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DYNAMICAL SYSTEMS ANALYSIS OF A MODEL DESCRIBING TASMANIAN DEVIL FACIAL TUMOUR DISEASE

Part of: Local and nonlocal bifurcation theory Asymptotic theory Stability theory Approximation methods and numerical treatment of dynamical systems

Published online by Cambridge University Press:  18 March 2013

N. J. BEETON  and
L. K. FORBES
N. J. BEETON*
Affiliation:
School of Zoology, University of Tasmania, Sandy Bay, Tasmania 7005, Australia
L. K. FORBES
Affiliation:
School of Mathematics and Physics, University of Tasmania
*
✉[email protected]
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Abstract

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A susceptible–exposed–infectious theoretical model describing Tasmanian devil population and disease dynamics is presented and mathematically analysed using a dynamical systems approach to determine its behaviour under a range of scenarios. The steady states of the system are calculated and their stability analysed. Closed forms for the bifurcation points between these steady states are found using the rate of removal of infected individuals as a bifurcation parameter. A small-amplitude Hopf region, in which the populations oscillate in time, is shown to be present and subjected to numerical analysis. The model is then studied in detail in relation to an unfolding parameter which describes the disease latent period. The model’s behaviour is found to be biologically reasonable for Tasmanian devils and potentially applicable to other species.

Keywords

Tasmanian devilepidemic modelnonlinear dynamicsstabilityHopf bifurcation

MSC classification

Secondary: 92D25: Population dynamics (general) 34D20: Stability 34E10: Perturbations, asymptotics 37M20: Computational methods for bifurcation problems 37M05: Simulation 37G15: Bifurcations of limit cycles and periodic orbits
Type
Research Article
Information
The ANZIAM Journal , Volume 54 , Issue 1-2 , October 2012 , pp. 89 - 107
Copyright
Copyright ©2013 Australian Mathematical Society 

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