Bifurcation analysis of nonlinear reaction-diffusion equations—II. Steady state solutions and comparison with numerical simulations

The steady state spatial patterns arising in nonlinear reaction-diffusion systems beyond an instability point of the thermodynamic branch are studied on a simple model network. A detailed comparison between the analytical solutions of the kinetic equations, obtained by bifurcation theory, and the results of computer simulations is presented for different boundary conditions. The characteristics of the dissipative structures are discussed and it is shown that the observed behavior depends strongly on both the boundary and initial conditions. The theoretical expressions are limited to the neighborhood of the marginal stability point. Computer simulations allow not only the verification of their predictions but also the investigation of the behavior of the system for larger deviations from the instability point. It is shown that new features such as multiplicity of solutions and secondary bifurcations can appear in this region.     

Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

This paper is concerned with the existence of travelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

The response of a spatially distributed neuron to white noise current injection

The depolarization of a passive nerve cylinder or dendritic tree in the equivalent cylinder representation is assumed to satisfy the cable equation. We consider in detail the effects of white noise current injection at a given location for the case of sealed end boundary conditions and for an initial resting state. The depolarization at a point is a Gaussian random process but is not Markovian. Expressions (infinite series) are obtained for the expectation, variance, spatial and temporal covariances of the depolarization. We examine the steady state expectation and variance and investigate how these are approached in time over the whole neuronal surface. We consider the relative contributions of various terms in the series for the expectation and variance of the depolarization at x=0 (soma, trigger zone, recording electrode) for various positions of the input process. It is found that different numbers of terms must be taken to obtain a reasonable approximation depending on whether the stimulus is at proximal, central or distal parts of the dendritic tree. We consider briefly the interspike time problem and see in an approximate way how spatial effects are important in determining the mean time between impulses.The research was supported by Canadian National Research Council Operating Grant No. A9259 and No. A4559. The authors are grateful to Howard James and Joan Lang who did most of the machine computation     

Pattern formation in generalized Turing systems

Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.Supported in part by NIH Grant # GM29123     

A model for resolving the plankton paradox: coexistence in open flows

1. Recent developments in the field of chaotic advection in hydrodynamical/environmental flows encourage us to revisit the population dynamics of competing species in open aquatic systems.
2. We assume that these species are in competition for a common limiting resource in open flows with chaotic advection dynamics. As an illustrative example, we consider a time periodic two-dimensional flow of viscous fluid (water) around a cylindrical obstacle.
3. Individuals accumulate along a fractal set in the wake of the cylinder, which acts as a catalyst for the biological reproduction process. While in homogeneous, well mixed environments only one species could survive this competition, coexistence of competitors is typical in our hydrodynamical system.
4. It is shown that a steady state sets in after sufficiently long times. In this state, the relative density of competitors is determined rather by the fractal nature of the spatial distribution of the advected species, and by their initial conditions, than by their competitive abilities. We argue that two factors, the strong chaotic mixing along a fractal set and the boundary layer around the obstacle, are responsible for the coexistence.     

Oscillatory viral dynamics in a delayed HIV pathogenesis model

We consider an HIV pathogenesis model incorporating antiretroviral therapy and HIV replication time. We investigate the existence and stability of equilibria, as well as Hopf bifurcations to sustained oscillations when drug efficacy is less than 100%. We derive sufficient conditions for the global asymptotic stability of the uninfected steady state. We show that time delay has no effect on the local asymptotic stability of the uninfected steady state, but can destabilize the infected steady state, leading to a Hopf bifurcation to periodic solutions in the realistic parameter ranges.     

Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension

We establish the existence of global classical solutions and non-trivial steady states of a one-dimensional attraction-repulsion chemotaxis model subject to the Neumann boundary conditions. The results are derived based on the method of energy estimates and the phase plane analysis.     

Numerical bifurcation analysis of the pattern formation in a cell based auxin transport model

Transport models of growth hormones can be used to reproduce the hormone accumulations that occur in plant organs. Mostly, these accumulation patterns are calculated using time step methods, even though only the resulting steady state patterns of the model are of interest. We examine the steady state solutions of the hormone transport model of Smith et al. (Proc Natl Acad Sci USA 103(5):1301–1306, 2006) for a one-dimensional row of plant cells. We search for the steady state solutions as a function of three of the model parameters by using numerical continuation methods and bifurcation analysis. These methods are more adequate for solving steady state problems than time step methods. We discuss a trivial solution where the concentrations of hormones are equal in all cells and examine its stability region. We identify two generic bifurcation scenarios through which the trivial solution loses its stability. The trivial solution becomes either a steady state pattern with regular spaced peaks or a pattern where the concentration is periodic in time.     

Siphons in Chemical Reaction Networks

Siphons in a chemical reaction system are subsets of the species that have the potential of being absent in a steady state. We present a characterization of minimal siphons in terms of primary decomposition of binomial ideals, we explore the underlying geometry, and we demonstrate the effective computation of siphons using computer algebra software. This leads to a new method for determining whether given initial concentrations allow for various boundary steady states.     

Periodic solutions: a robust numerical method for an S-I-R model of epidemics

 We describe and analyze a numerical method for an S-I-R type epidemic model. We prove that it is unconditionally convergent and that solutions it produces share many qualitative and quantitative properties of the solution of the differential problem being approximated. Finally, we establish explicit sufficient conditions for the unique endemic steady state of the system to be unstable and we use our numerical algorithm to approximate the solution in such cases and discover that it can be periodic, just as suggested by the instability of the endemic steady state. Received: 1 September 1995 / Revised version: 30 April 1997     

An avoidance learning situation. A neural net model

A simple avoidance situation is considered in terms of a neural net learning model. Data for the control situation can be represented by an expression having three parameters which determine the initial and the steady state activities together with the transient aspects. The introduction of a learning parameter then allows one to calculate satisfactorily the results obtained in the experimental situation in which shock is applied. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research Development Command under Grant No. AF AFOSR 370-63 and in part by the United States Public Health Service Grant RCA GM K6 18,420.     

On the time dependent diffusion of macromolecules through transient open junctions and their subendothelial spread. 2. Long time model for interaction between leakage sites

In Part 1 of this study (Weinbaum et al., 1988) a short time model has been proposed to describe the initial time dependent leakage of macromolecules at short distances (5 microns or less) from the exit of a transient open junction which the authors have hypothesized as a characteristic feature of endothelial cells in the process of turnover (Weinbaum et al., 1985). This open junction pathway has also been proposed (Weinbaum et al., 1988) to be the primary ultrastructural correlate of the 20 nm diameter large pore suggested by Renkin et al. (1977) using the predictions of cylindrical pore theory. The short time model in (Weinbaum et al., 1988), however, has major limitations in that it neglects the interaction between leakage sites, macromolecular entry through other pathways, the finite thickness of the vessel wall and the curvature of the cell perimeter. The longer time model developed herein will attempt to describe each of these features and also present an improved model and analytic solution for the steady state flux and uptake. In the previous steady state model developed by Weinbaum et al. (1985) the effect of the resistance of the transient open junctions and the non-isotropic diffusion in the underlying tissue due to the internal elastic lamina (IEL) were both neglected. New solutions are first presented which describe the effect of these important model refinements on the steady state macromolecular permeability of the major arteries. Time dependent solutions are then presented to predict the transient longer time labeling following the introduction of tracer macromolecules of varying size. These solutions and the corresponding short time solutions in Weinbaum et al. (1988) are the first solutions to our knowledge to describe the difficult time-dependent boundary value problem to determine how the channel exit concentration and flux at a leaky junction vary with time. This is accomplished by casting the boundary value problem in the form of an integral equation for the unknown flux at the cleft exit and then solving this problem using a specially designed numerical technique. The theoretical predictions are used to interpret the behavior of the localized leaks to HRP and albumin that have been reported in Stemerman et al. (1986) and our own recent experiments (Lin et al., 1988).     

带时滞互惠模型的稳定性和行波解的存在性

本文建立了一类空间非局部带时滞影响的互惠生物种群系统模型.前部分利用线性化方法证明了该模型的简单动力学行为,即证明了零平衡点和两个边界平衡点都是不稳定的,唯一的正平衡点是稳定的,同时还用Redlinger上下解方法得出了该模型的初边值问题存在唯一的正则解;后部分则证明了该反应扩散系统连接零平衡点和正平衡点的行波解的存在性.     

The Freter model: A simple model of biofilm formation

A simple, conceptual model of biofilm formation, due to R. Freter et al. (1983), is studied analytically and numerically in both CSTR and PFR. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid. One of these is stable for any particular set of parameter values and sharp and explicit conditions are given for the stability of each. The effects of adding an anti-microbial agent to the CSTR are examined.Supported by NSF Grant DMS 0107439 and UTA Grant REP 14748717Supported by NSF Grant DMS 0107160     

Time-dependent subpopulation induction in heterogeneous tumors

Time-dependent induction of clonal heterogeneity in the neoplastic micro-environment is analysed within the context of a competitive ecology. A model that describes a constant source for clonal emergence was analysed by Michelsonet al. (1987) as an extension of a model proposed by Jansson and Revesz (1974). The extended model has been termed the JRE Model. This paper extends these analyses to time-dependent emergence rates which may represent induction in the presence of a cytotoxic agent. If the analysis is constrained to the tumor micro-environment, and if the emergent subpopulation is drug resistant, then the model may describe the induction and emergence of drug resistant subclones in a growing neoplasm. Asymptotic closed form solutions are derived for a class of emergence rate functions which decay asymptotically to a constant mutation rate. This underlying mutation rate may represent spontaneous mutation to the resistant phenotype, and has been analysed stochastically (Coldmanet al., 1985). The asymptotic solutions to the time-dependent model approach the steady state solution for the JRE Model which represents the dynamics observed in the presence of a constant, spontaneous mutation rate. The clinical and biological implications of these results are discussed. Research support provided in part by Hungarian National Foundation for Scientific Research Grant No. 6032/6319 and ACS Grant IN45-Z and ACS PDT 243B.     

The role of initial inoculum on epidemic dynamics

Transient dynamics are important in many epidemics in agricultural and ecological systems that are prone to regular disturbance, cyclical and random perturbations. Here, using a simple host-pathogen model for a sessile host and a pathogen that can move by diffusion and advection, we use a range of mathematical techniques to examine the effect of initial spatial distribution of inoculum of the pathogen on the transient dynamics of the epidemic. We consider an isolated patch and a group of patches with different boundary conditions. We first determine bounds on the host population for the full model, then non-dimensionalizing the model allows us to obtain approximate solutions for the system. We identify two biologically intuitive groups of parameters to analyse transient behaviour using perturbation techniques. The first parameter group is a measure of the relative strength of initial primary to secondary infection. The second group is derived from the ratio of host removal rate (via infection) to pathogen removal rate (by decay and natural mortality) and measures the infectivity of initial inoculum on the system. By restricting the model to mimic primary infection only (in which all infections arise from initial inoculum), we obtain exact solutions and demonstrate how these depend on initial conditions, boundary conditions and model parameters. Finally, we suggest that the analyses on the balance of primary and secondary infection provide the epidemiologist with some simple rules to predict the transient behaviours.     

Pattern formation in an immobilized bienzyme system. A morphogenetic model.

Experimental and theoretical studies of a reaction-diffusion model of two immobilized enzymes participating in the cellular acid-base metabolism, namely glutaminase and urease, are presented. The system shows an unstable steady state at pH 6.0, where any perturbation will drive the system towards a more alkaline or more acidic pH, owing to the autocatalytic behaviour with respect to pH exhibited by both enzymes. When diffusion is coupled to reaction by means of immobilization, different patterns of the internal pH profile appear across the membrane. If the bienzymic membrane is subjected to a perturbation at its boundaries, of the same amplitude but in opposite directions, the internal pH evolves through an asymmetric pattern to attain a nearly symmetric distribution of pH. The pH value at the final steady state is more acidic or more alkaline than the initial state according to the initial and boundary conditions. The final nearly symmetric state is attained more rapidly when less enzyme is immobilized (1.8 x 10(-4) M.s-1 as against 3.3 x 10(-4) M.s-1 of total enzyme activity in the membrane volume). The experimental results agree rather well qualitatively with numerical predictions of the model equations.     

Global dynamics of a selection model for the growth of a population with genotypic fertility differences

A population growth model is considered for a one locus two allele problem with selection based entirely on fertility differences. A local stability analysis is carried out for the critical points — which include possible polymorphic states — of the resulting nonlinear differential equations. The methods of dynamical systems theory are applied to obtain limiting genotypic proportions for every initial state. Thus the results are global and there are no periodic solutions.Research for this paper was partially supported by the National Science and Engineering Research Council of Canada Grant NSERC A-8130Research for this paper was partially supported by the National Science and Engineering Research Council of Canada Grant NSERC A-4823Research supported by NSF Grant MCS 7901069. A portion of the work was carried out while the author was a Visiting Professor at the University of Utah, Salt Lake City, Utah     

The transient kinetics of uptake of galactosides into Escherichia coli.

The uptake of galactosides into Escherichia coli via the lactose permease was studied in the time range 0.01-10s by rapid mixing and quenched flow. An initial transient was observed under two conditions. Firstly, a lag in the approach to the steady state was observed at low galactoside concentrations (less than Km). Secondly, a burst of uptake was observed when anaerobic cell suspensions were mixed with aerobic substrate solutions. However, the cause of the burst of uptake appears to be a burst in the rate of respiration. The rate of galactoside uptake during this phase is 10-fold greater than during the steady state.     

A Simple Self-Maintaining Metabolic System: Robustness,Autocatalysis, Bistability

A living organism must not only organize itself from within; it must also maintain its organization in the face of changes in its environment and degradation of its components. We show here that a simple (M,R)-system consisting of three interlocking catalytic cycles, with every catalyst produced by the system itself, can both establish a non-trivial steady state and maintain this despite continuous loss of the catalysts by irreversible degradation. As long as at least one catalyst is present at a sufficient concentration in the initial state, the others can be produced and maintained. The system shows bistability, because if the amount of catalyst in the initial state is insufficient to reach the non-trivial steady state the system collapses to a trivial steady state in which all fluxes are zero. It is also robust, because if one catalyst is catastrophically lost when the system is in steady state it can recreate the same state. There are three elementary flux modes, but none of them is an enzyme-maintaining mode, the entire network being necessary to maintain the two catalysts.