In the following:
Semi-analytical solutions are developed for the diffusive Nicholson's blowflies equation. Both one and two-dimensional geometries are considered. The Galerkin method, which assumes a spatial structure for the solution, is used to approximate the governing delay partial differential equation by a system of ordinary differential delay equations. Both steady-state and transient solutions are presented. Semi-analytical results for the stability of the system are derived and the critical parameter value, at which a Hopf bifurcation occurs, is found. Semi-analytical bifurcation diagrams and phase-plane maps are drawn, which show the initial Hopf bifurcation together with a classical period doubling route to chaos. A comparison of the semi-analytical and numerical solutions shows the accuracy and usefulness of the semi-analytical solutions. Also, an asymptotic analysis for the periodic solution near the Hopf bifurcation point is developed, for the one-dimensional geometry.
H.Y. Alfifip, T.R. Marchant, and M.I. Nelson. Semi-analytical solutions for the 1- and 2-D diffusive Nicholson's blowflies equation. IMA Journal of Applied Mathematics, 79, Number 1, 175-199, 2014. http://imamat.oxfordjournals.org/content/79/1/175.abstract.html?etoc .
Objective. To examine the time, frequency and duration of
each direct care activity conducted by personal carers in
Australian residential aged care homes.
Methods. A time-motion study was conducted to observe 46 personal carers at two high-care houses in two facilities (14 days at Site 1 and 16 days at Site 2). Twenty-three direct care activities were classified into eight categories for analysis.
Results. Overall, a personal carer spent approximately 45% of their time on direct care, corresponding to 3.5 h in an 8-h daytime shift. The two sites had similar ratios of personal carers to residents, and each resident received 30 min of direct care. No significant differences between the two sites were found in the time spent on oral communication, personal hygiene and continence activities. Personal carers at Site 1 spent significantly less time on toileting and mobility activities than those at Site 2, but more time on lunch activity. Although oral communication took the longest time (2 h), it occurred concurrently with other activities (e.g. dressing) for 1.5 h.
Conclusions. The findings provide information that may assist decision makers in managing the operation of high-care residential aged care facilities, such as planning for task allocation and staffing.
S. Qianp, P. Yu, D.M. Hailey, Z. Zhang, P.J. Davy, and M.I. Nelson. Time spent on daytime direct care activities by personal carers in two Australian residential aged care facilities: a time-motion study. Australian Health Review, 38, 230--237, 2014. http://www.publish.csiro.au/?paper=AH13161
The activated sludge process is widely used to treat domestic and industrial wastewater. A significant drawback of this process is the production of `sludge'; the disposal of which can comprise a significant proportion of the total operating costs of a wastewater treatment plant.
We analyze the steady-state operation of a membrane bioreactor system (MBR) incorporating a sludge disintegration unit (SDU) to reduce sludge production. We provide a qualitative understanding of the model by finding analytically the steady-state solutions of the model and determining their stability as a function of the residence time.
In practice a target value of the mixed liquor suspended solids (MLSS) content within the membrane reactor is specified. Applying the mathematical technique of singularity theory we show that if the sludge disintegration factor is sufficiently high then the MLSS content is guaranteed to be below the target value. This model prediction, of key interest from a practical perspective, was not identified in the original investigation of this model, which relied upon numerical integration of the governing equations.
Keywords: Bioreactors; Cell growth; Mathematical modelling; Membrane bioreactor; Reaction engineering; Wastewater treatment.
M.I. Nelson and T.C.L. Yuem. A Mathematical Analysis of a Membrane Bioreactor Containing a Sludge Disintegration System. Chemical Engineering Communications, 201(10): 1384-1403, 2014. DOI: 10.1080/00986445.2013.809001.
The activated sludge process is one of the major aerobic processes used in the biological treatment of wastewater. A significant drawback of this process is the production of excess `sludge', the disposal of which can account for 50-60% of the running costs of a plant. We investigate how the volume and mass of excess sludge produced is reduced by coupling the bioreactor to a sludge disintegration unit.
Asma Alharbi p, Mark Ian Nelson, Annette Worthy, Harvinder Sidhu. Sludge formation in the activated sludge process with a sludge disintegration unit. In Mark Nelson, Tara Hamilton, Michael Jennings and Judith Bunder, editors, Proceedings of the 11th Biennial Engineering Mathematics and Applications Conference, EMAC-2013, volume 55 of ANZIAM J., pages C348--C367, August 2014. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/7803 .
We provide a detailed, and thorough, investigation into the concentration multiplicity and dynamic stability of a prototype non-linear chemical mechanism: quadratic autocatalysis subject to non-linear decay in a continuously stirred tank reactor. This model was previously investigated in the literature using numerical path-following techniques. The contribution of this study is the application of singularity theory and degenerate Hopf-bifurcation theory to obtain analytical representations of many of the features of interest in this system. In particular, we use these presentations to identify critical values of an unfolding parameter below which specified phenomenon are no longer exhibited
Ahmed Hussein Msmali, Mark I. Nelson and Maureen P. Edwards. Quadratic autocatalysis with non-linear decay. Journal of Mathematical Chemistry, 52(8): 2234-2258, 2014. http://dx.doi.org/10.1007/s10910-014-0382-5.
Semi-analytical solutions for autocatalytic reactions with mixed quadratic and cubic terms are considered. The kinetic model is combined with diffusion and considered in a one-dimensional reactor. The spatial structure of the reactant and autocatalyst concentrations are approximated by trial functions and averaging is used to obtain a lower-order ordinary differential equation model, as an approximation to the governing partial differential equations. This allows semi-analytical results to be obtained for the reaction-diffusion cell, using theoretical methods developed for ordinary differential equations. Singularity theory is used to investigate the static multiplicity of the system and obtain a parameter map, in which the different types of steady-state bifurcation diagrams occur. Hopf bifurcations are also found by a local stability analysis of the semi-analytical model. The transitions in the number and types of bifurcation diagrams and the changes to the parameter regions, in which Hopf bifurcations occur, as the relative importance of the cubic and quadratic terms vary, is explored in great detail. A key outcome of the study is that the static and dynamic stability of the mixed system exhibits more complexity than either the cubic or quadratic autocatalytic systems alone. In addition it is found that varying the diffusivity ratio, of the reactant and autocatalyst, causes dramatic changes to the dynamic stability. The semi-analytical results are show to be highly accurate, in comparison to numerical solutions of the governing partial differential equations.
M.R. Alharthip, T.R. Marchant and M.I. Nelson. Mixed quadratic-cubic autocatalytic reaction-diffusion equations: Semi-analytical solutions. Applied Mathematical Modelling, 38(21-22), 5160-5173. 2014. http://dx.doi.org/10.1016/j.apm.2014.04.027.