In aquatic microbial systems, high-magnitude variations in abundance, such as sudden blooms alternating with comparatively long periods of very low abundance (“apparent disappearance”), are relatively common. We suggest that in order for this to occur, such variations in abundance in microbial systems and, in particular, the apparent disappearance of species do not require seasonal or periodic forcing of any kind or external factors of any other nature. Instead, such variations can be caused by internal factors and, in particular, by bacteria–phage interaction. Specifically, we suggest that the variations in abundance and the apparent disappearance phenomenon can be a result of phage infection and the lysis of infected bacteria. To illustrate this idea, we consider a reasonably simple mathematical model of bacteria–phage interaction based on the model suggested by Beretta and Kuang, which assumes neither periodic forcing nor action of other external factors. The model admits a loss of stability via Andronov–Hopf bifurcation and exhibits dynamics which explains the phenomenon. These properties of the model are especially distinctive for spatially nonhomogeneous biosystems as well as biosystems with some sort of cooperation or community effects.

Bounds for Hardy differences, that is, improvements and reverses of the well-known Hardy inequality, are obtained.

The discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.

A relationship between the fractal geometry and the analysis of recursive (divide-and-conquer) algorithms is investigated. It is shown that the dynamic structure of a recursive algorithm which might call other algorithms in a mutually recursive fashion can be geometrically captured as a fractal (self-similar) image. This fractal image is defined as the attractor of a mutually recursive function system. It then turns out that the Hausdorff-Besicovich dimension D of such an image is precisely the exponent in the time complexity of the algorithm being modelled. That is, if the Hausdorff D-dimensional measure of the image is finite then it serves as the constant of proportionality and the time complexity is of the form Θ(nD), else it implies that the time complexity is of the form Θ(nD logpn), where p is an easily determined constant.

The aggregation–decomposition method is used to derive a sufficient condition for the equi-ultimate boundedness of large-scale systems governed by nonlinear ordinary differential equations.

We show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.

The effect of an enclosed air cavity on the natural vibration frequencies of a rectangular membrane is investigated. The modes specified by an even integer are not affected. For the odd-odd modes, the frequency equation is found via a Green's function formulation and is solved to first order in a parameter representing the effect of the cavity of the rectangular drum. The frequencies are raised, with the fundamental being most affected. In the case of degeneracies, each degenerate mode contributes to the frequency shift, but the degeneracy itself is not broken to first order.

Two-dimensional gravity-capillary solitary waves propagating at the surface of a fluid of infinite depth are considered. The effects of gravity and of variable surface tension are included in the free-surface boundary condition. The numerical results extend the constant surface tension results of Vanden-Broeck and Dias to situations where the surface tension varies along the free surface.

Fluid turbulence is often modelled using equations derived from the Navier–Stokes equations, perhaps with some semi-heuristic closure model for the turbulent viscosity. This paper considers a possible alternative hypothesis. It is argued that regarding turbulence as a manifestation of non-Newtonian behaviour may be a viewpoint of at least comparable validity. For a general description of nonlinear viscosity in a Stokes fluid, it is shown that the flow patterns are indistinguishable from those predicted by the Navier–Stokes equation in one- or two-dimensional geometry, but that fully three-dimensional flows differ markedly. The stability of linearized plane Poiseuille flow to three-dimensional disturbances is then considered, in a Tollmien–Schlichting formulation. It is demonstrated that the flow may become unstable at significantly lower Reynolds numbers than those expected from Navier–Stokes theory. Although similar results are known in sections of the rheological literature, the present work attempts to advance the philosophical viewpoint that turbulence might always be regarded as a non-Newtonian effect, to a degree that is dependent only on the particular fluid in question. Such an approach could give a more satisfactory account of the underlying physics.

The Wiener-Hopf technique is applied to the quasi-linear infiltration problem of flow from a shallow half-plane pond. Fully-saturated conditions hold immediately under the pond, while on the surface away from the pond the linearised evaporative loss is assumed to be proportional to the local relative permeability.

Evaporation from the non-wetted region increases the water flow from the pond into the soil, thereby coupling to the effects from capillarity. Linearised evaporation introduces an additional length scale and additional logarithmic expressions to those derived previously. The total rate of volumetric flow into the soil from the pond per unit length of perimeter, in addition to the usual gravity flow, increases somewhat slowly as evaporation increases. The most extreme case considered in this paper yielded an additional flow rate 63% greater than that obtained in the absence of evaporation.

The interaction between evaporation and capillarity is enhanced in poorlydraining soils, where the reduced ability to transmit liquid water need not be compensated by a corresponding reduction in evaporative losses. However, in freelydraining soils the interaction between evaporation and capillarity is probably small.

Competition between a finite number of searching insect parasites is modelled by differential equations and birth-death processes. In the one species case of intraspecific competition, the deterministic equilibrium is globally stable and, for large populations, approximates the mean of the stationary distribution of the process. For two species, both inter- and intraspecific competition occurs and the deterministic equilibrium is globally stable. When the birth-death process is reversible, it is shown that the mean of the stationary distribution is approximated by the equilibrium. Confluent hypergeometric functions of two variables are important to the theory.

Let M be an n-dimensional space-like hypersurface in a locally symmetric Lorentz space, with n(n−1)R=κH(κ>0) and satisfying certain additional conditions on the sectional curvature. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M, respectively. We show that if the mean curvature is nonnegative and attains its maximum on M, then:

1. (1) if H2<4(n−1)c/n2, M is totally umbilical;

2. (2) if H2=4(n−1)c/n2, M is totally umbilical or is an isoparametric hypersurface;

3. (3) if H2>4(n−1)c/n2 and S satisfies some pinching conditions, M is totally umbilical or is an isoparametric hypersurface.

We introduce a model for the execution of large market orders in limit order books, and use a linear combination of self-exciting Hawkes processes to model asset-price dynamics, with the addition of a price-impact function that is concave in the order size. A criterion for a general price-impact function is introduced, which is used to show how specification of a concave impact function affects order execution. Using our model, we examine the immediate and permanent impacts of large orders, analyse the potential for price manipulation, and show the effectiveness of the time-weighted average price strategy. Our model shows that price depends on the balance between the intensities of the Hawkes process, which can be interpreted as a dependence on order-flow imbalance.

We study the one-dimensional conservation law. We use a characteristic surface to define a class of functions, within which the integral version of the conservation law is solved in a simple and direct way. A simple algorithm for computing the unique solution is developed. The method uses the equal-area principle and yields the solution for any given time directly.

We study bound states in weakly deformed and heterogeneous waveguides, and compare analytical predictions using a recently developed perturbative method with precise numerical results for three different configurations: a homogeneous asymmetric waveguide, a heterogenous asymmetric waveguide and a homogeneous broken strip. We have found excellent agreement between the analytical and numerical results in all the examples; this provides a numerical verification of the analytical approach.

The effect of an isolated topographic bump in a two-layer fluid on a β-plane is investigated. An analytical solution is derived in terms of the appropriate Green's function for arbitrary topography of finite horizontal extent. It is found that the disturbances generated by the bump are composed of two fundamental modes which may be wave-like or evanescent. The wave-like modes are topographically induced Rossby waves which occur only when there is eastward flow in at least one of the layers. These waves are always confined to the downstream (eastward) side of the bump. Whereas previous studies of this type have concentrated on eastward flow over topography, the theory has been extended here to include a wide range of vertically sheared flows. Particularly important is the case of low level westward flow combined with upper level eastward flow, as it has direct application, for example, to the summertime atmospheric circulation over the sub-tropical regions of the continental land.masses. In this case a wave-like disturbance extends far downstream from the bump for sufficiently large shear, and is of smaller amplitude in the upper layer than in the lower layer because of the effects of the stratification. For small shears, the wave-like mode in the lower layer is small and the character of the disturbance is evanescent, confining it to the immediate neighbourhood of the bump. A stability analysis of the solutions shows that the disturbances may be baroclinically unstable for sufficiently large mean shear.

This is a short précis of a presentation on some of the recent advances in the area of extrapolation quadrature; given at David Elliott's 65th birthday conference in Hobart in February 1997.

Mealybug is an important pest of cassava plant in Thailand and tropical countries, leading to severe damage of crop yield. One of the most successful controls of mealybug spread is using its natural enemies such as green lacewings, where the development of mathematical models forecasting mealybug population dynamics improves implementation of biological control. In this work, the Sharpe–Lotka–McKendrick equation is extended and combined with an integro-differential equation to study population dynamics of mealybugs (prey) and released green lacewings (predator). Here, an age-dependent formula is employed for mealybug population. The solutions and the stability of the system are considered. The steady age distributions and their bifurcation diagrams are presented. Finally, the threshold of the rate of released green lacewings for mealybug extermination is investigated.

A simple algebraic method is presented to determine the necessary condition for the existence of a Hamiltonian circuit in a directed graph of n vertices. A search procedure is then introduced to identify any or all of the existing Hamiltonian circuits. The procedure is based upon finding a set of edges which will then be candidates for being parts of circuits of length n at any vertex of the graph.