This dissertation presents several applications of the multi-scaling (multi-timing) technique to the analysis of both single and two species population models where the defining parameters vary slowly with time. Although exact solutions in such cases would be preferred, they are almost always impossible to obtain when slow variation is involved. Numerical solutions can be obtained in these cases, however they are often time consuming and offer limited insight into what is causing the behaviour we see in the solution. Here an approximation method is chosen as it gives an explicit analytic approximate expression for the solutions of such population models. The multi-scaling method was chosen because the defining parameters are varying slowly compared to the response of the system. This technique is well-established in the physical and engineering sciences literature; however, it has rarely been applied in the area of population modelling. All single species differential equation population models incorporate parameters which define the model - for example, the growth rate r and the 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. Here we will demonstrate that for a Logistic model where the growth rate and carrying capacity both vary slowly with time, an analysis with multiple time scales leads to approximate closed form solutions that are explicit. These solutions prove to be valid for a range of parameter values and compare favourably with numerically generated ones.
| Identifer | oai:union.ndltd.org:ADTP/257101 |
| Date | January 2009 |
| Creators | Grozdanovski, Tatjana, Tatjana.grozdanovski@rmit.edu.au |
| Publisher | RMIT University. Mathematical and Geospatial Sciences |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | http://www.rmit.edu.au/help/disclaimer, Copyright Tatjana Grozdanovski |
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