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On a model for interference between searching insect parasites

Published online by Cambridge University Press:  17 February 2009

P. K. Pollett
P. K. Pollett
Affiliation:
Department of Mathematics, University of Queensland, Queensland 4072.
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Abstract

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The purpose of this paper is to study a stochastic model which assesses the effect of mutual interference on the searching efficiency in populations of insect parasites. By looking carefully at the assumptions which govern the model, I shall explain why the searching efficiency is of the same order as the total number, N, in the population, a conclusion which is consistent with the predictions of population biologists; previous studies have reached the conclusion that the efficiency is of order . The major results of the paper establish normal approximations for the distribution of the numbers of active parasites. These are valid at all stages of the process, in particular the non-equilibrium phase, where explicit analytic formulae for the state-probabilities are unavailable.

Type
Research Article
Information
The ANZIAM Journal , Volume 32 , Issue 2 , October 1990 , pp. 133 - 150
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Aim, S. E., “On the rate of convergence in diffusion approximations of jump Markov processes,” Thesis, University of Uppsala, 1978.Google Scholar
[2]Barbour, A. D., “On a functional central limit theorem for Markov population processes”, Adv. Appl. Probab. 6 (1974) 21–39.CrossRefGoogle Scholar
[3]Barbour, A. D., “Quasi-stationary distributions in Markov population processes”, Adv. Appl. Probab. 8 (1976) 296–314.CrossRefGoogle Scholar
[4]Barbour, A. D., “Density dependent Markov population processes”, preprint, Gonville and Caius College, Cambridge (1979).Google Scholar
[5]Diamond, P. M., “Stochastic models for interference between searching insect parasites”, J. Austral. Math. Soc. Ser. B 30 (1989) 268–277.CrossRefGoogle Scholar
[6]Dunstan, F. D. J. and Reynolds, J. F., “Applications of stochastic processes to chemical reaction kinetics”, Methods of Operations Research 33 (1978) 105–119.Google Scholar
[7]Dunstan, F. D. J. and Reynolds, J. F., “Normal approximations for distributions arising in the stochastic approach to chemical reaction kinetics”, J. Appl. Probab. 18 (1981) 263–267.CrossRefGoogle Scholar
[8]Gillespie, D. T., “Exact stochastic simulation of coupled chemical reactions”, J. Phys. Chem. 81 (1977) 2340–2361.CrossRefGoogle Scholar
[9]Hall, P., “On the roles of the Bessel and Poisson distributions in chemical kinetics”, J. Appl. Probab. 20 (1983) 585–599.CrossRefGoogle Scholar
[10]Hsu, J. and Wang, H., “Kinetics of bacterial adhesion - a stochastic analysis”, J. Theor. Biol. 124 (1987) 405–413.CrossRefGoogle ScholarPubMed
[11]Jordan, D. W. and Smith, P., Nonlinear ordinary differential equations, (Oxford University Press, Oxford, 1977).Google Scholar
[12]Kurtz, T. G., “Solutions of ordinary differential equations as limits of pure jump Markov processes”, J. Appl. Probab. 7 (1970) 49–58.CrossRefGoogle Scholar
[13]Kurtz, T. G., “Limit theorems for sequences of jump Markov processes approximating ordinary differential processes”, J. Appl. Probab. 8 (1971) 344–356.CrossRefGoogle Scholar
[14]Kurtz, T. G., “Strong approximation theorems for density dependent Markov chains”, Stochastic Process. Appl. 6 (1978) 223–240.CrossRefGoogle Scholar
[15]May, R. M., Stability and Complexity in Model Ecosystems, 2nd ed., (Princeton University Press, 1974).Google Scholar
[16]McQuarrie, D. A., “Stochastic approach to chemical kinetics”, J. Appl. Probab. A (1967) 413–478.CrossRefGoogle Scholar
[17]Quinn, B. G. and MacGillivray, H. L., “Normal approximations to discrete unimodal distributions”, J. Appl. Probab. 23 (1986) 1013–1018.CrossRefGoogle Scholar
[18]Parsons, R. W. and Pollett, P. K., “Quasistationary distributions for some autocatalytic reactions”, J. Statist. Phys. 46 (1987) 249–254.CrossRefGoogle Scholar
[19]Pollett, P. K., “On the problem of evaluating quasistationary distributions for open reaction schemes”, J. Statist. Phys. 46 (1989) 249–254.Google Scholar
[20]Pollett, P. K., “The generalized Kolmogorov criterion”, Stochastic Process. Appl. 33 (1989) 29–44.CrossRefGoogle Scholar
[21]van Kampen, N. G., Stochastic Processes in Physics and Chemistry, (North-Holland, Amsterdam, 1981).Google Scholar
[22]Whittle, P., “Statistical processes of aggregation and polymerization”, Proc. Camb. Phil. Soc. 61 (1965) 475–495.CrossRefGoogle Scholar
[23]Whittle, P., “Equilibrium distributions for an open migration process”, J. Appl. Probab. 5 (1968) 567–571.CrossRefGoogle Scholar