Multitype Branching Processes in Random Environment: Probability of Survival for the Critical Case

2018 ◽  
Vol 62 (4) ◽  
pp. 506-521 ◽  
Author(s):  
V. A. Vatutin ◽  
E. E. Dyakonova
Keyword(s):  
Random Environment ◽  
Critical Case ◽  
Branching Processes ◽  
Probability Of Survival ◽  
Multitype Branching Processes

On multitype branching processes in a random environment

1981 ◽  
Vol 13 (3) ◽  
pp. 464-497 ◽  
Author(s):  
David Tanny
Keyword(s):  
Random Environment ◽  
Stability Condition ◽  
Branching Processes ◽  
Classification Theorem ◽  
Regularity Conditions ◽  
Exponential Rate ◽  
Probabilistic Computation ◽  
Multitype Branching Processes ◽  
The Stability ◽  
Functional Iteration

This paper is concerned with the growth of multitype branching processes in a random environment (mbpre). It is shown that, under suitable regularity conditions, the process either explodes of becomes extinct. A classification theorem is given delineating the cases of explosion or extinction. Furthermore, it is shown that the process grows at an exponential rate on its set of non-extinction provided the process is stable. Criteria is given for non-certain extinction of the mbpre to occur, and an example shows that the stability condition cannot be removed. The method of proof used, in general, is direct probabilistic computation rather than the classical functional iteration techniques. Growth theorems are first proved for increasing mbpre and subsequently transferred to general mbpre using the associated mbpre and the reduced mbpre.

On multitype branching processes in a random environment

1981 ◽  
Vol 13 (03) ◽  
pp. 464-497 ◽  
Author(s):  
David Tanny
Keyword(s):  
Random Environment ◽  
Stability Condition ◽  
Branching Processes ◽  
Classification Theorem ◽  
Regularity Conditions ◽  
Exponential Rate ◽  
Probabilistic Computation ◽  
Multitype Branching Processes ◽  
The Stability ◽  
Functional Iteration

This paper is concerned with the growth of multitype branching processes in a random environment (mbpre). It is shown that, under suitable regularity conditions, the process either explodes of becomes extinct. A classification theorem is given delineating the cases of explosion or extinction. Furthermore, it is shown that the process grows at an exponential rate on its set of non-extinction provided the process is stable. Criteria is given for non-certain extinction of the mbpre to occur, and an example shows that the stability condition cannot be removed. The method of proof used, in general, is direct probabilistic computation rather than the classical functional iteration techniques. Growth theorems are first proved for increasing mbpre and subsequently transferred to general mbpre using the associated mbpre and the reduced mbpre.

Clone-selection and optimal rates of mutation

1973 ◽  
Vol 10 (4) ◽  
pp. 728-738 ◽  
Author(s):  
Ilan Eshel
Keyword(s):  
Survival Probability ◽  
Branching Processes ◽  
Intrinsic Rate ◽  
Term Survival ◽  
Probability Of Survival ◽  
Environmental Deterioration ◽  
Multitype Branching Processes ◽  
Quantitative Results ◽  
Biological Problem

The paper employs methods of multitype branching processes to evaluate the probability of survival of mutable clones under environmental conditions which are unfavorable to the original parent of the clone. When other factors are taken to be constant, the long-term survival probability of a clone is implicitly demonstrated as a function of the intrinsic rate of mutation carried by this clone. The existence of a mutation rate which maximizes clone survival probability is shown and the effects of environmental deterioration on this optimal rate are studied. Finally, rigorous quantitative results are obtained for the classical situation of a Poisson distribution of offspring numbers. These results are then applied to the biological problem of indirect selection (Eshel (1972)).

Clone-selection and optimal rates of mutation

1973 ◽  
Vol 10 (04) ◽  
pp. 728-738 ◽  
Author(s):  
Ilan Eshel
Keyword(s):  
Survival Probability ◽  
Branching Processes ◽  
Intrinsic Rate ◽  
Term Survival ◽  
Probability Of Survival ◽  
Environmental Deterioration ◽  
Multitype Branching Processes ◽  
Quantitative Results ◽  
Biological Problem

The paper employs methods of multitype branching processes to evaluate the probability of survival of mutable clones under environmental conditions which are unfavorable to the original parent of the clone. When other factors are taken to be constant, the long-term survival probability of a clone is implicitly demonstrated as a function of the intrinsic rate of mutation carried by this clone. The existence of a mutation rate which maximizes clone survival probability is shown and the effects of environmental deterioration on this optimal rate are studied. Finally, rigorous quantitative results are obtained for the classical situation of a Poisson distribution of offspring numbers. These results are then applied to the biological problem of indirect selection (Eshel (1972)).

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