Asymptotic behaviour of solutions of the diffusive Lotka-Volterra equations

A spatially discrete version of the diffusive Lotka-Volterra equations is considered. Asymptotical spatial homogeneity of solutions of the equations with equilibrium, periodic or zero flux boundary conditions is proved without regard to crowding effects. The proof does not require the assumption of equal diffusion coefficients and the restrictions on the dimension of space and on the initial data, which are necessary in the spatially continuous model.     

Pattern sensitivity to boundary and initial conditions in reaction-diffusion models

We consider Turing-type reaction-diffusion equations and study (via computer simulations) how the relationship between initial conditions and the asymptotic steady state solutions varies as a function of the boundary conditions. The results indicate that boundary conditions which are non-homogeneous with respect to the kinetic steady state give rise to spatial patterns which are much less sensitive to variations in the initial conditions than those obtained with homogeneous boundary conditions, such as zero flux conditions. We also compare linear pattern predictions with the numerical solutions of the full nonlinear problem.This work supported in part by U.S. Army Grant DAJA 37-81-C-0220 and the Science and Engineering Research Council of Great Britain Grant GR/c/63595     

Receptor-based models with hysteresis for pattern formation in hydra

In this paper, we propose a new receptor-based model for pattern formation and regulation in a fresh-water polyp, namely hydra. The model is defined in the form of a system of reaction-diffusion equations with zero-flux boundary conditions coupled with a system of ordinary differential equations. The production of diffusible biochemical molecules has a hysteretic dependence on the density of these molecules and is modeled by additional ordinary differential equations. We study the hysteresis-driven mechanism of pattern formation and we demonstrate the advantages and constraints of its ability to explain different aspects of pattern formation and regulation in hydra. The properties of the model demonstrate a range of stationary and oscillatory spatially heterogeneous patterns, arising from multiple spatially homogeneous steady states and switches in the production rates.     

Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel.

The synthetic channel [acetyl-(LeuSerSerLeuLeuSerLeu)3-CONH2]6 (pore diameter approximately 8 A, length approximately 30 A) is a bundle of six alpha-helices with blocked termini. This simple channel has complex properties, which are difficult to explain, even qualitatively, by traditional theories: its single-channel currents rectify in symmetrical solutions and its selectivity (defined by reversal potential) is a sensitive function of bathing solution. These complex properties can be fit quantitatively if the channel has fixed charge at its ends, forming a kind of macrodipole, bracketing a central charged region, and the shielding of the fixed charges is described by the Poisson-Nernst-Planck (PNP) equations. PNP fits current voltage relations measured in 15 solutions with an r.m.s. error of 3.6% using four adjustable parameters: the diffusion coefficients in the channel's pore DK = 2.1 x 10(-6) and DCl = 2.6 x 10(-7) cm2/s; and the fixed charge at the ends of the channel of +/- 0.12e (with unequal densities 0.71 M = 0.021e/A on the N-side and -1.9 M = -0.058e/A on the C-side). The fixed charge in the central region is 0.31e (with density P2 = 0.47 M = 0.014e/A). In contrast to traditional theories, PNP computes the electric field in the open channel from all of the charges in the system, by a rapid and accurate numerical procedure. In essence, PNP is a theory of the shielding of fixed (i.e., permanent) charge of the channel by mobile charge and by the ionic atmosphere in and near the channel's pore. The theory fits a wide range of data because the ionic contents and potential profile in the channel change significantly with experimental conditions, as they must, if the channel simultaneously satisfies the Poisson and Nernst-Planck equations and boundary conditions. Qualitatively speaking, the theory shows that small changes in the ionic atmosphere of the channel (i.e., shielding) make big changes in the potential profile and even bigger changes in flux, because potential is a sensitive function of charge and shielding, and flux is an exponential function of potential.     

Exact continuum solution for a channel that can be occupied by two ions

The classical Nernst-Planck continuum equation is extended to the case where the channel can be occupied simultaneously by two ions. A two-dimensional partial differential equation is derived to describe the steady-state channel. This differential equation is of the form of the generalized Laplace equation, but it has the novel feature that the boundary conditions are periodic. The finite difference solution takes approximately 8 s on a large computer. The equations are solved for the special case of a cylindrical channel with a fixed charge in the center. It is assumed that the forces on the ions result entirely from the sum of the Born image potential, the fixed charge potential, the interaction potential between the two ions, and the applied voltage. Approximate simple analytical expressions are derived for these potential terms, based on the assumption that the electric field perpendicular to the channel wall is zero. The potentials include the contribution from a diffuse charge (Debye-Huckel) reaction field in the bulk solution for the monovalent cation flux was obtained for channels with a radius of 4 A and lengths of 16 and 32 A and a fixed charge valence of -1 and -1.5. For these channels, a significant fraction (up to 90%) of the total resistance is contributed by the bulk solution and results were obtained for the case where the "channel" included 8 A of bulk solution at each channel end. These results for the two-ion channel were compared with the analytical solution for a one-ion channel. The one-ion channel is a fair approximation to the two-ion channel for a fixed charge of -1, underestimating the flux at high concentrations by approximately 30%. However, for a fixed charge of -1.5, the one-ion model is a poor approximation, with the two-ion flux about seven times that of the one-ion model at high concentrations. The absolute conductance and concentration dependence of these channels (with a fixed charge of -1) mimic the behavior of the large conductance K+ channel and the acetylcholine receptor channel.     

Dissipative structures in two dimensions.

Some further results on dissipative structures for one- and, more particularly, for two-dimensional bounded systems are presented. A model chemical network involving reactions and diffusion is investigated. The influence of the boundary conditions (no fluxes or fixed concentrations), the geometrical shape of the limits (circular or rectangular) and the size of the system on the variety of possible patterns is demonstrated. A comparison of the numerical results with bifurcation theory is outlined. Finally, the problem of multiplicity of stable ordered solutions which turns out to increase sharply from one to two dimensions, is discussed.     

Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems

The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an ‘intrinsic length’ which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.     

Current voltage curves of biomolecular lipid membranes.

The first part of this paper describes the current voltage curves of bimolecular membranes of oxidized cholesterol formed between two aqueous solutions of tetrabutylammonium chloride. These membranes are selectively permeable for cations and the membrane interfaces are electrically uncharged. The dependence of the membrane conductivity on the membrane potential can be described as the product of the conductivity at zero current ("zero conductivity") and a function called "overlinearity". The zero conductivity increases linearly with the concentration of tetrabutylammonium chloride. The overlinearity is independent of the concentration of tetrabutylammonium chloride. In the second part the Nernst-Planck and Poisson equations are integrated numerically for a three-phase system consisting of an aqueous electrolyte solution, a membrane and an aqueous electrolyte solution. Each phase is characterized by material constants. Appropriate boundary conditions cause the electric current to build up electrical double layers on both sides of the membrane. The opposing double layers with opposite electrical signs inject the soluble ions into the membrane. This ion injection accounts for the overlinearity of the current voltage curves, thus explaining the measured characteristics.     

Sedimentation Patterns of Rapidly Reversible Protein Interactions

The transport behavior of macromolecular mixtures with rapidly reversible complex formation is of great interest in the study of protein interactions by many different methods. Complicated transport patterns arise even for simple bimolecular reactions, when all species exhibit different migration velocities. Although partial differential equations are available to describe the spatial and temporal evolution of the interacting system given particular initial conditions, a general overview of the phase behavior of the systems in parameter space has not yet been reported. In the case of sedimentation of two-component mixtures, this study presents simple analytical solutions that solve the underlying equations in the diffusion-free limit previously subject to Gilbert-Jenkins theory. The new expressions describe, with high precision, the average sedimentation coefficients and composition of each boundary, which allow the examination of features of the whole parameter space at once, and may be used for experimental design and robust analysis of experimental boundary patterns to derive the stoichiometry and affinity of the complex. This study finds previously unrecognized features, including a phase transition between boundary patterns. The model reveals that the time-average velocities of all components in the reaction mixture must match—a condition that suggests an intuitive physical picture of an effective particle of the coupled cosedimentation of an interacting system. Adding to the existing numerical solutions of the relevant partial differential equations, the effective particle model provides physical insights into the relationships of the parameters that govern sedimentation patterns.     

Effects of dispersion on stability of multispecies prey-predator systems

The general multispecies prey-predator system with Gompertz's antisymmetric interactions is nonlinearly stable in the absence of dispersion and continues to remain stable with dispersion under both homogeneous reservoir and zero flux boundary conditions in a region containing the equilibrium state. It is proved that a general multispecies food-web model without antisymmetric interactions is stable in the absence of dispersion and remains stable with dispersion in the above-mentioned region.     

Density-dependent selection migration model with non-monotone fitness functions

This paper studies the classical single locus, diallelic selection model with diffusion for a continuously reproducing population. The phase variables are population density and allele frequency (or allele density). The genotype fitness depend only on population density but include one-hump functions of the density variable. With mild assumptions on genotype fitnesses, we study the geometry of the nullclines and the asymptotic behavior of solutions of the selection model without diffusion. For the diffusion model with zero Neumann boundary conditions, we use this geometric information to show that if the initial data satisfy certain conditions then the corresponding solution to the reaction-diffusion equation converges to the spatially constant stable equilibrium which is closest to the initial data.Research partially supported by NSF grant DMS-8920597Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Plasma equilibrium in axisymmetric poloidal magnetic field configurations in flux coordinates

A simple derivation is given of equilibrium equations in flux coordinates in the general case of an anisotropic-pressure plasma. The issue of how to formulate the boundary conditions for these equations is discussed for two types of configurations—a straight system and a system with an internal conductor. Examples of numerical solutions to the equilibrium problem for these configurations are presented.     

Bifurcations of nonlinear reaction-diffusion systems in prolate spheroids

Spontaneous pattern formation may arise in biological systems as primary and secondary bifurcations to nonlinear parabolic partial differential equations describing chemical reaction-diffusion systems subject to zero flux boundary conditions. Prepatterns are investigated, which arise in the three dimensional region of a prolate spheroid (elongated sphere). Pattern sequences and selection rules are established numerically. The results confirm previously recorded results of the spherical region upon which a prepattern theory of mitosis and cytokinesis is based. New results described here establish the emerging patterns as reliable prepatterns ensuring bipolarity during elongation of biological cells, as seen in anaphase of the process of mitosis.     

Properties of solution of the diffusion-reaction equation. II

Two theorems relating to properties of the solutions of the equations of continuity for the concentrations of the chemical species in a diffusion-reaction system are proved. The theorems concern boundary conditions under which the flux of a specified species can be guaranteed to be directed into the reaction region and the circumstances under which any two of the conditions (i) stationarity, (ii) flux equilibrium, and (iii) chemical equilibrium, imply the third. Application of these theorems to apparent active transport and to the properties of the differential equations for specific activities in a distributed tracer system are noted.     

An effective silencer design for artificially air conditioned environment

An effective silencer for an air conditioning duct is studied. A duct with an acoustically soft boundary is employed as an effective silencer. On the acoustically soft boundary the sound pressure is zero and it is impossible to realize such boundary in the air-borne sound field, because of the non-existence of a much lighter medium than the air. In this study, the arrangement of one-quarter wave-length acoustic tubes is employed as a soft boundary. This acoustic tube has frequency dependence, but the sound pressure becomes nearly zero at the tube mouth around the odd resonance frequency. The relation between the noise reduction efficiency and this acoustically soft boundary is examined experimentally and more than 40 dB noise reduction is obtained in a one-half octave band around the first resonance frequency. It is also made clear that more than one wave length of soft boundary is required to get enough reduction compared with the reduction obtained in the case of quite a long soft boundary.     

On a diffusive prey-predator model which exhibits patchiness

Spatial heterogeneity (patchiness) in certain predator-prey situations has been observed even though their environment appears homogeneous. As a model mechanism to explain this patchiness phenomenon we propose a predator-prey interaction system with diffusive effects. We show that when the diffusion of the prey is small compared with that of the predator the non-linearity which we call a hump effect in the prey interaction, is a key mechanism for the system to exhibit, asymptotically in time, stable heterogeneity in a bounded domain with zero flux boundary conditions. The model can reasonably be applied to certain terrestrial plant-herbivore systems.     

Characteristics of the flux dimension cue in apparent motion correspondence.

It has been shown that element flux and size (but not luminance) serve as correspondence cues in the apparent motion visual system. Results are now presented of a study of the characteristics of the flux cue. It was found that flux rather than luminance is used by the system even when the size of the elements is greater than the size limit of Ricco's law. There were interactions between the apparent motion processing of the size and flux dimensions, beyond the obvious dependence of flux on size: positively correlated size and flux differences between elements have a greater effect on correspondence than do negatively correlated differences. Finally, when comparing the fluxes of different elements, the apparent motion system uses relative flux (above or below background) rather than absolute flux (relative to zero).     

Spatial patterns in population conflicts

We consider a dynamical model for evolutionary games, and enquire how the introduction of diffusion may lead to the formation of stationary spatially inhomogeneous solutions, that is patterns. For the model equations being used it is already known that if there is an evolutionarily stable strategy (ESS), then it is stable. Equilibrium solutions which are not ESS's and which are stable with respect to spatially constant perturbations may be unstable for certain choices of the dispersal rates. We prove by a bifurcation technique that under appropriate conditions the instability leads to patterns. Computations using a curve-following technique show that the bifurcations exhibit a rich structure with loops joined by symmetry-breaking branches.     

Some problems of axisymmetric toroidal equilibria in orthogonal flux coordinates

Formulations of axisymmetric equilibrium problems are presented in orthogonal flux coordinates with boundary conditions that do not assume fixation of the boundary shape for both levitron-type traps with a plasma-embedded conductor and conductor-free configurations, including a tokamak. Illustrative examples of numerical solutions of these problems are presented. For traps with a conductor, it is demonstrated how the geometry of the equilibrium with an isodynamic magnetic surface can be found using flux coordinates.     

Ion acoustic cnoidal waves in a dusty plasma with a critical dust density

The propagation of nonlinear periodic ion acoustic waves in a dusty plasma is considered for conditions in which the coefficient in the nonlinear equation that describes the quadratic nonlinearity of the medium is zero. An equation that accounts for the cubic nonlinearity of the system is derived, and its solution is found. The dependence of the phase velocity of a cnoidal wave on its amplitude and modulus is determined. In describing the effect of higher order nonlinearities on the properties of a dust ion acoustic wave, two coupled equations for the first- and second-order potentials are obtained. It is shown that the nonlinear ion flux generated by a cnoidal wave propagating in a medium with a cubic nonlinearity is proportional to the fourth power of the wave amplitude.