Turing instability in pioneer/climax species interactions

Systems of pioneer and climax species are used to model interactions of species whose reproductive capacity is sensitive to population density in their shared ecosystem. Intraspecies interaction coefficients can be adjusted so that spatially homogeneous solutions are stable to small perturbations. In a reaction-diffusion pioneer/climax model we will determine the critical value of the diffusion rate of the climax species, below which the equilibrium solution is unstable to non-homogeneous perturbations. For diffusion rates smaller than this critical value, an equilibrium solution remains stable to spatially homogeneous perturbations but is unstable to non-homogeneous perturbations. A Turing (diffusional) bifurcation leads to the formation of spatial patterns in species' densities. Forcing, interpreted as stocking or harvesting of the species, can reverse the bifurcation and establish equilibrium solutions which are stable to small perturbations. The implicit function theorem is used to determine whether stocking or harvesting of one of the species in the model is the appropriate remedy for diffusional instability. The use of stocking or harvesting by a natural resource manager thus influences the long-term dynamics and spatial distribution of species in a pioneer/climax ecosystem.     

Spreading speeds and traveling waves in competitive recursion systems

This paper is concerned with the spreading speeds and traveling wave solutions of discrete time recursion systems, which describe the spatial propagation mode of two competitive invaders. We first establish the existence of traveling wave solutions when the wave speed is larger than a given threshold. Furthermore, we prove that the threshold is the spreading speed of one species while the spreading speed of the other species is distinctly slower compared to the case when the interspecific competition disappears. Our results also show that the interspecific competition does affect the spread of both species so that the eventual population densities at the coexistence domain are lower than the case when the competition vanishes.     

Traveling wave solutions in a plant population model with a seed bank

We propose an integro-difference equation model to predict the spatial spread of a plant population with a seed bank. The formulation of the model consists of a nonmonotone convolution integral operator describing the recruitment and seed dispersal and a linear contraction operator addressing the effect of the seed bank. The recursion operator of the model is noncompact, which poses a challenge to establishing the existence of traveling wave solutions. We show that the model has a spreading speed, and prove that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions by using an asymptotic fixed point theorem. Our numerical simulations show that the seed bank has the stabilizing effect on the spatial patterns of traveling wave solutions.     

Dynamics of prostate cancer stem cells with diffusion and organism response

We develop a systems based model for prostate cancer, as a sub-system of the organism. We accomplish this in two stages. We first start with a general ODE that includes organism response terms. Then, to account for normally observed spatial diffusion of cell populations, the ODE is extended to a PDE that includes spatial terms. Numerical solutions of the full PDE are provided, and are indicative of traveling wave fronts. This motivates the use of a well known transformation to derive a canonically related (non-linear) system of ODEs for traveling wave solutions. For biological feasibility, we show that the non-negative cone for the traveling wave system is time invariant. We also prove that the traveling waves have a unique global attractor. Biologically, the global attractor would be the limit for the avascular tumor growth. We conclude with comments on clinical implications of the model.     

基于植被指数的海南岛霸王岭热带森林地上生物量空间分布模拟

热带森林在全球碳循环方面扮演着重要的角色, 预测其生物量分布可以加深对碳循环过程的理解。然而目前基于植被指数模拟技术进行热带森林生物量分布的研究报道较少。该文以海南岛霸王岭林区热带森林为研究对象, 在基于遥感影像和135个公里网格样地调查的基础上, 分别选取归一化差异植被指数(NDVI)、短红外湿度植被指数(MVI5)、中红外湿度植被指数(MVI7)和比值植被指数(RVI)与总物种生物量、顶极种生物量和先锋种生物量做相关分析, 并利用逐步线性回归分析分别构建了基于植被指数的生物量回归模型; 利用残差图对模型的有效性进行检验。结果表明, MVI7和MVI5与总物种和顶极种生物量关系显著, 而NDVI和RVI对先锋种生物量具有较好的指示作用; 总物种、顶极种和先锋种生物量预测精度较高的区域分别占总面积的69.24%、73.98%和88.08%, 表明3个生物量模型均具有较好的拟合精度; 模拟结果表明总物种和顶极种生物量主要集中于研究区中部、北部和西南部区域, 而先锋种生物量无明显的分布规律, 是不均衡地散布于整个研究区域, 反映了群落组成结构、干扰历史、地形及气候因素等的影响。     

Spatial stability of traveling wave solutions of a nerve conduction equation.

A simplified FitzHugh-Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency wmax. The slower one is found to be unstable and the faster one to be stable, while that at wmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973. Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.     

一个具有年龄结构的单种群离散反应扩散模型的波前解

利用上下解方法研究了一个具有年龄结构的单种群离散反应扩散模型波前解的存在性,并证明了存在具有临界波速的波前解.     

Pioneer exclusion in a one-hump discrete pioneer-climax competitive system

One-hump maps as single species pioneer and climax ecological models confirm the expected pioneer exclusion in an uninterrupted two dimensional discrete pioneer-climax competition model. Other dynamic situations occur in such models. In particular a mutual exclusion principle is presented as a mathematical theorem.     

海南岛霸王岭热带天然林景观中主要木本植物关键种的潜在分布

在较大的空间尺度上生态位模型是预测物种潜在分布的有效途径之一。为了探讨在热带天然林景观中木本植物(限于乔木和灌木)主要关键种的潜在分布,在对海南岛霸王岭的热带天然林进行按公里网格样方调查的基础上,采用演替地位和最大潜在高度两个功能性指标对物种进行了功能群划分,并在功能群框架下运用优势度指数法进行了关键种的确定;采用基于地理信息系统(Geographic information system, GIS)的基于规则集合预测的遗传算法(Algorithm for rule-set prediction, GARP)生态位模型对主要关键种的地理分布进行了预测,并应用受试者工作特征分析进行了模型精度验证;应用多元线性回归分析对影响各关键种潜在分布的关键因子进行了确定。结果表明:除了顶极次林层乔木功能群和顶极主林层乔木功能群外,在先锋种功能群、顶极灌木种功能群和顶极超冠层乔木功能群中采用优势度指数法划分出的关键种较为理想;一般来讲,在进行预测的8个关键种中,除了先锋主林层乔木种海南杨桐(Adinandra hainanensis),其它3个先锋种毛稔(Melastoma sanquiueum)、银柴(Aporosa chinensis)和枫香(Liquidambar formosana) 在研究区北部、西部以及西南部均具有较高的发生概率,而顶极种除了顶极超冠层乔木种南亚松(Pinus merkusii)外,九节(Psychotria rubra)、高脚罗伞(Ardisia quinquegona)和海南椎(Castanopsis hainanensis)具有相似的潜在分布格局,在研究区中部、东南部和南部地区具有较高的发生概率;相关分析表明极端最低温、年均温、极端最高温、年均降水量、海拔和坡向6大因子是影响研究区关键种潜在分布的关键因子;精度检验表明,GARP模型对8个关键种的潜在分布预测效果均较好,而其中又以银柴和海南椎的预测精度最高。     

High population differentiation and unusual haplotype structure in a shade-intolerant pioneer tree species, Zanthoxylum ailanthoides (Rutaceae) revealed by analysis of DNA polymorphism at four nuclear loci

Differences in demographic history, life-history traits, and breeding systems affect nucleotide variation patterns. It is expected that shade-intolerant pioneer tree species have different patterns of genetic polymorphism and population structure than climax species. We studied patterns of nucleotide polymorphism at four putative starch pathway loci (agpSA, agpSB, agpL, and GBSSI) in Zanthoxylum ailanthoides , a shade-intolerant pioneer tree species that occupies forest gaps in warm-temperate forests of East Asia. Genetic diversity was lower within each population than among populations, and differentiation among populations was high across the loci ( F _ST = 0.32–0.64), as expected from the insect-pollinated breeding system and the metapopulation structure of this pioneer species. Numbers of haplotypes were smaller than those expected from the observed numbers of segregating sites. Single haplotypes accounted for more than 47% of all the sampled genes at the respective loci. These variation patterns were incompatible with neutral predictions for populations of a finite island model. Complex population dynamics, such as bottleneck and/or admixture, in the history of this pioneer tree species might have resulted in the observed patterns of genetic variation and population structure, which are different from those of climax wind-pollinated tree species, such as conifers. In contrast to the other loci investigated in this study, agpL showed nearly no variation in Z. ailanthoides (one singleton only), but there was some extent of variation in a closely related species, Zanthoxylum schinifolium . This suggests possibly a recent selective sweep at or near the locus in Z. ailanthoides .     

Is there a trade-off between species diversity and genetic diversity in forest tree communities?

The two most important components of biodiversity, species diversity and genetic diversity, have generally been treated as separate topics, although a coordination between both components is believed to be critical for ecosystem stability and resilience. Based on a new trait concept that allows for the assessment of genetic diversity across species, the relationship between species diversity and genetic diversity was examined in eight forest tree communities composed of different tree genera including both climax and pioneer species. It was intended to check whether a trade-off exists between the two diversity components as was found in a few studies on animal species.Using several isozyme-gene systems as genetic markers, the genetic diversity across species within each of the tree communities was determined by two measures, the commonly used intraspecific genetic diversity averaged over species and the recently developed transspecific genetic diversity per species. Both data sets were compared with the corresponding community-specific species diversity resulting in a positive relationship between the two diversity components. A statistically significant positive correlation was established between the transspecific genetic diversity per species and the species diversity for three isozyme-gene systems. Beyond that, consistent results were obtained using different parameters of the diversity measure which characterize the total, the effective and the number of prevalent variants. The number of prevalent variants reflected most significantly the non-randomness of the observed diversity patterns.These findings can be explained by the observation that the pioneer tree species reveal a by far higher genetic diversity than the climax tree species, which means that an increase in species diversity, due to the addition of several pioneer species at the expense of one or two climax species, goes along with an increase in the level of genetic diversity. Forest tree communities with the highest degree of species diversity exhibit therefore the highest transspecific genetic diversity per species. This result was discussed with regard to the particular composition and stability of forest tree communities.     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

Refinement and Pattern Formation in Neural Circuits by the Interaction of Traveling Waves with Spike-Timing Dependent Plasticity

Traveling waves in the developing brain are a prominent source of highly correlated spiking activity that may instruct the refinement of neural circuits. A candidate mechanism for mediating such refinement is spike-timing dependent plasticity (STDP), which translates correlated activity patterns into changes in synaptic strength. To assess the potential of these phenomena to build useful structure in developing neural circuits, we examined the interaction of wave activity with STDP rules in simple, biologically plausible models of spiking neurons. We derive an expression for the synaptic strength dynamics showing that, by mapping the time dependence of STDP into spatial interactions, traveling waves can build periodic synaptic connectivity patterns into feedforward circuits with a broad class of experimentally observed STDP rules. The spatial scale of the connectivity patterns increases with wave speed and STDP time constants. We verify these results with simulations and demonstrate their robustness to likely sources of noise. We show how this pattern formation ability, which is analogous to solutions of reaction-diffusion systems that have been widely applied to biological pattern formation, can be harnessed to instruct the refinement of postsynaptic receptive fields. Our results hold for rich, complex wave patterns in two dimensions and over several orders of magnitude in wave speeds and STDP time constants, and they provide predictions that can be tested under existing experimental paradigms. Our model generalizes across brain areas and STDP rules, allowing broad application to the ubiquitous occurrence of traveling waves and to wave-like activity patterns induced by moving stimuli.     

Propagation of HBV with spatial dependence

A mathematical model is proposed to simulate the hepatitis B virus (HBV) infection with spatial dependence. The existence of traveling waves is established via the geometric singular perturbation method. Numerical simulations show that the model admits non-monotone traveling profiles. Influences of various parameters on the minimum wave speed are also discussed.     

The long-time behavior of the solution to a non-linear diffusion problem in population genetics and combustion

The Fisher (1937) or Kolmogoroff-Petrovsky-Piscounoff (1937) equation exemplifies wave-like phenomena occurring in population genetics and combustion. In an earlier paper, we proposed an extension of this equation and obtained closed form traveling wave, stationary, and “symmetric” solutions. Employing the transformation properties of the extended equation, two integral invariants for the problem are obtained and two Lyapunov functionals, which characterize the evolution of the profile to a uniformly propagating traveling wave, are constructed. A generalization of this modified Fisher equation is proposed and we obtain its integral invariants, traveling wave solutions and wave speeds, as well as the Lyapunov functionals which describe its asymptotic evolution.     

Traveling wave solutions of a nonlinear reaction–advection equation

 We establish the existence of traveling wave solutions for a nonlinear partial differential equation that models a logistically growing population whose movement is governed by an advective process. Conditions are presented for which traveling wave solutions exist and for which they are stable to small semi-finite domain perturbations. The wave is of mathematical interest because its behavior is determined by a singular differential equation and those with small speed of propagation steepen into a shock-like solutions. Finally, we indicate that the smoothing presence of diffusion allows wave persistence when an advective slow moving wave may collapse. Received: 24 November 1997 / Revised version: 13 July 1998     

A Traveling Wave Model for Invasion by Precursor and Differentiated Cells

We develop and investigate a continuum model for invasion of a domain by cells that migrate, proliferate and differentiate. The model is applicable to neural crest cell invasion in the developing enteric nervous system, but is presented in general terms and is of broader applicability. Two cell populations are identified and modeled explicitly; a population of precursor cells that migrate and proliferate, and a population of differentiated cells derived from the precursors which have impaired migration and proliferation. The equation describing the precursor cells is based on Fisher’s equation with the addition of a carrying-capacity limited differentiation term. Two variations of the proliferation term are considered and compared. For most parameter values, the model admits a traveling wave solution for each population, both traveling at the same speed. The traveling wave solutions are investigated using perturbation analysis, phase plane methods, and numerical techniques. Analytical and numerical results suggest the existence of two wavespeed selection regimes. Regions of the parameter space are characterized according to existence, shape, and speed of traveling wave solutions. Our observations may be used in conjunction with experimental results to identify key parameters determining the invasion speed for a particular biological system. Furthermore, our results may assist experimentalists in identifying the resource that is limiting proliferation of precursor cells.     

A power function for forest structure and regeneration pattern of pioneer and climax species in patch mosaic forests

The DBH-class distribution in natural deciduous broad-leaved forests was elucidated with a power function. A power function (y=ax ^b, y: stem density, x: represents DBH class, a and b: constants) fits the distribution better than an exponential function (y = a exp bx). The parameter b in the power function is approximately –2. This means that the natural forests studied have a patch-mosaic structure and that tree cohorts regenerate from gaps. Parameter a implies the number of juveniles, and b means size-dependent mortality. The value of –2 for parameter b means that when trees in a given DBH class double their DBH, the density of the size class should decrease by one-fourth. This phenomenon results from self-thinning and is caused by horizontal space competition among trees, called the `tile model'. The parameter describing DBH-class distribution for a forest with self-thinning patches should be approximately –2. I call this the `–2 power law' for DBH-class distribution. In a typical natural forest dominated by deciduous broadleaf tree species, trees are recognized as pioneer or climax species by the parameters describing their regeneration patterns. When I applied the power functional model to the DBH-class distribution of each dominant species, in pioneer species parameter a was high and b was less than –2 (markedly less than zero), suggesting that there are many juveniles, but mortality is high. On the other hand, in climax species parameter a was low value and the value of b was larger (negative, but closer to zero), suggesting that there are not many juveniles, but mortality is low. A power-function analysis of DBH-class distribution can be used to clarify the patch mosaic structure of a forest, and to clarify the regeneration pattern of pioneer and climax species by applying the function for each species.     

Are buffers boring? Uniqueness and asymptotical stability of traveling wave fronts in the buffered bistable system

Traveling waves of calcium are widely observed under the condition that the free cytosolic calcium is buffered. Thus it is of physiological interest to determine how buffers affect the properties of calcium waves. Here we summarise and extend previous results on the existence, uniqueness and stability of traveling wave solutions of the buffered bistable equation, which is the simplest possible model of the upstroke of a calcium wave. Taken together, the results show that immobile buffers do not change the existence, uniqueness or stability of the traveling wave, while mobile buffers can eliminate a traveling wave. However, if a wave exists in the latter case, it remains unique and stable.