ANZIAM J. 47(EMAC2005) pp.C492--C506, 2007.
The logistic population model with slowly varying carrying capacity
J. J. Shepherd | L. Stojkov |
Abstract
Many single-species differential equation population models feature a carrying capacity---the limiting population supportable by the environment. For constant carrying capacities an exact solution may often be found, representing the evolving population in time. However, for time varying carrying capacities, exact solution is rarely possible, and numerical techniques must be used. We demonstrate that when the carrying capacity varies slowly with time, a multiple time scale analysis leads to approximate closed form solutions that, apart from being explicit, are comparable to numerically generated ones and which are valid for a range of parameter values.
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Authors
- J. J. Shepherd
- L. Stojkov
- School of Mathematical & Geospatial Sciences, RMIT University, PO Box 2476V, Melbourne, Victoria 3001, Australia. mailto:jshep@rmit.edu.au
Published January 4, 2007. ISSN 1446-8735
References
- Banks, R. B., Growth and Diffusion Phenomena : Mathematical Frameworks and Applications. Springer-Verlag, Berlin, Germany, 1994.
- Edelstein--Keshet, L., Mathematical Models in Biology, SIAM, USA, 2005.
- Holmes, M. H., Introduction to Perturbation Methods Springer, New York, 1995.
- Meyer, P., Bi-logistic growth, Technological Forecasting and Social Change 47, 89--102, 1994. doi:10.1016/0040-1625(94)90042-6
- Meyer, P. S., Ausubel, J. H., Carrying capacity: a model with logistically varying limits, Technological Forecasting and Social Change 61(3): 209--214, 1999. doi:10.1016/S0040-1625(99)00022-0
- Murray, J. D., Mathematical Biology I. An Introduction 3rd Ed, Springer--Verlag, Berlin, 2002.
- Nayfeh, A. H., Perturbation Methods John Wiley & Sons , 1973.