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ANZIAM J. 47(EMAC2005) pp.C541--C554, 2007.

Slow variation in the Gompertz model

T. Grozdanovski

J. J. Shepherd

(Received 21 December 2005; revised 15 December 2006)

Abstract

In many differential, single species, population models, the intrinsic model parameters are assumed to be given constants. Typical examples are the ``carrying capacity" and ``rate constant" arising in the logistic model. Such constant parameters usually allow exact solution of the model equations, to completely describe the evolving population. However, this assumption of constancy is not realistic since such parameters may exhibit a wide range of variation with time. This investigation considers a particularly useful population model---the Gompertz model---in the case where the model parameters vary slowly with time. Multi-timing methods construct an approximate expression for the population that has the advantages of being both explicit and giving results comparable to those obtained from numerical calculations.

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Authors

T. Grozdanovski

School of Mathematical and Geospatial Sciences, RMIT University, Melbourne, Australia. mailto:e61565@ems.rmit.edu.au

J. J. Shepherd

School of Mathematical and Geospatial Sciences, RMIT University, Melbourne, Australia. mailto:jshep@rmit.edu.au

Published January 5, 2007. ISSN 1446-8735

References

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2. Banks, R. B., Growth and Diffusion Phenomena: Mathematical Frameworks and Application, Springer--Verlag, Berlin, Germany, 1994.
3. Gompertz, B., On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Phil. Trans. Roy. Soc. London, 123, pp. 513--585. 1825.
4. Grozdanovski, T., Slow Variation in the Single Species Gompertz Population Model, Honours Thesis, School of Mathematical and Geospatial Science, RMIT University, 2005.
5. Grozdanovski, T., Nguyen, L. and Shepherd, J. J., A slowly varying harvesting model, in preparation.
6. Murdock, J. A., Perturbations Theory and Methods, Society for Industrial and Applied Mathematics, USA, 1999.
7. Stojkov, L. Population Modelling with Slowly Varying Carrying Capacities, Honours Thesis, Mathematics Department, RMIT University, 2003.