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.

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We are concerned here with the existence of at least one positive

continuous solution of the quadratic integral equation of fractional orders

x(t) = a(t) + Iαf(t; x(t)) · Iβg(t; x(t)); α; β 2 (0; 1]:

The maximal and minimal solutions are also proved. Some applications are

given.

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