The Holling type II harvesting term has the property that it is small for small population values, but grows monotonically with population growth, eventually saturating, at a constant value for very large populations. We consider here a population evolving according to a logistic rate, but harvested (predated) subject to a Holling type II harvesting term that varies slowly with time, possibly due to slow environmental variation. Application of a multitiming method gives us an approximation to the population at any time in two cases- survival to a slowly varying limit, and extinguishment to zero. The situation where there is a transition from survival to extinction is also analyzed, using a matched expansions approach. A uniformly valid approximate expression for the population, valid for all times is obtained. These results are shown to agree well with the results of numerical calculations.

We provide a validation of a formal approximate solution to the problem of the evolution of a slowly varying harvested logistic population. Using a contraction mapping proof, we show that the initial value problem for the population has an exact solution lying in an appropriately small neighbourhood of this approximate solution, under quite general conditions.

This work considers a harvested logistic population for which birth rate, carrying capacity and harvesting rate all vary slowly with time. Asymptotic results from earlier work, obtained using a multiscaling technique, are combined to construct approximate expressions for the evolving population for the situation where the population initially survives to a slowly varying limiting state, but then, due to increasing harvesting, is reduced to extinction in finite time. These results are shown to give very good agreement with those obtained from numerical computation.

The classic problem for a logistically evolving single species population being harvested involves three parameters: rate constant, carrying capacity and harvesting rate, which are taken to be positive constants. However, in real world situations, these parameters may vary with time. This paper considers the situation where these vary on a time scale much longer than that intrinsic to the population evolution itself. Application of a multiple time scale approach gives approximate explicit closed form expressions for the changing population, that compare favorably with those generated from numerical solutions.