什么是生物數(shù)學(xué)

HOPFBIFURCATIONINASYNAPTICALLYCOUPLEDFHNNEURONMODELWITHTWODELAYS

LipingZhang

CollegeofScience,NanjingUniversityofAeronauticsandAstronautics2010/07/30什么是生物數(shù)學(xué)？生物數(shù)學(xué)是生物學(xué)與數(shù)學(xué)之間的邊緣學(xué)科，用數(shù)學(xué)方法研究和解決生物學(xué)問(wèn)題，也對(duì)與生物學(xué)有關(guān)的數(shù)學(xué)方法進(jìn)行理論研究。對(duì)于今天的生物學(xué)者，數(shù)學(xué)的價(jià)值更應(yīng)該體現(xiàn)在建立在數(shù)量化基礎(chǔ)上的"模型化"。通過(guò)數(shù)學(xué)模型的構(gòu)建，可以將看上去雜亂無(wú)章的實(shí)驗(yàn)數(shù)據(jù)整理成有序可循的數(shù)學(xué)問(wèn)題，將問(wèn)題的本質(zhì)抽象出來(lái)。兩個(gè)最近的事例

SARS在對(duì)SARS的研究中，生物數(shù)學(xué)就發(fā)揮了作用。2003年春SARS暴發(fā)時(shí)，在有效的疫苗和抗病毒藥物研制出來(lái)之前，科學(xué)家最關(guān)心的是SARS流行的特征。兩個(gè)國(guó)際合作的研究小組使用了"SEIR"數(shù)學(xué)模型，對(duì)SARS的傳播趨勢(shì)進(jìn)行分析和預(yù)測(cè)，給有關(guān)部門提供了參考意見。AvianInfluenzaBirdflu

Avianinfluenzaisadiseaseofbirdscausedbyinfluenzavirusescloselyrelatedtohumaninfluenzaviruses.Transmissiontohumansinclosecontactwithpoultryorotherbirdsoccursrarelyandonlywithsomestrainsofavianinfluenza.Thepotentialfortransformationofavianinfluenzaintoaformthatbothcausesseverediseaseinhumansandspreadseasilyfrompersontopersonisagreatconcernforworldhealth.AvianInfluenzaBirdflu

生物數(shù)學(xué)幾個(gè)領(lǐng)域的基本介紹

種群動(dòng)力學(xué):種群的相互作用生物資源管理和綜合害蟲控制流行病動(dòng)力學(xué)藥物動(dòng)力學(xué)生物數(shù)學(xué)中的斑圖生物信息學(xué)生物數(shù)學(xué)已有一百年多年的歷史：?1798年Malthus人口增長(zhǎng)模型?1908年遺傳學(xué)的Hardy-Weinbe“平衡原理”?1925年Volterra捕食與被捕食模型?1927年KM傳染病模型?1973年許多著名的生物學(xué)雜志相繼創(chuàng)刊?現(xiàn)如今“生物信息學(xué)”的誕生是生物數(shù)學(xué)發(fā)展的里程碑時(shí)滯時(shí)滯對(duì)生物種群的影響一直是生物學(xué)家關(guān)心的問(wèn)題，時(shí)滯經(jīng)常出現(xiàn)在生物的活動(dòng)中。例如我們?nèi)粘Ｉ钪杏龅降囊曈X和聽覺的時(shí)滯現(xiàn)象、動(dòng)物血液再生原理，森林再生原理等?？紤]到種群密度變化對(duì)于增長(zhǎng)率的影響都不是瞬間發(fā)生的,而是與過(guò)去的生活狀態(tài)有關(guān),即有時(shí)間滯后的，還有動(dòng)物消化食物也需要一定的時(shí)間。在生物數(shù)學(xué)模型中如果引入時(shí)滯，相應(yīng)的動(dòng)力系統(tǒng)就變成了帶時(shí)滯的非線性動(dòng)力系統(tǒng)。由于時(shí)滯生物動(dòng)力系統(tǒng)的演化不僅依賴于系統(tǒng)的當(dāng)前狀態(tài)，還依賴于系統(tǒng)過(guò)去某一時(shí)刻或若干時(shí)刻的狀態(tài)，其運(yùn)動(dòng)方程要用泛函微分方程來(lái)描述，和常微分方程系統(tǒng)所描述的系統(tǒng)不同，時(shí)滯對(duì)系統(tǒng)的動(dòng)態(tài)性質(zhì)有很大的影響，時(shí)滯動(dòng)力系統(tǒng)一般有無(wú)窮多個(gè)特征值，解空間是無(wú)限維的，其理論分析往往很困難。目前，對(duì)于非線性時(shí)滯動(dòng)力系統(tǒng)尚沒有針對(duì)性特別強(qiáng)的研究方法，討論非線性常微分方程的方法，大多可以經(jīng)過(guò)改造用于非線性時(shí)滯微分方程的研究。例如研究生物動(dòng)力系統(tǒng)平衡點(diǎn)存在唯一性方法有：不動(dòng)點(diǎn)定理、M-矩陣和重合度理論等；平衡點(diǎn)局部穩(wěn)定性分析最基本的方法仍是考察特征方程根的變化，例如無(wú)害時(shí)滯不改變系統(tǒng)正平衡位置的漸近穩(wěn)定性，所以利用時(shí)滯為零時(shí)系統(tǒng)的漸近性去研究時(shí)滯不為零時(shí)系統(tǒng)正平衡位置的局部穩(wěn)定性，即用線性近似法研究研究平衡點(diǎn)的局部穩(wěn)定性問(wèn)題。對(duì)小時(shí)滯模型用平均法，對(duì)常數(shù)時(shí)滯以及連續(xù)時(shí)滯模型的全局穩(wěn)定性主要用Lyapunov方法。研究分岔現(xiàn)象的常見方法有：中心流形法、規(guī)范形理論、Lyapunov-Schmidt方法、攝動(dòng)法和多尺度法等。1.assumptions

Thebasicmodelmakesthefollowingassumptions:(H)

Themodelisgivenbythefollowingsystem:FanD,HongL.HopfbifurcationanalysisinasynapticallycoupledFHNneuronmodelwithdelays.CommunNonlinearSciNumerSimulat(2009),doi:10.1016/sns.2009.07.0252.StabilityforFHNneuronmodelwithonedelayObviously,E(0,0,0,0)isanequilibriumofsystem(1),linearizingitgivesThecharacteristicequationassociatedwithsystem(2)isgivenbyWhere,,Forand,Eq.(3)becomes(4)ByRouth-Hurwitzcriterionweknowthatif(H)issatisfiedthenallrootsofEq.(3)havenegativerealparts.3.BifurcationforFHNneuronmodelwithonedelayObviously,iv(v>0)isarootofEq.(4)ifandonlyif(5)Separatingtherealandimaginarypartsgives((6)Takingsquareonthebothsidesoftheequationsof(6)andsummingthemup,andlety=v2,whichleadsto:where(7)DenoteThenwehave(8)(9)Set(10)Let,then(10)becomes(11)where,Define(12)Withoutlossofgenerality,weassumethatEq.(7)hasfourpositiveroots,denotedby,,and,respectively.ThenEq.(6)hasthefourpositiverootswehave(13)DenoteWherek=1,2,3,4;ThenisapairofpurelyimaginaryrootsofEq.(4)withSimilartotheprovesof[8]weknowthatEq.(7)hasmorethanonepositiveroots.Thenthestabilityswitchmayexist.Summarizingtheabovediscussionswecanensurethestabilityinterval.(14)Theorem3.1Supposethat(H)issatisfiedandIftheconditions(a)(b),△△,and(c),△△,andthereexistsasuchthatandarenotsatisfied,thenthezerosolutionsofsystem(1)isasymptoticallystableforall.Ifoneoftheconditions(a),(b)and(c)of(1)issatisfied,thenthezerosolutionofsystem(1)isasymptoticallystablewhenIfoneoftheconditions(a),(b)and(c)of(1)issatisfied,and,thenthesystem(1)undergosaHopfbifurcationat(0,0,0,0)when4.StabilityandHopfbifurcationforFHN

neuronmodelwithtwodelayNowletbearootofEq.(2)Thenweget(16)WhereTakingsquareonthebothsidesoftheequationsof(14),weget(15)(15)IfEq.(15)haspositiveroot,withoutlossofgenerality,weassumeEq.(15)hasNpositiveroots,denotedby。NoticeEq.（12）weget(16)Define.LetbetherootofEq.(4)Satisfying.Bycomputation,wegetWhereSummarizingthediscussionsabove,wehavethefollowingconclusions.Theorem4.1Supposethat(H),holdandEq.(14)haspositiveroots.andhavethesamemeaningaslastdefinition.Weget(1)AllrootofEq.(4)havenegativerealpartsforandtheequilibriumofsystem(2)isasymptoticallystablefor.(2)Ifhold,thensystem(2)undergosaHopfbifurcationattheequilibriumE,when.5.StabilityanddirectionoftheHopfbifurcationIntheprevioussection,weobtainedconditionsforHopfbifurcationtooccurwhen.InthissectionwestudythedirectionoftheHopfbifurcationandthestabilityofthebifurcationperiodicsolutionswhen,usingtechniquesfromnormalformandcentermanifoldtheory.WeassumeLettinganddroppingthebarsforsimplification（17）Whereand(18)WhereFromthediscussioninSection2,weknowthatsystem(10)undergosaHopfbifurcationat(0,0,0)when,andtheassociatedcharacteristicequationofsystem(10)withhasapairofsimpleimaginaryroots.ResultBasedontheaboveanalysis,wecanseethateachgijcanbedeterminedbytheparameters.Thuswecomputethefollowingquantities:(29)Theorem5.1.In(29),determinesthedirectionofHopfbifurcation;if,thentheHopfbifurcationissupercritical(subcritical)andthebifurcationperiodicsolutionexistfor;determinesthestabilityofthebifurcationperiodsolution;bifurcatingperiodicsolutionarestable(unstable)if;anddeterminestheperiodofthebifurcatingsolution:theperiodincreases(decreases)if.normalformandcentralmanifoldtheoryNu