Slow nonlinear waves in magnetic flux tubes

A theory of weakly nonlinear slow waves in magnetic flux tubes is developed in the ideal MHD approximation. Fairly simple approximate dispersion relations are derived that are valid for waves of arbitrary wavelength. These dispersion relations make it possible to obtain a number of new model evolutionary equations for body and surface slow waves in magnetic flux tubes. It is established that there are two families of exact analytic solutions to the equations for weakly nonlinear slow waves. It is found that both the body and surface solitary waves can be in the form of either contractions or bulges running along the tube. A model Korteweg-de Vries-Burgers equation is derived and generalized to waves of arbitrary wavelength. It is shown that exact analytic solutions to these equations correspond to shock waves and hydraulic jumps (or bores) with nonoscillating fronts.     

Quantum theory of Cherenkov beam instabilities in plasma

A quantum theory of stimulated Cherenkov emission of longitudinal waves by an electron beam in an isotropic plasma is presented. The emitted radiation is interpreted as instability due to the decay of the de Broglie wave of a beam electron. Nonrelativistic and relativistic nonlinear quantum equations for Cherenkov beam instabilities are obtained. A linear approximation is used to derive quantum dispersion relations and to determine the instability growth rates. The mechanisms for nonlinear saturation of quantum Cherenkov beam instabilities are investigated, and the corresponding analytic solutions are found.     

Algal growth dynamics under light interaction: A nonlinear evolution problem

A mathematical model for interaction of algae with light is presented. This model consists of a couple of nonlinear integro-differential equations dealing with the evolution of algae concentration due to absorption of light and the evolution of light intensity due to absorption and scattering by algae and water. The model is formulated as a nonlinear abstract Cauchy problem in a Banach space suited to describe the physical features of the problem. By using techniques and results of semigroups theory, it is proved that the system has a unique global strict solution.Work performed under the auspices of G.N.F.M. (Gruppo Nazionale per la Fisica Matematica)     

Bifurcation analysis of nonlinear reaction-diffusion equations—I. Evolution equations and the steady state solutions

A model nonlinear network involving chemical reactions and diffusion is studied. The time evolution and bounds on the steady state solutions are analyzed. Spatially ordered solutions of the equations of the dissipative structure type are found by bifurcation theory. These solutions are calculated analytically and their qualitative properties are discussed.     

Stability analysis of a stochastic model for biomolecular selection

The technique of the probability generating function is used to derive the stochastic differential equations for a nonlinear model based on Eigen and Schuster's theory of biomolecular selection and evolution. The stabilities of various steady states are analyzed by using the linear stability approximation. The instability of a small starting population is investigated numerically. The minimum starting populations required for steady-state survival are then estimated for a wide range of parameters.     

Nonlinear ion acoustic waves in a quantum degenerate warm plasma with dust grains

A study is made of the propagation of ion acoustic waves in a collisionless unmagnetized dusty plasma containing degenerate ion and electron gases at nonzero temperatures. In linear theory, a dispersion relation for isothermal ion acoustic waves is derived and an exact expression for the linear ion acoustic velocity is obtained. The dependence of the linear ion acoustic velocity on the dust density in a plasma is calculated. An analysis of the dispersion relation reveals parameter ranges in which the problem has soliton solutions. In nonlinear theory, an exact solution to the basic equations is found and examined. The analysis is carried out by Bernoulli’s pseudopotential method. The ranges of the phase velocities of periodic ion acoustic waves and the velocities of solitons are determined. It is shown that these ranges do not overlap and that the soliton velocity cannot be lower than the linear ion acoustic velocity. The profiles of the physical quantities in a periodic wave and in a soliton are evaluated, as well as the dependence of the critical velocity of solitons on the dust density in a plasma.     

Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching

We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.     

Self-Organization of Microcircuits in Networks of Spiking Neurons with Plastic Synapses

The synaptic connectivity of cortical networks features an overrepresentation of certain wiring motifs compared to simple random-network models. This structure is shaped, in part, by synaptic plasticity that promotes or suppresses connections between neurons depending on their joint spiking activity. Frequently, theoretical studies focus on how feedforward inputs drive plasticity to create this network structure. We study the complementary scenario of self-organized structure in a recurrent network, with spike timing-dependent plasticity driven by spontaneous dynamics. We develop a self-consistent theory for the evolution of network structure by combining fast spiking covariance with a slow evolution of synaptic weights. Through a finite-size expansion of network dynamics we obtain a low-dimensional set of nonlinear differential equations for the evolution of two-synapse connectivity motifs. With this theory in hand, we explore how the form of the plasticity rule drives the evolution of microcircuits in cortical networks. When potentiation and depression are in approximate balance, synaptic dynamics depend on weighted divergent, convergent, and chain motifs. For additive, Hebbian STDP these motif interactions create instabilities in synaptic dynamics that either promote or suppress the initial network structure. Our work provides a consistent theoretical framework for studying how spiking activity in recurrent networks interacts with synaptic plasticity to determine network structure.     

Mathematical models in genetics

In this study, we present some of the basic ideas of population genetics. The founders of population genetics are R.A. Fisher, S. Wright, and J. B.S. Haldane. They, not only developed almost all the basic theory associated with genetics, but they also initiated multiple experiments in support of their theories. One of the first significant insights, which are a result of the Hardy–Weinberg law, is Mendelian inheritance preserves genetic variation on which the natural selection acts. We will limit to simple models formulated in terms of differential equations. Some of those differential equations are nonlinear and thus emphasize issues such as the stability of the fixed points and time scales on which those equations operate. First, we consider the classic case when selection acts on diploid locus at which wу can get arbitrary number of alleles. Then, we consider summaries that include recombination and selection at multiple loci. Also, we discuss the evolution of quantitative traits. In this case, the theory is formulated in respect of directly measurable quantities. Special cases of this theory have been successfully used for many decades in plants and animals breeding.     

Triphasic mixture model of cell-mediated enzymatic degradation of hydrogels

One critical component of engineering living tissue equivalents is the design scaffolds (often made of hydrogels) whose degradation kinetics can match that of matrix production by cells. However, cell-mediated enzymatic degradation of a hydrogel is a highly complex and nonlinear process that is challenging to comprehend based solely on experimental observations. To address this issue, this study presents a triphasic mixture model of the enzyme-hydrogel system, which consists of a solid polymer network, water and enzyme. On the basis mixture theory, the rubber elasticity theory and the Michaelis-Menton kinetics for degradation, the model naturally incorporates a strong coupling between gel mechanical properties, the kinetics of degradation and the transport of enzyme through the gel. The model is then used to investigate the particular problem of a single spherical enzyme-producing cell, embedded in a spherical hydrogel domain, for which the governing equations can be cast within the cento-symmetric assumptions. The governing equations are subsequently solved using an implicit nonlinear finite element procedure to obtain the evolution of enzyme concentration and gel degradation through time and space. The model shows that two regimes of degradation behaviour exist, whereby degradation is dominated either by diffusion or dominated by reaction kinetics. Depending on the enzyme properties and the initial hydrogel design, the temporal and spatial changes in gel cross-linking are dramatically impacted, a feature that is likely to strongly affect new tissue development.     

Triphasic mixture model of cell-mediated enzymatic degradation of hydrogels

One critical component of engineering living tissue equivalents is the design scaffolds (often made of hydrogels) whose degradation kinetics can match that of matrix production by cells. However, cell-mediated enzymatic degradation of a hydrogel is a highly complex and nonlinear process that is challenging to comprehend based solely on experimental observations. To address this issue, this study presents a triphasic mixture model of the enzyme–hydrogel system, which consists of a solid polymer network, water and enzyme. On the basis mixture theory, the rubber elasticity theory and the Michaelis–Menton kinetics for degradation, the model naturally incorporates a strong coupling between gel mechanical properties, the kinetics of degradation and the transport of enzyme through the gel. The model is then used to investigate the particular problem of a single spherical enzyme-producing cell, embedded in a spherical hydrogel domain, for which the governing equations can be cast within the cento-symmetric assumptions. The governing equations are subsequently solved using an implicit nonlinear finite element procedure to obtain the evolution of enzyme concentration and gel degradation through time and space. The model shows that two regimes of degradation behaviour exist, whereby degradation is dominated either by diffusion or dominated by reaction kinetics. Depending on the enzyme properties and the initial hydrogel design, the temporal and spatial changes in gel cross-linking are dramatically impacted, a feature that is likely to strongly affect new tissue development.     

Nonlinear theory of ion-acoustic waves in an ideal plasma with degenerate electrons

A nonlinear theory is constructed that describes steady-state ion-acoustic waves in an ideal plasma in which the electron component is a degenerate Fermi gas and the ion component is a classical gas. The parameter ranges in which such a plasma can exist are determined, and dispersion relations for ion-acoustic waves are obtained that make it possible to find the linear ion-acoustic velocity. Analytic gas-dynamic models of ion sound are developed for a plasma with the ion component as a cold, an isothermal, or an adiabatic gas, and moreover, the solutions to the equations of all the models are brought to a quadrature form. Profiles of a subsonic periodic and a supersonic solitary wave are calculated, and the upper critical Mach numbers of a solitary wave are determined. For a plasma with cold ions, the critical Mach number is expressed by an explicit exact formula.     

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.     

Nonlinear theory of the collective Cherenkov interaction of a dense relativistic electron beam with a dense plasma in a waveguide

A nonlinear theory of the instability of a straight relativistic dense electron beam in a plasma waveguide is derived for conditions of the stimulated collective Cherenkov effect. A study is made of a waveguide with a dense plasma such that the plasma wave excited by the beam during the instability can be escribed, with a good degree of accuracy, as a potential wave. General relativistic nonlinear equations are btained that describe the temporal dynamics of beam-plasma instabilities with allowance for plasma nonlinearity and the generation of harmonics of the initial perturbation. Under the assumption that the resonant interaction between the beam waves and the plasma waves is weak, the general equations are reduced to relativistic equations with cubic nonlinearities by using the method of expansion in small perturbations of the trajectories and momenta of the beam and plasma electrons. The reduced equations are solved analytically, the time scales on which the instability saturates are determined, and the nonlinear saturation amplitudes are obtained. A comparison between analytical solutions to the reduced equations and numerical solutions to the general nonlinear equations shows them to be in good agreement. Nonlinear processes caused by the relativistic nature of the beam are found to prevent stochastization of the system in the nonlinear stage of the well-developed instability. In contrast, a nonrelativistic electron beam is found to be subject to significant anomalous nonlinear stochastization.     

Structured population dynamics: continuous size and discontinuous stage structures

A nonlinear stochastic model for the dynamics of a population with either a continuous size structure or a discontinuous stage structure is formulated in the Eulerian formalism. It takes into account dispersion effects due to stochastic variability of the development process of the individuals. The discrete equations of the numerical approximation are derived, and an analysis of the existence and stability of the equilibrium states is performed. An application to a copepod population is illustrated; numerical results of Eulerian and Lagrangian models are compared.      

Spatial and spatio-temporal patterns in a cell-haptotaxis model

We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.     

Determination of sedimentation coefficients for small peptides.

Direct fitting of sedimentation velocity data with numerical solutions of the Lamm equations has been exploited to obtain sedimentation coefficients for single solutes under conditions where solvent and solution plateaus are either not available or are transient. The calculated evolution was initialized with the first experimental scan and nonlinear regression was employed to obtain best-fit values for the sedimentation and diffusion coefficients. General properties of the Lamm equations as data analysis tools were examined. This method was applied to study a set of small peptides containing amphipathic heptad repeats with the general structure Ac-YS-(AKEAAKE)nGAR-NH2, n = 2, 3, or 4. Sedimentation velocity analysis indicated single sedimenting species with sedimentation coefficients (s(20,w) values) of 0.37, 0.45, and 0.52 S, respectively, in good agreement with sedimentation coefficients predicted by hydrodynamic theory. The described approach can be applied to synthetic boundary and conventional loading experiments, and can be extended to analyze sedimentation data for both large and small macromolecules in order to define shape, heterogeneity, and state of association.     

From stochastic environments to life histories and back

Environmental stochasticity is known to play an important role in life-history evolution, but most general theory assumes a constant environment. In this paper, we examine life-history evolution in a variable environment, by decomposing average individual fitness (measured by the long-run stochastic growth rate) into contributions from average vital rates and their temporal variation. We examine how generation time, demographic dispersion (measured by the dispersion of reproductive events across the lifespan), demographic resilience (measured by damping time), within-year variances in vital rates, within-year correlations between vital rates and between-year correlations in vital rates combine to determine average individual fitness of stylized life histories. In a fluctuating environment, we show that there is often a range of cohort generation times at which the fitness is at a maximum. Thus, we expect ‘optimal’ phenotypes in fluctuating environments to differ from optimal phenotypes in constant environments. We show that stochastic growth rates are strongly affected by demographic dispersion, even when deterministic growth rates are not, and that demographic dispersion also determines the response of life-history-specific average fitness to within- and between-year correlations. Serial correlations can have a strong effect on fitness, and, depending on the structure of the life history, may act to increase or decrease fitness. The approach we outline takes a useful first step in developing general life-history theory for non-constant environments.