Feller’s Contributions to Mathematical Biology Ellen Baake∗ and Anton Wakolbinger† Abstract 5 1 This is a review of William Feller’s important contributions to mathematical biology. 0 The seminal paper [Feller1951] Diffusion processes in genetics was particularly influen- 2 tial on the development of stochastic processes at the interface to evolutionary biology, n and interesting ideas in this direction (including a first characterization of what is nowa- a J daysknownas“Feller’sbranchingdiffusion”)alreadyshapedupinthepaper[Feller1939] 1 (written inGerman)Thefoundations ofaprobabistic treatment ofVolterra’s theory ofthe 2 struggle for life. Feller’s article On fitness and the cost of natural selection [Feller1967] ] contains acriticalanalysisoftheconceptofgeneticload. O The present article will appear in: Schilling, R.L., Vondracek, Z., Woyczynski, W.A.: H TheSelected PapersofWilliamFeller. Springer Verlag. . h t a 1 Introduction m [ Feller had a persistent interest in biology. This is documented in numerous examples from 1 v mathematical genetics in his monograph [Feller1950, Feller1966], and by a couple of influ- 8 ential research papers at the interface of population biology and probability theory. Looking 7 2 back at these papers in historical perspective is highly rewarding: They are cornerstones of 5 biomathematics; they mirror the development of probability theory of their time; and at least 0 oneofthem([Feller1951])had lastingimpacton probabilitytheory. . 1 Feller’s important papers on the interface to biology are [Feller 1939], [Feller 1951], and 0 5 [Feller 1967]. The first one addresses general population dynamics, the other two are mainly 1 concerned with models in population genetics. The area of population dynamics is concerned : v with the growth, stabilisation, decay, or extinction of populations. Models of population dy- i X namics describe how the size of populations changes over time under given assumptions on r birthand death rates ofindividuals,whichmaydepend on thecurrent populationsizesincein- a dividualsinteract(e.g.compete)witheachother. Incontrast,populationgeneticsisconcerned withthegeneticcompositionofpopulationsundertheactionofvariousevolutionaryprocesses, suchasmutationandselection. Naturally,thereisnosharpboundarybetween thefields,aswe willalsoseein Feller’s contributions. Let usnowlookat them. 2 Feller and population dynamics In [Feller1939], a paper still in German entitled (in English translation) The foundations of a probabilistictreatmentofVolterra’stheoryofthestruggleforlife,Fellerpresentsasynthesisof ∗FacultyofTechnology,BielefeldUniversity,Box100131,33501Bielefeld,Germany †InstitutfürMathematik,Goethe-Universität,Box111932,60054FrankfurtamMain,Germany 1 two fundamental developmentsthat both started in 1931. On the one hand, Volterra presented hisbookLessonsaboutthemathematicaltheoryofthestruggleforlife[27];ontheotherhand, Kolmogorov published his seminal paper On analytical methods in probability theory [17]. Volterra’s book laid the foundations for the deterministic description of population dynamics in terms of systems of ordinary differential equations that model birth, death, and interaction of individuals. These models imply that populations are so large that random fluctuations can be neglected, and population sizes are measured in units so large that the size can be consideredacontinuousquantity. Kolmogorovpresentedthegeneralandsystematicformalism for the description of stochastic dynamics in terms of Markov chains in continuous time; in particular, he found the description for the evolution of probability weights and the transport of expectations in terms of differential equations, which we know today under the names of Kolmogorovforwardequationsand Kolmogorovbackward equations. In his 1939 paper, Feller ties these two fundamental developments together by applying Kolmogorov’s new formalism to some examples of Volterra’s population dynamics. We see here the birth of the stochastic description of population dynamics, which today has its firm placeinmathematicalbiology,andishighlydevelopedbothinanalyticaltermsandintermsof simulations. Feller’spaperisdevotedtothedescriptionofsinglepopulations(exceptfromasmallexcur- siontopredator–preymodelsintheend)andconsistsoftwolargeparts. Thefirstestablishesthe Kolmogorovforwardequations(KFE)fortheMarkovjumpprocesses(namely,birth-and-death processes)thatdescribefinitepopulations(remarkably,thereisnomentionoftheKolmogorov backward equations in this paper). The second part discusses a continuum analogue of such processes, a special case of which seems to be the first appearance of what today is called Feller’sbranchingdiffusion. It is remarkable to see (and a pleasure to read) that Feller notices some of the crucial rela- tionshipsbetweencorrespondingdeterministicandstochasticmodelsinthisearlypaper,which appearas acentral theme. For the sake of clarity, let us make explicit here the two fundamental limits of birth-death processes that are addressed in [Feller1939]. Consider a birth-death process K (t) with birth N rate nl and death rate nm when in state n, with N being the initial population size. Then, as the initial population size N tends to ¥ , the sequence of process (K (t)/N) , N = 1,2,..., N t≥0 converges indistributionto thesolutionofthedifferentialequation (1) x˙=(l −m )x, x(0)=1. This reflects a dynamical version of the Law of Large Numbers (see e.g. [19]). (Notably, due tothelinearity,theexpectationM(t):=E(K (t))satisfies(1)aswell.) Adifferentkindoflimit 1 emerges if one assumes that the individual split and death rates l and m depend on N and the process isnearlycriticalin thesensethat l =b +q /N and m =b +q /N, N 1 N 2 with q −q =: a . The Law of Large Numbers then says that the limit of the processes 1 2 (K (t)/N) is the constant 1. However, on a larger time scale the fluctuations become visi- N t≥0 ble: the sequence of processes (K (Nt)/N) converges in distribution to the solution of the N t≥0 stochasticdifferentialequation(5.1′)statedinparagraph3.1.1,whosediffusionequationis(4). Thisisaprototypeofadiffusionlimitforbirth-deathprocesses. In[Feller 1939],theselimiting proceduresarenotmadeexplicit(butsee[Feller1951]foramajorstepinthisdirection). Feller in 1939goes rather theotherway, in search for stochasticprocesses that correspond to agiven deterministicmodel. Let usnowexplainthemajorlinesofhisarticle. 2 2.1 Markov jump processes for population dynamics Inthefirstpart(Sections1–4),devotedtothestochasticdescriptionoffinitepopulations,Feller explains a variety of birth-and-death processes and sets up the Kolmogorovforward equations for them, i.e. he establishes the system of differential equations that describe how the proba- bilityweights forthe numberofindividualsaliveat timet evolveovertime. He starts with the simplelinear death process (where each individualdies at rate l , independently of all others), proceeds viathecorrespondingbirth process and thelinearbirth-and-deathprocess and finally arrivesatthegeneralbirth-and-deathprocess. Inanindividual-basedpicture,thelatterincludes interaction between individuals, so that the birth and/or death rates are no longer linear in the number of individuals. The case of logistic growth, which includes a quadratic competition term, serves as an important example; the case of ‘positive interaction’ (such as symbiosis) is nottreated explicitlyhere. Let uscommenton themajorinsightsofthispart. 2.1.1 Kolmogorov equations, their solutions, and relationship with deterministic de- scription. Feller notices that for a given net reproduction rate a per individual, by choosing l −m =a , oneobtainsavarietyoflinearbirth-deathprocesseswhoseexpectationvalueM(t)satisfiesone and the same ODE (1), whereas for a >0, there is exactly one linear pure birth process (l = a ,m = 0) with this property. Feller states this ambiguity explicitly when discussing logistic growth. Itsdeterministicversionisgivenbythedifferentialequation (2) m˙ =m(l −g m)=: f(m), which Feller also calls the Pearl-Verhulst equation. Here m is shorthand for m(t), the ‘deter- ministicversion’ofthepopulationsizeattimet,l denotesthepercapitanetreproductionrate in the absence of competition, and g is the competition parameter. Again, Feller notices that there are many possibilities in terms of birth-death processes that correspond to (2). They are parametrised in his Eq. (27), which describes the process with per capita birth at rate w −n n and per capita death at rate t −s n ifthere are currently n individuals. Here, wehave renamed g in Feller’sEq. (27)byn in ordertoachievecompatibilitywiththenotationin(2). FellerstartsoutbycalculatingtheexplicitsolutiontotheKFE ofthepurelineardeathpro- cess, that is, thenumberofinidividualsaliveat timet;hestates thisas theresult ofarecursive construction. With a typo corrected (el t −1 must be replaced by 1−e−l t in his formula (6)), thissameformulasaysthatthenumberofinidividualsaliveattimet hasabinomialdistribution with parameters N and e−l t if there are initiallyN individuals. (Today, after [Feller 1950], we would conclude this immediately, without solving systems of Kolmogorov forward equations, viatheprobabilisticargument that thereare initiallyN independentindividuals,each ofwhich dies at rate l and is therefore alive at time t with probability e−l t.) Likewise, the solution of the pure linear birth process with per capita birth rate l , which he gives in his Eq. (17), is the negative binomial distribution with parameters N and e−l t, which arises as the distribution of the sum of N independent random variables that are geometrically distributed with parameter e−l t. Again, this has a nice interpretation as the offspring of N independently reproducing ancestors. For thegeneral birth process, with arbitrary birth rates p , Feller notes that theKFE define n a probability distribution if and only if either only finitely many of the p are positive, or if n (cid:229) 1/p diverges; this is a standard textbook result today (usually presented in the general- n n isation to birth-and-death processes). Under the conditions stated, he also gives the explicit solutioninpassing. 3 2.1.2 Themomentsofthestochasticprocess, andtheirrelationshipwiththedeterminis- ticequation Fellerisparticularlyinterestedintheexpectation,variance,andothermomentsofthe(random) number of individuals alive at time t. In a trendsetting way, he does not calculate them from the explicit solution of the KFE, even where this is known; he rather uses the KFE to derive differentialequationsforthemoments. LetM(t)=(cid:229) kP (t)=E[K(t)]betheexpectednumber k k of individuals at time t. As stated above, Feller observes that, for the linear birth-and-death process, M(t) follows the differential equation for the deterministic population model, and hence the expectation of the stochastic process coincides with the deterministic solution. In contrast, forthe logisticmodel,he finds from thedifferential equationrelating thefirst and the second momentthat (3) M˙ < f(M) with f of Eq. (2). From this he argues that M is always less than the solution of the logistic equation. An alternativeway tosee(3)wouldbeto observethattheKFE gives d E[K(t)]=E[g(K(t)] dt whereg(k)=(cid:229) Q(k,n)n and n Q(n,n+1)=l n, Q(n,n−1)=g n2, Q(n,n)=−(l n+g n2), Q(k,n)=0 otherwise. As a matter of fact, it turns out that g(k)=l k−g k2, which is strictly concave, and hence (3) is a consequence of Jensen’s inequality. Since Feller does not consider models with positive interaction (such as symbiosis) in this part of the paper, he does not encounter the convex situation. 2.2 Diffusion equations for population dynamics Thesecondpartofthepaper(Sections5–8and10)isdevotedtothediffusionlimitofstochastic populationdynamics. WecannotresisttoquoteFeller’sthoughtsfromthebeginningofSection 5,formulatedinanalmostliteraryGerman,aboutthesubstantiallymorelithesomeprobabilistic treatment, in which the population size is no longer assumed as integer-valued, and where he alludesto similaritiesto theBrownian motion: Wir wenden uns nun der anderen von der in der Einleitung erwähnten wahr- scheinlichkeitstheoretischenBehandlungsweisendesWachstumsproblemszu,welche wesentlich geschmeidiger ist, und bei der die Grösse der Population nicht mehr ganzzahlig vorausgesetzt wird. Den Mechanismus des Vorgangs kann man sich hierähnlichwiebei der BrownschenMolekularbewegungvorstellen. Der Zustand der betrachtetenPopulation,d.h. ihre gesamteLebensenergie ist einer dauernden Veränderungunterworfen[...] Starting from the transition density, Feller calculates the infinitesimal drift a(x) and the infinitesimal variance b(x) (provided they exist). With remarkable intuition, and a clear view ofthebranchingproperty,hestatesthat,inthecaseofastochasticallyindependentreproduction oftheindividuals,a(x)and b(x) mustbeproportionaltox. Again,let usquotein German: 4 Nimmt man beispielsweisean, dass die Grösse der Populationkeinen Einflusshat auf die durchschnittliche Vermehrungsgeschwindigkeit der Einzelindividuen, d.h. dass diese untereinander stochastisch unabhängig sind [...], so müssen a(x) und b(x)offenbarproportionalzux sein[...] Thisgivesrisetohis equation(38),which reads as ¶ w(t,x) ¶ 2(xw(t,x)) ¶ (xw(t,x)) (4) =b −a . ¶ t ¶ x2 ¶ x Herew(t,·)isthedensityofpopulationsizeattimet,anda andb arepositiveconstants. This seems to be the first appearance of what became famous as Feller’s branching diffusion. We willcomeback tothisinSection 3.1. It is interesting to note that the diffusion process in this second part of the paper is not derivedfromthebirth-deathjumpprocesseswhichFellerhaspresentedinthefirstpart;maybe, at this early stage, the subtle rescaling required for this limit was not yet at his fingertips. A decade later, however, he had these techniques; see Section 3.1 on Diffusion processes in genetics. In 1939, Feller does allude to the birth-death processes, but theconnection is not yet clear. For example, he tells us that the a in (4) corresponds to the l encountered in the pure birth process. This is correct for the expected growth rate, but as a matter of fact a pure birth process cannot have the diffusion limit (4), since the paths of the former can only increase in time,whereas thepathsofthelatterhavefluctuationsinbothdirections. This issue reappears when Feller discusses the extinction probability of the diffusion pro- cess. He notes the important fact that this quantity increases with b , since it is tied to the fluctuations of theprocess, and at the sametimeemphasises as a sort of paradox that, evenfor a positivenet growth a >0, the diffusionprocess may die out with positiveprobability,while the population described by the deterministic differential equation (1), as well as a pure birth process,cannotdieout. (Thisparadoxisresolvedwhenonehasinmindthedifferentrescalings thatlead to (1)and(4).) Followingtheseconsiderationsofthelinearbirth-and-deathprocess,Fellerincludesdepen- dence between individuals in Sections 8 and 10. He presents two specific examples. The first is his Equation (51), which is the diffusion version of his Equation (7) and known today as Feller’sbranchingdiffusionwithlogisticgrowth[20,23]. Thesecondisthe(two-dimensional) diffusion describing a two-species model with predator-prey interactions, which now is also called Lotka-Volterraprocess, see Eq. (1.2) in [3]. As with the jump processes in the first part ofhispaper,Fellerisconcernedwiththemomentsofthediffusionprocessesandwritesdowna general recursionforthekthmomentsM . Inthetwo-speciesmodel,theinteractionispositive k (from the point of view of the predator), so we finally encounter the convex case, in which the expectationisgreater thanthesolutionofthecorrespondingODE. InSection9,Fellermakessomefinalremarksconcerningthedeterministiclimitofboththe birth-deathjumpprocessesandthebranchingdiffusion. Thesearebrief,heuristiccalculations, which hint at the convergence of the stochastic models to Volterra’s population models in the limit of infinite population size. Today, powerful laws of large numbers are availablefor large classesofsuchprocesses[8,Chap.11]. Theygofarbeyondthesimplelinearcasealludedtoin (1);rather,theyincludequitegeneralformsofdensitydependence. Thisleadsustothepresent stateofpopulationdynamics. 5 2.3 Afterthoughts Today, 75 years after [Feller 1939], stochastic population dynamics constitute a vibrant area of research, so wide that it is impossible to give an overview in a short paragraph. Suffice it to say that major questions raised by Feller continue to be ardent research themes. Above all, this is true of interactions within and between populations. Even simple models for the competition of two populations, whose deterministic limit can be tackled as an easy exercise, turn into hard problems when considered probabilistically. Specifically, diffusionmodels with interactionhavebecomeobjectsofintenseresearch, see, e.g., [6, ?] andreferences therein. Inthecontextofthiscommentary,itisparticularlynoteworthythataclassofmodelsknown underthenameofadaptivedynamicsbringstogetherecologicalaspects(onashorttimescale) and genetical aspects (on a longer time scale) and thus builds a bridge between population dynamicsandpopulationgenetics. Aniceoverviewofthistopicandmanyothersmaybefound inthemonographbyHaccou,Jagers,andVatutin[12]. LetusnowturntoFeller’scontributions topopulationgenetics. 3 Feller and population genetics As already laid out in Section 1, important processes in population genetics are those that describe the evolution of type frequencies, or in other words, of proportions of subpopulation sizes within a total population, whose size may vary as well. In this context, we may think of theindividualsas genes,where each geneis ofa certain type,say aorA. The foundations of mathematical population genetics were laid starting in the 1920s by Fisher, Wright, and Haldane. Their work mirrors the genetics of their time, today known as classicalgenetics. Ithadtorelyonthephenotypicappearanceofindividuals(colourofflower, surface structure of peas, body weight, milk yield ...). The molecular basis of genetics was stillunknown,sogeneshad tobetreated as abstractentities. When moleculargeneticsentered the labs in the 1960s, population genetics changed dramatically, with Kimura as a leading figure, see Section 3.2.1. The next (and, from a 2014 perspective, the last) big leap took place in 1982, when Kingman introduced the genealogical perspective via the coalescent process. Comprehensive overviews of population genetics theory are given in the textbooks by Ewens [9] and Durrett [5]; for coalescent theory in particular, we further recommend Berestycki [2] (from amathematicalpointofview)and Wakeley [28](froma morebiologicalperspective). With his contributions to population genetics, Feller thus was in the midst of an important line of development. We will comment on two of these articles. The first, Diffusion processes ingenetics[Feller1951],isalandmarkcontributiontowardsstochasticmodellingandanalysis via diffusion processes, and, as a matter of fact, reaches far beyond population genetics as such. Thesecond,On fitnessandthecostof naturalselection[Feller 1967], usesdeterministic modelling(and istherefore similarin spirittothe‘Volterraequations’). 3.1 Diffusion processes in genetics Feller’s article Diffusion processes in genetics [Feller 1951] appeared in the Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, which took place in 1950. The central role of [Feller 1951] is nicely put into perspective by the following quote from Thomas Nagylaki’s review [24] on Gustave Malécot and the transition from classical to modernpopulationgenetics: 6 Mathematicalresearchindiffusiontheoryinfluencedpopulationgeneticsonlygrad- ually. As described in more detail below, Wright was unaware of Kolmogorov’s (1931) pioneering paper, and Wright, Malécot, and Kimura were all apparently unacquainted with Khintchine’s (1933) book.[...] Thus, the mutually beneficial cross-fertilization between diffusion theory and population genetics did not start untilFellerpublishedhisseminal1951paper. In theintroductionofthatpaper, Feller setsthestageby writing: Relatively small populations require discrete models, but for large populations it ispossibletoapplya continuousapproximation,andthisleadsto processesofthe diffusiontype. Two diffusion processes are in the focus of the paper. One is what is nowadays called Feller’sbranchingdiffusion,theotheristheso-calledWright–Fisherdiffusion. Fellerdescribes thembytheirdiffusionequations(5.1)and(7.1),whicharetheKolmogorovforwardequations (orFokker–Planckequations)forthedensities,herecalledu(t,x),cf. Section2.2. Fellerwrites on pp.228–229: It is known that an essential part of Wright’s theory is mathematically equivalent to assuming a certain diffusion equation for the gene frequency (that is, the pro- portionof a-genes). In a footnoteon thesamepage, Feller giveshints tothe rootsof thisknowledgein thework of Kolmogorov,Fisher,Wright,and Malécot. 3.1.1 A foresight: Feller’sdiffusions assolutionsofstochasticdifferential equations Nowadays we do not hesitate to write the process described by Feller’s equations (5.1) and (7.1)as solutionsofstochasticdifferentialequationsinthesenseofItô: (5.1′) dZ = 2b Z dW +a Z dt, t t t t p (7.1′) dY = 2b Y(1−Y)dW +(g (1−Y)−g Y)dt, t t t t 2 t 1 t p where W is a standard Brownian motion. Feller legitimately resisted writing the processes in this form. In [Feller 1952], which grew out of Feller’s invited lecture at the International Congress ofMathematiciansintheyear1950,hewrites aboutItô’sStochasticAnalysis: This approach has the advantage that it permits a direct study of the properties of the path functions, such as their continuity, etc. In principle, we have here a possibility of proving the existence theorems for the partial differential equations [...] directlyfromthepropertiesof thepath functions. However, themethodisfor the time being restricted to the infinite interval and the conditions on [the diffu- sion and drift coefficients] a and b are such as to guarantee the uniqueness of the solution. So far, therefore, we cannot obtain any new information concerning the “pathological”cases. 7 3.1.2 Anemerging theme: Whathappens attheboundary? Indeed, the state spaces of (5.1) and (7.1) are not the ‘infinite interval’ (−¥ ,¥ ) but [0,¥ ) and [0,1], and it took 20 years until T. Yamada and S. Watanabe proved that the coefficients in the above stated SDEs are good enough to guarantee strong uniqueness of the solution, see [31] and also[30]. Acouplingargumentfrom StochasticAnalysisthenguarantees thatthesolution of (7.1′) converges in law to the unique equilibrium distribution whose density is the unique invariant probability density of (7.1), which is the Beta(g /b ,g /b )-density. Thus, although 2 1 for g < b the random path Y hits 0 with probability one (and similarly for g < b it hits 1 1 2 with probability one), these visits to the boundary do not lead (as conjectured by Feller on p. 239) to a non-vanishing accumulation of the masses concentrated at x=0 and x=1 which ismaintainedinthesteadystate,inotherwords,thecoefficient m inhisequation(7.3)isinfact equal to1. Questions like these may have been one source of motivation for Feller to initiate his groundbreaking studies on the boundary classification of diffusion processes, see his footnote on p. 234, where he speaks of boundary conditions of an altogether new type, and the one on p.229addedinproof,whereheannouncesthatasystematictheory,includingthenewboundary condition,is to appear in theAnnalsof Math. Feller’s classificationof boundariesis reviewed and commentedinSection 2ofMasatoshiFukushima’sessay inthisvolume. 3.1.3 The diffusionapproximationofthe Wright–Fisherchainand beyond As already indicated, another important aspect that is taken up in Feller’s paper is that of the diffusion approximation, i.e. the convergence of a sequence of (properly scaled) discrete pro- cessestothesolutionof(5.1)and(7.1),respectively. Intheformercasetheunderlyingdiscrete process is a Galton–Watson process, in the latter it is the Wright–Fisher Markov chain. The transitionprobabilitiesoftheWright–Fisherchainaregivenby(3.2),(3.4)and(3.5). Thediffu- sivemass-time-scalingisgivenby (8.5): aunit oftimeconsistsofN generations,and aunit of massconsistsofN (orhere2N)individuals,withN beingthetotalpopulationsize. Thescaling (8.4)oftheindividualmutationprobabilitesa ,a isthatofweakmutation,whichleadsinthe 1 2 scaling limitto the infinitesimalmean displacement a(x) and theinfinitesimalvariance2b(x), see (8.6) and (8.7). In the context of (7.1), the drift coefficient a(x) is due to the effect of mutation, and the diffusioncoefficient b(x) describes the strength of the fluctuations that come from the random reproduction. (In order to be consistent with (7.1), b should be replaced by i g in (8.4), (8.6) and (8.9)). The ‘convergence of generators’ which emerges from (8.6) and i (8.7)can beliftedto theconvergenceofthecorrespondingsemigroups,seee.g. thechapteron GeneticModelsinthemonograph[8]byEthierandKurtz. Theconvergencetheoremsin[8]complywithFeller’sprogrammaticproposal: Itshouldbe provedthat ourpassageto thelimitactuallyleads from(8.2)to (8.6), i.e. from theprobability weights of the Wright–Fisher chain to the probability densities the Wright–Fisher diffusion. To achieve this, Feller proposed an expansion into eigenfunctions (in particular he found the eigenvalues of the Wright–Fisher transition semigroup) and checked part of the convergence in Section 8 and Appendix I. Such a representation is not required in the systematic approach presented in [8]. Still, the approach via eigenfunctions is interesting in its own right, and has been extensivelyused inMathematicalBiology. At thebeginningofSection 9 (entitledOtherpossibilities)Feller writes: ThedescribedpassagetothelimitwhichleadstoWright’sdiffusionequation(7.1) is different from the familiar similar processes in physical diffusion theory where 8 the ratio D x/D t tends to infinity rather than to a constant. It rests entirely on the assumption (8.4) [of weak mutation]. We shall now see that any modification of this assumption leads to a non-singular diffusion equation of the familiar type (to normaldistributions). Indeed, for the scaling (9.1), (9.2) a = g e ,i = 1,2, Ne → ¥ , which corresponds to strong i i mutation,Fellerstatesalawoflargenumbers,i.e.aconvergenceofthetypefrequenciestothe g g equilibrium point 2 , 1 , and argues that the (properly scaled) process of fluctuations (cid:16)g1+g2 g1+g2(cid:17) around this equlibrium point converges to a process whose probability density satisfies the diffusionequation(9.10)(and thusisan Ornstein-Uhlenbeckprocess). 3.1.4 The diffusionapproximationofGalton–Watsonprocesses Thediffusionequation(5.1)appearedalreadyin[Feller 1939],seeEq.(4)inSection2.2. How- ever, as we have seen there, certain issues concerning the (scaling) limits of Galton–Watson processes had remained unrevealed in [Feller1939]. Towards 1950, Feller was ready to at- tack this. As to the diffusion approximation of a sequence of “nearly critical” Galton–Watson processes by (5.1), Feller gives a proof in Appendix II. His idea is to take the iterates f of n theoffspringgenerating function(which are knownto describe thegeneratingfunctions ofthe subsequentgenerationsizes)totheirscalinglimit. ThislimitturnsouttosatisfythePDE(12.9) (which,in turn,correspondsto (5.1)). Fellerwrites: We effect this passage to the limit formally: it is not difficult to justify these steps, since the necessary regularity properties of the generating functions f (x) were n establihedbyHarris[13]. Again, from today’s perspective, an alternative way is provided by the convergence of gener- ators, see [8]. In the very last lines of his Appendix II, Feller remains a bit sketchy when he writesthat the boundary condition u(t,0) follows from the fact that in the branching process theprobabilitymassflowingoutintotheorigintendstozero. In fact, for the solution Z of (5.1′) (with Z = 1, say), the probability mass flowing out into 0 the originis non-zero at any fixed timet, and thedensity of Z does not vanish near the origin. t Again, the desire to obtain clarity on questions like these may have been a motivation for Feller’s thenupcomingresearch on theboundarybehaviourofone-dimensionaldiffusions. 3.1.5 From two-type tocontinuum-type generalizationsofFeller’s branching diffusion In the introduction, Feller points out that serious difficulties arise if one wishes to construct populationmodelswith interactionsamongtheindividuals,and thatthesituationgrowsworse ifthepopulationconsistsofdifferent typesof individuals. Hethen writes: Infact,thebivariatebranchingprocessleadstosuchdifficultiesthatapparentlynot one single truly bivariate case has been treated in the literature. In the theory of evolutionthisdifficultyisovercomebytheassumptionofaconstantpopulationsize [...] In Section 10 the assumptions of constant population size is dropped and a trulybivariatemodelisconstructedwhichtakesintoaccountselectiveadvantages in a more flexible way. [...] The same limiting process which leads [...] to the diffusion equation of Wright’s theory can be applied to our new bivariate model andleadsto a diffusionequationintwo dimensions. 9 These two-dimensional Markov processes with branching property have been taken up and analysedinabroadercontextin1969inthepaper[30]whichcarriesthattitle. Alreadybefore, Watanabehadpublishedhisseminalpaper[29]whichestablishedFeller’sbranchingdiffusions with a continuum of types. This together with the poineering work of Don Dawson gave rise to a class of processes that were later called Dawson–Watanabe superprocesses ([6, 4]). A goodpartofPerkins’SaintFlourLecureNotes(part2of[4])isdevotedtosuperprocesseswith interactions, and thus is fully on the line of Feller’s program to construct population models withdifferenttypes ofindividualsand withinteractionsamongtheindividuals. 3.1.6 The inner lifeofFeller’s branching diffusion: excursions andcontinuum trees This is a good place to mention another fascinating development which is connected with Feller’sbranchingdiffusionsandisassociatedwiththenamesofDanielRayandFrankKnight (thelatterwas Feller’s doctoralstudentand Ed Perkins’PhD advisor). Thankstothebranchingproperty(andthetherebyimpliedinfinitedivisibility),therandom path Z of a Feller branching diffusion is a Poissonian sum of countably many “Feller excur- sions”z . Infact, eachofthemhasan“internallife”inthesensethatz isthesizeattimet ofa t continuum populationoriginating from one singleancestor. The genealogical tree of this pop- ulationcanbedescribedbyaBrownianexcursionh reachinglevelt,whichcanbeimaginedas the‘explorationpath’ofacontinuumrandomtreewhosemassaliveatlevelt isz . Thesecond t Ray–Knight theorem says that z can be represented as the local time spent by h at levelt. In t a discrete setting of Galton–Watson processes, this is ancticipated in Harris’ work [11] with its section on walks and trees. The correspondence between a Feller excursion z and an Itô excursionh is depicted onthefirst pageof[25], framed by pictures ofFellerand Itô, whomet inpersonatPrincetonin1954. See[25]formoreexplanations,andreferences togroundbreak- ingdevelopmentsthatdealtwiththegenealogicalstructurebehindFeller’sbranchingdiffusion, such as Aldous’ Continuum Random Tree (which plays in the world of random trees a similar role to that of Brownian motion in the classical invariance principle) and Le Gall’s Random Snake, which provides a representation of the Dawson–Watanabesuper-Brownian motionas a continuum-tree-indexedMarkovmotion. For more on excursions and excursion point processes in relation with Feller’s work, see Section 3.1ofthecontributionofM.Fukushima. 3.1.7 Frequencies inmultivariatecontinuum branching: conditioning and timechange AnotherinterestingquestionwhichFelleraddressesat theendofhisintroductionconcerns the relativefrequencies inabivariatemodelofbranchingdiffusions. Fellerwrites: [...] it is to be observed that in no truly bivariate case does the gene frequency satisfy a diffusion equation (Sections 6 and 10). In fact, if the population size is notconstant,thenthegenefrequencyisnotarandomvariableofaMarkovprocess. Thus, conceptually at least, the assumption of a constant population size plays a largerrolethanwould appearonthesurface. Indeed, as it turns out (and Feller may have been well aware of this), one way of passing from (5.1) to (7.1), say with a =g =g =0, is to consider two independent Feller branching 1 2 diffusions (solutions of (5.1′)) Z(1) and Z(2), conditioned to Z(1)+Z(2) = 1. Of course, this must be given a precise meaning, and this has been done in a much more general context in the papers by Etheridge and March [7] and Perkins [26]. The title of Perkins’ paper is 10