957
.
[6]
David G. Kendall.
Les processus stochastiques de croissance en biologie
,
1952
.
[7]
G. Reuter,et al.
Denumerable Markov processes and the associated contraction semigroups onl
,
1957
.