There is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a collection of noninteracting biased random walkers on a one-dimensional lattice is considered. The usual master equation approach requires that two continuum limits, involving three parameters, namely step length, time step and the random walk bias, approach zero in a specific way. We are interested in the case where the two limits are not consistent. New results are obtained using a Fokker–Planck equation and the results are highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to an accurate macroscopic description of stochastic local movement rules in agent-based models when the lattice spacing represents a physical object such as cell diameter.

Nanopore science, the study of individual nanoscale pores within thin membranes, is a fast-growing field which presents numerous interesting problems for physicists and applied mathematicians. Nanopores are most commonly applied to resistive pulse sensing (RPS) of individual particles suspended in aqueous electrolyte. The form of a resistive pulse is dependent on an array of experimental variables, including electrolyte characteristics, electrophoretic and convective transport, and (especially) pore and particle geometry. The level of analysis required depends on the application, but any broadly useful approach should be simple and flexible, due to the requirement for high data throughput and variations between different experimental systems and specimens. Here we review analytic methods for interpreting RPS experiments for particles in the approximate range 100 nm to 1 $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mu $m, focusing on calculation of resistance change as a function of the particle’s position. We detail a recently developed semi-analytical model and compare the modelled electric field with finite element results. The model can also be used to calculate particle motion, so that the experimental current–time history can be reconstructed. This approach is useful for a wide range of pore and particle geometries, and includes consideration of entrance effects. Tunable elastomeric pores with truncated linear cone geometry are used as a model system.

In this paper, we give general recommendations for successful application of the Douglas–Rachford reflection method to convex and nonconvex real matrix completion problems. These guidelines are demonstrated by various illustrative examples.

This note considers an established reaction–diffusion model for a combustion system, in which there are competing endothermic and exothermic reaction pathways. A combustion front is assumed to move at constant speed through the medium. An asymptotic theory is presented for solid fuels in which material diffusion is ignored, and it allows a simple and complete analysis of the approximate system in the phase plane. Both the adiabatic and nonadiabatic cases are discussed.

The steady response of a fluid with two layers of different density in a porous medium is considered during extraction through a point sink. Supercritical withdrawal in which both layers are being withdrawn is investigated using a spectral method. We show that for each withdrawal rate, there is a single entry angle of the interface into the point sink. As the flow rate decreases the angle of entry steepens until it becomes almost vertical, at which point the method fails. This limit is shown to correspond to the upper bound on sub-critical (single-layer) flow.

The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.

We propose and analyse a method based on the Riccati transformation for solving the evolutionary Hamilton–Jacobi–Bellman equation arising from the dynamic stochastic optimal allocation problem. We show how the fully nonlinear Hamilton–Jacobi–Bellman equation can be transformed into a quasilinear parabolic equation whose diffusion function is obtained as the value function of a certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence and uniqueness and derive useful bounds of classical Hölder smooth solutions. Furthermore, we construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit travelling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 index as an example of the application of the method.

A stochastic algorithm for bound-constrained global optimization is described. The method can be applied to objective functions that are nonsmooth or even discontinuous. The algorithm forms a partition on the search region using classification and regression trees (CART), which defines a region where the objective function is relatively low. Further points are drawn directly from the low region before a new partition is formed. Alternating between partition and sampling phases provides an effective method for nonsmooth global optimization. The sequence of iterates generated by the algorithm is shown to converge to an essential global minimizer with probability one under mild conditions. Nonprobabilistic results are also given when random sampling is replaced with points taken from the Halton sequence. Numerical results are presented for both smooth and nonsmooth problems and show that the method is effective and competitive in practice.

We examine the dynamics of fermentation process in a yeast cell. Our investigation focuses on the main branch pathway: pyruvate and acetaldehyde branch points. We formulate the kinetics for all enzymatic reactions as Michaelis–Menten models. Since the activity of an enzyme mainly depends on the conformational changes of the enzyme structure, the enzyme requires a certain period of time to reset its structure, until it is ready to bind substrates again. For this situation, a rate-limiting step exists, for which the catalytic process suffers a delay. Since all conversion processes are catalysed by enzymes, each reaction can experience a delay at a different time. To investigate how the delay affects the reaction processes, especially at the branch points, we propose that the rate-limiting step takes place at the first reaction. For this reason, a discrete time delay is introduced to the first kinetic model. We find a bifurcation diagram for the delay that depends on the rate of glucose supply and kinetic parameters of the first enzyme. By comparison, our analysis agrees with the numerical solution. Our numerical simulations also show that there is a certain glucose supply that may optimize ethanol production.

We study the sequence of monic polynomials orthogonal with respect to inner product

$$\begin{eqnarray*}\langle p, q\rangle = \int \nolimits \nolimits_{0}^{\infty } p(x)q(x){e}^{- x} {x}^{\alpha } \hspace{0.167em} dx+ Mp(\zeta )q(\zeta )+ N{p}^{\prime } (\zeta ){q}^{\prime } (\zeta ),\end{eqnarray*}$$

$\alpha \gt - 1$

$M\geq 0$

$N\geq 0$

$\zeta \lt 0$

$p$

$q$

In models of fluid outflows from point or line sources, an interface is present, and it is forced outwards as time progresses. Although various types of fluid instabilities are possible at the interface, it is nevertheless of interest to know the development of its overall shape with time. If the fluids on either side are of nearly equal densities, it is possible to derive a single nonlinear partial differential equation that describes the interfacial shape with time. Although nonlinear, this equation admits a simple transformation that renders it linear, so that closed-form solutions are possible. Two such solutions are illustrated; for a line source in a planar straining flow and a point source in an axisymmetric background flow. Possible applications in astrophysics are discussed.

This work focuses on finding optimal dividend payment and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections with delays. Starting from the classical Cramér–Lundberg process, using the dynamic programming approach, the value function obeys a quasi-variational inequality. With delays in capital injections, the company will be exposed to the risk of financial ruin during the delay period. In addition, the optimal dividend payment and capital injection strategy should balance the expected cost of the possible capital injections and the time value of the delay period. In this paper, the closed-form solution of the value function and the corresponding optimal policies are obtained. Some limiting cases are also discussed. A numerical example is presented to illustrate properties of the solution. Some economic insights are also given.

We propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal $L^2$-error estimate is derived for the semidiscrete approximation when the initial data is in $L^2$. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain.

A sporting league places every team into one of several divisions of equal size, and runs a round robin tournament for each division. Some teams are paired with another team, not necessarily in the same division, to share facilities. It is shown that however many teams are paired and whatever the pairings, it is always possible to schedule the fixtures in the minimum time, so that no two paired teams have home matches simultaneously.

It has been known for a long time that the equivariant $2+1$ wave map into the $2$-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.

Inequalities for spatial competition verify the pair approximation of statistical mechanics introduced to theoretical ecology by Matsuda, Satō and Iwasa, among others. Spatially continuous moment equations were introduced by Bolker and Pacala and use a similar assumption in derivation. In the present article, I prove upper bounds for the $k\mathrm{th} $ central moment of occupied sites in the contact process of a single spatial dimension. This result shows why such moment closures are effective in spatial ecology.

A new formula for Adomian polynomials is introduced and applied to obtain truncated series solutions for fractional initial value problems with nondifferentiable functions. These kinds of equations contain a fractional single term which is examined using Jumarie fractional derivatives and fractional Taylor series for nondifferentiable functions. The property of nonlocality of these equations is examined, and the existence and uniqueness of solutions are discussed. Convergence and error analysis for the Adomian series solution are also studied. Numerical examples show the accuracy and efficiency of this formula for solving initial value problems for high-order fractional differential equations.

Microbial competition for nutrients is a common phenomenon that occurs between species inhabiting the same environment. Bioreactors are often used for the study of microbial competition since the number and type of microbial species can be controlled, and the system can be isolated from other interactions that may occur between the competing species. A common type of competition is the so-called “pure and simple” competition, where the microbial populations interact in no other way except the competition for a single rate-limiting nutrient that affects their growth rates. The issue of whether pure and simple competition under time-invariant conditions can give rise to chaotic behaviour has been unresolved for decades. The third author recently showed, for the first time, that chaos can theoretically occur in these systems by analysing the dynamics of a model where both competing species grow following the biomass-dependent Contois model while the yield coefficients associated with the two species are substrate-dependent. In this paper we show that chaotic behaviour can occur in a much simpler model of pure and simple competition. We examine the case where only one species grows following the Contois model with variable yield coefficient while the other species is allowed to grow following the simple Monod model with constant yield. We show that while the static behaviour of the proposed model is quite simple, the dynamic behaviour is complex and involves period doubling culminating in chaos. The proposed model could serve as a basis to re-examine the importance of Contois kinetics in predicting complex behaviour in microbial competition.

We study the reflection of membrane-coupled gravity waves in deep water against a vertical barrier with a gap. A floating membrane is attached on both sides of the barrier. The associated mixed boundary value problem, which is not particularly well posed, is analysed. We utilize an orthogonal mode-coupling relation to reduce the problem to solving a set of dual integral equations with trigonometric kernel. We solve these by using a weakly singular integral equation. The reflection coefficient is determined explicitly, while having freedom to clamp the membrane with a spring of a certain stiffness on only one side of the vertical barrier. The physical problem is of capillary–gravity wave scattering by a vertical barrier with a gap, when the membrane density is neglected. In this case, the reflection coefficient is known up to an undetermined edge slope on either side of the barrier. The scattering quantity is computed and presented graphically against a wave parameter for different values of nondimensional parameters pertaining to the structures involved in the problem.

Although variance swaps have become an important financial derivative to hedge against volatility risks, closed-form formulae have been developed only recently, when the realized variance is defined on discrete sample points and no continuous approximation is adopted to alleviate the mathematical difficulties associated with dealing with the discreteness of the sample data. In this paper, a new closed-form pricing formula for the value of a discretely sampled variance swap is presented under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model. With the newly found analytical formula, not only can all the hedging ratios of a variance swap be analytically derived, the numerical values of the swap price can be efficiently computed as well.