In the following:
The Thomas model is a system of two reaction–diffusion equations which was originally proposed in the context of enzyme kinetics. It was subsequently realised that it offers a plausible chemical mechanism for the generation of coat markings on mammals. To that end previous investigations have focused on establishing the conditions for the Turing instability and on following the associated patterns as the bifurcation parameter increases through the instability.
In this paper we use modern ideas from the theory of dynamical systems to systematically investigate the formation of localised structures in this model. The Turing instability is found to exhibit a degeneracy at which it changes from being subcritical to supercritical (or vice versa). Associated with such degenerate points is a heteroclinic connection which is a crucial requirement for the generation of localised patterns.
Localised solutions containing a single `spike' are associated with the so-called Belyakov–Devaney transition. Localised solutions containing either multiple `peaks' or multiple `holes' are associated with homoclinic snaking bifurcations. These structures are investigated using continuation software. The snaking bifurcations are found to collapse when the system crosses the Belyakov–Devaney transition. The isolated spike solutions collapse at a codimension two heteroclinic connection to create the so-called collapsed snaking.
We show that the temporal stability of the localised solutions depends upon the presence of Hopf bifurcations which in turn are controlled by the diffusion ratio. For small values of the ratio only solutions in the spike region are destabilised by a Hopf bifurcation. For larger values of the diffusion ratio Hopf bifurcations destabilise most of the localised patterns.
Fahad Al Saadi, Annette Worthy, Haifaa Alrihieli and Mark Nelson. Localised spatial structures in the Thomas model. Mathematics and Computers in Simulation 194, 141-158 2022. https://doi.org/10.1016/j.matcom.2021.10.030.
Since the pioneering work of Turing, it has been known that diffusion can destablise a homogeneous solution that is stable in the underlying model in the absence of diffusion. The destabilisation of the homogeneous solutions leads to the generation of patterns. In recent years, techniques have been developed to analyse so-called localised spatial structures. These are solutions in which the spatial structure occurs in a localised region. Unlike Turing patterns, they do not spread out across the whole domain. We investigate the existence of localised structures that occur in two predator-prey models. The functionalities chosen have been widely used in the literature. The existence of localised spatial structures has not investigated previously. Indeed, it is easy to show that these models cannot exhibit the Turing instability. This has perhaps led earlier researchers to conclude that interesting spatial solutions can therefore not occur for these models. The novelty of our paper is that we show the existence of stationary localised patterns in systems which do not undergo the Turing instability. The mathematical tools used are a combination of Linear and weakly nonlinear analysis with supporting numerical methods. By combining these methods, we are able to identify conditions for a wide range of increasing exotic behaviour. This includes the Belyckov-Devaney transition, a codimension two spatial instability point and the formation of localised patterns. The combination of spectral computations and numerical simulations reveals the crucial role played by the Hopf bifurcation in mediating the stability of localised spatial solutions. Finally, numerical solutions in two spatial dimensions confirms the onset of intricate spatio-temporal patterns within the parameter regions identified within one spatial dimension.
Fahad Al Saadi, Annette Worthy, Ahmed Msmali and Mark Nelson. Stationary localised patterns without Turing instability. Mathematical Methods in the Applied Sciences 45(16), 9111-9129, 2022. https://doi.org/10.1002/mma.8295.
In the past, my mathematics students have frequently complained at any suggestion that they should communicate ideas through the medium of a written report. This article discusses student responses when they were asked to write a short report for the mayor of a (hypothetical) small town in response to the mayor's plan to eliminate a contagious disease by locking the town down for three weeks. I discuss the approaches that students took in constructing their reports and summarize some of the great ideas that they had. Many students could see the parallel between what they were asked to do in the assignment and concurrent discussions in the communities that they came from with regard to the spread of COVID-19. The idea that mathematicians might have to communicate ideas in the form of a written report was not dismissed out of hand. I also reflect on ways in which the learning experience could have been improved. This hinges on providing a mechanism by which an individual student has the opportunity to read the reports of all the other students.
We develop and investigate a model for sludge production in the activated sludge process when a biological reactor is coupled to a sludge disintegration unit (SDU). The model for the biological reactor is a slimmed down version of the activated sludge model 1 in which only processes related to carbon are retained. Consequently, the death-regeneration concept is included in our model which is an improvement on almost all previous models. This provides an improved representation of the total suspended solids in the biological reactor, which is the key parameter of interest. We investigate the steady-state behaviour of this system as a function of the residence time within the biological reactor and as a function of parameters associated with the operation of the SDU. A key parameter is the sludge disintegration factor. As this parameter is increased the concentration of total suspended solids within the biological reactor decreases at the expense increasing the chemical oxygen demand in the effluent stream. The existence of a maximum acceptable chemical oxygen demand in the effluent stream therefore imposes a maximum achievable reduction in the total suspended solids. This paper improves our theoretical understanding of the utility of sludge disintegration as a means to reduce excess sludge formation. As an aside to the main thrust of our paper we investigate the common assumption that the sludge disintegration processes occur on a much shorter timescale than the biological processes. We show that the disintegration processes must be exceptional slow before the inclusion of the biological processes becomes important.
S.S. Alsaeedp, M.I. Nelson, M. Edwards, and A. Msmali. A mathematical model for the activated sludge process with a sludge disintegration unit. Chemical Product and Process Modeling, 2022. https://doi.org/10.1515/cppm-2021-0064.
In this study, a uniformly distributed mathematical model for the self-heating process within compost piles is formulated and investigated. It consists of mass balance equations for oxygen and energy as well as the heat-generation processes due to both biological and chemical activities. This model is an extension of the study of Luangwilai et al. [2013] which consists of only biological activity. The singularity and degenerate Hopf bifurcation theories are used to determine the loci of different singularities: the isola, cusp, double-limit points and boundary limit set as well as the double-Hopf, generalized Hopf (Bautin) and Bogdanov–Takens bifurcations. In this investigation it is found that these loci separate the secondary parameter plane into twenty-two regions of different steady-state solution behaviors, whereas the model in the earlier study reported only eight regions. With more topological detail in the parameter space, a clearer understanding of the thermal behavior of a compost pile can be obtained, thus assisting compost operators in controlling the temperature within the pile more effectively: for example, achieving desirable elevated temperature range for ideal composting conditions via periodic solutions and S-shaped solution branch, or alternatively, understanding conditions, if not monitored carefully, that can also increase the likelihood of spontaneous ignition within the pile.
T. Luangwilai, H.S. Sidhu, and M.I. Nelson. Inclusion of biological and chemical self-heating processes in compost piles model: A semenov formulation. International Journal of Bifurcation and Chaos, 2230027, 2022. https://doi.org/10.1142/S0218127422300270.
Industrial compost piles contain large volumes of bulk organic materials. Normally, there are two main heat generation processes—oxidation of cellulosic materials and biological activity within the compost pile. Biological heating occurs at a lower temperature range, but it may `kick-start' the oxidation reaction. Nevertheless, biological heating is desirable and is a key component in composting operations. However, there are cases when the temperature within the compost piles increases beyond the ignition temperature of cellulosic materials which can result in spontaneous ignition. This investigation considers the self-heating process that occurs in a compost pile using a two-dimensional spatially-dependent model incorporating terms that account for self-heating due to both biological and oxidative mechanisms. The variation of temperature distribution within different pile geometries is examined.
T. Luangwilai, H.S. Sidhu, and M.I. Nelson. Understanding the factors affecting the self-heating process of compost piles: Two-dimensional analysis. In A. Clark, Z. Jovanoski, and J. Bunder, editors, Proceedings of the 15th Biennial Engineering Mathematics and Applications Conference, EMAC-2021, volume 63 of ANZIAM J., pages C15-C29, 2022. https://doi.org/10.21914/anziamj.v63.17119 .
We investigate a model for the treatment of a tumour through the application of a virus. In the original model it was assumed that the virus particles are released only at one time. Such a treatment strategy cannot eliminate a tumour, as the tumour-free steady-state solution is unstable except for pathological circumstances in which the tumour does not grow and/or the virus does not die. We extend the model by allowing the tumour to be treated by a continuous release of virus particles. We show that the scaled delivery rate has two threshold values: below the lower threshold the system evolves to a stable periodic solution; above the higher threshold the tumour is eradicated.
A.H. Msmali, M.I. Nelson, and F.S. Al Saadi. Treating cancerous cells with a continuous release of virus particles. In A. Clark, Z. Jovanoski, and J. Bunder, editors, Proceedings of the 15th Biennial Engineering Mathematics and Applications Conference, EMAC-2021, volume 63 of ANZIAM J., pages C195-C207, 2022. https://doi.org/10.21914/anziamj.v63.17108 .
The potential for materials undergoing oxidation reactions to spontaneously combust when they are stored in large stockpiles is well known. We consider an application in which such self-heating is desirable and investigate the use of inert hotspots as a means to promote thermal runaway. The size and location of the hotspot are found to have the largest effects on self-heating. Less pronounced are effects due a periodic ambient temperature. The advection velocity through the stockpile can have large effects.
M. Berry, M. Nelson, M. Moores, B. Monaghan, and R. Longbottom. Using inert hot-spots to induce ignition within industrial stockpiles. In A. Clark, Z. Jovanoski, and J. Bunder, editors, Proceedings of the 15th Biennial Engineering Mathematics and Applications Conference, EMAC-2021, volume 63 of ANZIAM J., pages C182--C194, 2022. https://doi.org/10.21914/anziamj.v63.17157 .
We consider a simple `toy model' for the spontaneous combustion of industrial compost stockpiles. The model is a scalar non-linear differential equation which can be analysed using techniques taught in an introductory subject on non-linear ordinary differential equations. This model was used as a case study in a third year subject. We discuss how students approached some of the questions. Could they transfer their prior knowledge about differential equations to an industrial case study? How would they cope with a problem which required both pen-and-paper and numerical calculations? Students used a variety of approaches but the worked solutions only showed one. It would be beneficial for students to see that there is not one correct method to solve such problems.
M.I. Nelson and F.I. Hai. Biological self-heating in industrial compost piles: an informal discussion of students applying prior mathematical skills within an industrial case study. In A. Clark, Z. Jovanoski, and J. Bunder, editors, Proceedings of the 15th Biennial Engineering Mathematics and Applications Conference, EMAC-2021, volume 63 of ANZIAM J., pages C208--C221, 2022. https://doi.org/10.21914/anziamj.v63.17076 .