On stochastic population models in genetics

1976 ◽  
Vol 13 (1) ◽  
pp. 127-131 ◽  
Author(s):  
F. P. Kelly
Keyword(s):  
Stationary Distribution ◽  
Population Model ◽  
Population Models ◽  
Transition Rates ◽  
Stochastic Population Models ◽  
Mutant Forms ◽  
Stochastic Population Model ◽  
General Stochastic

Trajstman (1974) has shown that two different population models used to study the number of mutant forms maintained in a population have a certain marginal stationary distribution in common. In this note a general stochastic population model is proposed which subsumes these two models and shows that their transition rates are also related.

On stochastic population models in genetics

1976 ◽  
Vol 13 (01) ◽  
pp. 127-131 ◽  
Author(s):  
F. P. Kelly
Keyword(s):  
Stationary Distribution ◽  
Population Model ◽  
Population Models ◽  
Transition Rates ◽  
Stochastic Population Models ◽  
Mutant Forms ◽  
Stochastic Population Model ◽  
General Stochastic

Trajstman (1974) has shown that two different population models used to study the number of mutant forms maintained in a population have a certain marginal stationary distribution in common. In this note a general stochastic population model is proposed which subsumes these two models and shows that their transition rates are also related.

Note on the permanence of stochastic population models

2019 ◽  
Vol 27 (2) ◽  
pp. 123-129
Author(s):  
Shashi Kant
Keyword(s):  
Biological Systems ◽  
Short Note ◽  
Time Frame ◽  
Population Models ◽  
Stochastic Permanence ◽  
Stochastic Population Models ◽  
Surface Time ◽  
Long Time ◽  
Definition Of ◽  
Stochastic Population Model

Abstract The concept of permanence of any system is an important technical issue. This concept is very significant to all kind of systems, e.g., social, medical, biological, population, mechanical, or electrical. It is desirable by scientists and investigators that any system under consideration must be long time survival. For example, if we consider any ecosystem, it is always pre-requisite that this system is permanent. In general language, permanence is just the persistent and bounded system in a particular surface time frame. But the meaning may vary with the type of systems. For example, deterministic and stochastic biological systems have different concepts of permanence in an abstract mathematical platform. The reason is simple: it is due to the mathematical nature of parameters, methods of derivations of the model, biological assumptions, details of the study, etc. In this short note, we consider the stochastic models for their permanence. To address stochastic permanence of biological systems, many different approaches have been proposed in the literature. In this note, we propose a new definition of permanence for stochastic population models (SPM). The proposed definition is applied to the well-known Lotka–Volterra two species stochastic population model. The note is closed with the open ended discussion on the topic.

On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains

1993 ◽  
Vol 25 (01) ◽  
pp. 82-102
Author(s):  
M. G. Nair ◽  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Absorbing State ◽  
Necessary And Sufficient ◽  
Forward Equations ◽  
Quasi Stationary Distribution

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992). In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.

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