Bifurcation of non-negative solutions for an elliptic system

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bifurcation词源

分岔的词源英文版The word "bifurcation" is derived from the Latin word "bifurcus," which means "having two forks." It is a term that is used to describe the process of dividing into two branches or parts. The word "bifurcation" is often used in the context of mathematics, physics, and biology.In mathematics, bifurcation is a term that is used to describe the occurrence of two solutions to a problem where there was previously only one. This can happen when a parameter in the problem is changed. For example, in the case of the quadratic equation y = ax^2 + bx + c, the discriminant b^2 - 4ac determines the number of solutions to the equation. If the discriminant is positive, there are two real solutions. If the discriminant is zero, there is one real solution. And if the discriminant is negative, there are no real solutions.In physics, bifurcation is often used to describe the behavior of systems that undergo phase transitions. For example, when a fluid undergoes a phase transition from a liquid to a gas, the fluid can either condense into a single drop or it can split into two or more drops. This behavior is known as a bifurcation.In biology, bifurcation is often used to describe the branching of organisms. For example, the branching of trees is a type of bifurcation. Bifurcation is also used to describe the branching of nerves and blood vessels.The word "bifurcation" is a versatile term that can be used to describe a wide variety of phenomena. It is a term that is used in a variety of fields, including mathematics, physics, and biology.中文版分岔的词源“分岔”一词源自拉丁语“bifurcus”，意为“有两个叉子”。

Hopf bifurcation

Hopf bifurcationFrom Wikipedia, the free encyclopedia (Redirected from Andronov-Hopf bifurcation )Jump to: navigation , search In the mathematical theory of bifurcations , a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf , and Aleksandr Andronov , is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane . Under reasonably generic assumptions about the dynamical system, we can expect to see a small-amplitude limit cycle branching from the fixed point.For a more general survey on Hopf bifurcation and dynamical systems in general, see [1][2][3][4][5].Contents[hide ]● 1 Overview r 1.1 Supercritical / subcritical Hopf bifurcationsr 1.2 Remarks r1.3 Example ● 2 Definition of a Hopf bifurcation ● 3 Routh–Hurwitz criterionr 3.1 Sturm seriesr 3.2 Propositions ● 4 Example● 5 References●6 External links [edit ] Overview[edit ] Supercritical / subcritical Hopf bifurcationsThe limit cycle is orbitally stable if a certain quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it isunstable and the bifurcation is subcritical.The normal form of a Hopf bifurcation is:where z , b are both complex and λ is a parameter. WriteThe number α is called the first Lyapunov coefficient.●If α is negative then there is a stable limit cycle for λ > 0:whereThe bifurcation is then called supercritical.●If α is positive then there is an unstable limit cycle for λ < 0. The bifurcation is called subcritical.[edit ] Remarks The "smallest chemical reaction with Hopf bifurcation" was found in 1995 in Berlin, Germany [6]. The same biochemical system has been used in order to investigate how the existence of a Hopf bifurcation influences our ability to reverse-engineer dynamical systems [7].Under some general hypothesis, in the neighborhood of a Hopf bifurcation, a stable steady point of the system gives birth to a small stable limit cycle . Remark that looking for Hopf bifurcation is not equivalent to looking for stable limit cycles. First, some Hopf bifurcations (e.g. subcritical ones) do not imply the existence of stable limit cycles; second, there may exist limit cycles not related to Hopf bifurcations.[edit ] ExampleThe Hopf bifurcation in the Selkov system(see article). As the parameters change, a limitcycle (in blue) appears out of an unstableequilibrium.Hopf bifurcations occur in the Hodgkin–Huxley model for nerve membrane, the Selkov model of glycolysis , the Belousov–Zhabotinsky reaction , the Lorenz attractor and in the following simpler chemical system called the Brusselator as the parameter B changes:The Selkov model isThe phase portrait illustrating the Hopf bifurcation in the Selkov model is shown on the right. See Strogatz, Steven H. (1994). "Nonlinear Dynamics and Chaos" [1], page 205 for detailed derivation.[edit ] Definition of a Hopf bifurcationThe appearance or the disappearance of a periodic orbit through a local change in the stability properties of a steady point is known as the Hopf bifurcation. The following theorem works with steady points with one pair of conjugate nonzero purely imaginary eigenvalues . It tells the conditions under which this bifurcation phenomenon occurs.Theorem (see section 11.2 of [3]). Let J 0 be the Jacobian of a continuous parametric dynamical system evaluated at a steady point Z eof it. Suppose that all eigenvalues of J 0 have negative real parts except one conjugate nonzero purely imaginary pair. A Hopf bifurcation arises when these two eigenvalues cross the imaginary axis because of a variation of the system parameters.[edit ] Routh–Hurwitz criterionRouth–Hurwitz criterion (section I.13 of [5]) gives necessary conditions so that a Hopf bifurcation occurs. Let us see how one can use concretely this idea [8].[edit ] Sturm series Let be Sturm series associated to a characteristic polynomial P . They can be written in the form:The coefficients c i,0 for i in correspond to what is called Hurwitz determinants [8]. Their definition is related to the associated Hurwitz matrix .[edit ] PropositionsProposition 1. If all the Hurwitz determinants c i ,0 are positive, apart perhaps c k,0 then the associated Jacobian has no pure imaginary eigenvalues.Proposition 2. If all Hurwitz determinants c i ,0 (for all i in are positive, c k " 1,0 = 0 and c k" 2,1 < 0 then all the eigenvalues of the associated Jacobian have negative real parts except a purely imaginary conjugate pair.The conditions that we are looking for so that a Hopf bifurcation occurs (see theorem above) for a parametric continuous dynamical system are given by this last proposition.[edit ] Example Let us consider the classical Van der Pol oscillator written with ordinary differential equations:The Jacobian matrix associated to this system follows:The characteristic polynomial (in λ) of the linearization at (0,0) is equal to:P (λ) = λ2 " μλ + 1.The coefficients are: a 0 = 1,a 1 = " μ,a 2 = 1 The associated Sturm series is:The Sturm polynomials can be written as (here i = 0,1):The above proposition 2 tells that one must have:c 0,0 = 1 > 0,c 1,0 = " μ = 0,c 0,1 = " 1 < 0.Because 1 > 0 and 1 < 0 are obvious, one can conclude that a Hopf bifurcation may occur for Van der Pol oscillator if μ = 0.[edit ] References1. ^ a b Strogatz, Steven H. (1994). Nonlinear Dynamics and Chaos . Addison Wesley publishing company.2. ^ Kuznetsov, Yuri A. (2004). Elements of Applied Bifurcation Theory . New York: Springer-Verlag. ISBN 0-387-21906-4.3. ^ a b Hale, J.; Ko ak, H. (1991). Dynamics and Bifurcations . Texts in Applied Mathematics. 3. New York: Springer-Verlag.4. ^ Guckenheimer, J.; Myers, M.; Sturmfels, B. (1997). "Computing Hopf Bifurcations I". SIAM Journal on Numerical Analysis .5. ^ a b Hairer, E.; Norsett, S. P.; Wanner, G. (1993). Solving ordinary differential equations I: nonstiff problems (Second ed.). New York: Springer-Verlag.6. ^ Wilhelm, T.; Heinrich, R. (1995). "Smallest chemical reaction system with Hopf bifurcation". Journal of Mathematical Chemistry 17 (1): 1–14.doi :10.1007/BF01165134. http://www.fli-leibniz.de/~wilhelm/JMC1995.pdf .7. ^ Kirk, P. D. W.; Toni, T.; Stumpf, MP (2008). "Parameter inference for biochemical systems that undergo a Hopf bifurcation". Biophysical Journal 95 (2):540–549. doi :10.1529/biophysj.107.126086. PMC 2440454. PMID 18456830. /biophysj/pdf/PIIS0006349508702315.pdf .8. ^ a bKahoui, M. E.; Weber, A. (2000). "Deciding Hopf bifurcations by quantifier elimination in a software component architecture". Journal of SymbolicComputation 30 (2): 161–179. doi:10.1006/jsco.1999.0353. [edit] External links● Reaction-diffusion systems● The Hopf Bifurcation● Andronov–Hopf bifurcation page at ScholarpediaCategories: Bifurcation theoryPersonal tools● Log in / create accountNamespaces● Article● DiscussionVariantsViews● Read● Edit● View historyActionsSearchInteractionToolboxPrint/exportLanguages● This page was last modified on 25 May 2011 at 02:56.● Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. See Terms of use for details. Wikipedia is a registered trademark of the W ikimedia Foundation, Inc., a non-profit organization.● Contact us● Privacy policy● About Wikipedia● Disclaimers●●。

非靶向代谢组学方法英语

非靶向代谢组学方法英语Non-targeted Metabolomics Methods in EnglishIntroductionNon-targeted metabolomics is an innovative approach in the field of metabolomics that aims to identify and quantify as many metabolites as possible in a given biological sample without any prior knowledge or bias towards specific metabolites. This method provides comprehensive insights into the global biochemical changes occurring in a biological system, such as a cell, tissue, or organism. In recent years, non-targeted metabolomics has gained immense popularity due to its ability to unravel intricate metabolic pathways and discover novel biomarkers for various diseases.Sample Collection and PreparationThe first step in non-targeted metabolomics is the collection and preparation of the biological sample. The choice of sample depends on the research question and can range from blood, urine, tissues, or even fecal samples. It is crucial to handle the samples with extreme care to avoid any degradation or contamination of metabolites. Sample preparation involves various techniques such as extraction, filtration, and derivatization, to enhance the stability and visibility of metabolites during subsequent analysis.Mass Spectrometry-Based AnalysisMass spectrometry (MS) is the key analytical technique used in non-targeted metabolomics. It detects and quantifies metabolites based on their mass-to-charge ratio (m/z) and abundance. Liquid chromatography-massspectrometry (LC-MS) and gas chromatography-mass spectrometry (GC-MS) are commonly used platforms for metabolite analysis. LC-MS is suitable for hydrophilic compounds, while GC-MS is preferred for volatile and thermally stable metabolites.Data Acquisition and PreprocessingOnce the samples are analyzed using MS, the raw data obtained needs to be processed and converted into a format suitable for downstream analysis. This step involves data acquisition, which includes peak picking, alignment, and normalization. Peak picking identifies and quantifies metabolite peaks in the acquired spectra, while alignment corrects any potential retention time variations. Normalization ensures that all samples are comparably represented, eliminating any technical biases.Statistical Analysis and IdentificationStatistical analysis is a crucial step in non-targeted metabolomics, as it helps in identifying significant metabolites and detecting patterns within the dataset. Multivariate statistical techniques, such as principal component analysis (PCA) and partial least squares-discriminant analysis (PLS-DA), are commonly used to visualize and interpret the data. Additionally, metabolite identification is performed by matching the acquired mass spectra with metabolite databases, such as the Human Metabolome Database (HMDB) and the Kyoto Encyclopedia of Genes and Genomes (KEGG), using tools like MassBank, MetFrag, or Metlin.Metabolic Pathway AnalysisOne of the key strengths of non-targeted metabolomics is its ability to unravel complex metabolic pathways. Pathway analysis tools, such as MetaboAnalyst, MetaboMiner, and Ingenuity Pathway Analysis (IPA), are used to identify significantly altered pathways and discover potential biomarkers. These analyses provide crucial insights into the underlying biochemical mechanisms and aid in understanding the disease pathogenesis or physiological responses.Challenges and Future PerspectivesDespite its numerous advantages, non-targeted metabolomics faces several challenges. Metabolite identification remains a major bottleneck due to the limited coverage of metabolite databases and the lack of standardization in data reporting. Additionally, the high complexity and dynamic range of metabolomes make it difficult to detect low-abundance metabolites accurately. Nevertheless, advancements in analytical techniques, bioinformatics, and collaborative efforts are steadily overcoming these challenges and driving the field forward.In conclusion, non-targeted metabolomics plays a vital role in understanding the complex metabolic dynamics within biological systems. Through the use of advanced mass spectrometry techniques, data analysis tools, and metabolite identification strategies, this approach has the potential to uncover novel biomarkers and therapeutic targets for various diseases. With continued advancements, non-targeted metabolomics is poised to revolutionize personalized medicine and contribute significantly to the field of biomedical research.。

常微分方程中的英文单词和短语

常微分方程中的英文单词和短语
常微分方程
Ordinary Differential Equtions
第一章基本概念
Chapter 1 Basic Concepts
常微分方程
阶线性非线性偏微分方程
通解特解初值条件初值问题柯西问题几何解释积分曲线线素线素场方向场等斜线对称形式奇异点通积分
boundary finite closed interval open set finite semi-open interval
Lagrange formula
l ocal finite covering theorem
bar area monotone decreasing sectorial area comparability solution minimum solution maximum solution sl ope
topological dynamical system
differential dynamical system
stability of solution Liapunov stability
asymptotic stability asymptotic stability field attraction domain globe asymptotic stability
第六章线性微分方程组
Chapter 6 Linear Differential Equations
存在和唯一性定理齐次线性微分方程组基本解组朗斯基行列式刘维尔公式解矩阵基解矩阵常数矩阵非齐次线性微分方程组常数列向量常数变异公式计算公式结构公式常系数线性微分方程组矩阵指数函数标准基解矩阵若尔当标准型矩阵函数实值解复值解代数余子式特征方程算子式算子多项式拉普拉斯逆变换

一类具有交叉扩散项的捕食-食饵模型的局部分歧

一类具有交叉扩散项的捕食-食饵模型的局部分歧容跃堂;董苗娜;何堤;王晓丽【摘要】研究一类带有交叉扩散项的捕食-食饵模型在齐次Dirichlet边界条件下分歧解的存在性.利用极大值原理和上下解法得到正解的先验估计,并借助Crandall-Rabinowitz分歧理论,得出局部分歧正解存在的充分条件.%The existence of bifurcation solutions for a predator-prey model with cross-diffusion under homogeneous Dirichlet boundary conditions is concerned.By the maximum principle,a priori estimate of positive solutions are obtained.Then by Crandall-Rabinowitz bifurcation theory,the sufficient conditions for the existence of positive solutions to a local bifurcation is proved.【期刊名称】《纺织高校基础科学学报》【年(卷),期】2016(029)004【总页数】7页(P443-449)【关键词】捕食-食饵;自扩散;交叉扩散;先验估计;局部分歧【作者】容跃堂;董苗娜;何堤;王晓丽【作者单位】西安工程大学理学院,陕西西安710048;西安工程大学理学院,陕西西安710048;西安工程大学理学院,陕西西安710048;西安工程大学理学院,陕西西安710048【正文语种】中文【中图分类】O175.26近年来,关于生物数学领域的捕食食饵模型的研究已经成为热点,尤其是对于种群扩散影响下的捕食模型,国内外学者均已取得了一些符合实际的研究成果.文献[1]研究了一类捕食模型的正常数平衡态解的稳定性及分歧;文献[2-3]利用极大值原理和分歧定理研究了一类捕食模型局部解的延拓;文献[4-7]利用分歧定理研究了模型在交叉扩散影响下的正解的存在性问题.在文献[8]中,作者提出了一类具有扩散项的捕食食饵模型,通过给出正解的先验估计及局部分歧解存在条件,进而得到该系统平衡态的全局分歧解及其走向;文献[9]则在上述基础上研究了该类模型在交叉扩散项影响下的分歧.在同时考虑交叉扩散和自扩散项时，本文将继续研究如下捕食-食饵模型在齐次Dirichlet边界条件下正解的存在性，即其中:Ω为RN中具有光滑边界∂Ω上的有界区域;u,v分别表示食饵和捕食者的种群密度;a,b,c,d,α,β都是正常数,m1,m3表示自扩散系数;m2,m4表示交叉扩散系数,反应函数是Bazykin研究捕食者的饱和不稳定性与食饵的稳定性时建立的功能反应函数,生物背景参见文献[10].本文将针对模型(1)的如下平衡态方程展开讨论.注:对于问题(2)的解(u,v),如果在Ω中,(u,v)中只有一个分量为0,则称其为半平凡解. 记}.定义中的范数为通常的Banach空间C1 (Ω)中的范数,令,则X是Banach空间. 首先,考虑特征值问题引理1[11] 假设为常数,则问题(3)的所有特征值满足λ1(p,q)<λ2(p,q)≤λ3(p,q)≤…→∞,相应的特征函数为φ1,φ2,….由文献[11]知λ1(p,q)是简单的且关于q(x)严格单调递增.为方便起见,简记λ1=λ1(0),相应的主特征函数φ1>0.再考虑边值问题引理2[11] (1) 如果a≤λ1,则u=0是问题(4)的唯一非负解;若a>λ1,则问题(4)的唯一正解为θa.(2) 如果c≤λ1,则v=0是问题(5)的唯一非负解;当c>λ1时,其存在唯一正解θc.因此,当a>λ1,问题(2)存在半平凡解(θa,0);当c>λ1,问题(2)存在半平凡解(0,θc).定义Z=(U,V),其中U=(1+m1u+m2v)u,V=(1+m3v+m4u)v,则即(u,v)≥0与(U,V)≥0之间存在一一对应的关系.现在,引入和问题(2)等价的半线性椭圆系统易知,当a,c>λ1时,问题(6)的两个半平凡解分别为,其中.引理3[12] 假设a>λ1,令,则L(a)的特征值均大于0.引理4 设c>λ1,则当cm4>d时,存在唯一的a=a*(c)∈(λ1,∞),满足,且a=a*(c)关于c严格单调递增.此外,∃ψ*≥0满足证明取.显然A(λ1,c)=λ1(-c)=λ1-c<0.由于当a→∞时θa→∞,故有.经计算得又因为与均严格单调递增,可知A(a,c)关于a严格单调递增.从而存在唯一的a=a*(c)>λ1,使得A(a*(c),c)=0.再对A(a*(c),c)=0两边关于c求导,得Aa(a*(c),c)·a*′(c)+Ac(a*(c),c)=0.由于Ac(a,c)<0,结合Aa(a,c)>0得知a*′(c)>0,即a=a*(c)关于c严格单调递增.类似可以证明以下引理.引理5 假设c>λ1,则当aβ>b时,就存在唯一的a=a*(c)∈(λ1,∞),满足,且a=a*(c)关于c严格单调递增.此外,∃φ*≥0满足现在,结合文献[12-13]中的方法给出系统(6)的正解存在的必要条件及先验估计.定理1 当a≤λ1,或者,则问题(6)没有正解.证明若问题(6)存在正解(U,V),由问题(6)中的第2个方程得两边同乘以V,分部积分得由Poincare不等式‖‖，可得,同理可证a>λ1.与已知条件矛盾,则定理1得证. 定理2 设且b-βa(1+αa)>0.若(U,V)是问题(6)的任意正解,则∀x∈Ω,有证明设∃x0∈Ω,使得(x).由于故有,则同理可得由(u,v)与(U,V)之间的关系知定理2成立.现在以a为分歧参数,参考文献[14-19],利用Crandall-Rabinowitz局部分歧定理,给出问题(6)发自半平凡解与的局部分歧正解的存在性.定理3 设且cm4>d,则为问题(6)的分歧点,且的领域内存在正解其中a*由唯一确定,ψ*>0满足证明令其中u,v均为(U,V)的函数.将问题(6)在(U,V)处Taylor展开为这里,偏导数为处的导数值,满足.同时对(U,V)求导,得令,则有显然T(a;0,0)=0.记关于在(a*;0,0)处的Frechlet导数是L(a*;0,0).经计算,L(a*;0,0)·(φ,ψ)=0等价于如果ψ≡0,那么由算子La*可逆知φ≡0,矛盾,所以ψ不恒为零.又,故有因此,算子L(a*;0,0)的核空间N(L(a*;0,0))=span{U0},U0=(φ*,ψ*)T,其中又令L*(a*;0,0)为L(a*;0,0)的自伴算子,类似可得由Fredholm选择公理知因此可得dimN(L(a*;0,0))=1,codimR(L(a*;0,0))=1.令,下面采用反证法证明假设∃(h,k)∈X,使得L1(a*;0,0)·(φ*,ψ*)=L(a*;0,0)·(h,k).经计算得那么有两边同时乘以ψ*,分部积分得由于cm4-d>0,且θa关于a严格单调递增,则上式左端大于0,矛盾.由Crandall-Rabinowitz局部分歧定理知,存在充分小的δ>0及C1连续曲线(a(s):Φ1(s),Ψ1(s)):(-δ,δ)→R×X满足a(0)=a*,Φ1(0)=0,Ψ1(0)=0,Φ1(s),Ψ1(s)∈Z 使得(a(s):(φ*+Φ1(s)),s(ψ*+Ψ1(s)))是T(a(s):的零点,其中X=Z⨁N(L(a*;0,0)),由于,因此可得到发自的局部分歧正解Γ*.同理可得到发自半平凡分支的局部分歧正解.定理4 设且aβ>b,则为问题(5)的分歧点,且的领域内存在正解a*由唯一确定,φ*>0满足充分小.这里(Φ2(s),Ψ2(s);a(s))是连续函数,满足a(0)=a*,Φ2(0)=0,Ψ2(0)=0,∫Ωψ2φ*dx=0,且RONG Yuetang,DONG Miaona,HE Di,et al.The local bifurcation for a kind of prey-predator model with cross-diffusion[J].Basic Sciences Journal of Textile Universities,2016,29(4):443-449.【相关文献】[1] 周冬梅,李艳玲.一类捕食模型正常数平衡态解的稳定性及分歧[J].科学技术与工程,2010,10 (23):5615-5619.ZHOU Dongmei,LI Yanling.Stability and bifurcation of positive constant steady-state solution for predator-prey model[J].Science Technology andEngineering,2010,10(23):5615-5619.[2] 李海侠,李艳玲.一类捕食模型正平衡解的整体分歧[J].西北师范大学学报:自然科学版,2006,42(2):8-12.LI Haixia,LI Yanling.Bifurcation of positive steady-state solutions for a king of predator-prey model[J].Journal of Northwest Normal University:Natural Science,2006,42(2):8-12. [3] 王妮娅,李艳玲.一类带收获率的的捕食模型的全局分歧和稳定性[J].安徽师范大学学报:自然科学版,2015,38(1):25-30.WANG Niya,LI Yanling.Global bifurcation and stability of a class of predator-prey models with prey harvesting [J].Journal of Anhui University:Natural Science Edition,2015,38(1):25-30.[4] KUTO K,YAMADA Y.Multiple coexistence states for a prey-predator system with cross-diffusion[J].J Differential Equations,2004,197(2):315-348.[5] 张晓晶,容跃堂,何堤,等.一类带有交叉扩散的捕食-食饵模型的分歧性[J].纺织高校基础科学学报,2014,27(3):322-326.ZHANG Xiaojing,RONG Yuetang,HE Di.Bifurcation for a prey-predator model with cross-diffusion[J].Basic Sciences Journal of Textile Universities,2014,27(3):322-326.[6] DUBEY B,DAS B,HASSAIN J.A prey-predator interaction model with self and cross-diffusion[J].Ecol Modelling,2002,141:67-76.[7] ZHANG Cunhua,YAN Xiangping.Positive solutions bifurcating from zero solution in a Lotka-Volterra competitive system with cross-diffusion effects[J].Appl Math J China Univ,2011,26(3):342-352.[8] 冯孝周,吴建华.具有饱和与竞争项的捕食系统的全局分歧及稳定性[J].系统科学与数学,2010,30(7):979-989.FENG Xiaozhou,WU Jianhua.Global bifurcation and stability for predator-prey model with predator saturation and competition[J].Journal of System Science and Mathematical Sciences,2010,30(7):979-989.[9] 何堤,容跃堂,张晓晶.一类具有交叉扩散的捕食-食饵模型的分歧[J].纺织高校基础科学学报,2015,28(4):426-430.HE Di,RONG Yuetang,ZHANG Xiaojing.Bifurcation for a prey-predator model with cross-diffusion[J].Basic Sciences Journal of Textile Universities,2015,28(4):426-430.[10] BAZYKIN A D.Nonlinear dynamics of interacting population[M].Singapore:World Scientific,1998.[11] 叶其孝,李正元,王明新.反应扩散方程引论[M].北京：科学出版社,2011:40-56.YE Qixiao,LI Zhengyuan,WANG Mingxin.Introduction of reaction-diffusionequations[M].Beijing:Science Press,2011:40-56.[12] 何堤,容跃堂,王晓丽,等.一类具有交叉扩散的捕食-食饵模型的局部分歧[J].西安工业大学学报,2015,35(11):872-876.HE Di,RONG Yuetang,WANG Xiaoli,et al.Local bifurcation for a prey-predator model with cross-diffusion[J].Journal of Xi′an Technological University,2015,35(11):872-876.[13] 容跃堂,何堤,张晓晶.带交叉扩散项的Holling Ⅳ捕食-食饵模型的全局分歧[J].纺织高校基础科学学报,2015,26(3):287-293.RONG Yuetang,HE Di,ZHANG Xiaojing.The global bifurcation for a prey-predator model with cross-diffusion and Holling Ⅳ [J].Basic Sciences Journal of TextileUniversities,2015,26(3):287-293.[14] 马晓丽,冯孝周.一类具有交叉扩散的捕食模型的正解的存在性[J].安徽大学学报:自然科学版,2011,35(5):26-31.MA Xiaoli,FENG Xiaozhou.The existence of positive solutions for a predator-prey model with cross-diffusion[J].Journal of Anhui University:Natural Science Edition,2011,35(5):26-31.[15] 马晓丽.一类具有交叉扩散的捕食模型的整体分歧[J].西安工业大学学报,2010,30(5):506-510. MA Xiaoli.Global bifurcation for a predator-prey model with cross-diffusion[J].Journal of Xi′an Technological University,2010,30(5):506-510.[16] 戴婉仪,付一平.一类交叉扩散系统定态解的分歧与稳定性[J].华南理工大学大学报:自然科学版,2005,33(2):99-102.DAI Wanyi,FU Yiping.Bifurcation and stability of the steady-state solutions to a system with cross-diffusion effect[J].Journal of South China University of Technology:Natural Science Edition,2005,33(2):99-102.[17] 柴俊平,李艳玲.带有交叉扩散项的捕食-食饵模型的全局分歧[J].纺织高校基础科学学报,2011,24(4):490-494.CHAI Junping,LI Yaning.Global bifurcation of a class of predator-prey models with cross-diffusion effect[J].Basic Sciences Journal of Textile Universities,2011,24(4):490-494. [18] WU J H.Global bifurcation of coexistence states for the competition model in the chemostat[J].Nonlinear Analysis,2000,39(7):817-835.[19] CRANDALL M G,RABINOWITZ P H.Bifurcation from simple eigenvalues[J].J Functional Analysis,1971,8(2):321-340.。

Bifurcation+Theory

Bifurcation Theory © John J. Tyson, 20091. Saddle-Node BifurcationSuppose we want to study the solutions of a pair of nonlinear ODEs as a particular parameter, p , is varied:d d (,;), (,;)d d (*,*;)0, (*,*;)0o o x y f x y p g x y p t tf x y pg x y p ====The steady state solution, at (x *, y *), depends on the chosen value of the parameter, p = p o . We want to know how the steady state changes as p is varied away from p o , so we introduce new variables that are deviations from the steady state:*, *, o x x y y p p ξηδ=−=−=−Then, the steady state equations can be approximated by:11121p 21222p 0 (neglecting 'higher-order' terms)0 (neglecting 'higher-order' terms)a a a a a a ξηδξηδ≈++≈++where a 11, a 12, a 21, a 22 are the elements of the Jacobian matrix, and a 1p , a 2p are partial derivatives of f (x,y;p ) and g (x,y;p ) with respect to p , evaluated at the steady state. These linear equations can be written in vector form1p 2p , where and a a ξδη⎛⎞⎛⎞===⎜⎟⎜⎟⎝⎠⎝⎠Jz b z b .This matrix equation has a non-trivial solution (Hint: Cramer’s Rule) as long as D =Det(J ) = a 11a 22–a 12a 21 ≠ 0. Hence, as long as D is non-zero, we can ‘continue’ the steady state by taking small steps, δ, in the parameter value. Whenever D becomes small, we simply rearrange the linear equations as follows,121p 11222p 21a a a a a a ηδξηδξ+=+=and now follow η and δ as we take small steps, ξ, in the x-direction. This trick allows us (i.e., AUTO) to ‘turn the corner’ at a saddle-node bifurcation point, where D = 0.As AUTO tracks a steady state solution for changing values of a parameter, p , it monitors D (p ), the determinant of the Jacobian matrix, and also Re{λ(p )} for all complex-conjugate pairs of eigenvalues. Whenever D (p c ) = 0 for some ‘critical’ value of p , AUTOrecords p c as a ‘limit point’ (LP), also known as a saddle-node (SN) bifurcation point. Whenever Re{λ(p c )} = 0, then AUTO records p c as a ‘Hopf bifurcation’ point (HB).A ‘one-parameter’ bifurcation diagram is a plot of the steady state value of a variable, say x *(p 1), as a function of the changing value of a parameter p . A typical bifurcation diagram might look like this:AUTO leaves little ‘labels’ (the circles in the diagram above) at bifurcation points, allowing the user to ‘grab’ a bifurcation point and follow it in a ‘two-parameter’bifurcation diagram. For example, we might grab one of the SN bifurcation points and follow it as two parameters, call them p and q , are varied. In this case, AUTO must solve 3 nonlinear algebraic equations in four unknowns (x *, y *, p and q ):(*,*;,)0, (*,*;,)0, (*,*;,)0f x y p q g x y p q D x y p q ===where D = Determinant of the Jacobian matrix at the steady state. We know that these three equations are satisfied simultaneously at a SN bifurcation point, say at p o and q o . To follow the SN as p and q are varied, AUTO linearizes the algebraic equations around the known solution11121p 1q 21222p 2q 12p q 0 0 0a a a a a a a a D D D D ξηδγξηδγξηδγ=+++=+++=+++where 1/, ..., /q D D x D D q =∂∂=∂∂. We can solve these equations, as before, for ξ, η and γ as functions of δ, and then plot o q q γ=+ as a function of o p p δ=+. A typical two-parameter bifurcation diagram for SN bifurcations looks like thisThe ‘cusp point’ is known as a ‘codimension-2’ bifurcation point because we must choose precise values of two different parameters, p and q , in order to reach the cusp.2. Hopf BifurcationIn 1942, E. Hopf proved a remarkable and powerful theorem about limit cycleoscillations in the vicinity of a critical point, p = p c , where Re{λ(p c )} = 0. (Let us assume that Re{λ(p )} < 0 for p < p c , and Re{λ(p )} > 0 for p > p c . If the opposite is true, then replace p by q = −p , and you have the assumed case.) In informal language, Hopf proved that for p in the vicinity of p c , there exists a one-parameter family of small-amplitudelimit cycle solutions for one of the following three cases: (i) p > p c . (ii) p < p c . (iii) p = p c . For cases (i) and (ii) the limit cycles are parameterized by their distance from thebifurcation point, i.e., | p − p c|. The amplitude and period of each limit cycle scales likec 2 A T b p p πω==+−where a and b are constants. (Case (iii) is rarely encountered, and we will not bother with it.)When the amplitude of the bifurcating limit cycles are plotted on a one-parameter bifurcation diagram, we get the following typical figures:Case (i) is called a super-critical Hopf bifurcation (creating stable limit cycles around an unstable steady state), and case (ii) is called a sub-critical Hopf bifurcation (creating unstable limit cycles around a stable steady state).Hopf bifurcation points can be ‘followed’ in a two-parameter bifurcation diagram, using exactly the same strategy outlined for following a SN bifurcation. In this case, the typical 1case (i), diagrammed above. For q = q 2, the one-parameter bifurcation diagram looks so:bifurcation point, called a ‘degenerate Hopf bifurcation’, where a locus of cyclic folds meets the locus of HB’s tangentially.If one follows a locus of HB’s on a two-parameter bifurcation diagram, then typically the line may (i) ‘run off to infinity’, or (ii) close in a loop, or (iii) end at a codimension-2‘Takens-Bogdanov’ bifurcation. A TB bifurcation point has a complex structure3. Homoclinic BifurcationsThe final sort of bifurcations that we must learn concern the ‘birth’ of limit cycles fromtrajectories that begin and end on saddle-type steady states.3a. Saddle-Loop (SL) BifurcationFirst, consider the sequence of changes to a vector field as a parameter p is changed:As p increases through p c, a limit cycle is born with large amplitude and infinite period.(At p = p c, it takes an infinite amount of time to go around the saddle loop.) Notice thequalitative difference between limit cycle creation by a Hopf bifurcation and by a saddle-loop bifurcation:Period AmplitudeFrequencyHopf Bifurcation infinitesimal finite finiteSaddle-Loop Bif’n finite infinitesimal InfiniteLike HB’s, SL bifurcations may give rise to either stable or unstable limit cycles.SL bifurcations are often found in conjunction with Hopf bifurcations, as in the exampleof a Takens-Bogdanov bifurcation above. If we were to compute a one-parameterbifurcation diagram for q = q1 (a little above the TB point), it would look like this:cnode’ point, and the unique trajectory that proceeds out of the saddle-node loops around to the other side and comes back into the saddle-node, forming an ‘invariant circle’. It takes an infinite amount of time to go around the invariant circle. For p > p c, the saddle-node point disappears and the invariant circle becomes a stable limit cycle, with finite amplitude and very long period, because there is a ‘slow’ part of the vector field in the region where the steady states used to be. As before, SNIC bifurcations can generate either stable or unstable limit cycles.SNIC bifurcations are created at a codimension-2 bifurcation point, when an SL locus merges tangentially into an SN locus. You will find such a bifurcation point in the two-parameter bifurcation diagram I used to illustrate the Takens-Bogdanov bifurcation.As I have said, SL and SNIC bifurcations can create either stable or unstable limit cycles. In the same way that supercritical HB changes to subcritical HB at a degenerate HB point, the stability of SL and SNIC limit cycles may change at a codimension-2 degenerate bifurcation point by throwing off a locus of cyclic fold (CF) bifurcations.4. SummaryAt a bifurcation point the solutions of a set on nonlinear ODEs undergo a dramatic, qualitative change in their characteristics: steady states appear or disappear, and limit cycles appear or disappear. Because the stable attractors of the ODEs determine, in large measure, the temporal behavior of the dynamical system, the bifurcations exhibited by a dynamical system determine, in large measure, the signal-response characteristics of the molecular control system. Because there are only a limited number of ‘generic’ bifurcations exhibited by nonlinear ODEs, there is only a small selection of bifurcations from which nature must build all the complex information processing capabilities of macromolecular reaction networks. We are now familiar with all the fundamental bifurcations of codimension-1 and -2. They are:Codimension-1Saddle-Node SN Coalescence of saddle and node to annihilate bothsteady statesHopf HB Steady state loses stability and gives rise to a small-amplitude limit cycle of finite periodCyclic Fold CF Coalescence of stable and unstable limit cycles toannihilate both oscillatory solutionsSaddle Loop SL Trajectory that loops from a saddle point back to itself,creating a large-amplitude limit cycle of infinite periodSaddle-Node Invariant-Circle SNIC Trajectory that loops from a saddle-node point back to itself, creating a large-amplitude limit cycle of infiniteperiodCodimension-2Cusp C Tangential coalescence of two SN loci; the dynamicalsystem is bistable inside the cuspDegenerateHopf DH Tangentialcoalescence of HB and CF loci, whichchanges the HB from supercritical to subcriticalTakens-Bogdanov TB Tangentialcoalescence of SN, HB and SL loci. Limitcycles exist in the small wedge between HB and SL. Neutral Saddle Loop NSL Tangential coalescence of SL and CF loci, whichchanges the stability of the bifurcating limit cycles Saddle-Node Loop SNL Tangential coalescence of SL and SN loci to create alocus of SNIC bifurcations。

神经外科5本杂志目录英文-中文 2022年10月

神外杂志英-中文目录(2022年10月) Neurosurgery1.Assessment of Spinal Metastases Surgery Risk Stratification Tools in BreastCancer by Molecular Subtype按照分子亚型评估乳腺癌脊柱转移手术风险分层工具2.Microsurgery versus Microsurgery With Preoperative Embolization for BrainArteriovenous Malformation Treatment: A Systematic Review and Meta-analysis 显微手术与显微手术联合术前栓塞治疗脑动静脉畸形的系统评价和荟萃分析mentary: Silk Vista Baby for the Treatment of Complex Posterior InferiorCerebellar Artery Aneurysms点评: Silk Vista Baby用于治疗复杂的小脑下后动脉动脉瘤4.Targeted Public Health Training for Neurosurgeons: An Essential Task for thePrioritization of Neurosurgery in the Evolving Global Health Landscape针对神经外科医生的有针对性的公共卫生培训:在不断变化的全球卫生格局中确定神经外科手术优先顺序的一项重要任务5.Chronic Encapsulated Expanding Hematomas After Stereotactic Radiosurgery forIntracranial Arteriovenous Malformations: An International Multicenter Case Series立体定向放射外科治疗颅内动静脉畸形后的慢性包裹性扩张血肿:国际多中心病例系列6.Trends in Reimbursement and Approach Selection for Lumbar Arthrodesis腰椎融合术的费用报销和入路选择趋势7.Diffusion Basis Spectrum Imaging Provides Insights Into Cervical SpondyloticMyelopathy Pathology扩散基础频谱成像提供了脊髓型颈椎病病理学的见解8.Association Between Neighborhood-Level Socioeconomic Disadvantage andPatient-Reported Outcomes in Lumbar Spine Surgery邻域水平的社会经济劣势与腰椎手术患者报告结果之间的关系mentary: Prognostic Models for Traumatic Brain Injury Have GoodDiscrimination But Poor Overall Model Performance for Predicting Mortality and Unfavorable Outcomes评论:创伤性脑损伤的预后模型在预测death率和不良结局方面具有良好的区分性，但总体模型性能较差mentary: Serum Levels of Myo-inositol Predicts Clinical Outcome 1 YearAfter Aneurysmal Subarachnoid Hemorrhage评论:血清肌醇水平预测动脉瘤性蛛网膜下腔出血1年后的临床结局mentary: Laser Interstitial Thermal Therapy for First-Line Treatment ofSurgically Accessible Recurrent Glioblastoma: Outcomes Compared With a Surgical Cohort评论:激光间质热疗用于手术可及复发性胶质母细胞瘤的一线治疗:与手术队列的结果比较12.Functional Reorganization of the Mesial Frontal Premotor Cortex in Patients WithSupplementary Motor Area Seizures辅助性运动区癫痫患者中额内侧运动前皮质的功能重组13.Concurrent Administration of Immune Checkpoint Inhibitors and StereotacticRadiosurgery Is Well-Tolerated in Patients With Melanoma Brain Metastases: An International Multicenter Study of 203 Patients免疫检查点抑制剂联合立体定向放射外科治疗对黑色素瘤脑转移患者的耐受性良好:一项针对203例患者的国际多中心研究14.Prognosis of Rotational Angiography-Based Stereotactic Radiosurgery for DuralArteriovenous Fistulas: A Retrospective Analysis基于旋转血管造影术的立体定向放射外科治疗硬脑膜动静脉瘘的预后:回顾性分析15.Letter: Development and Internal Validation of the ARISE Prediction Models forRebleeding After Aneurysmal Subarachnoid Hemorrhage信件:动脉瘤性蛛网膜下腔出血后再出血的ARISE预测模型的开发和内部验证16.Development of Risk Stratification Predictive Models for Cervical DeformitySurgery颈椎畸形手术风险分层预测模型的建立17.First-Pass Effect Predicts Clinical Outcome and Infarct Growth AfterThrombectomy for Distal Medium Vessel Occlusions首过效应预测远端中血管闭塞血栓切除术后的临床结局和梗死生长mentary: Risk for Hemorrhage the First 2 Years After Gamma Knife Surgeryfor Arteriovenous Malformations: An Update评论:动静脉畸形伽玛刀手术后前2年出血风险:更新19.A Systematic Review of Neuropsychological Outcomes After Treatment ofIntracranial Aneurysms颅内动脉瘤治疗后神经心理结局的系统评价20.Does a Screening Trial for Spinal Cord Stimulation in Patients With Chronic Painof Neuropathic Origin Have Clinical Utility (TRIAL-STIM)? 36-Month Results From a Randomized Controlled Trial神经性慢性疼痛患者脊髓刺激筛选试验是否具有临床实用性(TRIAL-STIM)？一项随机对照试验的36个月结果21.Letter: Transcriptomic Profiling Revealed Lnc-GOLGA6A-1 as a NovelPrognostic Biomarker of Meningioma Recurrence信件:转录组分析显示Lnc-GOLGA6A-1是脑膜瘤复发的一种新的预后生物标志物mentary: The Impact of Frailty on Traumatic Brain Injury Outcomes: AnAnalysis of 691 821 Nationwide Cases评论:虚弱对创伤性脑损伤结局的影响:全国691821例病例分析23.Optimal Cost-Effective Screening Strategy for Unruptured Intracranial Aneurysmsin Female Smokers女性吸烟者中未破裂颅内动脉瘤的最佳成本效益筛查策略24.Letter: Pressure to Publish—A Precarious Precedent Among Medical Students信件:出版压力——医学研究者中一个不稳定的先例25.Letter: Protocol for a Multicenter, Prospective, Observational Pilot Study on theImplementation of Resource-Stratified Algorithms for the Treatment of SevereTraumatic Brain Injury Across Four Treatment Phases: Prehospital, Emergency Department, Neurosurgery, and Intensive Care Unit信件:一项跨四个治疗阶段(院前、急诊科、神经外科和重症监护室)实施资源分层算法的多中心、前瞻性、观察性试点研究的协议26.Risk for Hemorrhage the First 2 Years After Gamma Knife Surgery forArteriovenous Malformations: An Update动静脉畸形伽玛刀手术后前2年出血风险:更新Journal of Neurosurgery27.Association of homotopic areas in the right hemisphere with language deficits inthe short term after tumor resection肿瘤切除术后短期内右半球同话题区与语言缺陷的关系28.Association of preoperative glucose concentration with mortality in patientsundergoing craniotomy for brain tumor脑肿瘤开颅手术患者术前血糖浓度与death率的关系29.Deep brain stimulation for movement disorders after stroke: a systematic review ofthe literature脑深部电刺激治疗脑卒中后运动障碍的系统评价30.Effectiveness of immune checkpoint inhibitors in combination with stereotacticradiosurgery for patients with brain metastases from renal cell carcinoma: inverse probability of treatment weighting using propensity scores免疫检查点抑制剂联合立体定向放射外科治疗肾细胞癌脑转移患者的有效性:使用倾向评分进行治疗加权的反向概率31.Endovascular treatment of brain arteriovenous malformations: clinical outcomesof patients included in the registry of a pragmatic randomized trial脑动静脉畸形的血管内治疗:纳入实用随机试验登记处的患者的临床结果32.Feasibility of bevacizumab-IRDye800CW as a tracer for fluorescence-guidedmeningioma surgery贝伐单抗- IRDye800CW作为荧光导向脑膜瘤手术示踪剂的可行性33.Precuneal gliomas promote behaviorally relevant remodeling of the functionalconnectome前神经胶质瘤促进功能性连接体的行为相关重塑34.Pursuing perfect 2D and 3D photography in neuroanatomy: a new paradigm forstaying up to date with digital technology在神经解剖学中追求完美的2D和三维摄影:跟上数字技术的新范式35.Recurrent insular low-grade gliomas: factors guiding the decision to reoperate复发性岛叶低级别胶质瘤:决定再次手术的指导因素36.Relationship between phenotypic features in Loeys-Dietz syndrome and thepresence of intracranial aneurysmsLoeys-Dietz综合征的表型特征与颅内动脉瘤存在的关系37.Continued underrepresentation of historically excluded groups in the neurosurgerypipeline: an analysis of racial and ethnic trends across stages of medical training from 2012 to 2020神经外科管道中历史上被排除群体的代表性持续不足:2012年至2020年不同医学培训阶段的种族和族裔趋势分析38.Management strategies in clival and craniovertebral junction chordomas: a 29-yearexperience斜坡和颅椎交界脊索瘤的治疗策略:29年经验39.A national stratification of the global macroeconomic burden of central nervoussystem cancer中枢神经系统癌症全球宏观经济负担的国家分层40.Phase II trial of icotinib in adult patients with neurofibromatosis type 2 andprogressive vestibular schwannoma在患有2型神经纤维瘤病和进行性前庭神经鞘瘤的成人患者中进行的盐酸埃克替尼II期试验41.Predicting leptomeningeal disease spread after resection of brain metastases usingmachine learning用机器学习预测脑转移瘤切除术后软脑膜疾病的扩散42.Short- and long-term outcomes of moyamoya patients post-revascularization烟雾病患者血运重建后的短期和长期结局43.Alteration of default mode network: association with executive dysfunction infrontal glioma patients默认模式网络的改变:与额叶胶质瘤患者执行功能障碍的相关性44.Correlation between tumor volume and serum prolactin and its effect on surgicaloutcomes in a cohort of 219 prolactinoma patients219例泌乳素瘤患者的肿瘤体积与血清催乳素的相关性及其对手术结果的影响45.Is intracranial electroencephalography mandatory for MRI-negative neocorticalepilepsy surgery?对于MRI阴性的新皮质癫痫手术，是否必须进行颅内脑电图检查？46.Neurosurgeons as complete stroke doctors: the time is now神经外科医生作为完全中风的医生:时间是现在47.Seizure outcome after resection of insular glioma: a systematic review, meta-analysis, and institutional experience岛叶胶质瘤切除术后癫痫发作结局:一项系统综述、荟萃分析和机构经验48.Surgery for glioblastomas in the elderly: an Association des Neuro-oncologuesd’Expression Française (ANOCEF) trial老年人成胶质细胞瘤的手术治疗:法国神经肿瘤学与表达协会(ANOCEF)试验49.Surgical instruments and catheter damage during ventriculoperitoneal shuntassembly脑室腹腔分流术装配过程中的手术器械和导管损坏50.Cost-effectiveness analysis on small (< 5 mm) unruptured intracranial aneurysmfollow-up strategies较小(< 5 mm)未破裂颅内动脉瘤随访策略的成本-效果分析51.Evaluating syntactic comprehension during awake intraoperative corticalstimulation mapping清醒术中皮质刺激标测时句法理解能力的评估52.Factors associated with radiation toxicity and long-term tumor control more than10 years after Gamma Knife surgery for non–skull base, nonperioptic benignsupratentorial meningiomas非颅底、非周期性良性幕上脑膜瘤伽玛刀术后10年以上与放射毒性和长期肿瘤控制相关的因素53.Multidisciplinary management of patients with non–small cell lung cancer withleptomeningeal metastasis in the tyrosine kinase inhibitor era酪氨酸激酶抑制剂时代有软脑膜转移的非小细胞肺癌患者的多学科管理54.Predicting the growth of middle cerebral artery bifurcation aneurysms usingdifferences in the bifurcation angle and inflow coefficient利用分叉角和流入系数的差异预测大脑中动脉分叉动脉瘤的生长55.Predictors of surgical site infection in glioblastoma patients undergoing craniotomyfor tumor resection胶质母细胞瘤患者行开颅手术切除肿瘤时手术部位感染的预测因素56.Stereotactic radiosurgery for orbital cavernous hemangiomas立体定向放射外科治疗眼眶海绵状血管瘤57.Surgical management of large cerebellopontine angle meningiomas: long-termresults of a less aggressive resection strategy大型桥小脑角脑膜瘤的手术治疗:较小侵袭性切除策略的长期结果Journal of Neurosurgery: Case Lessons58.5-ALA fluorescence–guided resection of a recurrent anaplastic pleomorphicxanthoastrocytoma: illustrative case5-ALA荧光引导下切除复发性间变性多形性黄色星形细胞瘤:说明性病例59.Flossing technique for endovascular repair of a penetrating cerebrovascular injury:illustrative case牙线技术用于血管内修复穿透性脑血管损伤:例证性病例60.Nerve transfers in a patient with asymmetrical neurological deficit followingtraumatic cervical spinal cord injury: simultaneous bilateral restoration of pinch grip and elbow extension. Illustrative case创伤性颈脊髓损伤后不对称神经功能缺损患者的神经转移:同时双侧恢复捏手和肘关节伸展。

_一类非线性奇性薛定谔方程的精确解

第二章动力系统和奇点........................................................................ 4
2.1 动力系统.................................................................................................................. 4
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个人学术水平评价指标———ｈ因子与ｐｈ因子概念介绍

８
西安工业大学学报第４０卷
ＳｐｒａｙｉｎｇａｎｄＲｅｌｅａｓｉｎｇＰｅｒｉｏｄｓｏｎＤｙｎａｍｉｃｓｏｆＧｅｎｅｒａｌｉｚｅｄＰｒｅｄａｔｏｒ?Ｐｒｅｙｍｏｄｅｌ［Ｊ］．ＩｎｔｅｒｎａｔｉｏｎａｌＪｏｕｒｎａｌｏｆＢｉｏｍａｔｈｅｍａｔｉｃｓ，２０１２，５（１）：１．［７］ＴＡＮＧＳＹ，ＬＩＡＮＧＪＨ．ＧｌｏｂａｌＱｕａｌｉｔａｔｉｖｅＡｎａｌｙｓｉｓｏｆａＮｏｎ?ＳｍｏｏｔｈＧａｕｓｅＰｒｅｄａｔｏｒ?ＰｒｅｙＭｏｄｅｌｗｉｔｈａＲｅｆｕｇｅ［Ｊ］．ＮｏｎｌｉｎｅａｒＡｎａｌｙｓｉｓ，２０１３，７６（１）：１６５．［８］ＷＡＮＧＳ，ＨＵＡＮＧＱＤ．ＢｉｆｕｒｃａｔｉｏｎｏｆＮｏｎｔｒｉｖｉａｌＰｅｒｉｏｄｉｃＳｏｌｕｔｉｏｎｓｆｏｒａＢｅｄｄｉｎｇｔｏｎ?ＤｅＡｎｇｅｌｉｓＩｎｔｅｒｆｅｒｅｎｃｅＭｏｄｅｌｗｉｔｈＩｍｐｕｌｓｉｖｅＢｉｏｌｏｇｉｃａｌＣｏｎｔｒｏｌ［Ｊ］．ＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃａｌＭｏｄｅｌｌｉｎｇ，２０１４，３９：１４７０．［９］ＢＡＫＥＨＫ．ＱｕａｌｉｔａｔｉｖｅＡｎａｌｙｓｉｓｏｆＢｅｄｄｉｎｇｔｏｎ?ＤｅＡｎｇｅｌｉｓＴｙｐｅＩｍｐｕｌｓｉｖｅＰｒｅｄａｔｏｒ?ＰｒｅｙＭｏｄｅｌｓ［Ｊ］．ＮｏｎｌｉｎｅａｒＡｎａｌ：ＲｅａｌＷｏｒｌｄＡｐｐｌｉｃａｔｉｏｎｓ，２０１０，１１：１３１２．［１０］ＹＡＮＧＪ，ＴＡＮＧＳＹ．ＨｏｌｌｉｎｇＴｙｐｅＩＩＰｒｅｄａｔｏｒ?
櫅殯
肖晨）
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简讯
个人学术水平评价指标 ———ｈ因子与ｐｈ因子概念介绍

慢性完全性闭塞的导丝技术介绍

directed forward probing Tip stiffness should penetrate even heavily calcified
CTO Guidewire Design
Anatomy of a CTO Guidewire Guidewire Operator Techniques
CTO Guidewire Categories Simplified “Lesion-Specific” CTO
Guidewire Use Algorithms
• Hydrophobic coating
• Increasing tip loads 3 -12 gm
• Excellent tip shape ability & shape retention
ASAHI CONFIANZA™ 9 – Tapered Tip Guide Wires
ASAHI CONFIANZA™ 9
Inflammation
Necrotic core Calcification
Chronic Total Occlusions
What’s Blocking up the Lumen? 1. Dense Fibrotic Tissue: COLLAGEN! 2. Calcification
Intraluminal Collagen
Hallmarks of a CTO Guidewire
• Tip styles - core-to-tip designs; sometimes tapered • Coils and covers - some favor increased radiopacity;
jointless coils for improved torque response; polymer covers for selected applications • Core tapers and materials - shorter tapers for improved torque response; generally stainless steel • Core diameters - larger for increased support and torque response • Coatings - hydrophilic for tracking (body) and hydrophobic for torque response (body and tip)

非线性波方程的奇异曲线和奇异行波

II
Abstract
The results show that the equation exist a new double compacton solutions. The new compacton solutions are different from the famous Rosenau-Hyman compacton solutions, because they are derived by a singular elliptic curve tangenting to the homoclinic orbits rather than the singular straight line in the phase space. In Chapter 6, we study the relasionship of the parabola singular curves and the singular traveling wave. The deformed Hunter-Zheng equation is specifically studied and corresponding travelling wave system possing the parabola singular curves is well derived. This is the first time that this type of equations has been found. Futher studying the periodic curves tangent to the parabola singular curves, we obtain the new singular periodic wave and analyze its dynamical behaviors. In Chapter 7, we summarize the work of the paper and put forward the research work in the future. Keywords: peakon solutions; compacton solutions; cuspon solutions; pseudo-cuspon solutions；periodic wave solutions

基因修饰细胞治疗产品非临床研究与评价技术指导原则(试行)

.20212一、前言 (1)二、适用范围 (1)三、总体考虑 (1)四、受试物 (2)五、动物种属/模型选择 (3)六、概念验证 (4)七、药代动力学 (5)八、非临床安全性 (6)（一）总体安全性考虑 (6)（二）基因表达产物的风险评估 (6)（三）插入突变风险评估 (7)（四）载体动员（Vector mobilisation）和重组风险评估8九、对特定类型基因修饰细胞产品的特殊考虑 (8)（一）基因修饰的免疫细胞（CAR或TCR修饰的T细胞/NK细胞） (8)（二）诱导多能干细胞（iPS）来源的细胞产品 (10)（三）基因编辑的细胞产品 (10)一、前言近年来，随着基因修饰技术的迅速发展，基因修饰细胞治疗产品已成为医药领域的研究热点。

由于基因修饰细胞治疗产品物质组成和作用方式与一般的化学药品和生物制品有明显不同，传统的标准非临床研究策略和方法通常并不适用于基因修饰细胞治疗产品。

为规范和指导基因修饰细胞治疗产品非临床研究和评价，在《细胞制品研究与评价技术指导原则》基础上，根据目前对基因修饰细胞治疗产品的科学认识，制定了本指导原则，提出了对基因修饰细胞治疗产品非临床研究和评价的特殊考虑和要求。

随着技术的发展、认知程度的深入和相关研究数据的积累，本指导原则将不断完善和适时更新。

二、适用范围本指导原则适用于基因修饰细胞治疗产品。

基因修饰细胞治疗产品是指经过基因修饰（如调节、修复、替换、添加或删除等）以改变其生物学特性、拟用于治疗人类疾病的活细胞产品，如基因修饰的免疫细胞（如T细胞、NK细胞、树突状细胞和巨噬细胞等）和基因修饰的干细胞（如造血干细胞、多能诱导干细胞等）等。

三、总体考虑非临床研究是药物开发的重要环节之一。

对于基因修饰细胞治疗产品，充分的非临床研究是为了：1）阐明基因修饰的目的、功能以及产品的作用机制，明确其在拟定患者人群中使用的生物学合理性；2）为临床试验的给药途径、给药程序、给药剂量的选择提供支持性依据；3）根据潜在风险因素，阐明毒性反应特征，预测人体可能出现的不良反应，确定不良反应的临床监测指标，为制定临床风险控制措施提供参考依据。

非富勒烯小分子受体不稳定单元

非富勒烯小分子受体不稳定单元英文回答：Non-fullerene small molecule acceptors (NF-SMAs) have attracted considerable attention as promising alternatives to fullerenes in organic solar cells (OSCs) due to their tunable optoelectronic properties, ease of synthesis, and low cost. However, the instability of NF-SMAs under ambient conditions remains a major challenge that hinders their practical application. The instability of NF-SMAs is primarily attributed to the presence of labile chemical bonds, such as C-C and C-H bonds, which are susceptible to attack by oxygen and moisture.Several strategies have been developed to improve the stability of NF-SMAs. One approach involves theintroduction of bulky side chains or steric hindrance around the labile bonds. This approach can effectively reduce the accessibility of these bonds to oxygen and moisture, thereby enhancing the stability of the NF-SMAs.Another approach involves the use of cross-linking agents to form covalent bonds between the NF-SMAs and the polymer matrix. This approach can prevent the NF-SMAs fromdiffusing out of the active layer and improve the long-term stability of the OSCs.The development of stable NF-SMAs is crucial for the commercialization of OSCs. By addressing the instability issues, NF-SMAs have the potential to become a viable alternative to fullerenes and enable the development of high-performance, low-cost OSCs.中文回答：非富勒烯小分子受体（NF-SMA）由于其可调的光电性能、易于合成和低成本，作为有机太阳能电池（OSC）中富勒烯的有希望的替代品而备受关注。

一类Wolbachia氏菌在蚊群传播的数学模型的动力学研究

1990 年以来，Hoffmann 和 Turelli 建立了世代不重叠的离散模型并做了解释与分析[4] [5]。2014 年， Zheng 等建立了如下时滞微分方程来研究 Wolbachia 传播动力学[6]：
DOI: 10.12677/aam.2021.105195
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应用数学进展
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注意到，模型中出生函数为线性函数。一般说来，随着种群规模的增加，其成年个体的出生率和未成年个体的幸存率将下降，这是由于种群内部的竞争引起的，而蚊群的内部竞争主要发生在未成年阶段，所以单位时间内成年个体不再是线性增长。另外，基于成年个体对资源的竞争假设，模型的死亡函数为二次函数。但是大量的事实表明，成蚊之间的竞争几乎可以忽略。基于以上考虑，本文建立一个新的时滞微分方程模型，出生和幸存函数为 Ricker 型函数，而死亡函数用线性函数。
立的随机微分方程模型，Li 等[12]建立的离散竞争模型，Huang 等[13]建立的时滞微分方程。
本文第二节主要分析了模型(2)的动力学性质，包括解的正性、有界性，平衡点的存在性和稳定性和

非自治广义Birkhoff系统的半负定矩阵梯度系统表示

非自治广义Birkhoff系统的半负定矩阵梯度系统表示王嘉航;张毅【摘要】A semi-negative definite matrix gradient system representation for a type non-autonomous gen-eralized Birkhoff system is studied.The condition under which a non-autonomous generalized Birkhoff system can be considered as semi-negative definite matrix gradient system is obtained.The characteristic of semi-negative definite matrix gradient system is then used to study the stability of non-autonomous gen-eralized Birkhoff system.Examples have been given to illustrate the applications of the results.%研究非自治广义Birkhoff系统的半负定矩阵梯度系统表示.给出了非自治广义Birkhoff系统成为半负定矩阵梯度系统的条件,利用半负定矩阵梯度系统的性质来研究解的稳定性.举例说明结果的应用.【期刊名称】《中山大学学报（自然科学版）》【年(卷),期】2018(057)003【总页数】4页(P60-63)【关键词】非自治广义Birkhoff系统;半负定矩阵梯度系统;稳定性【作者】王嘉航;张毅【作者单位】苏州科技大学土木工程学院,江苏苏州215011;河海大学土木与交通学院,江苏南京210098;苏州科技大学土木工程学院,江苏苏州215011【正文语种】中文【中图分类】O316梯度系统是一个数学系统，梯度系统是微分方程和动力系统研究中的重要问题[1-3]。

布鲁塞尔子的柱对称定态结构(二)——定态解的计算和分析

布鲁塞尔子的柱对称定态结构(二)——定态解的计算和分析刘雪萍;付美荣;张进军【摘要】本文在布鲁塞尔子的柱对称定态解构造的基础上,从布鲁塞尔子的反应扩散方程出发,利用稳定性分析和分支点理论详细地计算了布鲁塞尔子的柱对称定态解.计算结果表明,布鲁塞尔子的空间耗散结构呈柱对称,不仅随r变化,还受到z的调制;当第一分支点对应的参数kn'=k1,m'=1时,在柱的中心出现一个高浓度区.该研究结果对于了解演化着的生物化学和生命体系中的柱型结构具有一定的指导意义.【期刊名称】《山西师范大学学报（自然科学版）》【年(卷),期】2014(028)001【总页数】6页(P60-65)【关键词】布鲁塞尔子;柱对称结构;定态解【作者】刘雪萍;付美荣;张进军【作者单位】山西师范大学化学与材料科学学院,山西临汾041004;山西师范大学化学与材料科学学院,山西临汾041004;长治学院,山西长治046011;山西师范大学化学与材料科学学院,山西临汾041004【正文语种】中文【中图分类】O414.11968年，布鲁塞尔学派的Lefever和Prigogine[1,2]提出了布鲁塞尔子模型，在随后的几十年里，该模型被广泛应用于研究化学反应非线性现象中的化学反应稳定性问题[3～15].布鲁塞尔子模型由一对非线性反应扩散方程组成：(1)其中A，B是常数，X，Y是变数，D1和D2是扩散系数.该扩散方程的求解问题，是耗散结构理论的基本问题之一[2～6].深入地了解并解决这一问题将有助于我们更好地了解各种非线性化学现象，如多定态、化学振荡、化学波和化学混沌等等；同时还有助于生物稳定性的研究，比如细胞的分裂、胚胎的发展过程等等.在该扩散方程的求解问题上，前人已经做了大量的研究.Prigogine等人对该模型进行了比较深入的研究[3～7].1976年，Anchmuty和Nicolis详细地计算了布鲁塞尔子模型的一维时间周期解[8].接着Nicolis和Prigogine等人在一维的时空结构上讨论了这类现象[9].1983年，张纪岳等人在此基础上研究了布鲁塞尔子模型的球对称时空结构[10].1997年～1998年，张进军教授分别研究了布鲁塞尔子模型的轴对称结构和球破缺结构，并揭示了它在生物领域中的应用[12～14].在生物科学领域，大肠杆菌为两端钝圆的短小杆菌，本文在定态解构造的基础上，利用分支点理论详细地计算了布鲁塞尔子模型的柱对称定态结构的对称解，这对我们了解、分析大肠杆菌的分裂过程以及演化着的生物、生物化学等现象中的柱型结构有一定的指导意义.1 理论基础通过之前的分析我们已经知道，若方程(1)在柱面上满足固定边界条件(2)则适合这一边界条件的均匀定态解即热力学分支解为(3)通过线性稳定性分析[5]可得到方程(1)关于定态解(3)的线性化方程(4)其中(4)式是一个常系数线性方程组，在方程x=y= 0的边界条件下，该方程组的解(即L的本征函数)具有如下形式：(5)其中kn为零阶贝塞尔函数的第n个零点.将方程(5)代入方程(4)便得到特征方程：λ2-Tλ+Δ=0(6)其中对于取适当的由kn和m决定的特定的ωnm值，每当特征方程(6)至少有一个根λ的实部为正时，定态解(3)对ωnm的不均匀扰动是不稳定的，这样的扰动便会长大而且可能导致某种以ωnm为特征的有序结构，因此首先需要确定λ的实部为零的条件.从特征方程(6)中可得到：(7)根据分支解的存在性和稳定性定理得到结论(8)(9)它们合起来定义了B-ωnm稳定区和不稳定区的分界线.其中由(8)式确定的部分定义了稳定区和扰动能非振荡长大的不稳定区之间的的分界线，而由(9)式确定的部分定义了稳定区和扰动能振荡长大的不稳定区之间的的分界线.在前面的研究中，通过以上线性稳定性分析，我们已经得到了方程(1)的定态方程：(10)其中算子非线性部分h(x,y)=B/Ax2+x2y+2Axy.通过计算得到其第一分支点(11)并构造了求解方程(10)在第一分支点附近的定态解所需要的方程组…(12)式中an为已知函数，且a1=0…(13)(14)因此，在本文的研究中，我们将在这一构造的基础上，详细计算并分析布鲁塞尔子柱对称结构的定态解及其所蕴含的意义.2 定态柱对称解的计算通过详细的计算，我们得到方程在分支点Bc附近准至二级近似下的柱对称定态解为(15)式中，柱内浓度(16)当第一分支点对应参数kn′=k1,m′=1时，柱内浓度(17)图1是选用参数kn′=k1,m′=1，A=2，a=1，l=1，D1=0.16，D2=0.20，Bc=7.982,B=1.02Bc，经过计算得到ε=0.364 4的情况下，数值计算给出的浓度随空间位置的变化关系.图1(a)表示了在z一定的情况下浓度X随r的变化，图1(b)表示了在一定情况下浓度X随z的变化.图1 在参数kn′=k1，m′=1，A=2，a=1，l=1，D1=0.16，D2=0.20，Bc=7.982，B=1.02Bc下的二维浓度图Fig.1 Two-dimensional concentration figure with the parameters kn′=k1，m′=1，A=2，a=1，l=1，D1=0.16，D2=0.20，Bc=7.982，B=1.02Bc下的二维浓度图(a)z certain，x change with r (b)r certain，x change with z.图2是在和图1相同参数下的柱及柱内浓度变化的三维形貌图，其中，(a)为r=1，z=1的圆柱面，(b)、(c)、(d)为该圆柱沿x方向在x=-0.5，x=0，x=0.5的切片，(e)、(f)、(g)为该圆柱沿y方向在y=-0.5，y=0，y=0.5的切片，(h)、(i)、(j)为该圆柱沿z方向在z=-0.5，z=0，z=0.5的切片(在本文的三维图中，无特殊说明均这样表示).由图1可以看出：当第一分支点取kn′=k1，m′=1时，柱对称定态解的柱内浓度随r和z变化，呈柱对称结构(橄榄球结构)，在柱的中心出现了一个高浓度区.在r和z方向上，由中心到两侧浓度呈对称减小.图2 三维浓度图(参数和图1相同)Fig. 2 Three-dimensional concentration figure (The parameters are the same as in Fig. 1)(a) cylndrical surface at r=1，z=1, (b) slice at x=-0.5, (c) slice at x=0, (d) slice at x=0.5, (e) slice at y=-0.5, (f) slice at y= 0, (g) slice at y=0.5, (h) sliceat z=-0.5, (i) slice at z=0, (j) slice at z=0.5.图3是选用参数kn′=k1，m′=2，A=2，a=1，l=1，D1=0.016，D2=0.08，Bc=3.629，B=1.02Bc，经过计算得到ε=0.628 1的情况下，经过数值计算给出的浓度随空间位置的变化关系.图3(a)表示了在z一定的情况下浓度X随r的变化，图3(b)表示了在r一定的情况下浓度X随z的变化.图3 在参数kn′=k1，m′=2，A=2，a=1，l=1，D1=0.016，D2=0.08，Bc=3.629，B=1.02Bc下的二维浓度图Fig.3 Two-dimensional concentration figure with the parameters kn′=k1，m′=2，A=2，a=1，l=1，D1=0.016，D2=0.08，Bc=3.629，B=1.02Bc下的二维浓度图(a)z certain，x change with r (b)r certain，x change with z.图4是在和图3相同参数下的柱内浓度变化的三维形貌图.由图4可以看出：当第一分歧点为kn′=k1，m′≠1，与kn′=k1，m′=1的情况不同，在r方向对称，在z 方向不对称随m′的变化而变化，呈现出一定的有序结构.图4 三维浓度图(参数和图3相同)Fig.4 Three-dimensional concentration figure (The parameters are the same as in Fig. 3)(a) cylindrical surface that r equal to 1，z equal to 1, (b) slice at x=-0.5, (c)slice at x=0, (d) slice at x=0.5, (e) slice at y=-0.5, (f) slice at y=0, (g) slice at y=0.5, (h) slice at z=-0.5, (i) slice at z=0, (j) slice at z=0.5.通过以上分析，我们得出布鲁塞尔子柱对称定态解呈现出由kn′和m′共同决定的对称结构，这是由解的性质决定的.3 结论我们从布鲁塞尔子的反应扩散方程出发，在稳定性分析的基础上，应用分支点理论计算了在固定边界条件下柱对称结构的定态解.若第一分支点对应的控制参数临界值为Bc时，越过临界值柱内原来均匀对称的浓度发生对称性破缺变得不均匀，且随r和z变化，对于布鲁塞尔子柱对称结构，呈现出由参数kn′和m′共同决定的有序结构.研究布鲁塞尔子的柱型时空耗散结构对我们了解、分析大肠杆菌的分裂过程，探讨演化着的实际体系尤其是研究生物及生物化学中的柱型结构具有一定的指导意义.【相关文献】[1] Prigogine I, Lefever R. Symmetry breaking instabilities in dissipative systems. II [J]. J Chem Phys, 1968, (48):1695～1698.[2] Prigogine I. Structure dissipation and life in from theoretical physics to Biology [M]. Marois: Amsterdam, North-Holland, 1969.23～52.[3] Glansdorff P， Prigogine I.Thermodynamic theroy of structure, stability and fluctuations [M]. Wiley: New York, 1971.293～298.[4] Tyson J. Some further studies of nonlinear oscillations in chemical systems [J]. J Chem Phys, 1973, (58):3919～3930.[5] Auchmuty J F G， Nicolis G. Bifurcation analysis of nonlinear reaction-diffusion equations-I. Evolution equations and the steady state solutions [J]. Bull Math Biol, 1975, (37): 323～365.[6] Herschkowitz-Kaufman M. Bifurcation analysis of nonlinear reaction-diffusion equations-II. Steady state solutions and comparison with numerical simulations [J]. BullMath Biol, 1975, (37):589～636.[7] Marek M，Svobodová E. Nonlinear phenomena in oscillatory systems of homogeneous reactions-experimental observations [J]. Biol Chem, 1975, (3):263～273. [8] Auchmuty J F G， Nicolis G. Bifurcation analysis of reaction-diffusion equations-III. Chemical oscillations [J]. Bull Math Biol, 1976, (38):325～350.[9] Nicolis G， Prigogine I.Self-Organization in nonequilibrium systems: from dissipative structures to order through fluctuations [M]. Wiley: New York, 1977. 82～96.[10] 张纪岳，郭治安．布鲁塞尔子的球对称结构 [J]. 物理学报, 1983, (32): 1574～1585.[11] Kruel, Freund, Schneider. The effect of interactive noise on the driven Brusselator model [J]. J Chem Phys, 1990, 93: 416～427.[12] 张进军. 三分子模型的球对称破缺(Ⅰ) [J]. 山西师范大学学报(自然科学版), 1997, (11): 32～37.[13] 张进军. 三分子模型的球对称破缺(Ⅱ) [J]. 山西师范大学学报(自然科学版),1998, (12): 35～38.[14] 张进军.三分子模型球对称破缺在生物领域中的应用 [J].山西师范大学学报(自然科学版),1999,(13):25～27.[15] 李如生.非平衡态热力学和耗散结构 [M].北京: 清华大学出版社, 1986.215～239.。

一类具有交叉扩散的捕食-食饵模型正解的存在性

一类具有交叉扩散的捕食-食饵模型正解的存在性吕杨; 郭改慧; 袁海龙; 李书选【期刊名称】《《工程数学学报》》【年(卷),期】2019(036)006【总页数】9页(P658-666)【关键词】捕食-食饵模型; 交叉扩散; 正解; 存在性【作者】吕杨; 郭改慧; 袁海龙; 李书选【作者单位】陕西科技大学文理学院西安 710021; 西安交通大学数学与统计学院西安 710049【正文语种】中文【中图分类】O175.261 引言本文考虑如下具有交叉扩散的捕食-食饵模型其中Ω是RN 上的有界区域，∂Ω 是光滑边界；u, v 分别代表食饵与捕食者的密度.a, c, d,µ是正常数；α, β 和m 是非负常数；b 可能变号.当α = β = 0 且m > 0 时，系统(1)已被许多学者研究[1-10].特别地，文[2]利用分歧理论证明了正解的存在性.文[7]讨论了当某些参数充分大时，建立了系统正解的存在性、多解性、唯一性和稳定性.进一步，Du 和Lou[8]说明了当m 充分大时，奇异扰动系统存在Hopf 分歧，进而说明原系统存在Hopf 分歧.当α,β > 0 且m = 0 时，则系统(1)变成具有Lotka-Volterra 捕食-食饵模型，已经被许多生物数学家所探究[11-15].特别地，文[11]用分歧理论研究了正解的存在性，并证明共存区域随着β 增大而增大，随着α 增大而减小.进一步，文[14]讨论了当空间维数小于5 时，他们寻找到了一个与α, β 无关的先验估计.在文[16]，Wang 和Li 利用分歧理论说明当b ∈(b∗,b∗)，系统(1)至少有一个正解.特别地，当α 充分大时，他们以b 为分歧参数证明了全局分歧解在b = b∗从半平解(θa,0)出发连接到在b = (µ+1)λ1 的半平凡解(0,sϕ1)，从而形成一个有界解，其中s>0. 本文主要讨论当交叉扩散系数α 充分大时正解的极限行为.特别地，我们以a 为分歧参数，利用全局分歧理论说明极限系统的正解将随着分歧参数a 在正锥内到达无穷.令λ1(p)<λ2(p)≤λ3(p)≤···是下列问题的特征值其中我们知道λ1(p)是简单的、实的，λ1(p)关于p 是严格单调递增的.当p ≡ 0 时，我们记λ1(0)为λ1.进一步，我们记ϕ1 是λ1 对应的特征函数且满足ϕ1 >0,||ϕ1||2 =1.我们知道当a>λ1 时，下列问题存在唯一的正解θa，且a →θa 在a ∈(λ1,+∞)是连续的，θa 关于a 是单调递增的.进一步，θa 非退化，线性稳定.2 预备知识则我们有系统(1)可以写成引理1 令(u,v)是系统(1)的任意正解，则存在一个与α 无关的常数C1，使得证明令(u,v)是系统(1)的任意正解，则根据比较原理，我们知道其中U∗是方程在Ω 上且满足U∗|∂Ω =0 的解.从而说明 u, U 存在与α 无关的先验估计.对于V 的方程，我们有同理，我们说明v, V 存在与α 无关的先验估计.定义下面的引理说明a∗(b)的一些性质.由于证明过程比较简单，我们在此略去其证明而仅陈述其结果.引理2[11] 集合S 形成一个有界的曲线其中a=a∗(b)是b ∈(0,(µ+1)λ1]的正的连续函数，且满足下面的性质：(i) a=a∗(b)关于b ∈ (0,(µ +1)λ1)是严格单调递减的；(ii) a∗((µ +1)λ1)= λ1.在此，我们定义ϕ∗>0 满足下面的方程对p>N，我们定义Banach 空间X 和Y 如下通过Sobolev 嵌入定理，我们知道3 主要结论在本节中，我们主要考虑当α 充分大时，极限系统正解的存在性.通过引理1 我们知道，系统(1)或(2)存在与α 无关的先验估计.下面的引理表明当α 充分大时，系统(1)的任意正解收敛到下述系统(4)的正解. 引理3[16] 存在充分大的Λ，使得当α ≥Λ，且α=αi 时，(ui,vi)是系统(1)的任意正解，则上成立，其中是下面系统的正解我们通过分歧理论建立系统(4)正解的存在性，而这只需建立下面系统(5)正解的存在性.我们通过分歧理论说明当a ∈(a∗,∞)时，系统(5)至少存在一个正解.令则系统(4)可以写成当a>λ1 时，系统(5)存在半平凡解我们以a 为分歧参数，利用局部和全局分歧理论来建立系统(5)正解的存在性.令其中是的函数.通过在处Taylor 展开，令则显然f(θa,0)=−∆θa, g(θa,0)=0.令则其中令K 是−∆在齐次Dirichlet 边界条件下的逆算子，则定义T :R×X →X则是X 空间上的紧算子.令显然H(a;0,0)=0，且记的Frchet 导数为下面，我们利用局部分歧理论说明系统(5)在(a∗;θa∗,0)附近存在局部分歧解.定理1 设a > λ1，则(a∗;θa∗,0) ∈ R × X 是系统 (5)的分歧点，且存在δ > 0，在分歧点(θa,0)的邻域内存在如下形式的正解其中是光滑函数且满足证明令则令则如果矛盾.因此a=a∗, ker L1(a∗;0,0)=span{ψ∗,ϕ∗}，其中我们定义算子L1(a∗;0,0)的伴随算子为令则根据Fredholm 选择公理，我们知算子L1(a∗;0,0)的值域为因此R(L1(a∗;0,0))的余维数是1.为了在处应用局部分歧理论[17]，我们断言显然我们采用反证法.假设存在满足也就是在上面的等式两边同乘以ϕ∗，再积分，我们有矛盾.根据全局分歧理论，我们断言局部分歧解可以延拓为全局分歧解，并且该全局分歧解随着分歧参数a 在正锥内延伸到无穷.令假设σ ≥ 1 是算子T′(a)的特征值且其对应的特征函数为(ξ,η)，则则σ ≥ 1 是算子T′(a)的特征值的充要条件是：存在一些i = 1,2,···，使得a = ai(µ).进而，若a<a∗，则i(T(a,·),0)=1，如果a∗ <a<a2(1)，则i(T(a,·),0)= −1.根据全局分歧理论[18]，我们知道在R+×X 内，存在从(a∗;0,0)出发的连通分支C0，满足H(a;ξ,η) = 0，且在(a∗;0,0)附近，H(a;ξ,η)的所有零点都在定理1 中得到的那条分歧曲线令则C 是系统(5)从(a∗;0,0)分歧的解曲线，令在(a∗;θa∗,0)的小邻域内，解曲线C ⊂ P0.定理2 C −{(a∗;θa∗,0)}在正锥内随着分歧参数a 延伸到无穷.证明根据Rabinowitz 全局分歧理论[18]和更加一般的分歧理论[19,20]，我们断言全局分歧解C −{(a∗;θa∗,0)}必满足下列三个选择之一：(i) 全局分歧解C 连接半平凡解{a;θa,0}，其中 aa∗且I − T′(a)是不可逆的；(ii) 全局分歧解C 在R×X 是无界的；(iii) 存在a = 和W ∈\{0}，满足(;W) ∈C，其中是空间{(0,−∆ϕ∗)}的补空间，{(0,−∆ϕ∗)}由定理1 给出.如果则存在使得在Ω 的极限.由于由最大值原理我们知道，在Ω.假设令则根据二阶椭圆型方程的正则化理论，我们假设在C1 上成立且满足和在Lp 弱收敛.在上面系统的两边同时取极限，我们有根据最大值原理我们知道，>0 在Ω 上成立.因此由a∗的定义我们知道，有=a∗，矛盾.假设类似地，我们可以得到矛盾.因此，我们说明C −{(a∗;θa∗,0)} ⊂ P0.进而(i)不可能发生.由于在Ω 上ϕ∗>0，补空间不可能包含不变号的元素，从而(iii)也不可能发生.由Lp 估计和Sobolev 嵌入定理，我们知道存在正常数，使得因此，全局分歧解C 只能随着分歧参数a 在正锥内到达无穷.参考文献：【相关文献】[1]袁海龙，李艳玲.一类捕食-食饵模型共存解的存在性与稳定性[J].陕西师范大学学报(自然科学版)，2014, 42(1):15-18 Yuan H L, Li Y L.Coexistence of existence and stability of a predator-prey model[J].Journal of Shaanxi Normal University (Natural Science Edition), 2014, 42(1): 15-18[2]Blat J, Brown K J.Global bifurcation of positive solutions in some systems of elliptic equations[J].SIAM Journal on Mathematical Analysis, 1986, 17(6): 1339-1353[3]Brown K J.Nontrivial solutions of predator-prey systems with smalldiffusion[J].Nonlinear Analysis:Theory Methods & Applications, 1987, 11(6): 685-689[4]Casal A, Eilbeck J C, Lpez-Gmez J.Existence and uniqueness of coexistence states for a predator-prey model with diffusion[J].Differential & Integral Equations, 1994, 7(2): 411-439[5]Conway E D, Gardner R, Smoller J.Stability and bifurcation of steady-state solutions for predator-prey equations[J].Advances in Applied Mathematics, 1982, 3(3): 288-334[6]Dancer E N.On positive solutions of some pairs of differential equations[J].Transactions of the American Mathematical Society, 1984, 284(2): 729-743[7]Du Y, Lou Y.Some uniqueness and exact multiplicity results for a predator-preymodel[J].Transactions of the American Mathematical Society, 1997, 346(6): 2443-2475 [8]Du Y,Lou Y.S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model[J].Journal of Differential Equations, 1998, 144(2): 390-440[9]Du Y, Shi J.A diffusive predator-prey model with a protection zone[J].Journal of Differential Equations,2006, 229(1): 63-91[10]Du Y, Peng R, Wang M.Effect of a protection zone in the diffusive Leslie predator-prey model[J].Journal of Differential Equations, 2009, 246(10): 3932-3956[11]Kadota T,Kuto K.Positive steady states for a prey-predator model with some nonlinear diffusion terms[J].Journal of Mathematical Analysis & Applications, 2006, 323(2): 1387-1401[12]Kuto K, Yamada Y.Multiple coexistence states for a prey-predator system with cross-diffusion[J].Journal of Differential Equations, 2004, 197(2): 315-348[13]Kuto K.A strongly coupled diffusion effect on the stationary solution set of a prey-predator model[J].Advances in Differential Equations, 2007, 12(2): 145-172[14]Kuto K, Yamada Y.Coexistence problem for a prey-predator model with density-dependent diffusion[J].Nonlinear Analysis, 2009, 71(12): e2223-e2232[15]Nakashima K, Yamada Y.Positive steady states for prey-predator models with cross-diffusion[J].Advances in Differential Equations, 1996, 1(6): 1099-1122[16]Wang Y X, Li W T.Stationary problem of a predator-prey system with nonlinear diffusion effects[J].Computers & Mathematics with Applications, 2015, 70(8): 2102-2124 [17]Crandall M G, Rabinowitz P H.Bifurcation from simple eigenvalues[J].Journal of Functional Analysis,1971, 8(2): 321-340[18]Rabinowitz P H.Some global results for nonlinear eigenvalue problems[J].Journal of Functional Analysis,1971, 7(3): 487-513[19]Lpez-Gmez J.Spectral Theory and Nonlinear Function Analysis[M].Boca Raton: Chapman and Hall/CRC, 2001[20]Dancer E N.Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one[J].Bulletin of the London Mathematical Society, 2002, 34(5): 533-538。

李波-生物技术药物的非临床安全性研究

生物技术药物的非临床安全性研究李波中国药品生物制品检定所国家药物安全评价监测中心药物非临床安全性评价的目的z确定人体试验安全启始剂量和剂量递增方案z确定潜在的毒性靶器官, 以及毒性的可逆性z确定可毒性监测指标, 并可用于临床监测确定毒性监指标并用临床监生物技术药物分类z重组蛋白类：生长因子、细胞因子、激素、受体、重组蛋白类生长因子细胞因子激素受体酶类、凝血因子单克隆抗体类嵌合人源化z单克隆抗体类：嵌合、人源化z基因治疗类：基因转移产品、细胞治疗z疫苗类：疫苗、DNA疫苗疫苗类疫苗疫苗z组织器官移植产品：自身、异体生物技术药物安全性研究有关的技术指南ICH S6: 生物技术药物的非临床安全性评价指南主要内容:•选择相关的动物种属•剂量选择与研究周期确定•免疫原性•安全药理学、遗传毒性和致癌性试验•单克隆抗体药物的组织交叉反应研究•局部耐药性生物技术药物与小分子药品区别小分子药物生物技术药物•布洛芬环磷酰胺•胰岛素单克隆抗体小分子量•能具亲电子性•潜在活性代谢物大分子量•典型的非亲电子性•代谢物是小肽与氨基酸•化学合成•毒性常常为“非靶毒性”•发酵•与药理作用有关的毒性•多数不产生免疫原性•可产生免疫原性非临床安全性研究的比较生物技术药物z药理作用可用于相关动物小分子药物z代谢参数用于相关动物筛筛选z可以用单一种属选z要求啮齿和非啮齿类z有免疫原性z不要求遗传毒性z一般没有免疫原性z要求遗传毒性z生殖毒性z不要求代谢研究z要求生殖毒性z要求代谢研究z致癌实验z要求致癌实验相关动物种属的选择相关动物种属的择动物种属的选择非常重要与药理学作用相关动物种属•可能只有一种关联种属可供选择可能只有种关联种属可供选择药物的活性源于明确的靶点(受体)或抗原决定簇•般不能使用标准种属动物B l一般不能使用标准种属动物(大鼠和Beagle犬)•非人灵长类动物可能是相关动物不鼓励对非相关种属动物进行毒理学研究•不可靠的安全数据•无预期的非靶毒性相关动物选择的方法蛋白质氨基酸系列同源性比较体外受体结合试验组织交叉反应体外受体结合试验、组织交叉反应动物体内药理活性筛选例：高度保守的蛋白：例高度保守的蛋白G-CSF和EPO在很多动物上有活性例外：高度变异蛋白：和在很多动物同源性很低在很多动IL-2和IL-6在很多动物同源性很低，在很多动物上也存在药理活性相关动物选择的方法z非人灵长类动物与人类紧密相关，常表现相似药理作用z若不存在相关动物，可用表达人类药物靶标的转基因动物z有时没有体内动物模型，可使用受体/组织结合实验体外实验证明受体的相关性对灵长类对非啮齿啮齿选替代实受体有活类受体有对啮齿类受体选择替代实验系统，如：转基因动物性活性有活性转基因动物，同系物使用相关动物进行PK/TK、毒性实验支持临床验支持临床试验生物技术药物安全性评价中实验动物的选择相关动物选择：举例相关动物选择举例EPO--IgG Fcz EPO的分布信息：体外细胞试验，免疫组化技术FcR 的分布信息：体外细胞试验，免疫组化技术FcRz FcRBeagle 犬、猴犬、猴体内动物试验：大鼠、Beaglez体内动物试验：大鼠、z结果：结果Fc 分布信息缺乏分布信息缺乏Fc犬>猴> Beagle 犬体内抗体产生情况：大鼠> Beagle体内抗体产生情况：大鼠关注点：半衰期延长，红细胞升高带来的风险毒性试验周期/频率/途径z试验周期1、3或6 个月13或6个月z ICH S6 要求研究通常不超过6个月z给药频率给率z通常相当或高于临床拟给药频率z可以增加给药频率z给药途径z静脉或皮下z其他途径–肌肉注射, 关节内注射, 鞘内注射剂择剂量选择•相关问题•受毒性的限制，可能不能达到最大耐受量•由于药理作用，可能很难达到无明显作用水平剂量•剂量设计参考•靶点结合/饱和度•最大主要药效活性剂量•最高临床预期剂量ICH S6工作组建议最大剂量设置根据AUC可以•ICH-S6工作组建议最大剂量设置根据AUC，可以为人体拟用量的5倍药代动力学代z生物技术药物的ADME特性:吸收–途径常为静脉, 皮下和肌肉分布–大分子量蛋白主要存在于体循环内代谢–通常为蛋白降解物排泄受体调节的清除–免疫原免疫原性z生物技术药物通常对动物具有免疫原性z重复给药毒性实验中要对免疫原性进行评价，确定抗体形成是否影响药效、毒性和价确定抗体形成是否影响药效毒性和药物暴露z在动物身上出现的免疫原性，不能完全推测人体免疫原性免疫原性免疫原z产生抗体不是终止实验的充分理由，可以考虑提高剂量或增加给药频率以饱和抗体反应，以提高剂量或增加给药频率以饱和抗体反应以保证药物的暴露量如果抗药抗体的发生不是100%，可以通过增加每组中的动物数量以保持有效的研究z若在多数动物中抗体中和药物活性，可终止实验抗药性抗体（ADA）的影响抗抗体z增加清除和减小药物的摄取z能改变组织分布z减少药物的药理学活性z减少清除和增加半衰期/摄取z与内源性蛋白质的交叉反应抗药性抗体引起的毒性举例：形成免疫复合物在非临床研究中免疫复合物的形成内皮细胞炎症巨噬细胞活化和沉积是最常见的ADA 引起的毒性–通常无相关内皮细胞药物免疫复合物抗药性抗体联的人类风险非临床评价:损害的部位(肾, 肺, 关节)毒性，药物暴露和抗药性抗体之间的相关性荧光免疫组织化学免疫金标记和电子显微镜方法免疫毒性z生物技术药物常影响免疫系统对免疫系统的直接作用对免疫系统的直接作用：如：细胞因子类，IL-2引起血管渗漏综合征，IFNs、TNF引起流感样症状对免疫系统的间接作用：对免疫系统的间接作用如：自身抗体的产生，自身免疫性疾病的诱发和细胞免疫的改变免疫毒性z反复给药毒性实验，可进行免疫器官组织病理，免疫器官重量，血液学检查z体外免疫效应细胞实验（ex vivo）z自身免疫反应和过敏反应的预测方法缺乏常用免疫功能检测方法z免疫细胞表型分析：淋巴细胞；全血测定z天然免疫功能：自然杀伤（NK）细胞；巨噬细胞（Mфassay）功能；中性粒细胞功能中性粒细胞（Neutrophile）功能z获得性免疫（Adaptive immunity）功能-体液免疫(T淋巴细胞依赖性抗体反应) TDAR(T淋巴细胞依赖性抗体反应)TDARB细胞淋转化获得性免疫功能细胞介导免疫z获得性免疫功能－细胞介导免疫T淋巴细胞转化试验细胞毒性T淋巴细胞（CTL）试验迟发超敏反应（DHR）z宿主抵抗力试验：肿瘤攻击实验；细菌寄生虫攻击实验单次给药毒性实验z生物技术药物毒性较弱，一般很难得到致死剂量和发现严重毒性，单次给药毒性得到的信息很少，很难为后续实验提供依据z如果可能可与安全药理结合进行反复给药毒性z给药周期至少要达到临床拟给药时间，临床使用〈7天，给药周期2周，长期用药，给药周期6月给药频率应等于或大于临床给药频率若z给药频率应等于或大于临床给药频率，若临床给药频率按半衰期决定，也应根据半衰期进行遗传毒性z生物技术药物和其降解产物通常被认为不会直接与DNA或染色体相互作用z只有在含有有机连接物（organic linker）时，才考虑进行遗传毒性实验，处方敷料或有新杂质出现时，也应考虑z含有非天然氨基酸致癌性实验z一般不需要致癌性实验z有促进细胞增殖作用的，应关注其致癌性有促进细胞增殖作用的应关注其致癌性注意反复给药毒性中组织病理学结果体外细胞增殖实验z若仍有疑问，可考虑进行标准致癌实验若仍有疑问可考虑进行标准致癌实验（受试物在动物上要有活性和没有免疫原性）生殖毒理学研究殖毒学研究若非啮齿类动物中有药理活性，可以使用种非啮•若非啮齿类动物中有药理活性，可以使用一种非啮齿类动物•若仅在非人灵长类中有药理活性•生育力毒性可在一般毒性中研究•胚胎-胎儿毒性•围产期毒性•可以选择一种啮齿动物同源蛋白质用于生殖毒性研究基因治疗产品安全性研究z病毒载体：病毒复制能力的回复突变z转移基因在非靶组织分布，尤其生殖腺分布z基因组整合（插入突变），尤其生殖细胞整合z转移基因的异常表达，持续表达z载体、基因、表达产物对宿主免疫的影响载体基因表达产物对宿主免疫的影响基因治疗类相关动物选择z产品特点、临床适应症，给药方式z所选动物对基因产物和基因转运系统的生物反应与人体反应相关有复制能力病毒载体所选动物应对病z有复制能力病毒载体，所选动物应对病毒感染敏感，与人类野生型病毒感染的病理变化相似疫苗类非临床安全研究注意问题z毒性实验前要完成免疫原性研究，确定相关动物和免疫程序z考虑相关动物、免疫途径、剂量、暴露的频率和时程终点评价的时间频率和时程、终点评价的时间z重点考虑免疫毒性，根据疫苗的特性确定免疫学指标。