The asymptotic transectional/circumferential homogeneity of the solutions of reaction-diffusion systems in/on cylinder-like domains

It is often reported that an animal with spotty coat markings on its body has a tail with stripe-shaped pattern. In other various biological and chemical phenomena in/on cylinder-like domains, longitudinally periodic band patterns are observed much more often than the other non-uniform patterns. This paper mathematically explains these observations by proving that, in/on a long and narrow cylinder-like domain, any solution of reaction-diffusion system asymptotically loses its spatial dependence in the transectional/circumferential direction.     

Oscillatory reaction-diffusion equations on rings

We study the behavior of traveling waves in - systems on both homogeneous and inhomogeneous rings. The stability regions in parameter space of - waves were previously known [15, 19]; the results are extended here. We show the existence of Hopf bifurcations of traveling waves and the stability of the limit cycles born at the Hopf bifurcation for some parameter ranges. Using a Lindstedt-type perturbation scheme, we formally construct periodic solutions of the - system near a Hopf bifurcation and show that the periodic solutions superimposed on the original traveling wave have the effect of altering its overall frequency and amplitude. We also study the - system on an annulus ofvariable width, which does not possess reflection symmetry about any axis. We formally construct traveling waves on this variable-width annulus by a perturbation scheme, and find that perturbing the width of the annulus alters the amplitude and frequency of traveling waves on the domain by a small (order ²) amount. For typical parameter values, we find that the speed, frequency, and stability are unaffected by the direction of travel of the wave on the annulus, despite the rotationally asymmetric inhomogeneity. This indicates that the - system on a variable-width domain cannot account for directional preferences of traveling waves in biological systems.     

Some analytical results about a simple reaction-diffusion system for morphogenesis

Summary The reaction-diffusion system considered involves only one nonlinear term and is a gradient system. In a bifurcation analysis for the equilibrium states, the global existence of infinitely many solution branches can be shown by the method of Ljusternik-Schnirelmann. Their stability is studied. Using a Ljapunov functional it can be shown that the solutions of the time-dependent system converge to the equilibrium states.     

Algebraic structures generating reaction-diffusion models: The activator-substrate system

We shall construct a class of nonlinear reaction-diffusion equations starting from an infinitesimal algebraic skeleton. Our aim is to explore the possibility of an algebraic foundation of integrability properties and of stability of equilibrium states associated with nonlinear models describing patterns formation.     

Brucellosis, botflies, and brainworms: the impact of edge habitats on pathogen transmission and species extinction

Ecological interactions between species that prefer different habitat types but come into contact in edge regions at the interfaces between habitat types are modeled via reaction-diffusion systems. The primary sort of interaction described by the models is competition mediated by pathogen transmission. The models are somewhat novel because the spatial domains for the variables describing the population densities of the interacting species overlap but do not coincide. Conditions implying coexistence of the two species or the extinction of one species are derived. The conditions involve the principal eigenvalues of elliptic operators arising from linearizations of the model system around equilibria with only one species present. The conditions for persistence or extinction are made explicit in terms of the parameters of the system and the geometry of the underlying spatial domains via estimates of the principal eigenvalues. The implications of the models with respect to conservation and refuge design are discussed. Received: 10 June 1999 / Revised version: 7 July 2000 / Published online: 20 December 2000     

Reaction-diffusion approach to modeling of the spread of early tumors along linear or tubular structures

We consider a system composed of a tubular sheet of early tumor cells, occupying the surface of a structure existing in the organism. We assume that the cells have a potential for proliferation in response to a growth factor. This model can be thought of as representing an early stage (pre-in situ) of tumor evolution. A biomedical example of such process might be the atypical adenomatous hyperplasia in the lung. Destabilization of the equilibrium in such system represents initial invasion of cancer. We are looking for a transition from a slightly perturbed equilibrium state to uncontrolled and irregular growth. We examine a mathematical model of a population of cells distributed over a linear or tubular structure. Growth of cells is regulated by a growth factor, which can diffuse over the structure. Aside from this, production of cells and of the growth factor is governed by a pair of ordinary differential equations. Equation for the cell number follows from an accepted model of cell cycle. Equation for the bounded receptor particle number follows from a time-continuous Markov process. We demonstrate existence of the solutions of the complete model, using the method of invariant rectangles. We find conditions under which diffusion causes destabilization of the spatially homogeneous steady state, leading to exponential growth and apparently chaotic spatial patterns, following a period of almost constancy. This phenomenon may serve as a mathematical explanation of "unexpected" rapid growth and invasion of temporarily stable structures composed of cancer cells.     

食物链种群模型的格子Boltzmann方法模拟

用格子Boltzmann方法求解用反应一扩散方程组描述的食物链种群模型．我们用一维和二维方程组进行数值实验,模拟结果与现有的数值实验结果很好地吻合,反映了格子Boltzmann方法的高效性和稳定性,并就二维格子、Boltzmann格式,通过其等价的差分格式,由极值原理证明了该格式的稳定性．     

混合拟单调系统行波解的存在性

运用单调迭代方法,证明了混合拟单调系统的行波解的存在性．当反应扩散系。统的反应函数是混合拟单调函数时,如果选取一对合适的耦合上下解作为迭代初值,则迭代序列将收敛到一对拟解．而且在这对拟解之间存在系统的行波解．     

一类自催化反应扩散模型正解的唯一性与稳定性

主要考虑了一类三分子自催化反应扩散系统．在齐次Dirichlet和Robin边界条件下,当反应率c适当小,系统没有共存态;当c适当大,系统至少有一个共存态;当c充分大,系统有唯一渐近稳定的共存态．特别地,在一维空间上共存态是唯一的．在齐次Neumann边界条件下系统是一个简单系统．     

Modelling antibiotic- and anti-quorum sensing treatment of a spatially-structured Pseudomonas aeruginosa population

The bacterial cell to cell signalling system known as quorum sensing (QS) is essential for the regulation of virulence in many pathogens and offers a specific biochemical target for novel antibacterial therapies. Expanding on earlier work, in which consideration was given to the primary QS system (lasR system) in a homogeneous population of the common human pathogen Pseudomonas aeruginosa, we build a simple spatial model of an early-stage P. aeruginosa biofilm subject to treatment with topically applied anti-QS drugs (of two specific kinds) and conventional antibiotics. In the case of a slowly growing biofilm we show that both kinds of anti-quorum sensing drug are effective in reducing the level of the relevant signal molecule (3-oxo-C12-homoserine lactone; henceforth AHL), in each case obtaining an explicit bound on the steady-state AHL profile in terms of a prescribed surface drug concentration. Using numerical methods, we are also able to reproduce the hysteretic phenomena exhibited by the homogeneous model, in particular showing that for each kind of anti-QS drug there is a parameter regime in which a catastrophic collapse occurs in the steady-state AHL concentration as the surface drug concentration passes some critical value; an alternative way of interpreting this result is to say that, for a prescribed surface drug concentration, there is a critical biofilm depth such that treatment is successful until this depth is reached, but fails thereafter. In the thick-biofilm limit we show that the critical concentration of each drug increases exponentially with the biofilm thickness (or, conversely, that the critical depth increases logarithmically with surface drug concentration); this is dramatically different to the behaviour observed in the corresponding homogeneous model, where the critical concentrations grow linearly with bacterial carrying capacity, and thus highlights the relative difficulty of treating a large, spatially-structured population with diffusing antibacterials.