Density dependent harvesting of a logistic population in a slowly varying environment

Authors

  • Tatjana Grozdanovski
  • John J Shepherd
  • Andrew J Stacey

DOI:

https://doi.org/10.21914/anziamj.v51i0.2271

Keywords:

multi-scaling, harvesting, slow variation

Abstract

We apply a multiscale method to construct general analytic approximations for the solution of a harvested logistic system, where the system parameters vary slowly in time. Such approximations are a useful alternative to numerical solutions and are applicable to a range of parameter values. We consider two situations: subcritical harvesting, where the population survives; and supercritical harvesting, where it is driven to extinction. These approximations give excellent agreement with the numerical solutions of test cases. References
  • J. R. Beddington and R. M. May. Harvesting natural populations in a randomly fluctuating environment. Science, 197:463--465, 1977.
  • C. W. Clark. Mathematical Bioeconomics: The Optimal Management of Renewable Resources. 2nd Edn. Wiley-Interscience, 2005.
  • T. L. Cromer. Harvesting in a seasonal environment. Math. Comput. Model., 10(6):445--450, 1988.
  • T. Grozdanovski. Multi-Scaling Methods Applied to Population Models. Ph.D. Thesis, RMIT University, 2009.
  • T. Grozdanovski and J. J. Shepherd. Slow variation in the Gompertz model. ANZIAM J., 47:C451--C554, 2007. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/1061
  • T. Grozdanovski, J. J. Shepherd and A. Stacey. Multiscaling analysis of a logistic model with slowly varying coefficients. Appl. Math. Lett., 22:1091--1095, Elsevier, 2009. doi:10.1016/j.aml.2008.10.002.
  • T. Legovic and G. Peric. Harvesting population in a periodic environment. Ecol. Model., 24:221--229, 1984.
  • J. D. Murray. Mathematical Biology I: An Introduction. 3rd Edn. Springer--Verlag, Berlin, 2007.
  • S. Rosenblat. Population models in a periodically fluctuating environment. J. Math. Biol., 9:23--36, 1980.

Published

2010-02-26

Issue

Section

Proceedings Engineering Mathematics and Applications Conference