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

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.