A singular limit for an age structured mutation problem

The spread of a particular trait in a cell population often is modelled by an appropriate system of ordinary differential equations describing how the sizes of subpopulations of the cells with the same genome change in time. On the other hand, it is recognized that cells have their own vital dynamics and mutations, leading to changes in their genome, mostly occurring during the cell division at the end of its life cycle. In this context, the process is described by a system of McKendrick type equations which resembles a network transport problem. In this paper we show that, under an appropriate scaling of the latter, these two descriptions are asymptotically equivalent

The discrete fragmentation equations : semigroups, compactness and asynchronous exponential growth

In this paper we present a class of fragmentation semigroups which are compact in a scale of spaces defined in terms of finite higher moments. We use this compactness result to analyse the long time behaviour of such semigroups and, in particular, to prove that they have the asynchronous growth property. We note that, despite compactness, this growth property is not automatic as the fragmentation semigroups are not irreducible

Delayed stability switches in singularly perturbed predator-prey models

In this paper we provide an elementary proof of the existence of canard solutions for a class of singularly perturbed predator-prey planar systems in which there occurs a transcritical bifurcation of quasi steady states. The proof uses a one-dimensional theory of canard solutions developed by V. F. Butuzov, N. N. Nefedov and K. R. Schneider, and an appropriate monotonicity assumption on the vector field to extend it to the two-dimensional case. The result is applied to identify all possible predator-prey models with quadratic vector fields allowing for the existence of canard solutions

The role of a subsurface lime-fly ash barrier in the mitigation of acid sulphate soils

The effectiveness of using a sub-surface lime-fly ash barrier to reduce the oxidation of a pyritic soil layer and to improve groundwater and surface water quality was investigated for land affected by acid sulphate soils near Berry in southeastern NSW, Australia. Prior to the installation of the lime-fly ash barrier, groundwater and surface water analyses indicated a highly acidic environment. High concentrations of dissolved aluminium, total iron and sulphate in the groundwater were a result of falling groundwater tables and biotic oxidation. Traditional management techniques of ground water manipulation, via floodgates or weirs, would be rendered ineffective in arresting biotic oxidation where the pyrite layer is submerged. The study combined field and laboratory analysis in order to determine the feasibility of the lime-fly ash barrier at the study site. A comprehensive field study incorporated the installation of piezometers and observation wells to determine the level of the phreatic surface along with the monitoring of water quality parameters at the site of the lime-fly ash barrier, and also floodgate sites and the site of the self-regulating tilting weir. The installation of the lime-fly ash barrier was undertaken by the pumping of a slurry through boreholes via pressure pumping. The subsurface lime-fly ash barrier, as an acid sulphate soil remediation technique, was shown to significantly improve groundwater quality. Groundwater pH increased to values between 4.5 and 5.5. The concentration of the pyritic oxidation products, acidic cations Al³+ and Fetotal, basic cations Ca²+ and Mg²+ and anions Cl- and SO4² -, also, on average decreased in the groundwater after the installation of the lime-fly ash barrier. A comparison between the average roundwater table elevations before and after the installation of the barrier also indicated a perched water table, which would reduce the exposure of pyritic soil to oxygen, and in turn reduce pyritic oxidation and the generation of acidic products. The Lime-fly ash barrier is effective in remediating acid sulphate soils in areas in which floodgates and weirs cannot be installed. A comparison of the result shows that the lime-fly ash barrier had greater success in increasing the groundwater pH than the self-regulating tilting weir. The lime-fly ash barrier treats acid sulphate soils and the related environmental problems before they occur, whereas, the floodgates treat the pyrite oxidation products generated after they have been discharged into the flood mitigation drains. Significantly greater concentrations of Al³+, Fetotal and SO4 ²- were found in the groundwater at the floodgate sites

Chapman-Enskog asymptotic procedure in structured population dynamics

Complexity of many biological models often makes impossible their robust theoretical and numerical analysis and thus requires a systematical method of reducing the number of variables in the system in such a way that the dynamics of the simplified model approximates the original way in a reasonable way. Such an aggregation of variables is often done by ad hoc methods. In turns out that in many biological systems the well-known Chapman-Enskog asymptotic procedure is well suited for aggregation which then can be viewed as passing from a microscopic (kinetic) to a macroscopic (hydrodynamic) description of the system. We demonstrate this approach by applying it to an age- and space-structured population model

Key strategies for managing acid sulphate soil (ASS) problems on the southeastern coast of New South Wales, Australia

The acidification of Australian coastal waterways as a result of the oxidation of acid sulphate soil (ASS) containing appreciable quantities of sulphidic material (e.g. pyrite) has well recognised environmental, economic and social effects including the loss of fish, biodiversity and agricultural productivity as well as the corrosion of concrete and steel infrastructure by acidic drainage. Largescale artificial drainage and one-way floodgates in low-lying coastal floodplains has lowered the groundwater table, thus enhancing pyritic oxidation and increasing the distribution, magnitude and frequency of acid generation and release of toxic metals such as aluminium (Al3+) and iron (total Fe) from ASS. Engineering strategies implemented on the Shoalhaven Floodplain, southeast New South Wales, Australia have been designed to remediate ASS. These include: (1) fixed-level v-notch weirs, which raise the groundwater table above the pyritic layer and reduce the rate of discharge of acidic products from the groundwater into the drains; (2) modified two-way floodgates, which allow for tidal buffering of acidic drainage; (3) a subsurface alkaline horizontal impermeable lime-fly ash barrier, which prevents pyrite oxidation and neutralises acidic groundwater and (4) an alkaline permeable reactive barrier (PRB) using recycled materials, which significantly increases groundwater pH and reduces Al and Fe concentrations within and down-gradient of the PRB. A critical review of each of these strategies will outline their role in remediating ASS and their respective benefits and limitations

Coagulation and fragmentation processes with evolving size and shape profiles : a semigroup approach

We investigate a class of bivariate coagulation-fragmentation equations. These equations describe the evolution of a system of particles that are characterised not only by a discrete size variable but also by a shape variable which can be either discrete or continuous. Existence and uniqueness of strong solutions to the associated abstract Cauchy problems are established by using the theory of substochastic semigroups of operators