We investigate the effects of toxins on the multiple coexistence solutions of an unstirred chemostat model of competition between plasmid-bearing and plasmid-free organisms when the plasmid-bearing organism produces toxins. It turns out that coexistence solutions to this model are governed by two limiting systems. Based on the analysis of uniqueness and stability of positive solutions to two limiting systems, the exact multiplicity and stability of coexistence solutions of this model are established by means of the combination of the fixed-point index theory, bifurcation theory and perturbation theory.

The main purpose of this paper is to study the existence of travelling waves with a critical speed for an influenza model with treatment. By using some analysis techniques that involve super-critical speeds and an approximation method, the existence of travelling waves with the critical speed is proved.

In this paper, we study singularly perturbed Filippov systems. More specifically, our main question is to know how the dynamics of Filippov systems is affected by singular perturbations. We extend the Fenichel theory developed in Fenichel (J. Differ. Equ., 1979, Vol. 31, pp. 53–98) to these systems. In addition, the study of non-smooth constrained systems is considered.

This paper is concerned with the asymptotic propagations for a nonlocal dispersal population model with shifting habitats. In particular, we verify that the invading speed of the species is determined by the speed c of the shifting habitat edge and the behaviours near infinity of the species’ growth rate which is nondecreasing along the positive spatial direction. In the case where the species declines near the negative infinity, we conclude that extinction occurs if c > c*(∞), while c < c*(∞), spreading happens with a leftward speed min{−c, c*(∞)} and a rightward speed c*(∞), where c*(∞) is the minimum KPP travelling wave speed associated with the species’ growth rate at the positive infinity. The same scenario will play out for the case where the species’ growth rate is zero at negative infinity. In the case where the species still grows near negative infinity, we show that the species always survives ‘by moving’ with the rightward spreading speed being either c*(∞) or c*(−∞) and the leftward spreading speed being one of c*(∞), c*(−∞) and −c, where c*(−∞) is the minimum KPP travelling wave speed corresponding to the growth rate at the negative infinity. Finally, we give some numeric simulations and discussions to present and explain the theoretical results. Our results indicate that there may exists a solution like a two-layer wave with the propagation speeds analytically determined for such type of nonlocal dispersal equations.

Metastable transitions in Langevin dynamics can exhibit rich behaviours that are markedly different from its overdamped limit. In addition to local alterations of the transition path geometry, more fundamental global changes may exist. For instance, when the dissipation is weak, heteroclinic connections that exist in the overdamped limit do not necessarily have a counterpart in the Langevin system, potentially leading to different transition rates. Furthermore, when the friction coefficient depends on the velocity, the overdamped limit no longer exists, but it is still possible to efficiently find instantons. The approach, we employed for these discoveries, was based on (i) a simple rewriting of the Freidlin–Wentzell action in terms of time-reversed dynamics and (ii) an adaptation of the string method, which was originally designed for gradient systems, to this specific non-gradient system.

Recently, the free boundary problem of quasistationary Stokes flow of a mass of viscous liquid under the action of surface tension forces has been considered by R. W. Hopper, L. K. Antanovskii, and others. The solution of the Stokes equations is represented by analytic functions, and a time dependent conformal mapping onto the flow domain is applied for the transformation of the problem to the unit disk. Two coupled Hilbert problems have to be solved there, which leads to a Fredholm boundary integral equation. The solution of this equation determines the time evolution of the conformal mapping. The question of the existence of a solution to this evolution problem for arbitrary (smooth) initial data has not yet been answered completely. In this paper, local existence in time is proved using a theorem of Ovsiannikov on Cauchy problems in an appropriate scale of Banach spaces. The necessary estimates are obtained in a way that is oriented at the a priori estimates for the solution given by Antanovskii. In the case of small deviations from the stationary solution represented by a circle, these a priori estimates, together with the local results, are used to prove even global existence of the solution in time.

In the focussing problem for the porous medium equation, one considers an initial distribution of material outside some compact set K. As time progresses material flows into K, and at some finite time T first covers all of K. For radially symmetric flows, with K a ball centred at the origin, it is known that the intermediate asymptotics of this focussing process is described by a family of self-similar solutions to the porous medium equation. Here we study the postfocussing regime and show that its onset is also described by self-similar solutions, even for nonsymmetric flows.

A mathematical model for the hydrodynamic flow and fibre deflections in a wet-spinning process is presented. The model is characterized by two parameters: α which is a characteristic ratio of the hydrodynamic force to the tension force acting on a fibre, and β which is a characteristic ratio of the fibre drag to the fluid inertia. The model is applied to three situations: (i) the spin bath of the process where α ≪ 1 and β ≫ 1, (ii) the preheat bath where α is order 1 and β ≫ 1, (iii) a fibre spreading process where α is order 1 and β is order 1 or smaller. In each case, analytical results are obtained for the fluid flow and fibre deflections and compared with experiment.

We study the annihilation dynamics arising in the KPP-Fisher equation, proposed by Fisher in 1936 to model the propagation of a mutant gene and subsequently studied rigorously in the seminal work of Kolmogorov, Petrovskii and Piskunov. The approach is via a comparison theorem, where the comparison functions satisfy equations which are linearizable to the heat equation. In some sense, we have obtained a ‘linearization’ of the KPP-Fisher equation.

In 2008, the Defense Advanced Research Project Agency commissioned a database known as the Integrated Crisis Early Warning System to serve as the foundation for models capable of detecting and predicting increases in political conflict worldwide. Such models, by signalling expected increases in political conflict, would help inform and prepare policymakers to react accordingly to conflict proliferation both domestically and internationally. Using data from the Integrated Crisis Early Warning System, we construct and test a self-exciting point process, or Hawkes process, model to describe and predict amounts of domestic, political conflict; we focus on Colombia and Venezuela as examples for this model. By comparing the accuracy of fitted models to the observed data, we find that we are able to closely describe occurrences of conflict in each country. Thus, using this model can allow policymakers to anticipate relative increases in the amount of domestic political conflict following major events.

Based on the fact that the incubation periods of epidemic disease in asymptomatically infected and infected individuals are inevitable and different, we propose a diffusive susceptible, asymptomatically infected, symptomatically infected and vaccinated (SAIV) epidemic model with delays in this paper. To see whether epidemic disease can propagate spatially with a constant speed, we focus on the travelling wave solution for this model. When the basic reproduction number of the corresponding spatial-homogenous delayed differential system is greater than one and the wave speed is greater than or equal to the critical speed, we prove that this model admits nontrivial positive travelling wave solutions. Our theoretical results are of benefit to the prevention and control of epidemic.

Some new global results are given about solutions to the boundary value problem for the Euler–Lagrange equations for the Ginzburg–Landau model of a one-dimensional superconductor. The main advance is a proof that in some parameter range there is a branch of asymmetric solutions connecting the branch of symmetric solutions to the normal state. Also, simplified proofs are presented for some local bifurcation results of Bolley and Helffer. These proofs require no detailed asymptotics for solution of the linear equations. Finally, an error in Odeh's work on this problem is discussed.

The standard scales that are used to non-dimensionalize the temperature- and time-dependent Ginzburg–Landau (TTDGL) model developed by Schmid [27], eliminate temperature- dependent parameters, and thus do not allow for superconducting phenomena due to variations in temperature. In this study, a set of new scales is presented to non-dimensionalize the TTDGL model so that the resulting dimensionless system depends upon a temperature parameter as well. Moreover, some properties of solutions to TTDGL system as a function of temperature are explored. Numerical experiments illustrating the temperature-dependency of vortex nucleation in type-II superconductors as well as the transition to the Meissner state in type-I superconductors are presented.

Whether enjoying the lucid prose of a favourite author or slogging through some other writer’s cumbersome, heavy-set prattle (full of parentheses, em dashes, compound adjectives, and Oxford commas), readers will notice stylistic signatures not only in word choice and grammar but also in punctuation itself. Indeed, visual sequences of punctuation from different authors produce marvellously different (and visually striking) sequences. Punctuation is a largely overlooked stylistic feature in stylometry, the quantitative analysis of written text. In this paper, we examine punctuation sequences in a corpus of literary documents and ask the following questions: Are the properties of such sequences a distinctive feature of different authors? Is it possible to distinguish literary genres based on their punctuation sequences? Do the punctuation styles of authors evolve over time? Are we on to something interesting in trying to do stylometry without words, or are we full of sound and fury (signifying nothing)?

In our investigation, we examine a large corpus of documents from Project Gutenberg (a digital library with many possible editorial influences). We extract punctuation sequences from each document in our corpus and record the number of words that separate punctuation marks. Using such information about punctuation-usage patterns, we attempt both author and genre recognition, and we also examine the evolution of punctuation usage over time. Our efforts at author recognition are particularly successful. Among the features that we consider, the one that seems to carry the most explanatory power is an empirical approximation of the joint probability of the successive occurrence of two punctuation marks. In our conclusions, we suggest several directions for future work, including the application of similar analyses for investigating translations and other types of categorical time series.

This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. First, we study the asymptotic behaviour of the system and prove that it eventually converges to a stationary state. Next, we more closely investigate the behaviour of the system in the presence of multiple EAs. Under suitable assumptions and based on a small mutation variance asymptotic, we describe the existence of a long transient regime during which the pathogen population remains far from its asymptotic behaviour and highly concentrated around some phenotypic value that is different from the one described by its asymptotic behaviour. In that setting, the time needed for the system to reach its large time configuration is very long and multiple evolutionary attractors may act as a barrier of evolution that can be very long to bypass.

We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

We consider a coating flow of nematic liquid crystal (NLC) fluid film on an inclined substrate. Exploiting the small aspect ratio in the geometry of interest, a fourth-order nonlinear partial differential equation is used to model the free surface evolution. Particular attention is paid to the interplay between the bulk elasticity and the anchoring conditions at the substrate and free surface. Previous results have shown that there exist two-dimensional travelling wave solutions that translate down the substrate. In contrast to the analogous Newtonian flow, such solutions may be unstable to streamwise perturbations. Extending well-known results for Newtonian flow, we analyse the stability of the front with respect to transverse perturbations. Using full numerical simulations, we validate the linear stability theory and present examples of downslope flow of nematic liquid crystal in the presence of both transverse and streamwise instabilities.

A cross-diffusion system modelling the information herding of individuals is analysed in a bounded domain with no-flux boundary conditions. The variables are the species' density and an influence function which modifies the information state of the individuals. The cross-diffusion term may stabilize or destabilize the system. Furthermore, it allows for a formal gradient-flow or entropy structure. Exploiting this structure, the global-in-time existence of weak solutions and the exponential decay to the constant steady state is proved in certain parameter regimes. This approach does not extend to all parameters. We investigate local bifurcations from homogeneous steady states analytically to determine whether this defines the validity boundary. This analysis shows that generically there is a gap in the parameter regime between the entropy approach validity and the first local bifurcation. Next, we use numerical continuation methods to track the bifurcating non-homogeneous steady states globally and to determine non-trivial stationary solutions related to herding behaviour. In summary, we find that the main boundaries in the parameter regime are given by the first local bifurcation point, the degeneracy of the diffusion matrix and a certain entropy decay validity condition. We study several parameter limits analytically as well as numerically, with a focus on the role of changing a linear damping parameter as well as a parameter controlling the cross-diffusion. We suggest that our paradigm of comparing bifurcation-generated obstructions to the parameter validity of global-functional methods could also be of relevance for many other models beyond the one studied here.

Consider a Hele-Shaw cell with the fluid (liquid) confined to an angular region by a solid boundary in the form of two half-lines meeting at an angle απ; if 0 < α < 1 we have flow in a corner, while if 1 < α [les ] 2 we have flow around a wedge. We suppose contact between the fluid and each of the lines forming the solid boundary to be along a single segment that does not adjoin the vertex, so we have air at the vertex, and contemplate such a situation that has been produced by injection at a number of points into an initially empty cell. We show that, if we assume the pressure to be constant along the free boundaries, the region occupied by the fluid is the image of a rectangle under a conformal map that can be expressed in terms of elliptic functions if α = 1 or α = 2, and in terms of theta functions if 0 < α < 1 or 1 < α < 2. The form of the function giving the map can be written down, and the parameters appearing in it then determined as the solution to a set of transcendental equations. The procedure is illustrated by a number of examples.