Contextual models and the non-Newtonian paradigm

Biological systems exhibit a wide range of contextual effects, and this often makes it difficult to construct valid mathematical models of their behaviour. In particular, mathematical paradigms built upon the successes of Newtonian physics make assumptions about the nature of biological systems that are unlikely to hold true. After discussing two of the key assumptions underlying the Newtonian paradigm, we discuss two key aspects of the formalism that extended it, Quantum Theory (QT). We draw attention to the similarities between biological and quantum systems, motivating the development of a similar formalism that can be applied to the modelling of biological processes.

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0, 1) or (1, 2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.

Mathematical modelling of LiFePO4 cathodes

LiFePO4 is a commercially available battery material with good theoretical discharge capacity, excellent cycle life and increased safety compared with competing Li-ion chemistries. It has been the focus of considerable experimental and theoretical scrutiny in the past decade, resulting in LiFePO4 cathodes that perform well at high discharge rates. This scrutiny has raised several questions about the behaviour of LiFePO4 material during charge and discharge. In contrast to many other battery chemistries that intercalate homogeneously, LiFePO4 can phase-separate into highly and lowly lithiated phases, with intercalation proceeding by advancing an interface between these two phases. The main objective of this thesis is to construct mathematical models of LiFePO4 cathodes that can be validated against experimental discharge curves. This is in an attempt to understand some of the multi-scale dynamics of LiFePO4 cathodes that can be difficult to determine experimentally. The first section of this thesis constructs a three-scale mathematical model of LiFePO4 cathodes that uses a simple Stefan problem (which has been used previously in the literature) to describe the assumed phase-change. LiFePO4 crystals have been observed agglomerating in cathodes to form a porous collection of crystals and this morphology motivates the use of three size-scales in the model. The multi-scale model developed validates well against experimental data and this validated model is then used to examine the role of manufacturing parameters (including the agglomerate radius) on battery performance. The remainder of the thesis is concerned with investigating phase-field models as a replacement for the aforementioned Stefan problem. Phase-field models have recently been used in LiFePO4 and are a far more accurate representation of experimentally observed crystal-scale behaviour. They are based around the Cahn-Hilliard-reaction (CHR) IBVP, a fourth-order PDE with electrochemical (flux) boundary conditions that is very stiff and possesses multiple time and space scales. Numerical solutions to the CHR IBVP can be difficult to compute and hence a least-squares based Finite Volume Method (FVM) is developed for discretising both the full CHR IBVP and the more traditional Cahn-Hilliard IBVP. Phase-field models are subject to two main physicality constraints and the numerical scheme presented performs well under these constraints. This least-squares based FVM is then used to simulate the discharge of individual crystals of LiFePO4 in two dimensions. This discharge is subject to isotropic Li+ diffusion, based on experimental evidence that suggests the normally orthotropic transport of Li+ in LiFePO4 may become more isotropic in the presence of lattice defects. Numerical investigation shows that two-dimensional Li+ transport results in crystals that phase-separate, even at very high discharge rates. This is very different from results shown in the literature, where phase-separation in LiFePO4 crystals is suppressed during discharge with orthotropic Li+ transport. Finally, the three-scale cathodic model used at the beginning of the thesis is modified to simulate modern, high-rate LiFePO4 cathodes. High-rate cathodes typically do not contain (large) agglomerates and therefore a two-scale model is developed. The Stefan problem used previously is also replaced with the phase-field models examined in earlier chapters. The results from this model are then compared with experimental data and fit poorly, though a significant parameter regime could not be investigated numerically. Many-particle effects however, are evident in the simulated discharges, which match the conclusions of recent literature. These effects result in crystals that are subject to local currents very different from the discharge rate applied to the cathode, which impacts the phase-separating behaviour of the crystals and raises questions about the validity of using cathodic-scale experimental measurements in order to determine crystal-scale behaviour.

Mathematical modelling of tumour growth and interaction with host tissue and the immune system

In this thesis, three mathematical models describing the growth of solid tumour incorporating the host tissue and the immune system response are developed and investigated. The initial model describes the dynamics of the growing tumour and immune response before being extended in the second model by introducing a time-varying dendritic cell-based treatment strategy. Finally, in the third model, we present a mathematical model of a growing tumour using a hybrid cellular automata. These models can provide information to pre-experimental work to assist in designing more effective and efficient laboratory experiments related to tumour growth and interactions with the immune system and immunotherapy.

Data requirements of hedonic models in determining the impact of infrastructure charges on new housing costs

The use of hedonic models to estimate the effects of various factors on house prices is well established. This paper examines a number of international hedonic house price models that seek to quantify the effect of infrastructure charges on new house prices. This work is an important factor in the housing affordability debate, with many governments in high growth areas having user-pays infrastructure charging policies operating in tandem with housing affordability objectives, with no empirical evidence on the impact of one on the other. This research finds there is little consistency between existing models and the data sets utilised. Specification appears dependent upon data availability rather than sound theoretical grounding. This may lead to a lack of external validity with model specification dependent upon data availability rather than sound theoretical grounding.

Mathematical modelling of intramembranous bone formation during fracture healing

During fracture healing, many complex and cryptic interactions occur between cells and bio-chemical molecules to bring about repair of damaged bone. In this thesis two mathematical models were developed, concerning the cellular differentiation of osteoblasts (bone forming cells) and the mineralisation of new bone tissue, allowing new insights into these processes. These models were mathematically analysed and simulated numerically, yielding results consistent with experimental data and highlighting the underlying pattern formation structure in these aspects of fracture healing.

Quantifying uncertainty in parameter estimates for stochastic models of collective cell spreading using approximate Bayesian computation

Wound healing and tumour growth involve collective cell spreading, which is driven by individual motility and proliferation events within a population of cells. Mathematical models are often used to interpret experimental data and to estimate the parameters so that predictions can be made. Existing methods for parameter estimation typically assume that these parameters are constants and often ignore any uncertainty in the estimated values. We use approximate Bayesian computation (ABC) to estimate the cell diffusivity, D, and the cell proliferation rate, λ, from a discrete model of collective cell spreading, and we quantify the uncertainty associated with these estimates using Bayesian inference. We use a detailed experimental data set describing the collective cell spreading of 3T3 fibroblast cells. The ABC analysis is conducted for different combinations of initial cell densities and experimental times in two separate scenarios: (i) where collective cell spreading is driven by cell motility alone, and (ii) where collective cell spreading is driven by combined cell motility and cell proliferation. We find that D can be estimated precisely, with a small coefficient of variation (CV) of 2–6%. Our results indicate that D appears to depend on the experimental time, which is a feature that has been previously overlooked. Assuming that the values of D are the same in both experimental scenarios, we use the information about D from the first experimental scenario to obtain reasonably precise estimates of λ, with a CV between 4 and 12%. Our estimates of D and λ are consistent with previously reported values; however, our method is based on a straightforward measurement of the position of the leading edge whereas previous approaches have involved expensive cell counting techniques. Additional insights gained using a fully Bayesian approach justify the computational cost, especially since it allows us to accommodate information from different experiments in a principled way.

Mathematical modelling of the impaction and spreading of spray droplets on leaves

This thesis concerns the development of mathematical models to describe the interactions that occur between spray droplets and leaves. Models are presented that not only provide a contribution to mathematical knowledge in the field of fluid dynamics, but are also of utility within the agrichemical industry. The thesis is presented in two parts. First, thin film models are implemented with efficient numerical schemes in order to simulate droplets on virtual leaf surfaces. Then the interception event is considered, whereby energy balance techniques are employed to instantaneously predict whether an impacting droplet will bounce, splash, or adhere to a leaf.

Harvest strategy evaluations and co-management for the Moreton Bay Trawl Fishery

Non-Technical Summary Seafood CRC Project 2009/774. Harvest strategy evaluations and co-management for the Moreton Bay Trawl Fishery Principal Investigator: Dr Tony Courtney, Principal Fisheries Biologist Fisheries and Aquaculture, Agri-Science Queensland Department of Agriculture, Fisheries and Forestry Level B1, Ecosciences Precinct, Joe Baker St, Dutton Park, Queensland 4102 Email: tony.courtney@daff.qld.gov.au Project objectives: 1. Review the literature and data (i.e., economic, biological and logbook) relevant to the Moreton Bay trawl fishery. 2. Identify and prioritise management objectives for the Moreton Bay trawl fishery, as identified by the trawl fishers. 3. Undertake an economic analysis of Moreton Bay trawl fishery. 4. Quantify long-term changes to fishing power for the Moreton Bay trawl fishery. 5. Assess priority harvest strategies identified in 2 (above). Present results to, and discuss results with, Moreton Bay Seafood Industry Association (MBSIA), fishers and Fisheries Queensland. Note: Additional, specific objectives for 2 (above) were developed by fishers and the MBSIA after commencement of the project. These are presented in detail in section 5 (below). The project was an initiative of the MBSIA, primarily in response to falling profitability in the Moreton Bay prawn trawl fishery. The analyses were undertaken by a consortium of DAFF, CSIRO and University of Queensland researchers. This report adopted the Australian Standard Fish Names (http://www.fishnames.com.au/). Trends in catch and effort The Moreton Bay otter trawl fishery is a multispecies fishery, with the majority of the catch composed of Greasyback Prawns (Metapenaeus bennettae), Brown Tiger Prawns (Penaeus esculentus), Eastern King Prawns (Melicertus plebejus), squid (Uroteuthis spp., Sepioteuthis spp.), Banana Prawns (Fenneropenaeus merguiensis), Endeavour Prawns (Metapenaeus ensis, Metapenaeus endeavouri) and Moreton Bay bugs (Thenus parindicus). Other commercially important byproduct includes blue swimmer crabs (Portunus armatus), three-spot crabs (Portunus sanguinolentus), cuttlefish (Sepia spp.) and mantis shrimp (Oratosquilla spp.). Logbook catch and effort data show that total annual reported catch of prawns from the Moreton Bay otter trawl fishery has declined to 315 t in 2008 from a maximum of 901 t in 1990. The number of active licensed vessels participating in the fishery has also declined from 207 in 1991 to 57 in 2010. Similarly, fishing effort has fallen from a peak of 13,312 boat-days in 1999 to 3817 boat-days in 2008 – a 71% reduction. The declines in catch and effort are largely attributed to reduced profitability in the fishery due to increased operational costs and depressed prawn prices. The low prawn prices appear to be attributed to Australian aquacultured prawns and imported aquacultured vannamei prawns, displacing the markets for trawl-caught prawns, especially small species such as Greasyback Prawns which traditionally dominated landings in Moreton Bay. In recent years, the relatively high Australian dollar has resulted in reduced exports of Australian wild-caught prawns. This has increased supply on the domestic market which has also suppressed price increases. Since 2002, Brown Tiger Prawns have dominated annual reported landings in the Moreton Bay fishery. While total catch and effort in the bay have declined to historically low levels, the annual catch and catch rates of Brown Tiger Prawns have been at record highs in recent years. This appears to be at least partially attributed to the tiger prawn stock having recovered from excessive effort in previous decades. The total annual value of the Moreton Bay trawl fishery catch, including byproduct, is about $5 million, of which Brown Tiger Prawns account for about $2 million. Eastern King Prawns make up about 10% of the catch and are mainly caught in the bay from October to December as they migrate to offshore waters outside the bay where they contribute to a large mono-specific trawl fishery. Some of the Eastern King Prawns harvested in Moreton Bay may be growth overfished (i.e., caught below the size required to maximise yield or value), although the optimum size-at-capture was not determined in this study. Banana Prawns typically make up about 5% of the catch, but can exceed 20%, particularly following heavy rainfall. Economic analysis of the fishery From the economic survey, cash profits were, on average, positive for both fleet segments in both years of the survey. However, after the opportunity cost of capital and depreciation were taken into account, the residual owner-operator income was relatively low, and substantially lower than the average share of revenue paid to employed skippers. Consequently, owner-operators were earning less than their opportunity cost of their labour, suggesting that the fleets were economically unviable in the longer term. The M2 licensed fleet were, on average, earning similar boat cash profits as the T1/M1 fleet, although after the higher capital costs were accounted for the T1/M1 boats were earning substantially lower returns to owner-operator labour. The mean technical efficiency for the fleet as a whole was estimated to be 0.67. That is, on average, the boats were only catching 67 per cent of what was possible given their level of inputs (hours fished and hull units). Almost one-quarter of observations had efficiency scores above 0.8, suggesting a substantial proportion of the fleet are relatively efficient, but some are also relatively inefficient. Both fleets had similar efficiency distributions, with median technical efficiency score of 0.71 and 0.67 for the M2 and T1/M1 boats respectively. These scores are reasonably consistent with other studies of prawn trawl fleets in Australia, although higher average efficiency scores were found in the NSW prawn trawl fleet. From the inefficiency model, several factors were found to significantly influence vessel efficiency. These included the number of years of experience as skipper, the number of generations that the skipper’s family had been fishing and the number of years schooling. Skippers with more schooling were significantly more efficient than skippers with lower levels of schooling, consistent with other studies. Skippers who had been fishing longer were, in fact, less efficient than newer skippers. However, this was mitigated in the case of skippers whose family had been involved in fishing for several generations, consistent with other studies and suggesting that skill was passed through by families over successive generations. Both the linear and log-linear regression models of total fishing effort against the marginal profit per hour performed reasonably well, explaining between 70 and 84 per cent of the variation in fishing effort. As the models had different dependent variables (one logged and the other not logged) this is not a good basis for model choice. A better comparator is the square root of the mean square error (SMSE) expressed as a percentage of the mean total effort. On this criterion, both models performed very similarly. The linear model suggests that each additional dollar of average profits per hour in the fishery increases total effort by around 26 hours each month. From the log linear model, each percentage increase in profits per hour increases total fishing effort by 0.13 per cent. Both models indicate that economic performance is a key driver of fishing effort in the fishery. The effect of removing the boat-replacement policy is to increase individual vessel profitability, catch and effort, but the overall increase in catch is less than that removed by the boats that must exit the fishery. That is, the smaller fleet (in terms of boat numbers) is more profitable but the overall catch is not expected to be greater than before. This assumes, however, that active boats are removed, and that these were also taking an average level of catch. If inactive boats are removed, then catch of the remaining group as a whole could increase by between 14 and 17 per cent depending on the degree to which costs are reduced with the new boats. This is still substantially lower than historical levels of catch by the fleet. Fishing power analyses An analysis of logbook data from 1988 to 2010, and survey information on fishing gear, was performed to estimate the long-term variation in the fleet’s ability to catch prawns (known as fishing power) and to derive abundance estimates of the three most commercially important prawn species (i.e., Brown Tiger, Eastern King and Greasyback Prawns). Generalised linear models were used to explain the variation in catch as a function of effort (i.e., hours fished per day), vessel and gear characteristics, onboard technologies, population abundance and environmental factors. This analysis estimated that fishing power associated with Brown Tiger and Eastern King Prawns increased over the past 20 years by 10–30% and declined by approximately 10% for greasybacks. The density of tiger prawns was estimated to have almost tripled from around 0.5 kg per hectare in 1988 to 1.5 kg/ha in 2010. The density of Eastern King Prawns was estimated to have fluctuated between 1 and 2 kg per hectare over this time period, without any noticeable overall trend, while Greasyback Prawn densities were estimated to have fluctuated between 2 and 6 kg per hectare, also without any distinctive trend. A model of tiger prawn catches was developed to evaluate the impact of fishing on prawn survival rates in Moreton Bay. The model was fitted to logbook data using the maximum-likelihood method to provide estimates of the natural mortality rate (0.038 and 0.062 per week) and catchability (which can be defined as the proportion of the fished population that is removed by one unit of effort, in this case, estimated to be 2.5 ± 0.4 E-04 per boat-day). This approach provided a method for industry and scientists to develop together a realistic model of the dynamics of the fishery. Several aspects need to be developed further to make this model acceptable to industry. Firstly, there is considerable evidence to suggest that temperature influences prawn catchability. This ecological effect should be incorporated before developing meaningful harvest strategies. Secondly, total effort has to be allocated between each species. Such allocation of effort could be included in the model by estimating several catchability coefficients. Nevertheless, the work presented in this report is a stepping stone towards estimating essential fishery parameters and developing representative mathematical models required to evaluate harvest strategies. Developing a method that allowed an effective discussion between industry, management and scientists took longer than anticipated. As a result, harvest strategy evaluations were preliminary and only included the most valuable species in the fishery, Brown Tiger Prawns. Additional analyses and data collection, including information on catch composition from field sampling, migration rates and recruitment, would improve the modelling. Harvest strategy evaluations As the harvest strategy evaluations are preliminary, the following results should not be adopted for management purposes until more thorough evaluations are performed. The effects, of closing the fishery for one calendar month, on the annual catch and value of Brown Tiger Prawns were investigated. Each of the 12 months (i.e., January to December) was evaluated. The results were compared against historical records to determine the magnitude of gain or loss associated with the closure. Uncertainty regarding the trawl selectivity was addressed using two selectivity curves, one with a weight at 50% selection (S50%) of 7 g, based on research data, and a second with S50% of 14 g, put forward by industry. In both cases, it was concluded that any monthly closure after February would not be beneficial to the industry. The magnitude of the benefit of closing the fishery in either January or February was sensitive to which mesh selectivity curve that was assumed, with greater benefit achieved when the smaller selectivity curve (i.e., S50% = 7 g) was assumed. Using the smaller selectivity (S50% = 7 g), the expected increase in catch value was 10–20% which equates to $200,000 to $400,000 annually, while the larger selectivity curve (S50% = 14 g) suggested catch value would be improved by 5–10%, or $100,000 to $200,000. The harvest strategy evaluations showed that greater benefits, in the order of 30–60% increases in the tiger annual catch value, could have been obtained by closing the fishery early in the year when annual effort levels were high (i.e., > 10,000 boat-days). In recent years, as effort levels have declined (i.e., ~4000 boat-days annually), expected benefits from such closures are more modest. In essence, temporal closures offer greater benefit when fishing mortality rates are high. A spatial analysis of Brown Tiger Prawn catch and effort was also undertaken to obtain a better understanding of the prawn population dynamics. This indicated that, to improve profitability of the fishery, fishers could consider closing the fishery in the period from June to October, which is already a period of low profitability. This would protect the Brown Tiger Prawn spawning stock, increase catch rates of all species in the lucrative pre-Christmas period (November–December), and provide fishers with time to do vessel maintenance, arrange markets for the next season’s harvest, and, if they wish, work at other jobs. The analysis found that the instantaneous rate of total mortality (Z) for the March–June period did not vary significantly over the last two decades. As the Brown Tiger Prawn population in Moreton Bay has clearly increased over this time period, an interesting conclusion is that the instantaneous rate of natural mortality (M) must have increased, suggesting that tiger prawn natural mortality may be density-dependent at this time of year. Mortality rates of tiger prawns for June–October were found to have decreased over the last two decades, which has probably had a positive effect on spawning stocks in the October–November spawning period. Abiotic effects on the prawns The influence of air temperature, rainfall, freshwater flow, the southern oscillation index (SOI) and lunar phase on the catch rates of the four main prawn species were investigated. The analyses were based on over 200,000 daily logbook catch records over 23 years (i.e., 1988–2010). Freshwater flow was more influential than rainfall and SOI, and of the various sources of flow, the Brisbane River has the greatest volume and influence on Moreton Bay prawn catches. A number of time-lags were also considered. Flow in the preceding month prior to catch (i.e., 30 days prior, Logflow1_30) and two months prior (31–60 days prior, Logflow31_60) had strong positive effects on Banana Prawn catch rates. Average air temperature in the preceding 4-6 months (Temp121_180) also had a large positive effect on Banana Prawn catch rates. Flow in the month immediately preceding catch (Logflow1_30) had a strong positive influence on Greasyback Prawn catch rates. Air temperature in the preceding two months prior to catch (Temp1_60) had a large positive effect on Brown Tiger Prawn catch rates. No obvious or marked effects were detected for Eastern King Prawns, although interestingly, catch rates declined with increasing air temperature 4–6 months prior to catch. As most Eastern King Prawn catches in Moreton Bay occur in October to December, the results suggest catch rates decline with increasing winter temperatures. In most cases, the prawn catch rates declined with the waxing lunar phase (high luminance/full moon), and increased with the waning moon (low luminance/new moon). The SOI explains little additional variation in prawn catch rates (~ <2%), although its influence was higher for Banana Prawns. Extrapolating findings of the analyses to long-term climate change effects should be interpreted with caution. That said, the results are consistent with likely increases in abundance in the region for the two tropical species, Banana Prawns and Brown Tiger Prawns, as coastal temperatures rise. Conversely, declines in abundance could be expected for the two temperate species, Greasyback and Eastern King Prawns. Corporate management structures An examination of alternative governance systems was requested by the industry at one of the early meetings, particularly systems that may give them greater autonomy in decision making as well as help improve the marketing of their product. Consequently, a review of alternative management systems was undertaken, with a particular focus on the potential for self-management of small fisheries (small in terms of number of participants) and corporate management. The review looks at systems that have been implemented or proposed for other small fisheries internationally, with a particular focus on self-management as well as the potential benefits and challenges for corporate management. This review also highlighted particular opportunities for the Moreton Bay prawn fishery. Corporate management differs from other co-management and even self-management arrangements in that ‘ownership’ of the fishery is devolved to a company in which fishers and government are shareholders. The company manages the fishery as well as coordinates marketing to ensure that the best prices are received and that the catch taken meets the demands of the market. Coordinated harvesting will also result in increased profits, which are returned to fishers in the form of dividends. Corporate management offers many of the potential benefits of an individual quota system without formally implementing such a system. A corporate management model offers an advantage over a self-management model in that it can coordinate both marketing and management to take advantage of this unique geographical advantage. For such a system to be successful, the fishery needs to be relatively small and self- contained. Small in this sense is in terms of number of operators. The Moreton Bay prawn fishery satisfies these key conditions for a successful self-management and potentially corporate management system. The fishery is small both in terms of number of participants and geography. Unlike other fisheries that have progressed down the self-management route, the key market for the product from the Moreton Bay fishery is right at its doorstep. Corporate management also presents a number of challenges. First, it will require changes in the way fishers operate. In particular, the decision on when to fish and what to catch will be taken away from the individual and decided by the collective. Problems will develop if individuals do not join the corporation but continue to fish and market their own product separately. While this may seem an attractive option to fishers who believe they can do better independently, this is likely to be just a short- term advantage with an overall long-run cost to themselves as well as the rest of the industry. There are also a number of other areas that need further consideration, particularly in relation to the allocation of shares, including who should be allocated shares (e.g. just boat owners or also some employed skippers). Similarly, how harvesting activity is to be allocated by the corporation to the fishers. These are largely issues that cannot be answered without substantial consultation with those likely to be affected, and these groups cannot give these issues serious consideration until the point at which they are likely to become a reality. Given the current structure and complexity of the fishery, it is unlikely that such a management structure will be feasible in the short term. However, the fishery is a prime candidate for such a model, and development of such a management structure in the future should be considered as an option for the longer term.

Mathematical modelling of human corneal oxygenation and cell kinetics in corneal epithelial wound healing

Healthy transparent cornea depends upon the regulation of fluid, nutrient and oxygen transport through the tissue to sustain cell metabolism and other critical processes for normal functioning. This research considers the corneal geometry and investigates oxygen distribution using a two-dimensional Monod kinetic model, showing that previous studies make assumptions that lead to predictions of near-anoxic levels of oxygen tension in the limbal regions of the cornea. It also considers the comparison of experimental spatial and temporal data with the predictions of novel mathematical models with respect to distributed mitotic rates during corneal epithelial wound healing.

Analytical modelling of integrated solar drying system

The drying of fruit and vegetables is a subject of great importance. Dried fruit and vegetables have gained commercial importance, and their growth on a commercial scale has become an important sector of the agricultural industry. However, food drying is one of the most energy intensive processes of the major industrial process and accounts for up to 15 % of all industrial energy usage. Due to increasingly high electricity prices and environmental concern, a dryer using traditional energy sources is not a feasible option anymore. Therefore, an alternative/renewable energy source is needed. In this regard, an integrated solar drying system that includes highly efficient double-pass counter flow v-groove solar collector, conical-shaped rock-bed thermal storage, auxiliary heater, the centrifugal fan and the drying chamber has been designed and constructed. Mathematical model for all the individual components as well as an integrated model combining all components of the drying system has been developed. Mathematical equations were solved using MATLAB program. This paper presents the analytical model and key finding of the simulation.

Models of Dispersal and Diversification

Ecology and evolutionary biology is the study of life on this planet. One of the many methods applied to answering the great diversity of questions regarding the lives and characteristics of individual organisms, is the utilization of mathematical models. Such models are used in a wide variety of ways. Some help us to reason, functioning as aids to, or substitutes for, our own fallible logic, thus making argumentation and thinking clearer. Models which help our reasoning can lead to conceptual clarification; by expressing ideas in algebraic terms, the relationship between different concepts become clearer. Other mathematical models are used to better understand yet more complicated models, or to develop mathematical tools for their analysis. Though helping us to reason and being used as tools in the craftmanship of science, many models do not tell us much about the real biological phenomena we are, at least initially, interested in. The main reason for this is that any mathematical model is a simplification of the real world, reducing the complexity and variety of interactions and idiosynchracies of individual organisms. What such models can tell us, however, both is and has been very valuable throughout the history of ecology and evolution. Minimally, a model simplifying the complex world can tell us that in principle, the patterns produced in a model could also be produced in the real world. We can never know how different a simplified mathematical representation is from the real world, but the similarity models do strive for, gives us confidence that their results could apply. This thesis deals with a variety of different models, used for different purposes. One model deals with how one can measure and analyse invasions; the expanding phase of invasive species. Earlier analyses claims to have shown that such invasions can be a regulated phenomena, that higher invasion speeds at a given point in time will lead to a reduction in speed. Two simple mathematical models show that analysis on this particular measure of invasion speed need not be evidence of regulation. In the context of dispersal evolution, two models acting as proof-of-principle are presented. Parent-offspring conflict emerges when there are different evolutionary optima for adaptive behavior for parents and offspring. We show that the evolution of dispersal distances can entail such a conflict, and that under parental control of dispersal (as, for example, in higher plants) wider dispersal kernels are optimal. We also show that dispersal homeostasis can be optimal; in a setting where dispersal decisions (to leave or stay in a natal patch) are made, strategies that divide their seeds or eggs into fractions that disperse or not, as opposed to randomized for each seed, can prevail. We also present a model of the evolution of bet-hedging strategies; evolutionary adaptations that occur despite their fitness, on average, being lower than a competing strategy. Such strategies can win in the long run because they have a reduced variance in fitness coupled with a reduction in mean fitness, and fitness is of a multiplicative nature across generations, and therefore sensitive to variability. This model is used for conceptual clarification; by developing a population genetical model with uncertain fitness and expressing genotypic variance in fitness as a product between individual level variance and correlations between individuals of a genotype. We arrive at expressions that intuitively reflect two of the main categorizations of bet-hedging strategies; conservative vs diversifying and within- vs between-generation bet hedging. In addition, this model shows that these divisions in fact are false dichotomies.

Mathematical modelling of neurons and neural networks

The basic concepts and techniques involved in the development and analysis of mathematical models for individual neurons and networks of neurons are reviewed. Some of the interesting results obtained from recent work in this field are described. The current status of research in this field in India is discussed

Relative performances of textural models in estimating soil moisture characteristic

The soil moisture characteristic (SMC) forms an important input to mathematical models of water and solute transport in the unsaturated-soil zone. Owing to their simplicity and ease of use, texture-based regression models are commonly used to estimate the SMC from basic soil properties. In this study, the performances of six such regression models were evaluated on three soils. Moisture characteristics generated by the regression models were statistically compared with the characteristics developed independently from laboratory and in-situ retention data of the soil profiles. Results of the statistical performance evaluation, while providing useful information on the errors involved in estimating the SMC, also highlighted the importance of the nature of the data set underlying the regression models. Among the models evaluated, the one possessing an underlying data set of in-situ measurements was found to be the best estimator of the in-situ SMC for all the soils. Considerable errors arose when a textural model based on laboratory data was used to estimate the field retention characteristics of unsaturated soils.

Advances in the Mathematical Modelling of Hepatitis C Virus Dynamics

Mathematical models have provided key insights into the pathogenesis of hepatitis C virus (HCV) in vivo, suggested predominant mechanism(s) of drug action, explained confounding patterns of viral load changes in HCV infected patients undergoing therapy, and presented a framework for therapy optimization. In this article, I present an overview of the major advances in the mathematical modeling of HCV dynamics.