Transitions in density dependent harvesting of a logistic population in a slowly varying environment

We previously applied a multiscale method to construct general analytic approximations to the solution of a harvested logistic system, where the system parameters vary slowly in time and the harvesting was maintained at either subcritical or supercritical levels---representing survival or extinction of the population. This article extends these results by including an analytic approximation through the transition from subcritical harvesting to supercritical harvesting. This approximates the population as it is driven from a surviving population to extinction by over harvesting. These results compare favourably with numerical solutions. References G. Dahlquist and G. Soderlind. Some problems related to stiff nonlinear systems. In R. Glowinski and J. L. Lions, editors, Computing Methods in Applied Sciences and Engineering V, pages 57--74. North--Holland, Amsterdam, 1982. T. Grozdanovski. Multi-Scaling Methods Applied to Population Models. Ph.D. Thesis, RMIT University, 2009. T. Grozdanovski, J. J. Shepherd and A. Stacey. Density dependent harvesting of a logistic population in a slowly varying environment. ANZIAM J. 51:C32--C47, 2010. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/2271 R. E. O'Malley Jr. Singular Perturbation Methods for Ordinary Differential Equations. Springer-Verlag, New York, 1991