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A linear birth and death process under the influence of another process

Published online by Cambridge University Press:  14 July 2016

Prem S. Puri
Prem S. Puri*
Affiliation:
Purdue University
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Abstract

Let {X1 (t), X2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X2(t), t ≧ 0} may influence the growth of the process {X1(t), t ≧ 0}, while the process X2 (·) itself grows without any influence whatsoever of the first process. The process X2 (·) is taken to be a simple linear birth and death process with λ2 and µ2 as its birth and death rates respectively. The process X1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X2 (·) in the following manner: λ (t) = λ1 (θ + X2 (t)); µ(t) = µ1 (θ + X2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X1 (·), first conditionally given a realization of the process {X2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X1 (t) always exists as t→∞, except only when both λ1 > µ1 and λ2 > µ2. Also, certain problems concerning moments of the process, regression of X1 (t) on X2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

Keywords

BIRTH AND DEATH PROCESSESINTERACTING GROWTH PROCESSESMARKOV PROCESSESDURATION OF INTERACTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 1 , March 1975 , pp. 1 - 17
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Bailey, N. T. J. (1957) The Mathematical Theory of Epidemics. Hafner, New York.Google Scholar
[2] Barnett, V. D. (1962) The Monte Carlo solution of a competing species problem. Biometrics 18, 76–103.Google Scholar
[3] Bartlett, M. S. (1957) On theoretical models for competitive and predatory biological systems. Biometrika 44, 27–42.CrossRefGoogle Scholar
[4] Bartlett, M. S. (1960a) Stochastic Population Models in Ecology and Epidemiology. Methuen, London.Google Scholar
[5] Bartlett, M. S. (1960b) Monte Carlo studies in ecology and epidemiology. Proc. Fowth Berkeley Symp. Math. Statist. Prob. 4, 39–56.Google Scholar
[6] Becker, N. G. (1970) A stochastic model for two interacting populations. J. Appl. Prob. 7, 544–564.Google Scholar
[7] Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and their Applications. McGraw-Hill, New York.Google Scholar
[8] Chapman, D. G. (1967) Stochastic models in animal population ecology. Proc . Fifth Berkeley Symp. Math. Statist. Prob. 5, 147–162.Google Scholar
[9] Chiang, C. L. (1954) Competition and other interactions between species. Statistics and Mathematics in Biology. (Ed. Kempthorne, O., et al.) The Iowa State College Press.Google Scholar
[10] Dietz, K. (1966) On the model of Weiss for the spread of epidemics by carriers. J. Appl. Prob. 3, 375–382.Google Scholar
[11] Dietz, K. and Downton, F. (1968) Carrier-borne epidemics with immigration I — Immigration of both susceptibles and carriers. J. Appl. Prob. 5, 31–42.Google Scholar
[12] Gani, J. (1965) On a partial differential equation of epidemic theory. Biometrika 52, 617–622.Google Scholar
[13] Gani, J. (1967) On the general stochastic epidemic. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 4 271–279.Google Scholar
[14] Kendall, D. G. (1948) On generalised ‘birth-and-death’ processes. Ann. Math. Statist. 19, 1–15.Google Scholar
[15] Mertz, D. B. and Davies, R. B. (1968) Cannibalism of the pupal stage by adult flour beetles: An experiment and a stochastic model. Biometrics 24, 247–275.Google Scholar
[16] Neyman, J., Park, T. and Scott, E. L. (1956) Struggle for existence. The Tribolium model: Biological and Statistical aspects. Proc. Third Berkeley Symp. Math. Statist. Prob. 4, 41–79.Google Scholar
[17] Park, T. (1948) Experimental studies of interspecies competition. I. Competition between populations of the flour beetles Tribolium confusum Duval and Tribolium castaneum Herbst. Ecol. Monographs 18, 265–308.Google Scholar
[18] Puri, P. S. (1967) A class of stochastic models of response after infection in the absence of defense mechanism. Proc. Fifth Berkeley Sym. Math. Statist. Prob. 4, 551–535.Google Scholar
[19] Puri, P. S. (1966) On the homogeneous birth-and-death process and its integral. Biometrika 53, 61–71.Google Scholar
[20] Puri, P. S. (1968) Some further results on the homogeneous birth-and-death process and its integral. Proc. Camb. Phil. Soc. 64, 141–154.Google Scholar
[21] Puri, P. S. (1969) Some limit theorems on branching processes and certain related processes. Sankhya, ser. A 31, 57–74.Google Scholar
[22] Puri, P. S. (1968) Interconnected birth and death processes. J. Appl. Prob. 5, 334–349.CrossRefGoogle Scholar
[23] Puri, P. S. (1971) A method of studying the integral functionals of stochastic processes with applications: I. Markov chains case. J. Appl. Prob. 8, 331–343.CrossRefGoogle Scholar
[24] Puri, P. S. (1972) A method of studying the integral functionals of stochastic processes with applications III. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 3, 481–500.Google Scholar
[25] Weiss, G. H. (1965) On the spread of epidemic by carriers. Biometrics 21, 481–490.Google Scholar