Early development and quorum sensing in bacterial biofilms

 We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions. Received: 10 February 2002 / Revised version: 29 October 2002 / Published online: 19 March 2003 Key words or phrases: Bacterial biofilm – Quorum sensing – Mathematical modelling – Numerical solution – Asymptotic analysis – Travelling wave analysis     

Self-organization of large-scale ULF electromagnetic wave structures in their interaction with nonuniform zonal winds in the ionospheric E region

A study is made of the generation and subsequent linear and nonlinear evolution of ultralow-frequency planetary electromagnetic waves in the E region of a dissipative ionosphere in the presence of a nonuniform zonal wind (a sheared flow). Hall currents flowing in the E region and such permanent global factors as the spatial nonuniformity of the geomagnetic field and of the normal component of the Earth’s angular velocity give rise to fast and slow planetary-scale electromagnetic waves. The efficiency of the linear amplification of planetary electromagnetic waves in their interaction with a nonuniform zonal wind is analyzed. When there are sheared flows, the operators of linear problems are non-self-conjugate and the corresponding eigenfunctions are nonorthogonal, so the canonical modal approach is poorly suited for studying such motions and it is necessary to utilize the so-called nonmodal mathematical analysis. It is shown that, in the linear evolutionary stage, planetary electromagnetic waves efficiently extract energy from the sheared flow, thereby substantially increasing their amplitude and, accordingly, energy. The criterion for instability of a sheared flow in an ionospheric medium is derived. As the shear instability develops and the perturbation amplitude grows, a nonlinear self-localization mechanism comes into play and the process ends with the self-organization of nonlinear, highly localized, solitary vortex structures. The system thus acquires a new degree of freedom, thereby providing a new way for the perturbation to evolve in a medium with a sheared flow. Depending on the shape of the sheared flow velocity profile, nonlinear structures can be either purely monopole vortices or vortex streets against the background of the zonal wind. The accumulation of such vortices can lead to a strongly turbulent state in an ionospheric medium.     

A field theory of neural nets: II. Properties of the field equations

The field equation derived in Part I (Griffith,Bull. Math. Biophysics,25, 111–120, 1963a) is examined further. The stability of critical solutions is investigated and it is shown that, at least in certain cases, general solutions tend toward critical solutions. The relationship between the present field theory and a conventional matrix formulation is derived.     

Nonlinear Planetary Electromagnetic Vortex Structures in the Ionospheric F-Layer

A study is made of the dynamics of planetary-scale electromagnetic waves in the F-layer of the ionosphere. It is shown that, in this layer, a new branch of large-scale magneto-ionospheric wave perturbations is generated under the action of the latitudinal variations of the geomagnetic field, which are a constant property of the ionosphere. The waves propagate along the parallels with phase velocities of tens to hundreds of km/s. The pulsations of the geomagnetic field in the waves can be as strong as several tens of nT. A possible self-localization effect is revealed: these waves may form nonlinear localized solitary vortices moving either westward or eastward along the parallels with velocities much higher than the phase velocities of the linear waves. The characteristic dimension of a vortex is about 10⁴ km or even larger. The magnetic fields generated by vortex structures are one order of magnitude stronger than those in linear waves. The vortices are long-lived formations and may be regarded as elements of strong structural turbulence in the ionosphere. The properties of the wave structures under investigation are very similar to those of ultralow-frequency perturbations observed experimentally in the ionosphere at middle latitudes.     

Propagation of Turing patterns in a plankton model

The paper is devoted to a reaction-diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.     

Helical waves in the vortex electron anisotropic hydrodynamic model

Solutions to the vortex electron anisotropic hydrodynamic equations are investigated that describe nonlinear helical waves in an anisotropic magnetized plasma. The possibility of constructing such solutions is provided by the symmetry properties of the equations. An optimum family of one-dimensional subgroups of a symmetry group consistent with the equations is constructed that makes it possible to derive other, essentially different solutions.     

Rogue Waves: From Nonlinear Schr?dinger Breather Solutions to Sea-Keeping Test

Under suitable assumptions, the nonlinear dynamics of surface gravity waves can be modeled by the one-dimensional nonlinear Schrödinger equation. Besides traveling wave solutions like solitons, this model admits also breather solutions that are now considered as prototypes of rogue waves in ocean. We propose a novel technique to study the interaction between waves and ships/structures during extreme ocean conditions using such breather solutions. In particular, we discuss a state of the art sea-keeping test in a 90-meter long wave tank by creating a Peregrine breather solution hitting a scaled chemical tanker and we discuss its potential devastating effects on the ship.     

Propagation of Turing patterns in a plankton model

The paper is devoted to a reaction–diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.     

Effect of the shear flow in the generation and self-organization of internal gravity wave structures in the dissipative ionosphere

A linear mechanism for the generation and amplification of internal gravity waves and their further nonlinear dynamics in the stably stratified dissipative ionosphere in the presence of an inhomogeneous zonal wind (shear flow) is studied. For shear flows, the operators of linear problems are non-self-conjugate and the corresponding eigenfunctions are nonorthogonal. Therefore, the canonical modal approach is poorly applicable to study such motions. In this case, the so-called nonmodal mathematical analysis is more adequate. Dynamic equations and equations for the energy transport of internal gravity perturbations in the ionosphere with shear flows are derived on the basis of the nonmodal approach. Exact analytic solutions of linear and nonlinear equations are found. The growth rate of the shear instability of internal gravity waves is determined. It is revealed that perturbations grow in time according to a power law, rather than exponentially. The frequency and wavenumber of the generated internal gravity modes depend on time; hence, a wide spectrum of wave perturbations caused by linear effects (rather than nonlinear turbulent ones) forms in the ionosphere with shear flows. The efficiency of the linear mechanism for the amplification of internal gravity waves during their interaction with the inhomogeneous zonal wind is analyzed. A criterion for the development of the shear instability of such waves in the ionospheric plasma is obtained. It is shown that, in the presence of shear instability, internal gravity waves extract the shear flow energy in the initial (linear) stage of their evolution, due to which their amplitude and, accordingly, energy increase substantially (by an order of magnitude). As the amplitude increases, the mechanism of nonlinear self-localization comes into play and the process terminates with the self-organization of strongly localized solitary nonlinear internal gravity vortex structures. As a result, a new degree of freedom of the system and a new way of the evolution of perturbations in a medium with a shear flow appear. Inductive and viscous dampings limit the lifetime of vortex internal gravity structures in the ionosphere; nevertheless, their lifetime is long enough for them to strongly affect the dynamic properties of the medium. It is revealed on the basis of the analytic solution of a set of time-independent nonlinear dynamic equations that, depending on the velocity profile of the shear flow, the nonlinear internal gravity structures can take the form of a purely monopole vortex, a dipole cyclone-anticyclone pair, a transverse vortex chain, or a longitudinal vortex path against the background of the inhomogeneous zonal wind. The accumulation of such vortices in the ionosphere can result in a strongly turbulent state.     

Aggregative effects for a reaction-advection equation

The time course of a single population subject to logistic growth and drift towards regions of increasing population density is modelled by a quasilinear differential equation of the first order. The stationary solutions and the travelling waves are investigated. The existence of travelling waves with compact support is used to prove, among other properties, that populations initially concentrated in a finite region preserve this property for all future times.Visiting A. von Humboldt Fellow at the Universität Tübingen, Lehrstuhl für Biomathematik, Auf der Morgenstelle 28, D-7406 Tübingen 1     

A nonlinear treatment of the protocell model by a boundary layer approximation

The “protocell” is a mathematical model of a self-maintaining unity based on the dynamics of simple reaction-diffusion processes and a self-controlled dynamics of the surface. In this paper its spatio-temporal behaviour far from the stationary structure is investigated by means of a boundary layer approximation. It is shown in detail how a simplified and mathematically feasible equation can be derived from the original parabolic problem. It turns out that the known instability which is initiated in the linear region around the stationary structure is continued further in the direction to a division by nonlinear dynamics.     

Monotone travelling fronts in an age-structured reaction-diffusion model of a single species

 We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay. Received: 1 November 2001 / Revised version: 10 May 2002 / Published online: 23 August 2002 Mathematics Subject Classification (2000): 35K57, 92D25 Key words or phrases: Age-structure – Time-delay – Travelling Fronts – Reaction-diffusion     

Dynamic formation of oriented patches in chondrocyte cell cultures

Growth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. in J. Cell. Phys. 204:830–838, 2005). In particular, as chondrocytes approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al. in Physica D 89:346–367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.     

Traveling wave solutions from microscopic to macroscopic chemotaxis models

In this paper, we study the existence and nonexistence of traveling wave solutions for the one-dimensional microscopic and macroscopic chemotaxis models. The microscopic model is based on the velocity jump process of Othmer et al. (SIAM J Appl Math 57:1044–1081, 1997). The macroscopic model, which can be shown to be the parabolic limit of the microscopic model, is the classical Keller–Segel model, (Keller and Segel in J Theor Biol 30:225–234; 377–380, 1971). In both models, the chemosensitivity function is given by the derivative of a potential function, Φ(v), which must be unbounded below at some point for the existence of traveling wave solutions. Thus, we consider two examples: F(v) = lnv{\Phi(v) = \ln v} and F(v) = ln[v/(1-v)]{\Phi(v) = \ln[v/(1-v)]}. The mathematical problem reduces to proving the existence or nonexistence of solutions to a nonlinear boundary value problem with variable coefficient on \mathbb R{\mathbb R}. The main purpose of this paper is to identify the relationships between the two models through their traveling waves, from which we can observe how information are lost, retained, or created during the transition from the microscopic model to the macroscopic model. Moreover, the underlying biological implications of our results are discussed.     

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.     

Nonlinear dynamics of drift structures in a magnetized dissipative plasma

A study is made of the nonlinear dynamics of solitary vortex structures in an inhomogeneous magnetized dissipative plasma. A nonlinear transport equation for long-wavelength drift wave structures is derived with allowance for the nonuniformity of the plasma density and temperature equilibria, as well as the magnetic and collisional viscosity of the medium and its friction. The dynamic equation describes two types of nonlinearity: scalar (due to the temperature inhomogeneity) and vector (due to the convectively polarized motion of the particles of the medium). The equation is fourth order in the spatial derivatives, in contrast to the second-order Hasegawa-Mima equations. An analytic steady solution to the nonlinear equation is obtained that describes a new type of solitary dipole vortex. The nonlinear dynamic equation is integrated numerically. A new algorithm and a new finite difference scheme for solving the equation are proposed, and it is proved that the solution so obtained is unique. The equation is used to investigate how the initially steady dipole vortex constructed here behaves unsteadily under the action of the factors just mentioned. Numerical simulations revealed that the role of the vector nonlinearity is twofold: it helps the dispersion or the scalar nonlinearity (depending on their magnitude) to ensure the mutual equilibrium and, thereby, promote self-organization of the vortical structures. It is shown that dispersion breaks the initial dipole vortex into a set of tightly packed, smaller scale, less intense monopole vortices-alternating cyclones and anticyclones. When the dispersion of the evolving initial dipole vortex is weak, the scalar nonlinearity symmetrically breaks a cyclone-anticyclone pair into a cyclone and an anticyclone, which are independent of one another and have essentially the same intensity, shape, and size. The stronger the dispersion, the more anisotropic the process whereby the structures break: the anticyclone is more intense and localized, while the cyclone is less intense and has a larger size. In the course of further evolution, the cyclone persists for a relatively longer time, while the anticyclone breaks into small-scale vortices and dissipation hastens this process. It is found that the relaxation of the vortex by viscous dissipation differs in character from that by the frictional force. The time scale on which the vortex is damped depends strongly on its typical size: larger scale vortices are longer lived structures. It is shown that, as the instability develops, the initial vortex is amplified and the lifetime of the dipole pair components-cyclone and anticyclone-becomes longer. As time elapses, small-scale noise is generated in the system, and the spatial structure of the perturbation potential becomes irregular. The pattern of interaction of solitary vortex structures among themselves and with the medium shows that they can take part in strong drift turbulence and anomalous transport of heat and matter in an inhomogeneous magnetized plasma.     

Bifurcation analysis of an orientational aggregation model

 We consider an integro-differential equation for the evolution of a function f on the circle, describing an orientational aggregation process. In the first part we analyze generic bifurcations of steady-state solutions when a single eigenvalue changes sign. Lyapunov-Schmidt reduction leads to the bifurcation equation which is solved explicitly by formal power series. We prove that these series have positive radius of convergence. Two examples exhibit forward and backward bifurcations, respectively. In the second part we assume that the first and second eigenvalues become positive. Again we use Lyapunov-Schmidt reduction to arrive at the reduced bifurcation system from which we get the bifurcating branches as power series. We calculate the two most important parameters of the reduced system for two examples; one of them has interesting mode interactions which lead to various kinds of time-periodic solutions. Received: 23 April 2001 / Revised version: 29 October 2002 / Published online: 28 February 2003 Key words or phrases: Actin – Cytoskeleton – Orientational Aggregation – Bifurcation Analysis – Mode Interaction – Power Series Expansion     

A Nonlinear Stability Analysis of Vegetative Turing Pattern Formation for an Interaction–Diffusion Plant-Surface Water Model System in an Arid Flat Environment

The development of spontaneous stationary vegetative patterns in an arid flat environment is investigated by means of a weakly nonlinear diffusive instability analysis applied to the appropriate model system for this phenomenon. In particular, that process can be modeled by a partial differential interaction–diffusion equation system for the plant biomass density and the surface water content defined on an unbounded flat spatial domain. The main results of this analysis can be represented by closed-form plots in the rate of precipitation versus the specific rate of plant density loss parameter space. From these plots, regions corresponding to bare ground and vegetative patterns consisting of parallel stripes, labyrinth-like mazes, hexagonal arrays of gaps, irregular mosaics, and homogeneous distributions of vegetation, respectively, may be identified in this parameter space. Then those theoretical predictions are compared with both relevant observational evidence involving tiger and pearled bush patterns and existing numerical simulations of similar model systems as well as placed in the context of the results from some recent nonlinear vegetative pattern formation studies.     

Generation of infrasonic waves by low-frequency dust acoustic perturbations in the Earth’s lower ionosphere

It is shown that, during Perseid, Geminid, Orionid, and Leonid meteor showers, the excitation of low-frequency dust acoustic perturbations by modulational instability in the Earth’s ionosphere can lead to the generation of infrasonic waves. The processes accompanying the propagation of these waves are considered, and the possibility of observing the waves from the Earth’s surface is discussed, as well as the possible onset of acoustic gravitational vortex structures in the region of dust acoustic perturbations. The generation of such structures during Perseid, Geminid, Orionid, and Leonid meteor showers can show up as an increase in the intensity of green nightglow by an amount on the order of 10% and can be attributed to the formation of nonlinear (vortex) structures at altitudes of 110–120 km.