We provide theoretical insights into the activated sludge process by utilising the activated sludge model number 1 in a two reactor cascade with an ideal settling unit. The process configuration contains a settling unit and a recycle unit. We investigate how the removal of organic carbon and total nitrogen depend upon the hydraulic retention time as well as the process configuration, namely the impact of aeration, the presence of a settling unit, and recirculation of mixed liquor from the second reactor to the first reactor. As the hydraulic retention time increases from zero there are two critical values. Organic carbon can only be removed for values of the retention time higher than the first critical value. Removal of organic nitrogen occurs in conjunction with the removal of organic carbon.Heterotropic and autotrophic biomass are only viable for values of hydraulic retention larger than the first and second critical values, respectively. Inorganic nitrogen can only be significantly removed when autotrophic biomass are present. However, the removal of inorganic nitrogen is significantly influenced by the process configuration.

The stochastic nature of environmental factors that govern the behavior of fire, such as wind and fuel, exposes wildfire modeling to a degree of uncertainty. In order to produce more realistic wildfire predictions, it is, therefore, necessary to incorporate these uncertainties within wildfire models in a way that reflects the influence of environmental stochasticity on wildfire propagation. Otherwise, the risks of the potential danger of a given wildfire may be under-represented. Specifically, environmental stochasticity in the form of wind variability results in considerable uncertainty in the output of fire spread models. Here, we consider two stochastic wind models and their implementation in the spark fire simulator framework to capture the environmental uncertainty related to wind variability. The results are compared with the output from purely deterministic wildfire spread models and are discussed in the context of the potential ramifications for wildfire risk management.

Decision making is a human process that is a fundamental part of competition. As a realisation of decision making, Command and Control, or C2, has been studied in the literature for adversarial populations, yet these models do not explicitly capture competition. Our work here seeks to enrich such competition models by extending ecologically inspired population models to include decision dynamics through coupling with a nonlinear system of oscillators (the Kuramoto equation) to model the action-perception cycle in decision making. Through asymmetric competition models of increasing complexity, we highlight the importance of competitive agility and decision making processes necessary for a population to be successful.

The self-heating and spontaneous ignition process pose a fire risk for industrial biomass piles during storage. Most studies, from theoretical to numerical, pay more attention on the effect of pile size on self-heating and self-ignition, which in essence is due to chemical reactions. However, the effect of ambient humidity on the self-heating and spontaneous ignition process, which is due to physical process of water evaporation and vapor condensation, is not well understood. In fact, fire accidents and the related experimental studies have shown that the sudden increase of ambient humidity would cause the rapid increase of biomass pile temperature, leading to spontaneous ignition. In order to fill this knowledge gap, this work proposed a computational self-heating model, coupling heat and mass transfer process and both the microbial and chemical reactions. The processes of moisture evaporation, transportation and convective exchange of vapor were considered in this model to study the effect of humidity on self-heating process. The model was validated against the full-scale experiments of Zhanjiang Biomass Power Plant firstly. The numerical results show that the sudden increase of ambient humidity can lead to a quick increase of pile temperature due to the condensation process. Spontaneous ignition is highly dependent upon the heat generation evolution of the chemical reaction within the pile. The sudden increase of humidity could speed up the chemical reaction process, leading to a fire. This study helps understand the role of humidity on self-heating process during biomass storage.

The activated sludge process is the most widely used process for the biological treatment of domestic and industrial wastewaters.Wastewater treatment plants using the activated sludge process are in widely used in developed and developing countries.The activated sludge model number 1 (ASM #1) is an internationally accepted standard for activated sludge modelling.It describes nitrogen and chemical oxygen demand within suspended-growth treatment processes, including mechanisms for nitrification and denitrification.We analysed the biological treatment of a wastewater when a cascade of five reactors were used.Operating conditions were investigated in which the first reactor was not aerated.The second reactor could be either be aerated or not aerated.The remaining reactors were aerated.The process configuration included one settling tank and one recycle step.Both of these were placed after the final reactor in the cascade.We used the hydraulic retention time (HRT) as the bifurcation parameter and investigate how the total nitrogen concentration in the effluent (TN e ) stream depended upon the operation of the recycle step and the state of aeration in the second reactor.When the reactor configuration included an ideal settling tank, the total nitrogen concentration is defined aswhere the state variables on the right hand side are the concentration of soluble nitrate and nitrite nitrogen (S NO ), soluble ammonium nitrogen (S NH ), and soluble biodegradable organic (S ND ) respectively.We explored a number of different aspects of a five reactor cascade.1. We compared the performance of reactor configurations up to five reactors, with the first reactor not aerated.The qualitative behaviour was the same, with three regimes of HRT: for low HRT, the system was in a washout state, where the process failed, then a transition, or bifurcation, when the heterotrophic biomass species were present, and finally another transition, when both the heterotrophic and autotrophic biomass species were present.The final transition occurred at lower HRT as the number of reactors in the configuration was increased.There was also an improvement in the performance.2. We explored the effect of varying the HRT in each reactor and compared the performance with an equal HRT configuration.For the most of total HRT values, the equal HRT configuration offered close to the best performance.For a total HRT near the transition where the heterotrophic and autotrophic biomass species were present, the equal HRT configuration offered close to the worst performance.3. We explored the effect the configuration with a recycling step, compared to a configuration where there was no recycling step.We found that the configuration with a recycling step outperformed one without.However, the performance improvement occurred up to critical recycling rate.For higher recycling ratios, the performance deteriorated.4. Finally, we explored the effect of either aerating the first and second reactors, while aerating the other three reactors, not aerating the first reactor and aerating the second or not aerating the first and second reactors.We found that it was better to not aerate the first reactor and aerate the remaining reactors.

We consider a native species whose environment is stressed due to the presence of an ecosystem engineer, a specific type of invasive species that has the ability to convert native habitat into one which is more conducive to its own survival. The re-engineering of the native habitat results in an increase in the intra-species competition among the resident species. A key aspect of ecosystem engineer dynamics is that their invasion is limited by the size of the native habitat and their ability to transform it. It has long been recognised that invasive species play a role in declining biodiversity and the degradation of native habitat. We explore the effect of a single engineer species on a resident species and its habitat. The invading engineer species does not prey on the resident species but does compete for available habitat. The engineer species converts, modifies or re-engineers native habitat. The rate at which this conversion is performed determines whether or not colonisation will be successful. Over time the converted habitat degrades forming a decayed habitat that is not suitable to either species. The decayed habitat returns to its native state through a process of recovery. Once in its native state the habitat can be occupied by either the resident species or modified again by the ecosystem engineer. This recycling of the habitat is an important and novel feature in the model. We investigate the dynamics of this model. We assume the dynamics of both species is governed by a logistictype differential equation whose carrying capacity is equal to the size of its habitat. In the logistic equation, it is the carrying capacity that determines the population size. A reduction in habitat will drive the population to a smaller size, similarly an increase in habitat will increase the population size. An analysis of the model reveals three different approaches that may be used to control the invading species. The first is based on reducing the ability for the engineer species to convert native habitat. The model shows the existence of a minimum conversion rate below which the engineer species can not colonise the native habitat. Therefore conservation strategies should focus on reducing the engineer’s ability to convert habitat. A second strategy is a well known strategy, the harvesting of the invading species. And the third is based on ‘quarantining’ the decayed habitat for a certain period of time. Although quarantining does not alter the conversion rate it does however favour the resident species by not permitting the engineer species to become the dominant species. However, large quarantine periods cause oscillations within the system which may not be desirable. Finally, we propose directions for future work.

Traditional combat models, such as Lanchester’s equations, are typically limited to two competing populations and exhibit solutions characterized by exponential decay—and growth if logistics are included. We enrich such models to account for modern and future complexities, particularly around the role of interagency engagement in operations as often displayed in counterinsurgency operations. To address this, we explore incorporation of nontrophic effects from ecological modeling. This provides a global representation of asymmetrical combat between two forces in the modern setting in which noncombatant populations are present. As an example, we set the noncombatant population in our model to be a neutral agency supporting the native population to the extent that they are noncombatants. Correspondingly, the opposing intervention force is under obligations to enable an environment in which the neutral agency may undertake its work. In contrast to the typical behavior seen in the classic Lanchester system, our model gives rise to limit cycles and bifurcations that we interpret through a warfighting application. Finally, through a case study, we highlight the importance of the agility of a force in achieving victory when noncombatant populations are present.

Abstract We consider the numerical solution of competitive exothermic and endothermic reactions in the presence of a chaotic advection flow. The resulting behaviour is characterized by a strong dependence on the competitive reaction history. The burnt temperature is not immediately connected to simple enthalpy calculations, so there is a subtlety in the interplay between the major parameters, notably the Damköhler number, the ratio of the heats of exothermic and endothermic reactions, as well as the ratio of their respective activation energies. This paper seeks to explore the way these parameters affect the steady states of these reaction fronts and their stability.

The evaluation of structural collapse capacity is an integral part of the estimation of collapse risk in performance based earthquake engineering. Several methods and procedures have been proposed in the past to quantify collapse capacity. However, there is no clear consensus on which method is most appropriate. Recently physics-based collapse criteria have been proposed, but their effectiveness is yet to be compared with codes of practice. In order to ensure safety and consistency, different code/standard based recommendations enforce the use of thresholds predominantly in terms of the engineering demand parameter (EDP) for the performance level of collapse prevention. These thresholds serve as limit-state criteria to evaluate the collapse capacity of a structure. This paper compares several performance measures that are derived using different criteria to understand their impact on the final collapse risk estimates. Four different criteria are studied, two of which are based on standards, and the other two are physics-based, which use energy formulations. Three different ASCE 7-16 code conforming RC buildings are designed and analysed. The effects of modelling uncertainties and of the ground motion spectral shape have been considered to impart more confidence in the results. The collapse risk estimates are interpreted in terms of different collapse performance measures. The estimates derived from all the four criteria are compared. It was found that the collapse risk is significantly affected by the choice of the criterion. For the same collapse risk, the physics-based criteria allow higher ultimate deformation at collapse. On the other hand, the code/standard based criteria tend to be conservative as they censor the deformation response of the structure using upper bound thresholds. It was also found that the physics-based criteria could underestimate the collapse risk, yet they can still be employed to produce more economical designs.

The development of collapse fragility curves is an essential requirement for the assessment of the collapse risk of structures. These fragility curves depend on the structural collapse capacity that is evaluated in terms of either the intensity measure (IM) or the damage measure (DM); such as the engineering demand parameter (EDP). In turn, collapse capacity estimates are sensitive to the method employed for their assessment. Conventionally, the IM and DM rules are employed in conjunction with incremental dynamic analyses (IDA) to quantify the collapse capacity of a structure. However, this approach has been criticised for being subjective in nature, since it depends on the structural response approaching predetermined threshold values. Therefore, it does not relate to the actual dynamic instability. Although the selection of these thresholds stems from the results of experimental investigations and equivalent numerical models and serve as an indirect check for dynamic instability, the present study seeks to provide a mathematical basis for defining collapse criteria. A dynamical system approach is used to formulate a mathematical criterion for defining P-Delta instability induced seismic collapse in single-degree-of-freedom (SDOF) structures. The non-linear SDOF structure is considered as a non-autonomous, non-smooth system, and is studied as an ensemble of different sub-systems. Collapse is defined as the point when the dominant system eigenmode of the structure changes from stable to unstable, and remains unstable as the structure collapses. This approach can be applied to first mode governed multi-degree-of-freedom (MDOF) structures when studied as an equivalent SDOF structure. It is found that the dynamical systems approach results in higher deformations at collapse when compared to the conventional IM/DM rule based approach, suggesting the conservatism involved in the latter. However, it results in lower deformations at collapse when compared to the energy criterion, which relies on the occurrence of large deformations to predict collapse. Furthermore, the derived fragility curves show that the proposed approach yields lower probabilities of collapse when compared to the conventional method. Therefore, the proposed method can be used an alternative method for the performance design of structures.

We consider a system of reaction-diffusion equations describing combustion dynamics. The reaction is assumed to undergo two competitive reactions, one which is exothermic and one which is endothermic. The one-dimensional model has been shown to exhibit complex behaviour, from propagating combustion waves with a constant speed to period doubling cascades and the possibility of chaotic wave speeds. In this study, we extend the combustion model from one to two dimensions by exploring a model of an insulated strip with no heat loss and axially symmetric spread. In particular, we compare and contrast the behaviour of the systems in one and two dimensions. References J. D. Buckmaster and G. S. S. Ludford. Theory of Laminar Flames. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, 1982. doi:10.1017/CBO9780511569531. V. Gubernov, A. Kolobov, A. Polezhaev, H. Sidhu, and G. Mercer. Period doubling and chaotic transient in a model of chain-branching combustion wave propagation. P. Roy. Soc. A Math. Phy., 466:27472769, 2010. doi:10.1098/rspa.2009.0668. A. Hmaidi, A. C. McIntosh, and J. Brindley. A mathematical model of hotspot condensed phase ignition in the presence of a competitive endothermic reaction. Combust. Theor. Model., 14:893920, 2010. doi:10.1080/13647830.2010.519050. G. N. Mercer and R. O. Weber. Combustion waves in two dimensions and their one-dimensional approximation. Combust. Theor. Model., 1:157165, 1997. doi:10.1088/1364-7830/1/2/002. J. J. Sharples, H. S. Sidhu, A. C. McIntosh, J. Brindley, and V. V. Gubernov. Analysis of combustion waves arising in the presence of a competitive endothermic reaction. IMA J. Appl. Math., 77:1831, 2012. doi:10.1093/imamat/hxr072. S. D. Watt, R. O. Weber, H. S. Sidhu, and G. N. Mercer. A weight-function approach for determining watershed initial conditions for combustion waves. IMA J. Appl. Math., 62:195206, 1999. doi:10.1093/imamat/62.2.195.

Construction in earthquake prone areas is an expensive task, as the structural design warrants high material consumption. This is partly due to the fact that the seismic design philosophy is based on the concept of stability through energy dissipation. The energy dissipation is conventionally achieved by controlled plastifi-cation and hysteresis of the structural elements that require high quantities of reinforcements and sophisticated detailing, in both steel and concrete structures. Further, due to inherent uncertainty in the occurrence and the characteristics of earthquakes and also in the current simulation models, a conservative design is essential that can provide a sufficient margin of safety against failure. In the last two decades, however, there is a push towards developing performance designs. This new evolved philosophy of seismic design aims to quantify the uncertainties and the unknown aspects of the design to reduce the margins of safety, while sustaining equally high reliability. As a result, this leads to designs with low requirements of material consumption, thereby making them economic. Consequently, the complexity in the design process increases in almost every aspect, right from quantification of uncertainty by performing Monte-Carlo simulations to developing high-fidelity models that can incorporate all forms of non-linearity in the design. Moreover, to reduce the margins of safety, it becomes imperative to accurately estimate the point of failure or structural collapse capacity. However, currently under the Performance Based Earthquake Engineering (PBEE) framework, the collapse capacity is not evaluated corresponding the the actual dynamic instability in the structural system. Instead, it is estimated corresponding to subjective threshold values of engineering demand parameters, such as lateral deformation. Therefore, in the current paper, a novel-approach is presented that uses dynamical system theory for evaluat-ing dynamic instability in a structure that can be used to accurately estimate its collapse capacity. A P-Delta instability is the dynamic instability that occurs when gravity loads magnify the force demand due to the ge-ometry of the deformed structure, leading to high overturning moments on the base. This is widely studied under mainstream structural analysis. For simplicity, a single-degree-of-freedom (SDOF) system is studied. Therefore, the current work is targeted towards the structures that can be idealised as an SDOF system. The dynamic instability leading to “structural collapse” is defined when the real part of the dominant eigenvalue of the oscillator system becomes positive and remains positive until large deformations occurs. The current study uses harmonic excitations for evaluating dynamic instability and therefore acts as a precursor to a larger study aimed at evaluating mathematical instability in structures under the effects of seismic ground motions.

We consider non-adiabatic combustion waves arising from two-step competitive exothermic reaction schemes. A numerical method is employed to study the behaviour of this system and we show that the inclusion of heat loss can lead to a period-doubling route to the termination of the propagating flame front. The nature of oscillations becomes more complex with increasing loss of heat until the system can no longer sustain a propagating front. In other words, beyond some critical value of heat loss, extinction of the combustion reaction would occur. For the non-adiabatic case, particularly close to the extinction threshold, large excursions in temperature and wave speed above those observed for the adiabatic case can occur. Such behaviour close to extinction may have implications for safety or industrial processes.

We consider non-adiabatic combustion waves arising from a two-step exothermic system. Our previous work showed that in certain parameter regions, the combustion wave can evolve to the “fast” solution branch, the “slow” solution branch or diffuse to the ambient temperature (extinction wave). Here, we are interested to find critical initial temperature profiles which evolve to these three types of steady solutions. For a particular family of temperature profiles, we construct a weight function which can be used to predict which of these three types of waves an initial temperature profile will evolve to.

From a point source, landscape fires accelerate until they reach a quasi-equilibrium rate of spread. The rate at which a fire accelerates from its ignition affects the time first responders have to attack a fire in its initial stages when it is more easily suppressed. As such, knowledge of the rate of acceleration of a fire from ignition can be valuable from a fire management perspective. However, the majority of studies in wildland fire science have been dedicated to development of models for the quasi-equilibrium rate of spread attained by the fire after its acceleration phase. Comparatively little attention has been given to the development of models that specifically account for the growth phase of a fires development.The rate of acceleration depends on many factors including variations in ambient and induced wind speed and direction, variation in moisture content of the fuel, fuel stratification and slope variation. Present models of fire growth from a point ignition are expressed as deterministic algebraic equations, thereby neglecting variability. The numerous variables involved make predictions of rate of spread from a point source very difficult.In this paper we consider two approaches to model the acceleration phase of a fire. The first considers fitting a sigmoidal (logistic) function to experimental data using a nonlinear regression procedure. In the second approach we propose the use of stochastic differential equations to investigate the growth of a fire to quasi-equilibrium. In addition to providing a more realistic portrayal of the time series data relating to fire growth, this second approach allows for better discrimination of the mechanisms driving the growth phase of fire spread.The models are assessed by appealing to observations of experimental fire growth. Specifically the data relate to fires growing from a point ignition under the influence of a uniform wind. The results indicate that both approaches can provide an accurate representation of the observed data, but that the approach based on stochastic differential equations yields 95% prediction bounds that are narrower than those obtained from the nonlinear regression. The difference in prediction bounds indicates that the way stochasticity is incorporated into fire growth models has implications for how models inform decisions about the likelihood of a fire self-extinguishing before it reaches quasi-equilibrium, and the magnitude of the rates of spread it is likely to exhibit during the initial stages of growth.

Firefighting is a hazardous occupation that requires the wearing of appropriate protective clothing which must be designed to be flame- and heat-resistant while also allowing for a firefighter’s ease of movement. To increase the thermal protection provided by a firefighting suit and decrease the likelihood of the firefighter receiving skin burns, we propose incorporating a layer of a phase-change material (PCM) as well as air gaps in its structure. We investigate the distribution of heat through the layers of a firefighting suit and skin to determine whether this approach will be successful when different suit configurations are exposed to a range of fire scenarios with external heat fluxes between \(5\,{\rm kWm}^{-2}\) and \(84\,{\rm kWm}^{-2}\). We use a one-dimensional model of heat transfer which we solve numerically to determine the length of time each suit configuration allows a firefighter to be exposed to heated conditions before suffering irreversible thermal skin damage. Thermal damage to the skin is known to occur when the temperature in the basal layer exceeds \(44\)°C. Our earlier research indicated that the combination of air gaps and a PCM layer reduces the likelihood of skin burns, and that the most effective position of a PCM in a suit is near the outer layer. This current work considers a number of different PCM compounds for providing additional thermal protection while ensuring that the extra weight required is feasible for a firefighting suit. We found that of the PCMs studied, \({\rm MgCl_2\cdot 6H_2O}\) with an overall thickness of 0.17 mm gave the best improvement in the time until thermal skin damage (of between \(13\%\) and \(19\%\)), depending on the fire scenario.

Abstract: In ecology, the theory of alternative stable states predicts that ecosystems can exist under multiple “states”. Ecosystems may transition from one stable state to another, in what is known as a state shift. Typically such shifts are considered as instantaneous and isolated non-interacting events resulting from environmental shocks, whose dynamics resemble the dynamics of a ball in a “potential well”. More often however, ecological systems are subject to continuous variation due to environmental drivers such as rainfall, temperature, among others, and these interact with the ecosystem dynamics to alter the potential well. Such drivers are known to impact intra-species interactions.

The plight of the reindeer population on the Pribilof Islands, Bering Sea, is an interesting example of resource depletion and population collapse. To U.S. government officials, the abundance of lichen on the Pribilof Islands, together with the absence of large grazing animals, and the lack of natural predators appeared ideal conditions for the introduction of reindeer in 1911. Once introduced, the reindeer population grew rapidly followed by a precipitous decline, with very few remaining by 1951. It is generally believed that grazing pressure by the reindeer in combination with markedly warmer and drier climatic conditions caused a rapid and sustained reduction of lichen. We use a variant of the Lotka-Volterra model and apply it to available data to model the reindeer population in response to changes in the availability of lichen.

A system of coupled partial differential equations which models a complex system of a solid fuel, endothermically pyrolysing to a combustible gas, which in turn exothermically reacts with oxygen, is studied.We use a numerical method based on the Crank-Nicholson finite difference scheme to solve the governing equations in order to investigate the dynamics of this model.It has previously been shown that there exist solutions to the model which exhibit oscillatory propagating combustion waves.In this work, we extend the original study to explore the parameter space to locate regions where both steady propagating waves (single valued wave speed) and also pulsating waves exist.Parameter space where no propagating waves are possible (extinction region) is also determined.