J.J. Shepherd

Multi-scaling analysis of a logistic model with slowly varying coefficients

All single-species differential-equation population models incorporate parameters which define the model - for example, the rate constant, r, and carrying capacity, K, for the Logistic model. For constant parameter values, an exact solution may be found, giving the population as a function of time. However, for arbitrary time-varying parameters, exact solutions are rarely possible, and numerical solution techniques must be employed. In this work, we demonstrate that for a Logistic model in which the rate constant and carrying capacity both vary slowly with time, an analysis with multiple time scales leads to approximate closed form solutions that are explicit, are valid for a range of parameter values and compare favourably with numerically generated ones.

Harvesting a logistic population in a slowly varying environment

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.

Perturbation analysis of the helical flow of non-Newtonian fluids with application to a recirculating coaxial cylinder rheometer

This paper analyzes the flow of non-Newtonian fluids between infinitely long coaxial cylinders, when the inner cylinder rotates with given angular velocity @W, and a given axial flow rate, Q, is superimposed on this rotational motion. Such helical flow is of significance in the modelling of the action of a recirculating coaxial cylinder rheometer, where the axial flow is superimposed on the standard cup-and-bob rheometer to allow accurate rheological measurements involving fluids that are settling in nature. Such fluids are encountered in many mineral, food, and chemical processes of industrial significance. The analysis presented here applies the perturbation approach to analyze the above flows in two situations of physical interest: 1.the case of low axial flow rates; and 2.the case where the radial separation of the cylinders is small. In each case, the appropriate perturbation parameter is identified, and appropriate expressions for the velocity field are obtained for non-Newtonian fluids of interest. More significantly, this analysis allows the construction of approximate forms of the Reiner-Rivlin relation for this flow, which relates the angular velocity @W to the observed torque, M, at the inner cylinder, through the rheological parameters defining the fluid. Subsequent measurement of @W and M allows these parameters to be determined for a given fluid model. Where possible, the findings of the perturbation analysis are compared directly with experimental measurements involving a model of such a recirculating rheometer.

Survival to extinction in a slowly varying harvested logistic population model

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.