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Equilibrium states of two stochastic models in mathematical ecology Yu, Feng 2005

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Equilibrium States of Two Stochastic Models in Mathematical Ecology by Feng Yu A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) The University of British Columbia January 2005 © Feng Yu, 2005 Abstract This work deals with two problems arising in mathematical ecology. The first problem is concerned with diploid branching particle models and its behavior when rapid stirring is added to the interaction. The particle models involve two types of particles, male and female, and branching can only occur when both types of particles are present. We show that if the branching rate is sufficiently large, this particle model has a nontrivial stationary distribution, i.e. one that does not concentrate all weight on the all-0 state, using a compari-son argument due to R. Durrett. We also show extinction for small branching rates, thereby establishing the existence of a phase transition. We then add two different rapid stirring mechanisms to the interactions and show that for the particle models with rapid stirring, there also exist nontrivial stationary distribution(s); for this, we analyze the limiting P D E and establish a condition on the PDE that guarantees existence of nontrivial stationary distributions for sufficient fast stirring. The second problem deals with a model of sympatric speciation, i.e. speciation in the absence of geographical separation, originally proposed by U. Dieckmann and M . Doebeli in 1999. We modify their original model to obtain several constant-population particle models. We concentrate on a continuous-time model that converges to a deterministic dynamical system as the number of particles becomes large. We establish various results regarding whether speciation occurs by studying the existence of bimodal stationary distributions for the limiting dynamical system. Contents Abstract ii Contents iii Acknowledgements v 1 Introduction and Overview 1 I Existence of Nontrivial Stationary Distribution for the Diploid Branching Particle System with Rapid Stirring 2 2 The Particle Models 3 2.1 Diploid Branching Particle Model 4" 2.2 Description of the Particle Model with Rapid Stirring 7 2.3 Convergence to a PDE for Lily-pad Stirring 9 2.4 Convergence to a PDE for Individual Stirring 11 3 Results on the Diploid Branching Particle Model 14 3.1 Existence of Stationary Distributions 14 3.2 Extinction for Sufficiently Small X/5 16 3.3 Survival for Sufficiently Large X/5 16 4 Convergence Theorem for Individual Stirring 22 5 Existence of Invariant Stationary Distribution For Lily-pad Stirring 31 5.1 Lower Bounds: Existence of d\ and D\ in Condition (*) 32 5.1.1 Analysis of the ODE (5.2) 36 5.1.2 Analysis of the PDE (5.1) 49 5.2 Upper Bounds: Existence of d2 and D2 in Condition (*) 60 II Stationary Distributions of A Model of Sympatric Speciation 61 6 A Model on Sympatric Speciation 62 6.1 Introduction 62 iii 6.1.1 The Dieckmann-Doebeli Model 62 6.1.2 A conditioned Dieckmann-Doebeli model 64 6.1.3 A Moran Model with Competitive Selection . . . 68 6.2 The Particle Model 69 6.2.1 The Strong Selection Model 70 6.2.2 The Weak Selection Model 73 7 The Selection-Mutation Equation 76 7.1 Mild Competition: b close to 1 78 7.2 Intense Competition: b close to 0 81 7.2.1 Study of A One-dimensional System 82 7.2.2 Large enough fi 85 7.2.3 Small fi: Existence of Mike Stationary Measure 88 7.2.4 Small ii: Existence of Bimodal Stationary Measure 90 8 Stationary Distributions 97 Appendix A A Result on the Conditioned Dieckmann-Doebeli Model 109 iv Acknowledgements I would like to thank the external examiner, Robert Adler, and the university examiners, Michael Doebeli and Jim Zidek, for their feedback, suggestions, and generous comments. I am also grateful for the help and guidance provided by members of my supervisory com-mittee, Martin Barlow and John Walsh, and my research supervisor, Ed Perkins, through all my years at UBC. Without them, this would never have been possible. F E N G Y U The University of British Columbia January 2005 Chapter 1 Introduction and Overview This work consists of two parts, each of which involves a class of models arising from a problem qf mathematical ecology. In the first problem, we study diploid branching particle system models. This class of particle systems differs from the "usual" models that one normally finds in the literature, in that there are two types of particles, modelling the male and female populations, and branching (i.e. birth of new particles) requires the presence of both male and female particles. In the second problem, we study various particle models that are all related to a sympatric speciation model proposed in [Dieckmann and Doebeli 1999]. In both particle models, we are mainly concerned with the equilibrium behaviour. More specifically, we show that the stationary distributions of the particle models have desirable properties, e.g. nontriviality (i.e. does not concentrate all weight on the all-0 state) in the first problem and bimodality in the second problem. 1 Par t I Existence of Nontrivial Stationary Distribution for the Diploid Branching Particle System with Rapid Stirring Chapter 2 The Particle Models In this part of our work, we consider a type of particle systems that can be used to model sexual reproduction of a certain species. This work was inspired in part by [Dawson and Perkins 1998]. In that paper, the following system of stochastic partial differential equations is the object of study: ^(t,x) = ^Au(t,x) + (ju(t,x)v(t,x))1/2W1(t,x) ^(t,x) = ±Av(t,x) + (7u(t,x)v(t,x))1/2W2(t,x), (2.1) where A = VJ 4 J^? is the Laplacian, 7 > 0, and Wi(t,x) (i = 1,2) are independent space-time white noises on R + x K. One can associate u(x,t) and v(x,t) with the male and female populations of "particles" (respectively) at spatial location x and time t. Loosely speaking, (2.1) says that individual male or female particles moves around according to Brownian motion, but branching is only possible when both male and female particles are present at the same spatial location. Notice that at spatial locations where the female population is 0, the branching rate for the male population is also 0, therefore the male population does not "die" and the only effect on the male population at those spatial locations is the diffusive effect of the heat kernel A . This behaviour is not very realistic, since one would expect a "natural" death rate of male particles even without the presence of any female particles. In this work, we study a model involving a finite number of male and female particles with more (somewhat) "realistic" behaviour. The model we study involves two types of particles, male and female, residing on the integer grid S = Zd or eZd. More specifically, each site x £ S contains two nests, one for the male particle and the other for the female particle. Each nest can be inhabited by at most 1 particle, either male or female. Let E = {0,1} and F = E x E be the set of possible states at each site in S. For x G S, we write where ^(x) denotes the number (0 or 1) of male particles at site x, and £,2(x) denotes the number of female particles at site x. We define the interaction neighbourhood Af = {0,yi,...,yN}, 3 and the neighbourhood of x M x = x + H . For example, TV = {0, — 1,1} if we have nearest-neighbour interaction on Z . Let Ci(x,m,f) denote the rate at which nest m (rn = 1,2) of site x flips to state i (i = 0,1), and assume Ci(x,m, £) depends only on the neighbourhood Afx, i.e. a(x,m,P) = hi,m(f(x),f(x + yi),...,£,(x + yN)) for some function /it,m : FN+1 —> R + . The death rate Co is constant, c o ( ^ m , 0 = ( 0 ) Q t h e r w i s e , (2-2) while the birth rate ci(x,m,£) is positive only if both male and female particles can be found in M x . For example, the diploid branching particle model we consider in Chapter 2.1 a bit later has _ J Xni(x,l , ^ i ^ O ^ z . O , i f ? m ( x ) = 0 , . C l ( x ' m ' ° = ^ - otherwise ' ( 2 ' 3 ) where nrn,(x,f) = \{z&Nx:(m'(x + z) = l}\, i.e. at rate A, each pair of male and female particles in M x give birth to a particle at nest m of site x if that nest is not already occupied. A more stringent condition, as in the particle model with rapid stirring we consider in Chapter 2.2 a bit later, is to require both parent particles to reside at the same site, i.e. where ni+2(x,0 - \{z G Afx :t\x + z) = 1 and f2(x + z) = 1}|. This more stringent condition should not alter the behaviour of the particle system if one allows a larger A than in (2.3), but it does help to simplify the analysis somewhat. 2.1 Diploid Branching Particle Model We first describe the model with birth and death rates as in (2.2) and (2.3), for which we will establish the existence of nontrivial stationary distribution(s) and consequently a phase transition later in Chapter 3. First, we restate the model in words: 1. Birth. For each nest (x,m) and each pair (zi,z2) S JVX x Mx such that ^ (^ i ) = 1 and f2(z2) = 1, where z\ and z2 need not be distinct, with rate A, a child of (z\,z2) is born into nest m of site x if (a;, m) is not already occupied. 4 2. Death. Each particle dies at rate 5. We can think of this particle system as a generalized spin system, generalized in the sense that the phase space at each site is {0, l } 2 rather than {0,1}. One can refer to Chapter 3 of [Liggett 1985] for a detailed introduction on classic spin systems. We observe that the "all-0" state (i.e. ^(x) — £,2{x) = 0 for all a;) is an absorbing state; therefore the probability measure that concentrates only on the "all-0" state is a trivial stationary distribution. We say a stationary distribution is nontrivial if it does not concentrate only on the "all-0" state. A major goal of this work is to establish the existence of nontrivial stationary distributions for various particle systems. This interacting particle system involving the birth and death mechanisms described above can be constructed using a countable number of Poisson processes [Durrett 1995]. Without any loss of generality, we assume A and 8 to be < 1, since we can just slow down time by max(A,<5) if either A > 1 or S > 1. Define We assume c* < oo. Let {T^hm : n > 1} be the arrival times of independent rate c* Poisson processes, and {U%,l'm : n > 1} be independent uniform random variables on [0,1]. At time t = T * ' i , m , nest (x,m) flips to state i if LW> m < a(x,m,£t_)/c*, and stays unchanged Since the number of Poisson processes is infinite, there is no first flip and the exis-tence and uniqueness of the process from this construction is not completely trivial. One can, however, use Theorem 2.1 of [Durrett 1995] to find a small to such that the spatial grid S can be divided into an infinite number of components, each of which is finite and no two of which interact during time [0,£n]. This allows construction of the process up to time to, and by iterating this procedure, we can construct the process for all t. One can also see easily that this construction is unique. Alternatively, one can explicitly write down the generators Ql and Q2 associated with the particle system with death rates (2.2) and birth rates (2.3) and (2.4) respectively: otherwise. E SC(x)(f(Z - 5x,m) ~ /(0) (x,m)eSx{l,2} + E ^ X ( ! / )^W(l-rCa:)) ( /« + ^ ,m)-/(0) (2.5) G2f(0 E - Sx,m) - /(0) (x,m)eSx{l,2} + £ xe(y)e(y)(i-C(x))(m + sx,m)-f(0) (2.6) 2/G.A4 5 where / has compact support and Sx,m is a function on 5 x {1,2} that is one at (x, m) and zero everywhere else, and apply Theorem B3 in [Liggett 1999] (Theorem 1.3.9 in [Liggett 1985] only gives the Markov property) to see that Q is a Markov generator and therefore determines a unique ({0,1} 2) 2 ' Feller Markov process. An important consequence of the construction using Poisson processes described in the last paragraph is that the semigroup Ttf{£o) — .E ? < ) /(£t) corresponding to the particle system is a Feller semigroup. As in Corollary 2.3 of [Durrett 1995], one can show that if £o ~~* £o> then for t < to, ' / (£") ~* ^ ° / ( £ t ) since S consists of components that are finite and do not interact with each other during [0,*o]. One can then iterate this for as many times as one likes. Summarizing results from the three previous paragraphs, we have the following theorem: Theorem 2.1.1. There exists a unique Feller process £t constructed as before with genera-tor (2.5) or (2.6). One can represent this construction graphically, for which we give an example with S = Z and N = {-1,0,1} in figure 2.1. Let m G {1,2}, x,y,z G S, {Rn'm,n > 1} be independent Poisson processes with rate <5, and {T£'m'y'z,n > 1}, with y,z G Afx, be independent Poisson processes with rate A. At space-time points ((x,m), R*,m), we draw a symbol <5 to indicate that the particle (if any) residing at (x,m) is killed at time R%m. At space-time points ((x,m),T%'m'y'z), we draw arrows from ((y, l),T ,*>m ' 2 / , z) and ((z, 2),T£'m'y'z) to {(x, m), T£>m<y<z) to indicate that a birth event will occur at nest (x, m) if (x, m) is not already occupied and nests (y, 1) and (z, 2) are both occupied at time T£'m'v'z. In figure 2.1, the bottom line represent the state (occupied or empty) of nests at t = 0. We use thick lines to represent occupied (wet) nests, and thin lines to represent empty nests. Without any birth or death event, the state of a nest remains unchanged as t increases. At a death event, i.e. at points marked by S, a thick line is changed to a thin line, while a thin line remains unchanged. And at a birth event, the state of nests at the origins of the two arrows pointing at (x, m) is checked - if they are both occupied, then a thin line at (x, rn) is changed to a thick line, while a thick remains unchanged; otherwise, nothing happens. In Chapter 3, we will use this graphical construction to establish the existence of a nontrivial stationary distribution for the diploid branching particle model if A/<5 is sufficiently large, and extinction if A/<5 is sufficiently small. The particle system £ with generator (2.5) or (2.6) is attractive in the sense that £ is monotonic in initial conditions. One can check that if £o(x) < £o(x) for all x G S, where (0,0) < (0,1) < (1,1) and (0,0) < (1,0) < (1,1) but (0,1) g (1,0), then ^(s) < &(a;) for all x and t. This is true since every birth or death event preserves the inequality <. For example, if £t-(a:) = (0,0) and tft_(x) = (0,1), and at time t there is a male birth event at site x, then £,t{%) = (1>0) and tft(^ ) = ( l j l ) , so the inequality £t(x) < £t(z) n a s been maintained. Similarly, one can check that the particle system £ is increasing in the birth rate A and decreasing in the death rate S, by coupling the random variables T^'* ; m and U%'l'm involved in the constructions in the obvious way. Because of this monotonicity, along with the existence of nontrivial stationary distributions for sufficiently large A/5 and extinction 6 1 i i • 1 • S ' i T 6 1 1 1 5 • [• 1 1 1 I - 2 -1 0 1 2 Figure 2.1: Graphical representation: the solid lines represent nests (x, 1), while the dotted lines represent nests (x, 2). Thick lines indicate occupied (wet) nests, while thin lines indicate empty nests. for sufficiently small X/S which we will establish a bit later in this work, we may conclude that there is a phase transition in the behaviour of the particle system £. 2.2 Description of the Particle Model with Rapid Stir-ring If we add rapid stirring to the particle system, i.e. we scale space by e and "stir" neighbouring particles at rate e~2 in addition to performing the birth and death mechanisms, then the particle system converges to the solution of a reaction-diffusion PDE as e —> 0 (see Theorems 8.1 and 8.2 in [Durrett 1995] and the beginning of Chapter 2.3). This PDE represents the "mean-field" behaviour of the particle system and is usually easier to analyze than the particle system itself. As promised earlier, we will establish later in Chapter 3 that there is a phase transition for the diploid branching particle model (i.e. without rapid stirring), but obtaining any reasonable estimates on exactly where this transition occurs seems to be difficult. One advantage of adding rapid stirring mechanisms is that one can get a pretty good idea where the phase transition occurs in the rapidly stirred particle model by 7 analyzing the limiting PDE, or simulating this PDE on a computer. Moreover, this convergence establishes a connection between the particle model and P D E systems, which is of independent interest. Since many PDE's arise out of natural systems, this connection justifies the study of the PDE. The underlying stochastic system can also yield information about the PDE; for example, in our case, as we will see in Chapter 2.3, the monotonicity of the particle system will lead to the mono tonicity of the PDE. Information about the PDE will similarly yield information about the particle model. In Chapter 5, we will establish condition (*) on the PDE (see page 31), which will tell us that there exist nontrivial stationary distributions for the particle system with sufficiently small e. For the particle models with rapid stirring, we work with S = eZd, and denote the corresponding process by £ e . We also assume the birth and death rates in (2.2) and (2.4) (i:e. generator Q2 in (2.6)) with 5 = 1, while the neighbourhood Af is nearest neighbour: Here we use the I^-norm: ||y|| = ^2k=iDk- In addition to the transitions in the diploid branching model, we introduce spatial movement of particles between neighbouring sites called "rapid stirring". We consider two rapid stirring mechanisms in this work, one called "lily-pad" stirring, and the other called "individual" stirring: • Lily-pad Stirring. For each x,y E eZd with | | x - y | | i = e, £e(x) = (£c,1{x)>£e'2(x)) a n d £e(y) = (£ e , : L(y), £ e ' 2(y)) are exchanged at rate e~2. • Individual Stirring. For each i e {1,2} and x,y G eZd with ||a; - y| | i — e, £e'l(x) and £6'l(y) are exchanged at rate e~2. Just as in the particle model without rapid stirring described in 2.1, one can con-struct the particle model with either lily-pad stirring or individual stirring using a countable number of Poisson processes. Or alternatively, one can write down the generator explicitly and again apply Theorem B3 in [Liggett 1999] to establish: Theorem 2.2.1. There exists a unique Feller process £t with generator QL for the particle model with Lily-pad stirring or generator Q1 for the particle model with individual stirring: A / - = { y : | M | = 0 o r e}. (2.7) x,ytZ'l,\\x-y\\i=e m G {1,2 }, re, j/e Z''•, 1  a; - j/111 = e where £{x,m'), ifz = y i{y,m'), ifz = x and £(z,m'), if{z,m')^(x,m),(y,m) £(x,rn), if (z,m') = (y,m) Z{y,m), if(z,m') = (x,m) 8 For lily-pad stirring, instead of thinking of a site that consists of two nests as in the diploid branching model, we can view each site as having 4 states in F = {0,1}2 = {(0,0),(0,1),(1,0),(1,1)}. We restate the dynamics of the particle model in terms of these four states. At any site x G eZd, only the following transitions are possible: (0,0) <-> (0,1), (0,1) <-> (1,1), (0,0) <-> (1,0), and (1,0) <-> (1,1), i.e. only one particle is born or dies at a particular time. The rates of these transitions are as follows: C(o,o)(z, a = 1 if = (0,1) or ?(x) = (1,0), c (o,i)(z,a = c ( i , o ) ( x , £ e ) = 1 if ?(x) = (1,1), cm)(x,C)=c(lfi)(x,^)=Xn1+2(x,e) if£e(aO = (0,0), C(i,i)(z,^) = A n i + 2 ( x , £ £ ) if £ £ (z ) = (0,1) or = (1,0). In words, each particle, male or female, dies at rate 1. If site x is occupied by both a male and a female particle, then with rate A, it gives birth to a male (respectively female) child-particle at a neighbouring site provided that the neighbouring site is not already occupied by a male (respectively female) particle. For individual stirring, we still view the particle system with nests (x, m) G e Z d x {1,2} and each nest assuming one of two states in E = {0,1}. The difference between these two stirring mechanisms is that lily-pad stirring forces male and female particles at a site to move together, but individual stirring allows indepen-dent movement of male and female particles. Every exchange of particles, in both lily-pad stirring and individual stirring, is monotonicity preserving, thus neither stirring mechanism disrupts the monotonicity property of the particle system. 2.3 Convergence to a P D E for Lily-pad Stirring Consider the particle system with lily-pad stirring with its generator given by (2.7). For i G F, define ui(t,x) = P(&(x) = i) then Theorem 8.1 in [Durrett 1995] shows that if gi(x) is continuous and uf(0, ar) = a*(or) for all i, then uAt,x) = \imu\(t,x) £—»0 exists and satisfies the following system of PDE's: - Au( 0 jo) + U( 0,i) + U( i ) 0 ) - 2Adu(0,o)U(i,i) = A u ( 0 ) 1 ) + u ( M ) - u ( 0 ) 1 ) + Ad(u(o,o) - «(o , i ) )w(i , i ) = A u ( l i 0 ) + U ( M ) - u ( 1 ) 0 ) + Ad(u ( 0,o) - U(i,o))w(i,i) = - 2 ^ 1 , 1 ) + A d ( u ( 0 , i ) + U ( i , o ) ) « ( i , i ) - (2.9) d"(o,o) dt d^(o,i) dt tfa(l.O) dt <9«(i,i) dt 9 Obvious ly , must lie i n [0,1] for a l l i and t since i t is a l imi t of probabi l i t ies . W e want to s tudy the long t ime behaviour of (2.9). T h e system (2.9) is 3-dimensional i f one takes into account the condi t ion «(o,o) + u(0,i) + w(i,o) + u(i,i) = 1- W e first do two transformations on the 3-dimensional parameter space (u(o,o)>u(o,i)>u(i,o)>u(i,i)) t ° ob ta in a monotone 2-dimensional system, wh ich w i l l be easier to analyze. F i r s t , define UQ — W(o,o); u i = u(o,i) + «(i,0)> and U2 — u^i), then (uo,ui,u2) satisfies: duo dt du\ ~w dU2 ~dt = Auo + u\ — 2XduoU2 = Au\ + 2u2 — u\ + Xd(2uo — Ui)u2 = Au2 - 2u2 + Xduiu2. (2.10) T h e above system can be wr i t t en as the l i m i t i n g P D E under rap id s t i r r ing of another par t ic le sys tem Ce, s t i l l on S = eLd, w i t h state space F = { 0 , 1 , 2 } , and transi t ions 0 <-> 1 and 1 <-> 2 at rates ' c 0 (x ,C £ ) = l , i£Cix) = l Cl(x,Ce)=2 iiC(x) = 2 c 1 ( x , C £ ) = 2 A n 2 ( x , C £ ) i f C £ ( z ) = 0 c 2 ( x , C £ ) = A n 2 ( x , C £ ) i f Ce(ai) = 1, where n2(x,e) = \{zeM:C(x + z) = 2}\. U n d e r this model , monoton ic i ty s t i l l holds: i f Co(x) ^ Co(x) f ° r a l l x (here the order ing of F is the usual one: 0 < 1 < 2), then Q(x) < Q(x) for a l l x and t, since every t rans i t ion s t i l l preserves the inequal i ty < . L e t (uf(0,x), w 2 ( 0 , x ) ) = (gi(x),g2(x)) and (u\(0, x), u2(0, x)) = (<h(x), g 2 ( x ) ) be two sets of i n i t i a l d is t r ibut ions such that gi + g2 < <h + 52 and g 2 < g 2 everywhere. T h e n for a l l x , 4 ( 0 , x) < fi|(0,x) u f ( 0 , x ) + u 2 ( 0 , x ) < u\(0,x) + fZ|(0, x ) , so i t is possible to set up two in i t i a l condit ions Co and Co> such that P(Co(x) = i) = u\(Q,x) and P(CQ(X) = z) = u - ( 0 , x ) , i = 1,2, and Co0*0 ^ (0(2) holds for a l l x and UJ. Since Ct O^ ) ^ Ct (X) f ° r a u * a n c l xi t r i e monoton ic i ty proper ty of C implies P(Q(X) > 1) < P(Cte(^) > 1) and P(C ( e(x) > 2) < P(C t £(x) > 2), i.e. for a l l t and x , u\(t,x) < u2(t,x) u\{t, x) + u2(t,x) < u\(t,x) + u2(t,x). N o w we t ransform the parameter space a second t ime by defining (a, fi) = (u\ + u2,1*2) and w r i t i n g c = Xd, then (at, fit) is monotone i n the i n i t i a l cond i t ion since u\(t,x) —> w , ( i , x ) . 10 In particular, if (1 - a, a — (3,(3) and (1 - a, a — (3,(3) are both solutions to (2.10) with cto{x) < ao(x) and (3Q(X) < fio{x) for all x, then at{x) < at(x) and (3t(x) < p\(x) for all x and t. Straightforward calculation shows that (a, (3) satisfies the following system: ^ = Aa + (2c-2ca + l)(3-a d-l = A(3+(c(a-(3)-2)(3. Since (111,1*2) €E [0, l] 2, (a, (3) lies in the triangular region • U = {(u,v) : 0 < u,v < l,u > v} (2.11) for all t > 0. We change variables from (a, (3) to (u, v) and summarize this paragraph in the following lemma: Lemma 2.3.1. The PDE Au+ (2c(l -u) + l)v-u Av + (c(u -v)- 2)v (2.12) is monotone in initial conditions that lie in 1Z = {(u,v) : 0 < u,v < l,u > v}, i.e. if there are two initial conditions (uo,Vo) £ 1Z and (UQ,VQ) £ TZ, with uo < UQ and VQ < vo everywhere, then ut < ut and vt < vt everywhere, for all t; furthermore, both (ut,vt) and (ut,Vt) lie in TZ for all t. In Chapter 5, we will analyze (2.12) to establish the following theorem: Theorem 2.3.2. If X/5 is sufficiently large and e is sufficiently small, then there exists a nontrivial translation invariant stationary distribution for the diploid branching particle model with lily-pad stirring with generator (2.7). 2.4 Convergence to a PDE for Individual Stirring Unlike lily-pad stirring, Theorem 8.1 in [Durrett 1995] cannot be directly applied to get convergence to a PDE system for individual stirring. We can, however, follow the ideas used in the proof of that theorem 8.1 to establish a corresponding result, Theorem 4.0.5. We consider the particle model with individual stirring described in Chapter 2'.2. For i £ E, define ulm%x) = P{Q(x,m) =i). Then Theorem 4.0.5 implies that if O j i T O : R —> [0,1] is continuous and u|m(0, a;) = gi,m(x), then Ui,m(t,x) = \imu£im(t,x) au dt dv di 11 exists and satisfies the following system of PDE's: j ^ - — Aui , i - + 2cU 0 , l W l , l U l , 2 0 2 ^ - = Au 0 ,2 + l i i i 2 - 2 c U n , 2 M l , l U i i 2 j ^ 2 - = A u i , 2 - U l , 2 + 2 c W 0 , 2 M l , l W l , 2 = Aun.l + U l , l - 2cUo , lWl , lMl ,2 where we define c = A d 2 . Since u 0 , i + w i , i = Wn,2 + i t i ) 2 1, it suffices to study the PDE for u i , i and ui^'-dt Ait i , i - ui, i + 2c(l - ui,i)ui,iui,2 Aui,2 - Ul,2 + 2c ( l - Ul ,2)wi,lWl ,2- (2.13) dt Notice that if we start with a symmetric initial condition, i.e. o^i = ^ , 2 , then the solution to (2.13) is also symmetric. And if we define u = uifi = u i , 2 , then we obtain the following PDE for u: This PDE has been analyzed in [Durrett and Neuhauser 1994] as their sexual reproduction model (example 3 on page 291). In fact, it is not difficult to see that if u i , i = wi,2 then choosing the "father" from the male population is exactly the same as choosing the "father" from the the female population, hence it is quite natural for this reduction to occur. Theorem 4 of [Durrett and Neuhauser 1994] states that if c > 2.25 then the sexual reproduction model of Durrett and Neuhauser has nontrivial stationary distribution(s). Although this theorem does not directly apply to our particle system £6 with 2 types of particles because of the difference in stirring mechanisms, one can nevertheless trace through the proof of Lemma 3.3 of [Durrett and Neuhauser 1994] while making obvious changes, to establish a similar result: • Let 0 < pi < po < 1 be the two nonzero roots of f(u). Define f3 = (po — Pi)/10 and Ik — 2Lkei + [-L, L)d. If e is small, L is large, and £6(0) has density at least pi + (3 of both male particles and female particles in 7o, then for sufficiently large T, with high probability £ e(T) will have density of at least po — 6 in Ix and 7_i. This result can then be fed into a comparison argument, comparing the particle system with oriented percolation, as on page 312 of [Durrett and Neuhauser 1994] or in the proof of Theorem 3.3.2 later on, to establish the existence of nontrivial stationary distribution(s) for the particle system £ e under individual stirring with sufficiently small e. Then we have the following theorem: — = Au + f{u), f(u) = - u + 2c(l -u)u2. (2,14) 12 Theorem 2.4.1. If X/5 is sufficiently large and e is sufficiently small, then there exists a nontrivial translation invariant stationary distribution for the diploid branching particle model with individual stirring with generator (2.8). We will not explicitly write down the details of the proof, but instead refer the interested reader to [Durrett and Neuhauser 1994] for details. 13 Chapter 3 Results on the Dip lo id Branching Particle M ode l In this chapter, we assume the model with generator (2.5) described in Chapter 2.1, i.e. the particle system with birth and death mechanisms, but no stirring. We briefly restate the model to remind the reader: the rate at which nest m of site x flips to state i, Ci(x, m, £), is J 5, i f f" (x) = l f Am(x, c0(x,mti) = [ 0 ) o t h e r w i s e , c1{x,m,Q = [Qt otherwise where nm-(x,0 = \{z£Afx:en'(x + z) = l}\, and the J\fx contains the site x and its 2d nearest neighbours. The goal is to establish the existence of a phase transition. 3.1 Existence of Stationary Distributions We first establish that stationary distributions exist. Define £0(x) = (1,1) for all x. Let T t/(£o) = Et°f(£t) be the semigroup corresponding to the particle system, then Tt is a Feller semigroup by Theorem 2.1.1. We begin with a lemma. Lemma 3.1.1. For any A, B C S = 7Ld, the function t -» P (?t(x) = 0 Vx € A, ft(y) = 0 V j / 6 B ) (3.1) is increasing. Proof. Let ao = £ s and /3n = £s f ° r a n arbitrary fixed s. Then £ 0(x) > an (a;) and £o( x) ^ 0o(%)- Let (at,/?t) be the state at time t of the particle system that started 14 with initial condition (an,/3o)- Then by the fact that the particle system is monotone in initial conditions, we have —1 —2 £t(x) > at{x) and £t(x) > (3t(x) for all t and x. Thus by the Markov property of £, P = 0 Vx G A, | 2 (y ) = 0 Vy G fi) < P (at(a:) = 0 Va: G A, f3t(y) = 0 Vy G B) = p(?s+t(x) = 0\/x€A, fa+t(y) = 0VyeB). This implies that the function in (3.1) is increasing in t. • Theorem 3.1.2. As t —> oo, £t =4> £ 0 0 . TTie Zimii is a stationary distribution that stochas-tically dominates all other stationary distributions and called the upper invariant measure. Proof. For arbitrary subsets A, B, C = {xi,...,xm}, and D = {yi,..., yn} of S, we write P (ft(z) = 0 Vz G i , £(«;) = 0 V » e S , £ ( * ) = 1 V X G C , g(y) = 1 Vy G £>) (m+n \ JJ EA , where Ei = {ft(z) = 0 V 2 G A U {a;,}, ft(w) = 0 Vu> G 5} if i = 1,..., m and ^ = {^( z) = 0 Vz G A, | 2 (w) = 0 V t « € B U {yi-m}} if i = m + 1,..., m + n. We can use the inclusion-exclusion formula on P(U™^ra-Ei), i.e. ( m+n \ m+n U E i I = E p(Ei) - E W n Ej) + . . . + ( - i ) m + r i + 1 P ( ^ n . . . nE,). ' i=l / i=l i<j Every term in the above expansion is in the form of P(£(Z) = 0VZG; ft(w)=0\/WG~), which is increasing in t by Lemma 3.1.1. Therefore P(ft(z) = 0 Vz G A, ft(w) = 0 VUJ G B, ft(x) = 1 Va; G C, ft{y) — 1 Vy G D) converges for all A, B , C, and D, i.e. all finite dimensional distributions converge. Thus a weak limit ( £ 0 0 , ^ 0 0 ) exists and it follows from a standard result that since Tt is a Feller 15 1 2 1 2 semigroup, (Z^,^) is a stationary distribution. We can also easily see that ( £ 0 0 , ^ 0 0 ) dom-inates all other stationary distributions: let (£o>£o) he another stationary distribution and (C1,!2) be the process with initial condition (£o>£o), then (£t,£2) has the same distribution as (£o,£o) f ° r all i > 0 and (£ _ 1 ,£ 2 ) dominates ( £ \ £ 2 ) because of monotonicity, therefore dominates ( f 0 \ « g ) . • Theorem III.2.3 in [Liggett 1985] establishes the previous theorem for spin systems with state space {0,1} S, therefore it does not directly apply to our case. One can, however, easily adapt the proof of that theorem to this case, and obtain a slightly different proof. 3.2 Extinction for Sufficiently Small X/6 Theorem 3.2.1. If AIA^ 2 < 8, then the particle system £ has no nontrivial stationary distribution. Proof. We compare a modification of the particle system £ with the contact process. We recall that the contact process C on Zd has two states 0 and 1 at every site x, and has the following dynamics: an(x,C), i fC (x )=0 0, otherwise where n(x, () = | { z € Nx '• C(z) = 1}|. Theorem 2.6 of [Durrett 1995] states that if a\Af\ < 8, then the contact process has no nontrivial stationary distribution. We modify the mechanism of the particle model £ as follows: for the males, when (x, 1) G Sx {1} seeks out a pair of parents, say (xi,l) and (x2,2), it is no longer required that £ 2 ( x 2 ) = 1, but ^{xi) must still be 1. Correspondingly, when a female nest (a;, 2) G S x {2} seeks out a pair of parents (2:1,1) and (x 2 ,2) , it is only required that £ 2 (rc 2 ) = 1. We denote this modified process £ = ( £ \ £ 2 ) . The result of the modification is that £ : and £ 2 are now decoupled, and £ J behaves exactly the same as the contact process with birth rate a = \\M\. Furthermore, by Theorem III.1.5 in [Liggett 1985], the modified process ( £ \ c ; 2 ) stochastically dominates the original process ( £ \ £ 2 ) . If a\Af\ < 8, then £ has no nontrivial stationary distribution and (£t,£t) converges weakly to the all-0 state as t —> 00 for any initial condition. Thus ( £ t , £ 2 ) also converges to the all-0 state for any initial condition if a\M\ = \\Af\2 < 8, as required. • The proof above also shows that if J2x£o(x) + £o(x) i s finite, then the population dies out in finite time a.s. if A|Af | 2 < 5, since the contact process £ x or £ 2 has this property. 3.3 Survival for Sufficiently Large X/S We use the idea of Chapter 4 of [Durrett 1995], i.e. we compare the particle system to an oriented percolation process. First, we define the oriented percolation process W. Let Co = {(x, n) G Z 2 : x + n is even, n > 0} 16 and make Co into a graph by drawing oriented edges from (x, n) to (x + 1, n + 1) and from (x, n) to (x — 1, n+1). Site (x, n) is said to be a parent of sites (x + l,n + l) and (x — 1, n+1). Notice that any site (x, n) with n ^ 0 has two parents. We think of n as the time variable. Given random variables CJ(X, n) that indicate whether site (x, n) is open (1) or closed (0), we say that (y, I) can be reached from (x, m) if there is sequence of points x — zm,..., zi = y such that \zk — Z f c - i l — 1 for m < k < I and to(zk, k) — 1 for m < k < I; in this case, we write (x ,m)-> (i/.Z)-We say that w(x, n) (n > 1) is "M-dependent with density at least 1 — 7" if whenever (a;*, n*), 1 < i < I, is a sequence with | | ( X J , n*) — ( X J , n,-)||oo > M for i ^ j , we have P(w(xi,nj) = 0 for 1 < i < I) < j1. (3.2) Given an initial condition Wo C 2Z = {x : (x,0) S Co}, the process Wn = {2/: (x> 0) —* n ) f ° r some x G Wo}. gives all sites that can be reached from a site in Wo at time n. We say sites in Wn are wet. Theorem 4.2 of [Durrett 1995] states: Theorem 3.3.1. Let W% be an M-dependent oriented percolation with density at least 1 — 7 starting from the initial configuration WQ in which the events {x G Wg }, x G 2Z, are independent and have probability p. If p > 0 and 7 < 6 - 4 ( 2 M + 1 ) , then lim inf P(0 G j y 2 p J > 19/20. (3.3) This theorem shows that if the density of open sites 1 — 7 is sufficiently close to 1 and we start with a Bernoulli initial condition for Wo, then the probability that 0 is wet at time t does not go to 0 as t —> 00. Notice that the right hand side of the estimate (3.3) is a constant that does not depend on p. We will construct an oriented percolation process that is stochastically dominated by the particle system £, such that existence of nontrivial stationary distribution for the oriented percolation process implies existence of nontrivial stationary distribution for the particle system. Theorem 3.3.2. If X/5 is sufficiently large, then the particle system £ with generator (2.5) has a nontrivial stationary distribution. Proof. We follow the method of proof as in Chapter 4 of [Durrett 1995]. Since scaling time by an factor of 1/8 does not change the behaviour with respect to stationary distributions, we may assume without any loss of generality that 5=1. We will select an event G^a measurable with respect to the graphical representation in [—1,1] x Z d _ 1 x [0, T), i.e. measurable with respect to the filtration generated by all the Poisson arrivals {R*'m} and {T%'m,y'z} used to construct the particle model in Chapter 2.1 that arrive at any sites in [—1,1] x Z d _ 1 during 17 the time interval [0,i). For any 7 > 0 no matter how small, there is A and T and an event G&, with P ( G f o ) > l - 7 , so that on G & , if. &(0,0 0) - (1,1), then fr(l,0,... ,0) - ^ T ( - l , 0 , . . . , 0) = (1,1). One can achieve this by choosing T so small that the probability of any death occurring at any nests of sites (-1,0, . . . , 0), (0,0,..., 0), and (1,0,..., 0) is less than j/2; then one can choose A large enough so that the probability of having birth events from ((0,0,..., 0), 1) and ((0,0,..., 0), 2) to each of the four nests at sites (-1,0, . . . , 0) and (1,0,.. . , 0) during [0,T) is larger than 1 — 2. In other words, if we define the event G&, = {There are no death event during [0, T) at sites (-1,0, . . . , 0), (0,0,.. . , 0), or (1,0,... , 0); and there are birth events from ((0,0,..., 0), 1) and ((0,0,..., 0), 2) to each of the four nests at sites (-1,0, . . . , 0) and (1,0,...,0) during [0,T)}, then Gg0 satisfies the requirement and P(G^a) > 1 - 7 for some A and T. Gg0 is the "good event" that will ensure male and female particles get born at sites x — 1 and x + 1 provided site x is inhabited by both a male and a female particle. See figure 3.1 for an illustration of this event. 1 • 1 • 1 1 1 1 N y l • 1 • / ^ — ^ l V X 1 1 s ^ l ' ^ — ^ l l -1 0 1 Figure 3.1: Graphical representation of the event Gg 0: at least 4 birth events and no death events. We start with a configuration with events {£o(x) = 1} a n d {£o(x) = 1} all indepen-dent and having probability p\ and p2, respectively. Let Xn = {x: (x, n) £ C0, ZnT{x, 0, . . . , 0) = (1,1)}. 18 Before denning the oriented percolation process Wn, we first define Vn that will turn out to be slightly larger than WN but nevertheless dominated by XN. We define Vn inductively. First, we set VQ = XQ and leave w(-,0) undefined. Now assume that Vo, Vi , . . . , Vn, and CJ(X, I) with I < n — 1 have been defined such that VQ C XQ, ..., Vn C Xn. If x G Vn then we set 1, if G<r_ (i„T) occurs in the graphical representation ( ^ ( T - « . , ( $ , . T ) ' s translated by — xe\ in space and . ^ ^ —nT in time) ' 0, otherwise which ensures that LJ(X, n) = 1 with probability more than 1 — 7 if x € Vn. For completeness, if a: ^ Vn, then we set co(x,n) equal to an independent random variable that is 1 with probability 1 — 7 and 0 with probability 7. Now we define Vn+i to consist of all sites [x, n + 1) with either x — 1 G Vn and OJ(X — 1,'n) = 1 (3.5) or x+leVn and w(x + l,n) = 1. . (3.6) For (#', n) &Vn C X n , the definition of XN means that £nT(x', 0 , . . . , 0) = (1,1); if u(x', n) = 1, then the "good event" occurs in the space-time rectangle [x' - 1, x',+ 1] x 1 d ~ l x [nT, (n + 1)T), which, since £nr(z', 0, . . . , 0) = (1,1), implies that i(n+i)T{x' - 1,0,..., 0) = (1,1) and Zin+1)T{x' + 1,0,..., 0) = (1,1). Therefore the conditions (3.5) and (3.6) ensure that any (x,n + 1) G Vn+\ is a member of Xn+i, hence Vi j + i C Xn+\. By induction, for all n G Z + , we have Vn C X„. The.w(x, n) thus defined is a 2-dependent oriented percolation on £ 0 with density at least 1 — 7; notice that {w(0,0) — 0} and {w(2,0) = 0} are dependent events since both require that no death occur during [0, T) at nest ((1,0,..., 0), 1) or ((1,0, . . . , 0), 2), but {w(0,0) = 0} and {w(4,0) = 0} are clearly independent. Also, {w(xi,ni) = 0} and {u)(x2,n2) = 0} are independent for any x\ and x2 provided n\ ^ n2. Thus if (xi,rii), 1 < i < I, is a sequence with | | ( X J , rij) — (xj,nj)\\oo > 2 for i ^ j , then the events { O ; ( X J , rij) = 0} are all independent, which implies P(u(xi, ni) = 0 for 1 < i < I) < 7 7 , as required by (3.2). Notice that even though Vn+\ clearly depends on Vn, u(xi,n + 1) and u>(x2,n) as defined by (3.4) are indeed independent since they relate to independent Poisson arrivals in disjoint space-time rectangles. u(x, n) 19 • G occurs O G does not occur (-4,0) (-2,0) (0,0) (2,0) (4,0) Figure 3.2: Illustration of Vn and Wn: all sites in Wn are connected to (2,0) via a sequence of solid lines, while sites in Vn\Wn are connected to some site in Wn via a dotted line. The shaded rectangle indicates the space-time region that affects whether G^0 occurs Now we define Wn = {(x,n) : (x,n) G Vn and ui(x,n) = 1}, then Wn C Vn and Wn is a 2-dependent oriented percolation with density at least 1 — 7. If 7 is sufficiently small, then by Theorem 3.3.1, lim inf P(0 G W2n) > 19/20. n—>oo Since the particle system <f dominates Vn, which in turn dominates Wn, we have l iminfP($ 2 n T (0,0, . . . ,0) = (1,1)) > 19/20. (3.7) Tl —>00 Thus if we start with initial configuration (,Q(X) = (1,1) for all x (in this case, p = 1 for Theorem 3.3.1), then by Theorem 3.1.2, f t => 1^. And (3.7) implies that P( | 0 0 (x ) = ( l , l ) ) > 19/20 fo ranyxGZ^, i.e. the upper invariant measure <f TC is nontrivial. • Remark 3.3.3. Using the same idea of comparing the particle system f to an oriented percolation process, one can show that if the initial condition is finite, e.g. £o(0,0, . . . , 0) = (1,1) and 0 everywhere else, then for sufficiently large X, I iminfP(£ n r (x ,0 , . . . ,0 ) = (1,1) for some x) > 19/20 20 for some T. Here we use the following fact (see Theorem 4-1 of [Durrett 1995]) about Co = {(y,n): (0 ,0)- ( j / , n)}: Ifl< 6 " 4 ( 2 M + 1 ) 2 , then P(\C0\ < oo) < 1/20. (3.8) Remark 3.3.4. In fact, the proof of Theorem 3.3.1 also establishes the following: if the initial population has a positive density of male and female particles everywhere and X/5 is sufficiently large, then at sufficiently large times, the density of (1,1)-sites (i.e. sites with both a male and female particle) is larger than 9/10. Similarly, using (3.8), the following is also true: if the initial population is nonzero and X/5 is sufficiently large, then at sufficiently large times, the probability of survival (i.e. existence of a site with both a male and a female particle) is at least 9/10. 21 C h a p t e r 4 Convergence Theorem for Individual Stirring In this chapter, we establish the convergence result for the individual stirring model as promised in Chapter 2.4. We work in a slightly more general setting and consider random processes £t£ : eZd x {1,2, . . . , M} —> {0,1,.. . , K — 1}. We call each x G eZd a site, and each (x, m) G eZd x {1,2,. . . , M} a nest. There are M nests at each site. We think of the set of spatial locations Zd x {1,2, . . . , M} as consisting of M "floors" of Zd. Let M" = {0,«/i,...,eyjv} be the interaction neighbourhood of site 0 and r^- = max.xe/f \\x\\i be its radius. The process fj evolves as follows: 1. B i r t h and Death. The state of nest (x,m) flips to i, i = 0,...,«"— 1, at rate d(x, m, f) = hiim(t(x, m),P(x + ezi, m i ) , . . . , £(x + ezL, mL)), where L is a positive integer, z\,..., zi £ Af, m i , . . . , m ^ G {1,2, . . . , M}, and fcJira:{0,l,..,K-l}l+1^i(Cl+ with hitm(i,...) = 0 and K compact. 2. Rapid Stirring. For each m G {1,2, . . . , M } and x, y G eZ d with \\x — y\\i = e, tfe(a;,m) and £ £(y,m) are exchanged at rate e - 2 . This individual stirring model differs from the Lily-pad stirring model described in Chapter 2.2 in that the stirring action between corresponding nests at neighbouring sites are now independent. More specifically, exchanges are allowed between neighbouring nests on 22 the same floor only, i.e. between (0,1) and (e, 1) but not between (0,1) and (e, 2). We will show, in the theorem below, that this individual rapid stirring action between corresponding nests "decouples" (in the limit e —> 0) the dependence between all nests, at neighbouring sites and even at the same site. As an example, for d = 1, in the particle model with individual stirring with gener-ator (2.8), we have K = 2, M = 2, L = 4, Af = {0, -e, e}, c (x rn £) - { ^ = 1 ' ' 1 0 , otherwise ' and / C\ J A(£ (x - e , l )C(a ; - e ,2 )+ .£ (x + e,2)C(a; + e,l)), i f £ ( x , m ) = 0 C l (x,m ,0 = | 0 > o t h e r w i s e In particular, we should define (zi,mi) = (-1,1), (z2,m2) = (-1,2), (z 3, m 3) = (1,1), (z 4, m 4) = (1, 2), and hi = / i ^ m as fto(ao,ai,Q!2,a3,Q!4) = <5ao, and hi(ao,cti,ct2, a 3 , a 4 ) = A(a ia 2 + a 3 a 4 ) ( l - ao), where ao = £(x,m), ct\ = £(x + ez\,m\) = £(x — e, 1), eic. Theorem 4.0.5. Suppose {^(x, m), (x,m) € eZ d x {1,2,. . . , M}} are independent and let ueim(t,x) = P{£t(x,m) = i). If u<rm(0,x) = 5i,m(x) = gi(x,m) and gt : Rd x {1,2, . . . , M} —> [0,1] is continuous, then for any smooth function <fi with compact support, as e —> 0, E ^,(2/)1{«(S/,»n)=i} / <t>(y)ui,m(t,y) dy, (4.1) w/iere •Uii„l(i,x) is the bounded solution of = &ui,m + fi,m(u), Ui>m(0, X) = Oj(x, m), fi,m(u) = (Ci(0, m, Ol(£(0, m) ± i))u - £ (Cj-(0, m, £)1(£(0, m) = i))u , and (</>(£))« denotes the expected value of </>(£) under the product measure in which state j at nest m has density UjtTn, i.e. £(x,m), with x € eZd and 1 < m < M, are independent with P(£(x, m) = j) =• Uj,m. 23 Proof. We follow the program used in the proof of Theorem 8.1 in [Durrett 1995]. We first define a dual process for the particle system in part (a), then in part (b) we show that the dual process is almost a branching random walk. We will not explicitly write down parts (c) and (d) of the proof, which almost exactly resemble parts (c) and (d) of the proof of Theorem 8.1 in [Durrett 1995]. But to summarize these two parts, part (c) establishes that the dual process converges to a branching Brownian motion as e —> 0 and defines a candidate limit Ui>m(t,x) of u\m(t,x), and part (d) shows that this candidate limit satisfies the PDE in the statement of the theorem. Finally, part (e), which we write out explicitly, showsthat convergence described in (4.1) does occur, in addition to the convergence of the mean u\m(t,x) to u i > m ( i , x ) . a. Denning the dual process. The dual process associated with nest (x, m) and a fixed time t is a random process I*'m,t(s), s G [0,t], where lf'lh't(s)& | J {(x,m) : x eeZd,me {1,2,.. . ,M}} k (4.2) fcez+ consists of a finite number of particles residing at nests (x, m) G eZ d x {1,2, . . . , M}. The events that influence the behaviour of the dual process J^" 1,* at time s are those Poisson arrivals in the original process £ 6 that occur at time t — s. We start with #*•'(()) = {(£,m)} and evolve If 'm ' '(s) until s = t, which corresponds to rolling back the clock in the original process f e from time t back to time 0. The process i f >m'* is constructed such that: in order to know the value of £ e(x, m) at time t, it suffices to do a computation using values of f e at time 0 and nests (x,m) G /*•*•'(*). We call the influence set of !*•*•'({)). In order to define the dual process, we need to give a graphical construction for the particle system f £ similar to the one given in Chapter 2.1. We define a family of uniform [0,1] random variables {U%,m,t : n > 1} and two families of independent Poisson processes that respectively correspond to the birth/death flips and the rapid stirring mechanism: { T z , m , i . n > !} a t r a t e c * = sup € ) i n £ i c i ( a : . T O . 0 and {S*'y'm : n > 1} at rate e~2, with m £ {1,2,..., M} and ||a; — y| | i = e. Notice that c* < oo since max i i m | | / i t , m | | o o < oo. At T^ ' m ' 1 , we check all nests (x,m), (x + tz\,mi),..., (x + ezL,rni) and use U%'m'1 to decide whether nest (x, m) should flip to state i based on the function h^m. And at S*,v'm, we exchange values of ££(x,m) and f 6(j/,m), the corresponding nests at two neighbouring sites. The evolution of the dual process If'm't(s) depends on the Poisson arrival times r y x . m . i : n > 1} and \s^m : n > 1} in the following way: 1. If (x, m) G I f ' ^ i s - ) and T ^ ' m , i = t - s , then we set If = /?•*•*(*-) U {(x + e« i ,mi ) , . . . , (a; + ezL,mL)}. (4.3) Therefore each particle (x,m) G 7f'm''(s) gives birth to L new particles at rate c*. 24 Figure 4.1: Illustration of the dual process: there are two birth/death events here, each giving birth to four additional particles, two on each floor; the first birth event is from a male (m = 0) nest, while the second birth event is from a female (m = 1) nest. 2. If (x,m) e I f - ^ ( s - ) and either S%'y'm = t-sor S%x>m = t-s, then we set If^t(s) = {lf^t(s-)\{(x,m)})u{(y,m)}. 25 Thus two dual processes 7f' r f l'£(s) and /f '^ ' -^s) evolve independently from each other except at time s= someT*'™'* where (x,m) G ^^(s) n if' ,T™'''(s) or at time s= someS^-" 1 where (x,m) G lf'lh't(s) and (y,m) G /f '" ' ' ' '(s), or (y,m) G Jf ' m ' £(s) and (x,m) G i f •rfl'-t(s) . An equivalent way of describing the dual process is to define the pair {X?>™(s), *(s)) for 0 < s < t, where X?<™(s) is the ordered set Xf'™(s) = (Xf-™'°(s),... ,X £ * , * , * : "' r o ( 0 ) ~ 1 (s)) and each X| , 7 h ' J ' ( s ) G eZ d x {1,2,. . . , M } . For s = 0, define the number of particles /Cf ™(0) = n for some n G Z + and the location of particles X | , m j ( 0 ) = ( x . ) m . ) for 0 < j < n, such that /Cj''""(0)-1 /*,«*,*(()) = ( J { X f ' ^ ( 0 ) } , (4.4) j=0 where J*-A-t(0) is defined in (4.2). Typically, £*'*•(()) = 1 and X*>*'°(0) = {(x,m)}, but we use the more general initial condition since we will define X*'m'i inductively. Each particle in X£'m,i, 0 < j < JC*'m, jumps to a neighbouring nest on the same "floor" of eLd x {1,2, . . . , M} as dictated by the stirring mechanism (i.e. 5-arrivals), until a T-arrival of type TX>M'-7 with 0 < j < £*'™(s-) at some (x,m) G Xf™(s- ) . At time s of this T-arrival, we set £*•*(*) = L + £ * , A f > - ) and Xe (s) = (a; + tzk,mk) for 0 < < L, while leaving all existing Xf^'^s unchanged. Observe that /C* ' ' " (s) -1 7 * . * . * ( S ) = ( J { X f ™ ( s ) } , 3=0 26 therefore the relation (4.4) is maintained for all s G [0,t]. Afterwards, the ./Cfm(s) particles jump according to the S-arrivals until the next T-arrival at some (x,m) G {Xf ' r n ' J ' ( s / ) , 0 < A new particle may be born at a nest where a particle already resides; if this happens, we say that a collision occurs, and call the new particle fictitious. We prescribe this fictitious particle to give birth to L particles at rate c* and jump to a neighbouring nest on the same "floor" at rate e~2, independently from all other particles, fictitious or not; furthermore, all offsprings of a fictitious particle are defined to be fictitious as well. If the number of collisions is 0, then there are no fictitious particles. As will be seen in the next paragraph , this is in fact the case in the limit e —> 0. Note that the stirring mechanism does not cause particles at different nests to "collide", i.e. end up at the same nest, because only exchanges between nests occur under the stirring mechanism. b. Characterizing the dual process. Having finished defining the dual process, we proceed to show that with high probability, the dual process can be coupled to a branching random walk and the movement of one dual process is independent from the movement of another dual. This part of the proof is quite similar to part (b) of the proof of Theorem 8.1 of [Durrett 1995]. First, we couple X f ' m ' J to independent random walks Y*>m'i that start at the same location at the time of birth of X ^ m ^ and jump to a randomly chosen neighbour at rate 2de~2. We define the distance between two particles on two different "floors" to only depend on the site location: and say Xf ' " 1 ' - 3 is crowded if for some k ^ j, ||Xf'" 1 '- ' - X*'™'k\\i < r%j. Recall that reN is the radius of the interaction neighbourhood N. When X f ' m ' - 7 is not crowded, we define the displacements of Y*'™'* to be equal to those of X f ' A ' J . But when X f ' m ' j is crowded, we use independent Poisson processes to determine the jumps of Y^m^. To estimate the difference between Xf ' " 1 ' - 7 and Y*'m,i, we need to estimate the amount of time X f ' m ' J is crowded. If j ^ k, then V/ = X f ' m ' J ' ( s ) — X f ' m ' f c ( s ) is stochastically larger than Wg, a random walk that jumps to a randomly chosen neighbour in eZd x {1} at rate 4de - 2 (see page 175 of [Durrett 1995] for details on how to couple these two processes so that one is stochastically larger than the other). Strictly speaking, V/ is only defined after the j t h and kth particles of X f ' m are created, but as we shall see below, we are only interested in an upper bound on the occupation time of Vse and W | in a ball, so the fact that these particles may only start to exist at some positive time only helps matters. Thus for any integer M > 1, v^e = \{s < t : \\Vse\\i < Me}\is stochastically smaller than w™e = \{s<t: | |W s £ | | i < Me}\. Well known asymptotic results [Durrett 1995] for random walks imply that if te~2 > 2 then In the estimates above, d = 1 is the worst case, so we use Ew^le < CMet1/2 if te 2 > 2 from now on. j < K**(S>-)}. IK*. i,mi) - (x2,?n2)||i = ||xi - x 2 | | i , (4.5) 27 The amount of time X f ' m ' J is crowded in,[0,i], denoted Xe(*)> c a n he estimated as follows: oo E[xUt)} = Y,E^m^{t) = K]P(K^{t) = K) oo where we pick M large enough so that Me > r^, e.g. M = imaxn<i<jv In what follows, C... is a constant whose value may change from line to line. Then E[xi{t)} < E[w™e]E[Kf™{t)} = ec*LtE[w™€\ < CMec'Ltet1'2 if te"2 > 2. To see the middle equality above, we observe that the branching mechanism of the dual process described in (4.3) occurs at rate c* for both fictitious and nonfictitious particles, and every time a branching event occurs, one particle is replaced by L particles; therefore the mean number of branches at time t is ec L t . It follows that E[xi(t)} < CMec'Ltet1'2 + 2e2 < CMec'Lte(l +11'2). (4.6) This means that the expected number of births from Xf ' m j while there is some other X * ' m ' k in Af + Xf'™'! is smaller than CLtMec'ue{l+t1'2). . (4.7) Thus P(at least one collision during [0,t]) < P(at least one collision during [0, t] |£f*(t) < e-°-5)P{lC*'™(t) < e" 0 5 ) +P(/Cf > e-°- 5) < E[# of collisions during [Q,t)\K*>™{t) < £-°- 5]P(/Cf A ( i ) < e"0'5) +e°-5E[)C*'™(t)} (4.8) < C L , M e c * L t e 0 - 5 ( l + t 1 / 2 ) + e 0 V * i t < CL,Mec'Lte°-5(l+t^2), which —> 0 as e —> 0. We use (4.7) and the condition ]C*'m(t) < e~ 0 5 to bound the first expectation in (4.8), and we bound P(/C*'"l(t) < e - 0 5 ) in (4.8) above by 1. This shows that the probability of at least one collision within a single dual during time [0,t] tends to 0 as e —> 0. Furthermore, the same argument shows that the probability of at least one collision between two different duals for nests (x,rh) ^ (x',rh') during time [0, t] also tends to 0 as e —» 0. For that, we observe that the estimate (4.5) is independent of the initial condition W0E; so in particular, this estimate still holds even if \\WQ ||I = 0, which is the case when for example one considers two duals for two nests (x, rn) and (x, rh!) at the same site x. Hence two different duals are asymptotically independent in the limit e —> 0. 28 The estimate (4.6) also leads to the following estimate on the difference between Xx,m,j a n d y £ , m , j ( s e e p a g e 1 7 6 i n purrett 1995] for details): P i max \\X?'m>i(s) - y / ' m ' J ( s ) | o o > < CeUA(l + i 1 / 2 )e ' iO<s<t \l \pc*Lt (4.9) This shows that with high probability, the movements of all the particles in a dual can be coupled to independent random walks, in addition to being independent from the movement of any other dual. c. and d. Defining a candidate limit and showing the limit satisfies the PDE. We will not write down the details of these two parts of the argument, since they are almost exactly the same as parts (c) and (d) of the proof of Theorem 8.1 in [Durrett 1995]. From estimate (4.9), it is not too difficult to see that the dual process converges to the branching Brownian motion as e —> 0. We can then define the candidate limit Ui ) m ( t , x) (of u\m(t,x) = P(^(x,m) = i)) using the limiting branching Brownian motion as the dual process. Part (d) then establishes that the candidate limit Ui>rn(t,x) satisfies the integral from of the PDE in the statement of the theorem. e. The particle systems converge. So far we have established that ui,m(t> x) *• ui,m{t, x), i.e. the expected value converges. It remains to establish (4.1). For this, we define for a bounded function <f> with compact support supp(</>) of diameter D, yGeZd Then Em,<t>)\ ed Y, <l>(y)P(ZUy,rn) = i)=ed £ <t>(y)*i,mfrv) yEeZ'1 yeeZd <t>(y)ui,m(t,y) dy by bounded convergence. Now we compute the variance of (£ Var[(£,</>>] (4.10) „2d E ^(J')(1{«(w,m)=t} " P(Ct(y,m) = t)) <yeeZd e2d m2^W,iv,m)=r}-P{Ct{y,m) = i)f y£eZd + E e2d £ <f>(y)<f>(z) < D X (^fffo.m)^} - P{Ct{y,m) = »))(l{«(z,m)=<} - P{$(z,m) = i))] 2<"IJ»llL S U P C o v[^'(w,m)=i},l{«(*,m)=i}] y ,z£eZdnsupp(<f>) ,yytz +e 2dii y€eZdnsupp(<f>) 29 We observe that l{^(y,m)=i} 1S a random variable taking values in {0,1} and therefore has variance < 1/4. Also, part (b) of the proof shows that cov£ = sup C o v l l ^ ^ y ^ ) ^ } , l{ ? f ( Z i m ) = i }]-> 0 y,z£eZd,y^z as e —> 0. The argument leading to the asymptotic independence of two duals in part (b) works for any two nests, so Cov[l{£t(y<m)=iy, l{||(z,m)=i}] S o e s t o z e r o uniformly for all y ^ z . Now we have the following estimate on Var[(£j, <p)\: Var[<£,0>] < X?M||0||^cove + ^ed||</>||L, which —> 0 as e —> 0. Thus by (4.10) and Chebyshev's inequality, we have < P > S < - J4>(y)ui,m(t,y) dy > S d Y ^(y)ui,m(*>2/) - / <t>{y)ui,m{t,y) dy yeeZ,'1 (etA)-zd Y ^y)<m(t,y) yeeZ'1 d Y ^(y)ulm(t,y) - J(p{y)ui,m(t,y) dy d Y ^(yK,m(^y) - / 4>(y)ui,m(^y) dy < p 5 > 2 +p 4Vax[(ff, 52 y€eZd + P y€eZd > as e —> 0, and the theorem follows. • 30 C h a p t e r 5 Existence of Invariant Stationary Distribution For Lily-pad Stirring In this chapter, we establish the existence of nontrivial stationary distribution of the particle system with lily-pad stirring as promised by Theorem 2.3.2. First, we rewrite (2.12) in the statement of Lemma 2.3.1: We show that for sufficiently large c, the solution to (5.1) with initial condition u'o = f,vo = 9, f > 9 satisfies the following condition: (*) There are constants 0 < Di < d\ < d2 < D2 < 1, L, and T so that if VQ{X) G {D\, D2) for x G [—L,L] then VT(X) G (di,d2) for x G [—3L,3L]. According to Chapter 9 of [Durrett 1995], this is a sufficient condition for the existence of nontrivial invariant stationary distribution for the particle system with sufficiently fast stirring, so Theorem 2.3.2 will follow once condition (*) is established. Recall that Theo-rem 3.3.2 establishes that the diploid particle model without rapid stirring has a nontrivial stationary distribution if the birth rate A is sufficiently large. If one traces through the proof, however, one will find that "sufficiently large" in that argument means that A is larger than a number on the order of 6 1 0 0 , which is not too informative on where exactly the critical A for the phase transition is. On the other hand, one can get a far better idea of exactly for what A condition (*) holds. For this proof, we also establish condition (*) for sufficiently large c (recall that c = Ad), but here "sufficiently large" means that c is "only" larger than a number on the order of 100. We assume dimension d—1; extension to d > 1 is straightforward. The proof consists of two parts: the first part, Chapter 5.1, establishes the existence of constants di du ~&t dv di Au + (2c(l -u) + l)v-u Av + (c(u - v ) - 2)v. (5.1) 31 and D\, and the second part, Chapter 5.2, establishes the existence of constants d.2 and £) 2; the second part will be easy once the first part has been established. Theorem 9.2 in [Durrett 1995] establishes condition (*) for a specific predator-prey system with phase space {0,1,2} at each site. The critical fact used in the proof is that the associated ODE system (i.e. the dynamical system that results when one has constant initial conditions) has only one interior equilibrium point and has a global Lyapunov function. The phase portrait of the ODE associated with (5.1), however, shows that it has two interior equilibrium points, one of which is always a saddle point. See figure 5.3 for two examples. Thus it does not look likely that the ODE system associated with (5.1) has a readily iden-tifiable global Lyapunov function. The method we use to establish condition (*) for (5.1) is ad hoc, but does seem to apply to a wide variety of reaction-diffusion systems where the reaction part of the system is 2-dimensional (or even 3-dimensional), i.e. ^ = Au + f(u) where / : M 2 -> M 2 . As established in Lemma 2.3.1, the PDE (5.1) is monotone in initial conditions that lie in TZ = {{u,v) : 0 < u,v < l,u>v}. This fact is critical for the proof of existence of constants di and D\ in condition (*) above - it enables us to bound the initial condition VQ(X) below by a function, say vQ(x), both vo and UQ having values < Di for x G [—L,L], such that if the solution vt(x) to (5.1) with initial condition v${x) satisfies the condition vT(x) > dx Vx G [-3L,3L], then VT(X) > d\ Vx G [—3L, 3£] as well. We will also need results regarding the ODE associated with (5.1): (2c(l -u) + \)v-u (c(u-v)-2)v. (5.2) The above ODE system is also monotone in initial conditions, since if the initial condition for the PDE system in (5.1) is constant in x, then the solution (ut,vt) to (5.1) also remains constant in x for all time and therefore satisfies the ODE system in (5.2). 5.1 Lower Bounds: Existence of d\ and D\ in Condition (*) ' First we recall the definition of the region TZ from (2.11): TZ = {(u, v) : 0 < u, v < 1, u > v}. ' If the initial condition (UQ, VQ) lies in TZ, then so does the solution (ut, vt). We will establish the existence of constants d\, D\, L, and T using the nonlinear Trotter product formula (Proposition 15.5.2 from [Taylor 1996]): (ut,vt)= lim ( e ^ > A ^ / " ) n (/,<?). (5.3) n—>oo \ / du ~di dv 'di 32 Here the convergence occurs in the space BC^R) , the space of functions whose first deriva-tives are bounded and continuous on IR and extend continuously to the compactification R via the point at infinity; the norm used here is + dx In (5.3), es*(f,g) gives the (independent) evolution of / and g for time s according the heat equation du ~di dv dt Au Av, and Fs{f,g) gives the pointwise evolution of (/, g) for time s according to the ODE in (5.2), i.e. for all x, if (u0(x),v0(x)) = (f(x),g(x)) then Jrs(f,g)(x) = (ua(x),vs(x)) where (u(x),v(x)) evolves according to (5.2). Note that both e s A and Ts are monotone in initial conditions, therefore so is esAJ-"s. To establish the existence of constants Di and d\ in condition (*), it suffices to show that for any initial condition (uo,vo) with VQ dominating the function Di / [_ £ i i ] (x) , for sufficiently large T, (UT,VT) is such that VT dominates the function d\I^-zL,zL)(x)-1 -I 0 I (a) The function h -L-l -L -L + l 0 L-l (b) The function /o L + l Figure 5.1: The functions h and /o-Let # 2(R) = { / G L 2 (R) : | | / | | = / ( l + f2)|/(OI2<*£ < °°} d e n o t e t h e S o b o l e v space with parameter 2, where / denotes the Fourier transform of / . Equivalently, H2(M) consists of L2-functions with L2-second derivatives. We first define / n G H2(R) that will be the "shape" of the initial conditions (UO,VQ): IC 1. / 0 (x) = 1 for x G [-L + I, L - I}; IC 2. f0{x) = 0 for x G (oo, - L - l ] U [ L + I, oo); IC 3. f0(x) = h(x+L) for x G [-L-l, -L+l] and f0{x) = h(L-x) for x G [L-l, L+l], 33 where h G H2(R) is the following function: h(x) 1 (x+l\2 2\ I I ' 1 -1, X < -I -l<x<0 i ' * = ^ 2 0<x<l x > I l I l — x V2 2"v ; ) i (5.4) In the above definition, the choice of L is arbitrary as long as L > I, but we will later on choose / small such that A / 0 is large in [-L-/, —L + l]L)[L-l,L + l}. We call the intervals [-L — I, —L +1} and [L — l,L + l] the "transition regions". We observe that h is continuous at x = 0, with if -I < x < 0 if 0 < x < I so the graph of h in the plane is symmetric about the point (0, | ) and also, | A / 0 | < ^ (5.5) everywhere. We pick the initial condition to be UQ = ao/o,«o = bofo with (ao/o(0), &o/o(0)) = (ao, bo) € TCQ, where 1Zo is the region for the top tip of the line segment O(ao,bo) to be Figure 5.2: Shaded region is TZo, and 71 will be defined in (5.9). 34 defined later in (5.19). See figure 5.2 for an illustration of Tio and the line segment O(a0, bo). We will define a family of parallel piecewise-linear curves ABC(uo,vo) (see figure 5.3) that satisfy the following requirements: A B C 1. ABC(u0,v0) lies in ft0; A B C 2. ABC(uo,vo) passes through the points A, B, C, and (UQ,VO); A B C 3. A lies on the line u = v, B lies on the line u = v + 0.1, and C lies on the line u = 1; A B C 4. ABC(UO,VQ) = AB(UO,VQ) U BC(U0, vo), where line segments AB(u0,vo) connects A and B, and BC(UQ,VO) connects B and C; A B C 5. AB(uo,vo) makes an angle of — 9 (0 < 9 < arctan(0.05)) with the positive w-axis and BC(uo,Vo) is a horizontal line segment. We will establish the following: Proposition 5.1.1. If c is sufficiently large and (5o,6o) £'{(u,v) £ 71 : 0.55 <v< 0.8}, then for sufficiently small s, we have e s Ar(d 0 /o ,&o/o) > (asfsXfs), where fo is defined in (IC 1-3) on page 34, > means that > holds in each component, and as, bs, and fs satisfy the following conditions: 1. as and bs are constants depending on s, such that the curve ABC(as,bs) lies above the curve ABC(do,bo), and the vertical distance separating them is at least 62s. In par-ticular, since ABC(do,bo) lies above the horizontal line v — 0.5, so does ABC(as,bs). 2. fs(x) ( 1, x £ [-L + I - 5is,L - I + 5xs} h(x + L + 5\s), x £ [-L - I — 5is, —L + I — Sis] . . h{L + 5is-x), x£[L-l + 5is,L + l + 5is\ . [ ' } I 0, x £ (00, - L - l - Sis] U [L + 1 + Sis, 00) i.e. fs is fo with each of the two transition regions is translated by 5\s away from the origin. 3. by and 52 are positive constants independent of s and (do,6o)-The proof of the above proposition requires a few lemmas and will be deferred until the end of Chapter 5.1.2. Proposition 5.1.1 states that esAJrs(dofo,bofo) is bounded below by (asfs,bsfs), where (as,bs) moves 82s above ABC(ao,bo). Furthermore, by (5.6), the region where the value of (asfs,bsfs) (and hence esAJ-s(aofo,bofo)) is equal to or above (as,bs) has expanded by Sis, both to the left and to the right, while the transition regions of (dsfs,bsfs) are shifted left or right by 5is but maintains exactly the same profile as in 35 the initial condition. Thus by the monotonicity of esAJ73, we can iterate e s A . F s enough times and obtain information about the evolution of the PDE (5.1) for large time. From the construction of the piecewise linear curves ABC, requirement (5) implies that B(0.8,0.8) on AB(0.8,0.8) has u-coordinate > 0.75. Therefore Corollary 5.1.2 below is an easy consequence of Proposition 5.1.1. Let nv(uo,v0) =v0, (5.7) and [x\ = max{z e Z : z < x}. Corollary 5.1.2. If c and T are sufficiently large, and v0{x) > 0.55 for x G [-L + l,L — l] then 7rv((esA^s)^^(u0,v0)(x))>0.7 for x G [—3L, 3L] and sufficiently small s. In other words, the constants D\ and di in condition (*) are picked to be Di = 0.55 and di = 0.7. Note that we restrict VQ(X) = 60/0(2:) to be > 0.55 for x G [—L + l, L — l] in the above corollary because Prop 5.1.1 only works for (ao,6o) G {(u, v) G TZ : 0.55 < v < 0.8}. Also note that the "£" in condition (*) is picked to be L — I. 5.1.1 Analysis of the O D E (5.2) We first characterize the phase portrait of the ODE. We carry this out for sufficiently large c. See figure 5.3 for phase portraits with c = 5 and c = 25. Define n{u,v) = (nl(u,v),n2{u,v)) = {{2c(l-u) + l)v-u,{c{u-v)-2)v), (5.8) such that the solution to the ODE (5.2), {us,vs) = !F^(uo,vo), flows along the vector field n. Define the curves 7 1 , 7 2 , and 73: 7 1 = {(u,v) :ue [0,1],,= 1 + 2 c " . 2 e u } , (5-9) 72 = | ( « , « ) : u G [ 0 , l ] , w = u - ^ | , (5.10) 73 = { ( « , » ) : u e [ 0 1 l ] , u = t)}. (5.11) We have n\ = 0 on 71 and 772 = 0 on 7 2 . An easy calculation shows that for all c, (0,0) and (1,1) pass through 7 1 . We observe that T ? I is a linear function in u for fixed v, so 771 > 0 to the left of 71 and n\ < 0 to the right of 7 1 . By similar reasoning, we also have r\2 < 0 to the left of 7 2 , while 772 > 0 to its right. The two intersection points of 71 and 7 2 , 36 (a) c = 5 0.3 0.4 0.5 0.6 U 0.7 0.8 0.9 (b) c = 25 Figure 5.3: Phase space of the ODE are the only equilibrium points of r? in the interior of TZ, with O = (0,0) £ dTZ being the third equilibrium point. Elementary computation shows that O and P+ are stable, but P_ is a saddle point, thus one would expect any point that lies significantly above P_ to flow toward P+ under 77. Elementary calculations also show the following: P + - * ( l , l ) - asc-^oo, (5.12) P_ —> (0,0) as c —> 00, -p-1 > 00 as c —» 00, where P_ > u and P - ^ denote the u- and u- coordinates of P_, respectively. We will need some crude estimates of 771 and 772- First of all, since u — v > 0 everywhere in TZ, we have 772 > -2v. (5.13) If the point (u, v) is at least 5 to the right of 72, then since 772(14, v) = 0 on 72, 772(14,7;) > 5cv, (5-14) Similar reasoning shows that if the point (u,v) is at least 5 to the left of 71, then r)i(u,v) > 25cv. (5.15) Now the horizontal distance between 71 and 73 at a fixed v is (1 + 2 C ) T J d(v) = v. w 1 + 2 C 7 J 37 Simple calculations show that 2 C ( 2 C T 2 + 2v - 1 ) 4c(l + 2c) d (u) = ^ ^ — ^ — and a (u) = — 7 - —r^. v ' ( l + 2cu)2 v ; ( l + 2cu)3 Notice that d"(v) < 0 if v,c> 0, so for v G [0,1], d(v) is a strictly concave function, and u = ^ ( v T + 2 3 - l ) 2c is the unique point where d(v) attains its maximum for v G [0,1]. We observe that v —> 0 as c —> 0 0 . (5.16) If c is sufficiently large such that the horizontal line v = e lies above the line v = v but below the line v = 0.8, then for v G [e, 0.8], the minimum of d(v) occurs at v = 0.8, and as c —> 0 0 . This shows that for arbitrary e > 0 and sufficiently large c, the minimum horizontal distance between 71 and 73 for v G [e,0.8], is larger than 0.19. This fact will be needed a bit later on in the proof of Lemma 5.1.3. We also observe that since d(v) is strictly concave for v G [0,1], the curve 71 written as u = u(v) is also strictly concave for v G [0,1]; then (5.16) and the fact 1 + 2 c - V i + 2c U I T ; ) — > 1 as c —» 0 0 v ; 2c imply that 7x - » {(u,v) : v = 0,u G [0,1]} U {(u,u) :u = l,vG [0,1]} as c -> 0 0 . (5.18) This finishes the characterization of the phase portrait of ODE (5.2). These facts, which are admittedly tedious and not much fun to establish, will all be required later in the proof of Lemmas 5.1.3 and 5.1.5. We define the region (see figure 5.2) il 4- 2c)v 7l0 = {(u, v)eK:u< - 0-04,0.55 < v < 0.8}, (5.19) Recall that 71 defined in (5.9) is the curve v = 1 + 2 " _ 2 c M or u = ^ f ^ , and therefore the region 7l0 lies at least 0.04 to the left of the curve 71 . By (ABC 5), the line segment AB forms a small negative angle with the positive u-axis, so by requiring v > 0.55 in the definition of 7lo, we can be sure that all of ABC(ao, bo) lies above the horizontal line v = 0.5 if (ao,6o) G TZo- For any point (u,v) G TZQ, we have u < 2u and (u,v) lies at least 0.04 to the left of 71 . Since our initial condition has form (an/o, 6 0 / 0 ) , the set of values in each "transition region" {(a0fo(x),bofo(x)) :xe [L-l,L + l]}, 38 forms a line segment with endpoints O and (an,fro) in the (u,w)-plane. We require that the tip of this line segment (an, bo) lies in the region TZo. We do not need to worry about the case where the initial condition for the PDE (5.1) is such that (ao,&o) G TZ D {0.55 < v < 0.8} lies to the right of TZQ- If we want to establish condition (*) for that initial condition, then by the monotonicity of the PDE (5.1), it is sufficient to pick a'0 < ao such that (a'0, bo) G TZo and prove condition (*) for the initial condition (a0/o,6o/o)- Therefore we only consider (a 0, b0) in TZ0. Assuming (an,&o) lies in TZQ, a part of the line segment O(ao,bo) still lies below the horizontal line v = 0.55. To study the evolution of the whole line segment under Tn, we will a bit later consider two cases: 1. e < v < 0.8, and 2. 0 < v < e, where we will pick e = 0.24 in Chapter 5.1.2. We will construct piecewise linear curves ABC(vo,vo), v0 G [0.55,0.8], with A = (VQ, vo), B, and C satisfying the requirements laid down in (ABC 1-5) on page 35, in the proof of the following two lemmas. See figures 5.5 and 5.7 for an illustration of each lemma. Lemma 5.1.3. (Case 1) If (an,&o) ^es o n AB(vo,vo) = AB(ao,bo) with ao — bo < 0.09 and 0.55 < bo < 0.8, then for sufficiently small s, there exist as and bs with bs > 0.5, and a positive number K independent of s such that AB(as,bs) = AB(VO,VQ) and f°(aa0, ab0) > ((1 + Ks)aas, (1 + Ks)abs^ (5.20) for all a G [0,1]. Moreover, the constant K can be chosen to be arbitrarily large if c is also allowed to be arbitrarily large. Remark 5.1.4. Using some easy geometric considerations, one can say the following: if e is fixed and c is allowed to be arbitrarily large, then there exists an arbitrarily large positive number K depending on e but independent of s, such that if a £ [f-A] then ((1 + Ks)aas, (1 + Ks)abs) - {aas, abs) > (j^Ks, Ks^j , (5.21) and if a £ [0, £-) then ((1 + Ks)aas,{l + Ks)abs)-(aas,abs)> (0,0). (5.22) Lemma 5.1.5. (Case 2) If (ao,bo) lies on ABC(vo,vo) = ABC(ao,bo) with 0.08 + bo < ao < ^\+2cv ~ 0-04 and 0.55 < &o < 0.8, then for sufficiently small s, there exists a positive number K such that, if a £ [^,1], then T°(aao,abo) - (aa0,ab0) > (j^-Ks,Ks^ , (5.23) and if a £ [0, then ^(aao, abo) — (aao,abo) > (—2aaos, —2abos). (5.24) Moreover, the constant K can be chosen to be arbitrarily large if c is also allowed to be arbitrarily large. 39 In case 1 above (Lemma 5.1.3), (as, bs) changes with s, but in case 2 (Lemma 5.1.5), (a3,bs) remains fixed and equal to (ao,&o), thus (as,bs) is not explicitly defined. As will be seen later on, I is picked small so that the lower part of the "transition region" (of (ao/o, 6o/o)) moves up at a sufficiently large speed under the heat kernel to cancel out the downward movement as described in (5.24). But the heat kernel pushes down the top part of the "transition region", so K (and thus c) is picked large to cancel out that effect. And finally s is picked small so that the movement caused by is small. In case 1, we assume (ao, bo), the top tip of the line segment formed by the "transition region", lies to the left of the line u = v + 0.09, while in case 2, we assume that (ao,b0) lies to the right of u = v + 0.08. There is a thin strip, i.e. 0.08 < u - v < 0.09, where both cases apply, so we can apply either case 1 or case 2 there. Let (UQ,VQ) be a point on the line segment O(ao,bo). Intuitively, we would like to view "progress" as an increase in u-coordinate, which is measured by 7r„(f*(uo,t)o) — (UQ,VQ)). In case 1, however, it is not always possible for the ^coordinate to increase. So instead, we measure progress with respect to the family of parallel lines AB, each of which makes a small negative angle with the positive u-axis, and thus allowing the i>-coordinate to decrease slightly while still making "progress" with respect to AB. More specifically, we compare !F^{UQ,VQ) not with (UQ,VQ) = ( ^ O O ) f^o), hut with (us,vs) = (^a3, ^bs). We show that with respect to the lines AB, the entire line segment makes progress with respect to the lines AB when moving under rj. The u-coordinate actually increases very rapidly, so we move very quickly into where case 2 applies. In case 2, we compare P%(uo,vo) directly with («o,t>o), a n d show that the part of the line segment with u-coordinate > e makes progress, but the part of the line segment with u-coordinate < e actually makes small negative progress. This negative progress will be compensated by positive progress made under evolution according to the heat kernel (to be shown in Chapter 5.1.2). Proof of Lemma 5.1.3. We first define Tli = {{u,v) ETZ-.u-v £[0,0.1},v e[e,0.8}}, (5.25) Tl[ = {{u,v)£Ki: u - « 6 [0,0.09]}. (5.26) Tl'3 = ' {(u,v) G Tl : u-v G [0,0.1],u < 2v,v G [0,e)}. • (5.27) Later in the proof of Lemma 5.1.5, we will also define the follow three regions, which we include here for easy reference (See figure 5.4). Cl 4- 2r)v H2 = {(u,v) £TZ:v + 0.02 < u < ——r1- -0 .04 ,v£ [e,0.8]}, 1 + 2cv (1 + 2c)v 1Z'2 = Uu,v) G7l2 :v + 0.08<u< \ - 0.04, v > 0.55}, 1 4- 2eu 7l3 = {(u,v) G 71: u < 2v,v G [0,e)}. Notice that K[ C 7lu 71'2 C 7Z2, and 7l3c7l3. To study the evolution of J-v of a line segment OP that passes through the origin O and has its top tip P in the region 7Z[ C\7Zo, we will define a new vector field £ = (£1,^ 2) for (u, v) G Hi U 7l3, such that £1 < r)i and £2 < 2^ everywhere in Tli U Tl3, which means that •F|(u,v) <P°{u,v) (5.28) 40 t u u (a) Hi and 723 (b) 1Z2 and 72.3 Figure 5.4: Various regions. in both u- and u- coordinates, for small s and all (tt, v) G 724 U7£3. Notice that 72.1 i s required to stay a finite distance left of the line u — v = 0.1, the right edge of the parallelogram %\\ this is such that we can still control how much (u, v) G Tt\ moves under T^, even if it leaves 72-1 and enters the strip 1Z\\R!X. Also, we define £ in 72-i U 72.3 because this is the region where the line segments OP lie. We only consider the region 72.3 for (u, v) close to the origin, rather than the region {(u, v) G 72.: u - v G [0,0.1], v G [0, e)}, since top tip of the line segment bs will be > 0.5, which implies that u < 2v if (u, v) G OP. Step 1: Defining f. We first define £ on the diagonal line 73 n {(u,v) : v < 0.8} = {(u,v) : u = v,Q < v < 0.8}, where 73 is defined in (5.11). Let 9Q < \ arctan(0.05) be a small angle such that a line passing through [VQ,VQ) with VQ > 0.55 and making an angle of -20n with the positive u-axis intersects the vertical line u = 1 above the horizontal line v = 0.5. Define (5 = 0.1, (5.29) and pick Fi to be large but F i < Sc. (5.30) We also define £(0.8,0.8) = (0.8Fi,0.8(-2)), (5.31) 4 1 u Figure 5.5: Illustration of Lemma 5.1.3 (Case 1) where F\ is large enough such that £(0.8,0.8) makes an angle of —di with the positive w-axis, and 0 < 0i < do- This can be done for sufficiently large c. We define ^(v,v) = ^(0.8,0.8) = (vFu-2v), for v € [0,0.8]. This defines £ on the line segment 73 f l (TZi U 1Z'3). Finally, along straight lines that make angles of —B\ with the positive u-axis, denoted A'B'(v,v), we define £ to be equal to u), i.e. for all (u',v') G A'B'(v,v), we define (,(u',v')=(,(v,v). Here A' = (v,v) is the point where A'B'(v,v) intersects the line u = v and £ has already been defined, and B' is the intersection point of A'B'(v,v) and the right/bottom edge of the 4-gon TZ\ U 1Z'3, i.e. either the line u = v + 0.1 or the line u = 2v. Thus we have defined £ on all points in TZi UTZ'3. To summarize, £ in TZi is defined in a way such that: 42 1. On the line u = v where 0 < v < 0.8, ^=(vFuv(-2)), (5-32) such that f makes a small angle of —0\ with the positive it-axis. 2. £ is constant along lines that start at a point on the line u = v, and make angles of —6\ with the positive it-axis, where Oi < 00 < ^ arctan(0.05). (5.33) Step 2: Verifying £i < Tji. We divide into two sub-cases: 1. Tlr- e < v < 0.8; 2. TZ3: 0 < v < e. We first deal with sub-case 1. By the discussion following (5.17), the minimum horizontal distance between 71 and 73, defined in (5.9) and (5.11), is at least 0.19 for v G [e, 0.8] and sufficiently large c. Therefore the region 72.1 is more than 5 = 0.1 left of 71 n {(u, v) : e < v < 0.8}. Thus by (5.15), 771 > 25cv (5.34) in 7 i i . On the line segment 73 n IZi — {(u,v) : u = v,v G [e, 0.8]}, £ is defined by (5.32). Condition (5.30) then implies £1 = vFi < 5cv. Thus (5.34) shows that £1 < 771 on the line segment 73 fl 7?i. We also need to verify that £1 < 771 everywhere in TZ\. For that, recall that £ in 72-1 is constant along line segments A'B'(v,v), each of which makes an angle of —6\ with the positive it-axis, so we estimate how much the u-coordinate can decrease along the line segments A'B', to make sure that £ < 77 even on the line it — v + 0.1. We observe the following: since tan#i < 0.05, the amount by which the u-coordinate decreases, from the point A'(VO,VQ) on the line u = v to the point B' on the line u — v + 0.1, is 0.1tan#i < I < where we will pick e = 0.24 in Chapter 5.1.2. Thus even for (UQ,VQ) lying on the line u = v + 0.1, we still have A'B'(uo,vo) = A'B'(vi,vi) (i.e. (i>i,i>i) lies on the line u = v) for some v\ with v\ < 2VQ. Because £ is constant along A'B'(uo, VQ), we have €i(uo,vQ) = v\Fi < 2vQFi < 25cv0, where we use the requirement (5.30) in the last inequality. Hence for all (uo,"o) G TZ\, £1(1*0,uo) < Vi(u0,vo) by (5.34). 43 Now we deal with sub-case 2. For v < e < \ and u < 2v, we have 771 = (2c(l -u) + l)v-u> (2c(l - 2v) + l)v - 2v = (2c(l - 2v) - l)v > cv (5.35) if c is sufficiently large. This estimate applies to both Case 1 (this lemma) and a bit later on Case 2 (Lemma 5.1.5), and shows that on 73 D Tl'3 = {(u, v) : u = v, v G [0, e)}, £x = vFi < Scv = 0.1ci> < 771, where we use (5.29), (5.30), and (5.35) in the second, third, and fourth steps, respec-tively. For the rest of Tl'3, we make the observation that the u-coordinate of any point on A'B'(vi,vi) is larger than the u-coordinate v2 of the intersection point of A'B'(vi,v\) and the line u = 2v, which we obtain by solving v — v\ = — tan#i(2i; — v\), i.e. v2 = l f f i^V 1 > ^ T h e r e f o r e f o r a n y ( uo,«o) G AWivuVi), £i(uo,vo) = V1F1 < O.lcui < 0.2ci;o < 771, from which we conclude that £1 < r/i in , Step 3: Verifying £ 2 < %• This is considerably easier than verifying £1 < r/ i . From (5.32), £ 2 = -2v on the line segment 73 n (Tli U 1Z'3), so £ 2 <.r/2 by (5.13). Along A1 B'(vi,v\), the ^-coordinate decreases. So for any (UQ,VQ) G A'B'(vi,v\), £ 2 ( ^ 0 , wo) — -2vi < -2v0 < m-Thus £ 2 <-772 in Tlx U71'3. Step 4: Defining as, bs, and AB. The vector field £ is defined such that any point (UO,VQ) G Tlx U 7?.3 moves under £ at a constant speed (linear in i>o) along the line A'B'(UQ,VO). Thus any line segment OP lying in Tlx U7£3 remains a line segment (i.e. does not become a curve) under the flow £, and if (ux,Vx) and (u2, v2) are two points on such a line segment, then the ratio ] ^ | | ^ ' ^ | remains constant. We define j4B(tii,Di), 0 < v\ < 0.8, to be the line segment that makes an angle of —26Q with the positive u-axis and connects points A = (vx,vx) G 73 and B, with B lying on the right/bottom boundary of the 4-gon Tlx U Tl'3. Recall from (5.33) that #o is a small angle and A'B'(ao,bo) makes an angle of —Ox with the positive 'u-axis, where 0 < #1 < #o- Thus the part of A'B'(ao,bo) to the right of the point (ao,6o) lies strictly above AB(ao,bo), and the angle between A'B'(ao,bo) and AB(ao, bo) is at least #0 by the choice of #o and #1 in (5.33). Also, we define (as, bs) = AB(a0, b0) n O^ | (o 0 , b0), where (ao, bo) G Tio H Tl[ is the top tip of any line segment that we consider for this lemma. We collect various facts for later use: 1. J-£ moves the point (ao, bo) to the right. 2. The part of A'B'(ao, bo) to the right of the point (ao, bo) lies strictly above AB(ao, bo). 3. (as,bs) lies on AB(ao,bo). 44 4. ^|(o 0,6o) lies on A'B'(a0,b0) 5. (as,bs) and F|(ao,&o) both lie on 0F|(ao,&o)-The above facts imply that bs is a lower bound for ^ ( ^ ( a n , bo))-We now estimate the speed at which F|(ao, &o) separates from (as, bs). First of all, since A'B'(ao, bo) makes a negative angle with the positive u-axis, the intersection point (vi,vi) of A'B'(ao,bo) and the line u = v must lie above (a0,b0), i.e. vi > b0. This means that |£N,V>I = = 1(^1^1,-2^01 > boy/F? + 22 . Let F2(b0) = boyjF? + 22. Since /l'73'(ao, &o) and AB(ao,bo) make angles of - 0 i and -20o with the positive u-axis, respectively, where 0 < 9\ < 8o, the angle 62 between A'B'(ao, bo) and AB(ao,bo) is larger than 6Q- Let a = sF2(&o) be the distance between (ao,b0) and F|(ao,&o), and /? be the distance between (an,&o) and (a s,6 s), then the Euclidean distance 7 between F|(a0,fro) and (a s,6 s) (the thick line in figure 5.6) is 7 = \Ja2 + /?2 — 2a/3cos02, which attains the minimum usin0 2 when /3 = a cos 82, therefore 7 > sF2(bo) sin0 o. Since s is small and (an, 60) lies in TZo H (in particular, to the left of the line u — v = 0.09), T^(ao,bo) lies in {(«,!;) £ TZ : u — v £ [0,0.1],v > 0.5}, thus the smallest angle between OF|(ao,fro) (portion of which is (a s, 6s).F|(ao, 60)) and the positive u-axis is greater than arctan^| > | . Therefore the vertical distance between .F|(ao,&o) and (as,bs) is at least sF2(6o) sin 0O sin | . Similarly, F|(ao, bo) lies in TZ, so the largest angle between OF|(ao, 60) and the positive u-axis is less than j , hence the horizontal distance between these two points is at least sF2(bo) sin0o cos j, which is larger than sF^tbo) sin0o cos More precisely, F|(a 0,bo) - (as,bs) > \^sF2(b0) sin0 o cos- , sF2(b0) sin0 o sin-J = (^bo^F? + 2 2 sin0O, ^bo^F? + 2 2 sin0 o) Since 6S < 60, the above inequality implies J7l(a0,b0)-{as,bs) > (^~bs^F2 + 2 2 sin0 o, ^bs^F2 + 2 2 s in0 o ) . (5.36) By the choice of a.small 0o in (5.33), the entire line segment AB(ao,bo) lies above the horizontal line v = 0.5 if (ao,6o) G TZo n 7^ -1- Thus (as,bs) lies above the horizontal line u = 0.5, which means that as < 2bs. Thus (5.36) implies J rl(ao,b0)-{as,bs) > (^as JF2 + 2 2 s ind 0 , ^b , y /F 2 + 2 2 sin0 O ) . (5.37) 45 Figure 5.6: Another illustration of Lemma 5.1.3 (Case 1) If we define then (5.28) and (5.37) verifies condition (5.20) for the point (ao,bo). We recall from (5.30) that F i can be chosen to be arbitrarily large if we allow c to be large. Therefore K can also be chosen to be arbitrarily large if 0Q is fixed. For any other point (aao,abo) on the line segment O(ao,bo), a £ [0,1], notice that from (5.32), £(aao,abo) = a£(ao,&o), i.e. £ is linear on the line O(ao,bo). Also, the entire line segment O(a0,bo) lies in Tli UTZ'3. Thus by (5.28), condition (5.20) holds for all a €[0,1]. • Proof of Lemma 5.1.5. We define H2 = {(u,v) eTZ:v +0.02 <u< -0.04,v€ [e,0.8]}, (5.38) TZ'2 = {(u, v) G 7l2 : v + 0.08 < u < ^ | ^ - 0.04, v > 0.55}, (5.39) TZ3 = {{u, v) G Tl: u < 2v, v G [0, e)}. (5.40) For this lemma, the region TZ'2 is the region for P, the top tip of the line segment OP that connects the origin O and the point P. Since we pick e = 0.24 later in Chapter 5.1.2, any line segment OP, with P G 71'2, lies entirely in 7l2 U H3. We follow the same steps as in the proof of Lemma 5.1.3 (Case 1). Step 1: Defining As in the the proof of Case 1, we will define a vector field £, 46 Figure 5.7: Illustration of Lemma 5.1.5 (Case 2): ((aao,abo) = (—2aao,—2abo) and C(a0,&0) = (a0F3,b0F3). such that Ci < r/i and (2 < V2 everywhere in U i?3- First, we define C(1,1) = (F 3 ,F 3 ) , where we pick F3 large but with 0 < F 3 < 0.01c. (5.41) It is convenient to define £ at the point (1,1), even though this point is not even in 1Z2 LiR3. For (u,v) G IZ2, we define C(u,v) = (uF3,vF3). (5.42) And for (u,v) £ 1Z3, au,v) = (u(-2),v(-2)). (5.43) 47 Notice that £ is discontinuous across the horizontal line v = e. Step 2: Verifying C < n. The region Tl2 is at least 0.01 to the right of 72 (the line u = v + 2/c) for sufficiently large c. By (5.14), we have 772 > O.OICTJ (5.44) for all points in 1Z2. The region 1Z2 is also at least 0.04 left of 71 , where 71 is the curve of u =  {\X\C^. Thus by (5.15), we have 771 > 0.08c?j > 0.08ce = 0.08c(0.24) > 0.01c. (5.45) for all points in R2. By (5.41) and (5.42), X{u,v) < (0.01cu,0.01cu) < (0.01c,O.OICTJ), which implies, by (5.44) and (5.45), C{u,v) < r)(u,v) for (u, v) G TZ2. Recall from (5.18) that the curve 71 approaches the degenerate curve {(u,v) :v = 0,ue [0,1}}U{(U,V) :u = l,vG [0,1]} as c —> 0 0 . Therefore IZ3 stays to the left of 71 if c is sufficiently large, and by the discussion below (5.11) regarding the sign of 771, 771 > 0 for (u,v) G IZ3. The definition of C in (5.43) says that Ci < 0 for (u,v) G IZ3, therefore Ci(u,v) < 771 (u,u) for (it, v) G 7^3. Furthermore, (5.13) implies that C,2(u, v) < T]2(u, V). Thus for all (u, v) G TZ3. C(u,v) <ri(u,v). Step 3: Denning BC. We define BC[VQ,VQ) to be the horizontal line segment v = v\ starting at point B(VQ,VQ) = (v\ 4- 0.1, v\) on the line u = v + 0.1 and ending on the vertical line u = 1; notice that (VQ,VO) itself does not lie on the line segment BC(VQ,VO). This definition of BC(VQ,VQ) means that B(VQ,VO) = (vi + 0.1, ui) is the right end point of the line segment AB(vo,vo), which was defined in Step 4 of the proof of Lemma 5.1.3. We also define tf=y. (5-46) Notice that K can be made arbitrarily large since F 3 (picked in (5.41)) is allowed to be arbitrarily large. The vector field £ is defined such that any point (u,v) G 1Z2 moves in the direction > 0(u,v), i.e. C is a dilation for points in 1Z2. But any point (u,v) G IZ3 moves in the direction of (u, v)0, i.e. C is a contraction for points in TZ3. Thus any line segment OP with 48 P £ 7Z'2 immediately splits into two line segments under £; the two line segments, however, lie on the same straight line through the origin O. For the top tip of the line segment P = (ao, bo) £ 72-2, we have ^c(a 0 , bQ) - {ao,b0) > {sa0F3,sbQF3), (5.47) since the fact that both u- and v- coordinates increases under £ in 72-2 implies that saol^ and sboF3 are lower bounds for the increase in u- and v- coordinates, respectively. Since b0 > 0.5 if (a0,6o) £ TV2, (5.46) and (5.47) implies that Tsc{aoM) - {a0,b0) > (s^-boF3:sb0F^ > (^Ks,Ks^j . (5.48) This verifies (5.23) for P = (ao,&o) £ 722. For any other point (aao,abo) on OP that lies in 72.2 (i.e. a £ [^ ,1]), linearity in the definition ((aao,a6o) = ct((ao,bo) in (5.42) implies (5.23). The verification of (5.24) for points in 72-3 is similar. Recall from (5.43) the definition of ( in 72.3: C(u,v) = (-2u,-2v). Let P — (ao,6o) £ 722 and a £ [0, ^ ) . Then both crao and abo decrease under £, initially at speed 2aao and 2abo, respectively. The speed of decrease immediately becomes smaller than 2aao and 2abo (respectively) after the initial movement. Thus 2aao and 2a&o are upper bounds of the speed of decrease: F|(aao,a6o) — (aao,a6o) > (—2aaos, — 2abos), as required by (5.24). • To summarize the results in Lemmas 5.1.3 (Case 1) and 5.1.5 (Case 2), if the top tip of the line segment OP at time 0, P = [ao, bo) with 60 > 0.55, lies to the left of the line u = v + 0.09, then we use Case 1 to define (as,bs) £ AB(ao,&o) and £ < rj where (as,bs) is below but to the right of (ao,6o), such that £ moves (ao,&o) at an arbitrarily large speed below and to the right of (ao,6o)> but above (as,b3). Once the top tip of the line segment has moved to the right of the line u = v + 0.08 but to the left of the curve u = ^ 2 c ^ — 0-04, (or it lies between those two at time 0 to start with), then using Case 2, where (as,bs) = (ao, 60), we define ( < n such that ( moves (ao, 60) above and to the right > of (ao, bo), in fact, along the same direction as O(ao, bo), again at an arbitrarily large speed. Finally, if (ao, bo) lies to the right of the curve u = ^ 2 c u " — 0-04, then we move the initial condition to the left of this curve and apply Case 2. 5.1.2 Analysis of the P D E (5.1) Now we use the results obtained in the previous section about the evolution of the ODE (5.2), together with some results on the heat equation, to study the evolution of the PDE (5.1). First, we need to characterize how values in the transition region evolve according to the heat equation. We will establish two technical lemmas to that end. 49 Lemma 5.1.6. If I is fixed and f = fo is as defined in (IC 1-3) on page 34, then for x e { - L - l - s , - L - ± ) U ( L + ^ L + l + s) _ and s small, we have esAf(x) > f{x) + ^ . (5.49) Proof. First, we shift / right by L such that f(x) = h(x) for x G [-1,1] and f(x) = h(2L-x) for x G [2L - 1,2L + I}. Thus Af(x) = Ah(x) for x G {-I, I) and Af{x) = Ah(2L - x) for x G (2L — 1,2L + I), where { 0, x <-I -*,'o<?<1°- <5'50) 0 x > I We make the following observation to aid our computation: if u = esAh gives the evolution of the heat equation with initial condition h, then A u = A(esAh) gives the evolution of the heat equation with initial condition Ah, i.e. Define then A{esAh) = esA(Ah). (5.51) k(x) = Af(x), Ah(x), x G ( - / , / ) k(x)={ Ah{2L-x), x G (2L — l,2L + I) . " (5.52) 0 otherwise By equation (5.5.10) in [Taylor 1996], the solution of the heat equation can be expressed in terms of an integral. /OO -i 2 -=e-^k(x-y)dy. (5.53) -oo V47TS Using the above formula and the expression of k in (5.52), we can estimate esAk(x) for 50 x £ (—§Z, —I] and s small: -X -V— e •»•' J-x V 47TS J-x -1+2L V47TS /: yj-x-l V47TS J-x V47TS J - x V47TS J - x - i + 2 L V^ITS -X+1+2L e - ! L - > + / ~r=dy v+2L V47TS > > — dy > I f f°° 1 _ Z Z"00 1 _ z \ e *'dy , (5.54) where in the last step we use the fact that x £ {— |Z, —Z] implies |x| = —a: > Z. We can take s to be sufficiently small such that ff° ej^=dy < 10 5 / 3 , then with a substitution of variable in the first integral in (5.54), we obtain es*k{x ) > i f / " — e 4 dy - 10 1 / 1 „ 5 . /•(I'l-O/V- ! ^ N / ^0 If x £ (-Z — s, — Z], then \x\ — I < s < ^ /s if s < 1, and (5.55) implies ^ W > i Q - 1 0 - - j ( 1 - i = e - ^ ) > i , . On the other hand, for x £ (—Z, —255) and s small, we also have e^k(x) > i j (5.55) (5.56) (5.57) since for x £ {—1,1), k{x) = Ah{x) is a step function with discontinuity at 0, where Ah is given in (5.50). Estimates (5.56) and (5.57) on the behaviour of k under the heat kernel implies that for s small and x £ {—I — s, — 255)1 = {A{e^f))(x) = (e^(A/))(x) > ± where we use (5.51) in the second equality. This establishes (5.49) for x £ {—L — Z — s, —L — 2gg). Verification of (5.49) for x £ (L + ^gg, L + I + s) is similar. • 51 Lemma 5.1.7. Let t > 0 be fixed and fi = fo be as defined in (IC 1-3) on page 34, i.e. A W ( h(x + L), x £ {-L-l,-L + l) 1, x £ [-L + l,L-l] h(L-x), x£ (L-l,L + l) ( 0 otherwise (5.58) Let (ar) + mt, - L - l - t < x < L + l + t otherwise (5.59) where m > 0. Then there exist positive constants 8\ and 52 depending on m but independent oft such that if f2(x) (1 + S2t)h(x + L + 8tt), x £ (-L - I - Sit, -L + l- Syt) 1 + 82t, x£[-L + l- Syt, L-1 + Sit] (l + 52t)h(L + 5it-x), x £ (L-l + 8it,L + l + Sit) 0, otherwise (5.60) then f2 < f3. Proof. Without any loss of generality, assume m < 1. Let M = 1 A sup^gjj |/i(ar)|, then M = 1 Al/l = l/l since I will be picked to be < 1 in (5.62) a bit later. Define ( fl(x+^), X £ ( - L - l - ^ , - L + l - ^ ) 9i(x) = { /i(0), ar £ [-L + I • mt T i i mt i " 3 M ' ^ l ' r ZM\ h(x-i^), xE(L-l+3%,L + l + ^ ) ( 0, otherwise then any small piece of the curve of g\ is f\ shifted by either 0, f p , or —j^, with ^ < t. In particular, the two transition regions in g\ are the two transition regions in fi shifted by o r ~3M> a n d the "middle" region (i.e. the region sandwiched between the two transition regions) in g\ is the middle region of / i expanded left and right by Since M = sup x e R |/i(ar)|, we have 9i(x) ~ fi(x) < mt for all x £ (-L — I — ^,L +1 + j^), therefore f3 > g\ everywhere and in particular, since h(%) — fi(x) = mt for x £ (-L — I — t,L + I + t), we have /3(a;) - g\(x) > 2mt (5.61) for x £ (-L — I mt T _ i _ / _ i _ mt ' 3 M ' n ' 1 ' 3M' Next we define mt i . . I + -3- )9i(x). 52 Then mt , . mt h\x) ~ 9i{x) = -j-9l(x> - ~Y since g\(x) < 1 everywhere. The above inequality and (5.61) imply that hix) < f3{x) for x £ (-L - I - j]fi,L + I + jjH). Then Si = ^ and <52 = ^ satisfy the requirement of / 2 in (5.60), and the proof is complete. • For the remainder of this section, we will establish Proposition 5.1.1. We assume that the initial condition of the PDE (5.1) is (ao/o, °ofo), where fo is as defined in (IC 1-3) on page 34 and (an, fro) = (ao>fro) lies in the region TZo defined in (5.19)., By Remark 5.1.4 and Lemma 5.1.5, we can pick (as,bs) £ ABC(ao,bo), with (as,bs) — (ao,fro) if ao — bo > 0.08 (i.e. Lemma 5.1.5/Case 2), such that estimates (5.20)-(5.24) regarding the evolution of the ODE (5.2) are valid. We will use this, together with Lemma 5.1.6 at the beginning of this section, to show that there is a positive constant m such that for x £ (-L — I — s,L + l + s) and sufficiently small s, e s A^(a0/o,fro/o)(a;) - (asfo,bsfo)(x) > (asms,bsms). Finally we will apply Lemma 5.1.7 to complete the proof of Proposition 5.1.1. We divide this task into proving two lemmas, which correspond to the two cases in Lemmas 5.1.3 and 5.1.5, respectively. Before we proceed, we first pick (5.62) and c = 0.24 < 0.5fc (5.63) Lemma 5.1.8. (Case 1) Recall that TZ'XUTZ'3 = {(u,v) £ TZ : u — v £ [0,0.1] and u < 2v for v £ [0, e)} and TZ[ n TZo = {(u, v) £ TZ : u - v £ [0,0.09] and v £ [0.55,0.8]}. If {(a0fo(x), b0fo(x)) : x G[-L-l,-L + I}} C TZ[ U TZ'3 and (o0, fro) £ R[ n TZ0, where TZ0, TZ[ and TZ'3 are defined in (5.19), (5.25), and (5.26) respectively, and fo is defined in (IC 1-3) on page 34, then the conclusion of Proposition 5.1.1 holds. Proof. By Lemma 5.1.3 and Remark 5.1.4, for sufficiently small s, we can pick K and K large enough such that K > | , (5.64) 53 and a point (as,ba) G TZ with bs > 0.5 such that (a a/ 0(0),6 s/(0)) G AB(a o/o(0), 6o/o(0)) and ^(oofo(x),boMx)) > ((1 + Ks)asf0{x),{l + Ks)bsf0{x)) (5.65) for all x; furthermore, if bsfo(x) > e, (1 + Ks)bsfo(x) - bj0(x) > Ks, (5.66) and if b3fo(x) G [0, e), (1 + Ks)bsf0(x) - bsfo{x) > 0. (5.67) Here, (a s/o,6 s/o) is the function to which we compare ^(ao/o,6o/o) to see how much "progress" we are making in increasing the w-coordinate. Forx G [ -L - , L + 200] =• [ - L - ^ , - L + l]U(-L+l,L-l)U[L-l,L+^], where the intervals [-L — — L + l\ and [L — I, L + ^ j ] are in the transition region, we have b.f0{x) > 0.5/0 ^ " 2 b o j > 0-5/o ^ - i o o J = ™h V 100 therefore by (5.63), bsfo{x) > e. (5.68) Therefore by (5.66) we have, for x G \—L — L + (l+Ks)bsf0(x)-bsf0(x)>Ks. For x G [-L-l, - L - 25o)U (L+ L + l] where bsfo is possibly smaller than e, by (5.67), we have {l+Ks)bsfo{x)-bJo{x)>0. To summarize, combining (5.65) and the two inequalities above, we have 7r w (F> 0 /o , bof0))(x) > (1 + Ks)bsf0(x) (5.69) and { > Ks, x e [-L - goo,L + 2gg] >0, xe[-L-l,-L-^)U(L + ^,L + l] .(5.70) = 0, x ^ [ - L - Z , L + Z] As stated in (5.5), |A/o| < p. Therefore the heat operator esA applied to bsfo may decrease its value by at most ps. More precisely, esAbsf0 - bafo > ~^s. (5.71) 54 (l+Ks)bs jt i_ I I 200 -L + l 0 L-l L + 200 L + l Figure 5.8: The effect of the heat kernel on the function (1 + Ks)bsfo- The arrows indicates whether (l+Ks)bsfo(x) increases or decreases. The effects illustrated here are lower bounds. In [-L — j o g , L + j gg] , the function decreases, which is why we need the first line (5.70) to be > Ks to cancel out this decrease. everywhere. We can use (5.70) and (5.71) to obtain estimates on esAJ-%(aofo,bofo). We estimate the "progress" made after applying the heat kernel: by (5.69) and the monotonicity of the heat kernel esA, 7 r „ ( e s A ^ ( o o / o , 6 o / o ) - (asfo,bsfo))(x) > esA(l + Ks)bsf0(x) - bsf0(x). F O T X G [-L-goo, L-r-joo], esA(l + Ks)bsf0(x)-bsf0(x) = (14- Ks) {esAbsf0(x) - bsf0(x)) 4 ((1 4- Ks)bsf0(x) - bsf0(x)) by (5.71), the first line of (5.70), and the fact K > $ > ^ . Therefore > -(l+Ks)^s+3fs irv(es J^(a0/o,6o/o) - {asfo,bsf0))(x) > -^s (5.72) for sufficiently small s. On the other hand, for x € (—L — l — s, —L — -^)L)(L+-^Q,L + 1 + S), by Lemma 5.1.6 we have, e s A ( l 4 Ks)bJ0(x) - (1 4- Ks)bsf0(x) > (1 4 Ks)^s > ^s. Therefore by (5.65) and the above inequality, for x € (—L—l—s, —L—^Q)[J(L+-^,L+1+S), nv(esAF°(aofo,b0fo)-(asfo,bsfo))(x) > esA(l + Ks)bsf0(x) - bsf0(x) > (l + Ks)bsfQ(x) + ^s-bsf0(x) > 5PS' where the last line is due to the second and third lines of (5.70). Hence for x G (-L — I — s,L + I + s), nv(esAF°(a0fo,bofo))(x) > bs (j0(x) + ^ s ) 55 Then Lemma 5.1.7 implies that there exist positive constants 5X and 52 independent of s such that irv(ea*J*(aofo,bof0))(x) > bsf2(x), (5.73) where f2 is defined in (5.60) f2(x) ( (1 + 52s)h(x + L + Sis), x G (-L - I - Sis, -L + I - 6is) 1 + S2s, x G [-L + I - 5is,L - I + Sis] (1 + 52s)h(L + 5XS-X), xe{L-l + <5iS, L + 1 + Sis) (. 0, otherwise Similarly, the estimates in (5.72) to (5.73) also hold for the u-coordinate of e s A J r * ( a o / o , fro/o) — (asfo,bsfo), if bs on the right hand side of each inequality is replaced by as. So for all x G (—L — I — s, L + I + s), we have esAF°(a0fo, b0fo)(x) > (asf2{x), bsf2{x)), as required. In particular, (as,bs) in the statement of Proposition 5.1.1 should be ((1 + 52s)as,(l + 52s)bs). . • Lemma 5.1.9. (Case 2) Recall that (1 + 2c)v TZ'2 = {{u, v) G 7Z2 '• v + 0.08 < u < \ + 2 ( ! v - ° - 0 4 ' v > ° - 5 5 l ' and (1 + 2c)v 7e2u7e3 = Uu,v) eK :v+ 0.02 <u< \ „ -0.04 1 + 2C?J for v G [e, 0.8] and u <2v for v G [0, e)}. If {(ao/o(x),bofo{x)) : x G [-L — l,—L + I}} C 7l2 U 72.3 and (ao,bo) G R'2, where TZ2, R'2 and 72-3 are defined in (5.38), (5.39), and (5-40) respectively, and fo is defined in (IC 1-3) on page 34, then the conclusion of Proposition 5.1.1 holds. Before we prove this lemma, we observe that the union of all the regions where (ao, bo) may lie is (72/j n72-o) U72. 2 , which is exactly 72-o as defined in (5.19). If (ao, bo) G 724 ^  72-0, then the part of the line segment O(ao, bo) {O(ao, bo) consists of values of (ao/o, bofo)) above y — e lies in 72'!. On the other hand, if (ao,bo) G TZ'2, then the part of the line segment O(ao, bo) above y = e lies in 722. But in both these cases, for the part of the line segment O(ao,bo) below y = e, if suffices to consider 72-3 = {(u, v) G TZ : u < 2v,v G [0,e)}, because the top tip of O(ao,6o) in the (u,u)-plane lies above the horizontal line v = 0.5, where u < 1 < 2v. The sufficiency of restricting to {(u,v) G 72. : u < ^\+2^J ~ 0-04} has been discussed below (5.19) on page 39. Proof of Lemma 5.1.9. Under this case, the line segment formed by {(aofo{x), bofo{x)) : x G [—L — I, — L + l}} lies in 72.2Ui?3, i.e. the portion of the line segment 56 above the horizontal line v = e lies in TZ2 and right of the line u — v = 0.02, and the portion below v = e lies in TZ3. Furthermore, the top tip (ao,&o) lies in TZ'2, i.e. to the right of the line u — v = 0.08 and above the horizontal line v = 0.55. For x G \—L — ^ , L + ygg], we have 6o/o(x)>0.55/0(-L-4)>0.5/,(-4) > e by (5.63). Therefore, by Lemma 5.1.5, we can construct functions g2 and 53: fco/oix) + Ks, x £ [ - L - j ^ , L + '-100) w ' 100' -5 2 (* ) = < 60/0(^(1-25), i e ( - £ - l I - L - i L ) u ( L + 4 L + J) , '(5-74) 0, otherwise and b0fo(x)+Ks, x G 255 ,1+ 255] S s ^ H 6 0 /o(x)(l-2s) , x G ( - L - Z , - L - 5 0 0 ) U ( L + 2 ^ 0 ^ + 0 -0, otherwise such that both (j£g2,g2) and (^53,53) are lower bounds of J"*(a0/o, &o/o)- Notice that 53 < 92 everywhere, and g2 has discontinuities at — L — Z/100 and L + ^ /100, while g 3 has discontinuities at - Z , - Z/200 and L + Z/200. See figure 5.9 for graphs of g2 and 03. Now we construct the function 04: 9 A { X ) = { I 6 | " £ - f £ f f t , , , (5.75) I 53 (x), x G ( - c o , - L - T o o ] u [ L + T o T j , o o ) furthermore 04 is required to be C°°, monotone in [ - L — — ^]U[L+ j ^ , L + -^}, and lying between g2 and 03, with | A 5 4 ( x ) | < I (5.76) everywhere. For x £ [ - L - y^] U [L + J$Q,L + 255], the last requirement is automatic by (5.5); but for x G [ - L - 255, ioo]u[-^+ ioo">2o7)]' i t ; c a n D e achieved for sufficiently small s. Notice that since 54 < 02, (f^54,04) is a lower bound of Tfaofo, &o/o)-Furthermore, for x G [—L — 255, L + T^Q], S4(z) - 6o/o(a:) = Ka. - (5.77) Here (ao/o,fro/o) is the function to which we compare ^(ao/0,^0/0) to see how much "progress" we are making increasing the u-coordinate. We now turn to evolution according to the heat equation. First we deal with x ^ [—L ~ 25b"'-^  + 200]• ^ o r t m s , we use 03 as the lower bound for Trv(j-^(aofo> bofo))- We observe that 03 dominates (1 — 2s)fro/o, therefore monotonicity of the heat kernel implies e s A 53 > e s A ( ( l - 2s)60/o) = (1 - 2s)b0esAf0. 57 bo + Ks (a) The function g2 bo + Ks (b) The function g3 bo + Ks (c) The function 54 Figure 5 .9: The functions g2, <?3, and 54; dotted lines denote the function 6n/o-By Lemma 5.1.6, e s A/ 0(:r) > fo(x) + ^2 for alia; G (-L-l-s, -L-^)U(L+ ^,L+l + s), therefore esAg3(x) > (1 - 2s)b0 (f0(x) + ^) . Since 60 > 0.55, we have (1 — 2s)6n > 0.5 for sufficiently small s. Also recall that we pick I = in (5.62) such that JT = oV Thus the inequality above can be written as esAg3(x) > (1 - 2s)b0f0(x) + 0 . 5 ^ = b0f0(x) + (3 - 2b0fo(x))s. Finally, since 3 — 26o/o > 3 — 2 = 1, we have esAg3(x) >b0f0(x) + s. (5.78) for all x G (-L -I- s,-L- ^) U (L + ^,L + l + s). For x G \—L - 2oo,L + 25Q], we use g 4 as the lower bound for 7r^(J"*(a0/o,b0fo))-By (5.76), the heat operator e s A may decrease values of g4(x) by at most j?s, i.e. esAgA{x)-9i{x)>-^s. (5.79) Therefore for x G [-L — L + ^gg], we have 2 e s A g 4 (x) - b0fo(x) = (esAgA(x) - g4(x)) + (g4(x) - b0f0(x)) > --^s + Ks, where we apply (5.79) to e s Ag 4(x) - g4(x) and (5.77) to gA{x) - bofo{x). Thus for x G [~L- 2M>L+ 4®}' we have 1 T2 esAg4(x) - b0f0(x) > y^s since i f is chosen to be larger than 3// 2 in (5.64). The estimates in (5.78) and (5.80), together with the fact that g3 and g 4 are lower bounds of 7r w (^(ao/o, 6o/o))(x), imply that there is a positive constant m, such that for x G ( - L - Z - s, L + I + s), 7rv(esAF°(a0fo,b0fo))(x) > b0fo{x) + s > 6 0(/ 0(x) + a). As in Lemma 5.1.8, we apply Lemma 5.1.7 to obtain the estimate 7r ,(e 5 A^(a 0/o,6o/o))(x) > 6 0/ 2(x), (5.80) where f2 is defined in (5.60). For the u-coordinate of esAJ-^(aofo,bofo), we can obtain estimates (5.78) to (5.80) if we replace bo on the right hand side of each inequality by ao. So we conclude that for all x G {-L -l-s,L + l + s), esAf°(a0fo,bofo)(x) > (a 0/ 2(x),b 0f 2(x)), as required. In particular, (as,bs) in the statement of Proposition 5.1.1 should be ((1 + S2s)a0, (1 + S2s)bQ). • Proof of Proposition 5.1.1. The proposition follows from Lemmas 5.1.8 and 5.1.9, and the discussion below (5.19) on page 39 regarding the sufficiency of restricting the region for (a0,b0) to TIQ. • 59 5.2 Upper Bounds: Existence of d2 and D2 in Condition (.*) We establish the following proposition, which, together with Corollary 5.1.2, verifies condi-tion (*) on page 31. As in Corollary 5.1.2, the "L" in condition (*) is picked to be L — I. Proposition 5.2.1. If c is sufficiently large, then there exist constants d2 < D2 < 1 and T such that if v0(x) < D2 for x € \—L + l,L-l] then vt(x) < d2 for x € [-3L, 3L], where (ut,vt) solves the PDE (5.1). Proof. Because of the monotonicity of the PDE (5.1), it suffices to pick a uniform initial condition wo = some u, vo = D2, and show that at time T, UT = some u, VT < d2. Therefore we need only concern ourselves with the ODE (5.2). We can bound n2(u,v) defined in (5.8) for any v > 1 — \ as follows: n2{u, v) = (c(u — v) — 2)v < i c 1 .2)V = - V < - { 1 - - } <0 c, if c > 1. Thus for any two numbers D2 and d2 that satisfy I > D2 > d2 > I — \, there exists T, such that if VQ = D2, then VT < d2. • 60 P a r t II Stationary Distributions of A Model of Sympatric Speciation 61 C h a p t e r 6 A Model on Sympatric Speciation 6.1 I n t r o d u c t i o n Understanding Speciation is one of the great problems in the field of evolution. According to Mayr [Mayr 1963], speciation means the splitting of a single species into several, that is, the multiplication of species. It is believed that many species originated through geographically isolated populations of the same ancestral species [Dieckmann and Doebeli 1999]. This phenomenon is relatively easy to understand. In contrast, sympatric speciation, in which new species arise without geographical isolation, is theoretically much more difficult. 6.1.1 The Dieckmann-Doebeli Mode l Dieckmann and Doebeli [Dieckmann and Doebeli 1999] proposed a general model for sym-patric speciation, for both asexual and sexual populations. We will describe their model for the asexual population first. Each individual in the population is assumed to have a quantitative character (phenotype) x G IR determining how effectively this individual can make use of resources in the surrounding environment. A typical example is the beak size of a certain bird species, which determines the size of seeds that can be consumed by an individual bird. The function K : R —> R+ (carrying capacity) is associated with the sur-rounding environment, where Kx denotes the number of individuals of phenotype x that can be supported by the environment. For example, since birds with small beak size (say x\) are more adapted to eating small seeds than birds with large beak size (say x2, x2 > xi), KXl will be larger than KX2 if the surrounding environment produces more small seeds than ( — M 2 large seeds. In the Dieckmann-Doebeli model, Kx is taken to be cexp(— 2Jz ). Moreover, every pair of individuals compete at an intensity determined by the phenotypical distance of these two individuals. More specifically, an individual of phenotype x\ competes with an individual of phenotype x2 at intensity C X l _ X 2 , where Cx = exp(—^r). Therefore each individual in the population interacts with the environment via the carrying capacity K, and interacts with the population via the competition kernel C. 62 Let Nx(t) denote the number of individuals with phenotype x at time t. At any time, an individual of phenotype x gives birth at a constant rate, and dies at a rate proportional to L£d^Mh.^ i . e . inversely proportional to the rr-carrying capacity, but proportional to the intensity of competition exerted by the population on phenotype x, the numerator (C * N.(t))x = J2yCx-yNy(t) being how much competition (from every individual in the population) individuals with phenotype x suffer. In addition, every time an individual gives birth, there is a small probability that a mutation occurs and the phenotype of the offspring is different from that of the parent; in this case, the phenotypical distance between the offspring and the parent is then random and assumed to have a Gaussian distribution. Since the number of individuals of a certain phenotype increases via the birth mech-anism at a linear rate, but decreases via the death mechanism at a quadratic rate, extinction of all phenotypes will occur in finite time with probability one, i.e. N = 0 eventually. For large initial populations, however, extinction will happen far enough into the future that interesting behaviour does arise before the population becomes extinct. Monte-Carlo simulations, shown in figure 6.1, give a fairly good idea of the be-haviour of the Dieckmann-Doebeli model for asexual populations. If the initial popula-tion is monomorphic (t = 1 in figure 6.1), i.e. concentrated near a certain phenotype XQ (T.NN°\o) ~ ^o) ' t r i en the entire population first moves (t = 30,100,200 in figure 6.1) to-ward x, the phenotype with maximum carrying capacity. If ac > OK (this includes the case ac = oo, i.e. equal competition between all phenotypes), then the population stabilizes near phenotype x. But if ac < ax, then the monomorphic population concentrated at phenotype x splits into two groups, one group concentrating on a phenotype < x, while the other concentrating on a phenotype > x (t = 330,370,400,500 in figure 6.1). In the latter case, one can say that one species has evolved into two distinct species. t 30 t = 100 t = 200 £ = 300 £ = 330 £ = 370 £ = 400 £ = 500 £ = 600 A Figure 6.1: Simulation of the Dieckmann-Doebeli model with E = [—50,50] fl Z, ax — /600. V1000, and ac We now give a qualitative description of the Dieckmann-Doebeli model for sexual populations. Each individual in a sexual population is assigned three diploid genotypes with (say) five diallelic loci each. The first set of loci determines the ecological character x (i.e. phenotype in the asexual model). The second set of loci determines the marker trait, which 63 is ecologically neutral, i.e. individuals with different marker traits but the same ecological character have exactly the same birth and death rates. The third set of loci determines mating probabilities m; if m > 0, then such individuals prefer to mate with individuals of similar phenotypes; if m = 0, then such individuals have no preference; and if m < 0, then such individuals prefer to mate with individuals of a distant phenotype. In addition, |m| determines the strength of this preference. The birth rates and the death rates are calculated the same way as in the asexual model; in particular, only information from the first set of loci is used to calculate these rates, as this is the only genotype that determines the phenotype of the individual. Dieckman and Doebeli considered two cases in their sexual model: 1. mating depends on the ecological character; and 2. mating depends on the ecologically neutral marker trait. For example, in the second case, individuals with m > 0 prefers to mate with individuals of similar marker traits. Monte-Carlo simulations show that case 1 of the sexual model exhibits very similar behaviour to the asexual model, i.e. speciation if ac < &K and no speciation if ac > OK-A caveat: if ac < &K, then only individuals, who prefer to mate with individuals of similar phenotypes, survive after the population splits into two groups. Hence in the end, there are two distinct groups of individuals who refuse to mate with individuals from the other group. For case 2 of the sexual model, Monte-Carlo simulations indicate that ac < O~K is not enough for speciation to occur. In this case, ac < can is needed, where c < 1 is a constant. As the sexual model exhibits similar behaviour to the asexual model, we will con-centrate on the analysis of the simpler asexual model. It is our hope that understanding the asexual model will give some insights in explaining the behaviour of the sexual model as well. 6.1.2 A conditioned Dieckmann-Doebeli model Although the Dieckmann-Doebeli model for asexual populations is considerably less com-plicated than their model for sexual populations, it still seems too complicated for rigorous analysis. Thus we will attempt to simplify the model while preserving its key ingredients. Henceforth we refer to the Dieckmann-Doebeli model for asexual populations simply as the Dieckmann-Doebeli model. Before we describe our simplified Dieckmann-Doebeli model, we first introduce the concepts of fitness and selection. Selection occurs when individuals of different genotypes leave different numbers of offspring because their probabilities of surviving to reproductive age are different [Burger 2000]. If we define fitness to be a measure of how likely a particular individual produces offspring that will survive to reproductive age, then individuals with higher fitness should have higher probability of being selected for reproduction. Along these lines, it is natural to define fitness of a phenotype as the difference between the birth rate and the death rate of individuals of this phenotype. It is also natural to require the fitness function to be bounded between 0 and 1. The key feature of the Dieckmann-Doebeli model is that each individual has a fit-ness that depends on both the carrying capacity associated with its phenotype and the 64 configuration of the entire population. More specifically, the fitness of a phenotype x is an increasing function of Kx, the carrying capacity, but a decreasing function of (C * N)x, the competition it suffers. Here Nx is the number of individuals of phenotype x. In the Dieckmann-Doebeli model, the number of individuals can fluctuate with time. As mentioned before, since the birth rate is linear but the death rate is quadratic, extinction will occur in finite time with probability one, which makes it somewhat meaningless to analyze the equilibrium behaviour of the system. We make the assumption that the number of individuals AT is fixed over time, reflecting a constant carrying capacity of the overall population. The mechanism by which we achieve this is to require that death of an individual and birth of its single offspring occur at the same time, called replacement sampling in Moran particle models [Dawson 1993, Chapter 2.5]. This way, the number of individuals remains constant, and analyzing the behaviour of the population is then equivalent to analyzing the empirical distribution where xn, n = 1... N, denotes the phenotype of the n t h individual in a population of size Before we describe our simplified Dieckmann-Doebeli model, we say a few words about our terminologies and notations: we refer to individuals in a population as "particles", and sometimes refer to a phenotype as a "site". We consider multiple models, both discrete-time and continuous-time; for discrete-time models, we use V to denote the fitness function; but for continuous-time models, we use m instead. Our simplified discrete-time Dieckmann-Doebeli model is as follows: 1. E = [—L,L] n Z is the phenotype space, and ^ , 7 ^ G V(E) is a probability measure 2. K : E —> [0,1] is the carrying capacity, and C : Z —> R+ is the competition kernel; 3- Vx(TT) is the fitness of phenotype a; in a population with empirical distribution 7r (sometimes we notationally suppress the dependence on n); we define two possible fitness functions below; 4. A is a Markov transition matrix associated with mutation, with A(y,x) denoting the probability of a particle of phenotype y mutating to a particle of phenotype x; 5. At every time step t £ Z + , the entire population is replaced by a new population of N particles, each particle chosen independently, according to the distribution p.(t,nN): In (6.1), the denominator J2Z 7 rz'(0Vz(7r J V(^)) is simply the normalization factor such that Y^x Px(t, nN) = 1. In words, at every time step, the entire population dies and is replaced by 71=1 N. on E; (6.1) 65 a new population, each individual x choosing an individual x' from the original population as its parent with a probability proportional to its fitness Vx>, after which the new individual x undergoes mutation according to A. We consider two fitness functions: Kx V j 2 ) « = = r ^ . (6.2) Each of the two fitness function defined above is an increasing function of Kx and a de-creasing function of (C * n)x. resembles more closely the original Dieckmann-Doebeli model, but it has the disadvantage of being in a more complicated form than and it is also not differentiable. By Theorem 1 in [Del Moral 1998], which we state below, {n?,t G [0,T]} =4> {nt,t G [0,T]} as ./V —> co, where => denotes weak convergence and irt evolves according to the following deterministic dynamical system: s(t + i ) = $ > ( y , * ) ^ iry{t)Vy(n{t)) WMt)Y (6.3) Theorem 6.1.1. Suppose E is compact and M is a Feller-Markov transition kernel on V(E), i.e. M : V(V(E)) -> V(V{E)). Let M{N) = MCN, where CN is a Markov transition kernel on V(E) given by CNF(ir) = / F\^-YSXj\ 7T{dxx)... ir{dxN), fF(7r)  J^F^J^S^nid !) ..* i.e. a probability measure n replaced with an empirical measure nN formed by N particles chosen independently according to TT. Then as N —* oo. This theorem is easy to understand: take n = 1, then it says that the mapping MCN converges to M, i.e. changing the input measure of M by an empirical measure of iV particles makes almost no difference if N is large. Analyzing the dynamical system (6.3) is not easy, partly because it is of a compli-cated form that is nonlinear in n, and we cannot find any Lyapunov function that associates with (6.3). A continuously differentiable function V : U —> R is called a Lyapunov function if V is nondecreasing (or nonincreasing) along orbits. For a discrete-time dynamical sys-tem such as (6.3), this means that V(n(t + 1)) - V(n(t)) > 0 (or < 0) for all t > 0. For a continuous-time dynamical system, it means that dtV(-ir(t)) > 0 (or < 0). Simulations of (6.3), however, seem to display some interesting behaviour, which we will describe after carrying out some non-rigorous analysis of (6.3). 66 Without mutation, any site x with irx = 0 at any time r will stay 0 for all t > r. Mutation enables individuals of phenotype x to be born in future generations even if there are no individuals of phenotype x in the present generation. But if we start with a polymorphic initial measure, i.e. 7rx(0) ^ 0 for all x, then adding small mutation to the system should not cause significant changes in the behaviour of (6.3). Therefore we assume that A = I and 7r(0) is polymorphic. In this case, (6.3) can be simplified to TTx{t + 1) = Ez^(t)Vz(n(t))- . Thus if A = I, then # is a stationary distribution of (6.3) if and only if nx = -TrxVx(w) (6.4) c for some constant c. Condition (6.4) is equivalent to Vx(ir) = c for all x where TT(X) ^ 0, (6.5) Let K and C be in the form considered by Dieckmann and Doebeli, i.e. Kx = exp(—x2/2aj<) and Cx = exp(—x2/2ac). If V = V^2\ then condition (6.5) means that Kx = c(C * fr)(x) for all x where TT(X) ^ 0, which seems to indicate that if ac < ax, then TT should be close to W(0, a2K — ac). On the other hand, if V = then f is a stationary distribution if 1 — -^"> c £-* v * [s a strictly positive constant. Notice that if K and C are both Gaussian-shaped with KQ = Co = 1 then 7r = A/^O, a2K — ac) makes 1 - c*-,irz c o n s t a n t . furthermore, this constant is strictly positive since (C * 7r)(0) < Ko = 1 if ac < &K-Therefore for both and V^2\ the dynamical system (6.3) should have Gaussian-shaped stationary distributions if ac < O~K- hi simulations carried out, by Dieckmann and Doebeli [Dieckmann and Doebeli 1999], however, ac < OK is the case that leads to specia-tion, i.e. the stationary distribution supposedly has two sharp well-separated peaks, which contradicts the analysis carried out in the previous paragraph. Simulations of (6.3) with V = shown in Figure 6.2, reveal that if 7r(0) ~ 5o, initially the population does split into two groups and begins to move apart, but as t —* oo, the empirical measure converges to a Gaussian-shaped hump. This suggests the possibility that in the original Dieckmann-Doebeli model, conditioning on the population surviving long enough for convergence to equilibrium to occur (recall that in the original Dieckman-Doebeli model, extinction occurs in finite time), speciation is also a transitory phenomenon, rather than an equilibrium phe-nomenon. Simulations of (6.3) with V = V^2\ shown in Figure 6.3, does not even display transitory speciation behaviour. Instead, the initial spike at 0 simply widens to a Gaussian hump centred at 0. Hence the particular form of the dependence on K and C * 7r seems to affect whether or not speciation occurs. From the simulations and non-rigorous analysis above, it seems that the dynamical system in (6.3) does not have a bimodal stationary distribution if both K and C are taken to be Gaussian-shaped. If K and C are taken to be rectangular (i.e. Kx = l { | a ; | < L } and 67 t = 0 £ = 10 t = 20 t. = 30 t = 80 t. = 200 t = 400 £ = 800 t = 1200 t = 1600 f. = 2000 t = 3000 Figure 6.2: Simulation of (6.3) with E = [-149,149] f)Z, aK = 60, ac = 55, and V = V™. i = 0 £ = 40 i = 80 i = 200 t = 400 i = 3000 Figure 6.3: Simulation of (6.3) with E = [-149,149] n Z, aK = 60, cr c = 55, and V = V ( 2 ) . C X = l{|a; |<M} f° r some integers L and M) , however, results from Appendix A shows that there exist bimodal stationary distributions. More specifically, Theorem A.0.14 says that if un is a convergent sequence of symmetric stationary distributions for the conditioned Dieckmann-Doebeli model with mutation parameter \in with \xn —> 0, then vJLij] —> 0, where I = M — L + 1; in words, the mass in the middle gets very small as the mutation parameter approaches zero. 6.1.3 A M o r a n Mode l with Competitive Selection As discussed earlier, the dynamical system (6.3) cannot be easily associated with a Lya-punov function, which makes analyzing its behaviour difficult. Keeping in mind that the essential ingredient of the original Dieckmann-Doebeli model is that the fitness function is an increasing function of KX and a decreasing function of (C * it)x, we define the fitness mx(ir) to have the following form: mx(ir) = K X Y ^ B X - Z K Z T T Z ; (6.6) z where the "cooperation" kernel B can be taken to be 1 - C . We assume B is symmetric. In the original Dieckmann-Doebeli model, pairs of individuals with small phenotypical distance compete at a higher intensity than pairs of individuals with large phenotypical distance; in our model, pairs of individuals with small phenotypical distance cooperate at a lower intensity than pairs of individuals with large phenotypical distance. To make our formulation 68 cleaner, we also adopt a continuous-time model. The advantage of adopting mx in (6.6) as fitness and using a continuous time model is that the mean fitness of the population fn~n = Y Kxrrix = ^ T T X K X B X - Z K Z T T Z X x,z is a Lyapunov function [Burger 2000] for the dynamical system dtTTx = irx(mx - mn). (6.7) This assertion can be verified by the following calculation: dtfnn = 2 E mxdtTTx x = 2 ^ ] m 1 7 r I ( m I - m I ) x = 2YmxiTx{mx - m x ) - 2 ^ m^x{mx -mv) X X = 2 ^ 7 r I ( m I - m T f , (6.8) x where in the second line we use the fact. Y mnirx(mx - run) = ro2 - ro2 ^ wx = m 2 - m 2 = 0. X, X Since dtfn^ > 0 for any TT, mv is a Lyapunov function for the dynamical system (6.7), and in particular, the mean fitness m , increases at a rate proportional to the variance of the fitness. We call (6.7) the selection-only equation, as it does not have a part that corresponds to mutation. In Chapter 6.2.1, we will derive (6.7) as the deterministic limit of particle systems as the number of particles tends to infinity. 6.2 The Particle Model We introduce two particle models, one with "strong selection" that yields a deterministic limit, and another with "weak selection" that yields a stochastic limit. We work on space E = [ - L , L] n Z. Let L A = {(P-L, • • • ,Po, • • • ,PL) • Pi > 0 Vi and ^ pt = 1} i=-L be the space of probability measures on E , i.e. A = V ( E ) . Members of A are usually denoted by TT, TT, TTN, etc. We endow A with the following metric: d(ir, TT) = m a x \ T T ( X ) — TT{X)\. X Let K : E —> [0,1] be the carrying capacity function, and B : Z —+ [0,1] be the cooperation kernel, with B Z = 0 meaning that sites separated by phenotypical distance z do not coop-erate at all (i.e. compete at full intensity), and B Z = 1 meaning that they cooperate at full 69 intensity (i.e. do not compete at all). We assume B to be symmetric. The fitness of site x in a population with distribution 7r is defined as mx{ir) = K X Y B X - Z K Z T T Z . z If one abuses notation by writing iv" as a diagonal matrix, B as a matrix, and TT as a vector, then the vector formed by m. (TT) can be written as K B K T T . The mean fitness of a population with distribution TT is defined as mv = y^TTXKxBX-ZKzTTz. x,z If one abuses notation again, then m f f can be written as a quadratic form TT*KBKTT. Throughout this work, we will use symmetric or house-of-cards mutation, which means that the rate fixy = py at which phenotype x mutates to phenotype y depends on y only. This is a common assumption in population genetics [Burger 2000], and it is precisely this assumption that allows one to explicitly write down a Lyapunov function for the selection-mutation equation (to be defined in (6.10)). As a further simplification, we assume that jj,y = fi is constant in y, which makes the proofs a bit cleaner. Let XN(t) = {X^it),... ,X$(t)), t G R+, be an N-particle system, with X*(t) G E for all t and i. Define the empirical measure 6.2.1 The Strong Selection Model For our model with strong selection, the particle system undergoes the following: • Selection: At rate Nmvn, a particle, say X^, is chosen at random from the TV particles / N \ N and killed; at the same time, a new particle is born at x with probability m ^ J 7 1* . Since * = 1, m Z is a probability distribution; • Mutation: At rate N(2L + l)/j, a particle, say Xf, is chosen at random from the AT particles and killed; at the same time, a new particle is born at a site y with probability l 2L + 1-A particle at x gets replaced via selection by a particle at y at rate NTT^my(TTN)TTy , and gets replaced via mutation by a particle at y at a rate of Nfin^. Let K = s u p ^ ^ ^ ] K X , so that mx(Tr) = KX^TZ B X - Z K Z T T Z < K^2,zKTTZ = K 2 . Let l(dx) denote the Lebesgue measure on M + . The process described above can be constructed using a Poisson point process AN(dt,dx,dy,d^,de) on R+ x {(x, y) G E2 : x ^ y} x [0,1] x {1,2}, with intensity measure \N(AxBxCxD) = l(A)(#B)(#C)k(D), 70 where # denotes the counting measure, D C [0,1] x {1,2}, and k = I x (NK25\ + Np52). For all x,y £ E2 with x ^ y, jumps of AN(dt,x,y, [0,1], {1}) give possible times at which a particle at y may be replaced by a particle at x by the selection mechanism, while jumps of AN(dt, x, y, [0,1], {2}) give possible times at which a particle at y may be replaced by a particle at x by the mutation mechanism. The strong selection model can be expressed in terms of the following formula for TT^(t): rf(s-)mx(*N(s-))7cZ(s-) K2 ^{s-)my{-KN{s-))^{s-y K2 AN(ds,x,dy,d£,, 1) AN (ds,dy,x,d£, 1) 1 + N J J l(Z<rf(s-))AN(ds,x,dy,d{;,2) j% j l{i<^{s-))KN(ds,dy,x,d^2) (6.9) A solution to (6.9) exists because the total jump rate is finite for a fixed N. The two integrals inside the first set of brackets corresponds to selection, i.e. a particle at x gets replaced by a particle at y at rate N-Ky mx(TTN)TT^ due to selection. In particular, AN(ds, x, dy, d£, 1) in the first integral accounts for the killing of a particle at y and a new particle being born at x, and AN(ds, dy,x, d£, 1) in the second integral accounts for the killing of a particle at x and a new particle being born at y. The two integrals inside the second set of brackets corresponds to mutation, i.e. a particle at x gets replaced by a particle at y at rate Np,TTy due to mutation. Proposition 6.2.1. As N —> oo, the processes TTn converge weakly to a deterministic process TT that takes values in V(E) and obeys the following system of ODE's: dTTTX = 7vx(m(x, TT) - mv) + p(l - (2L + 1)TTX (6.10) Proof. First, we rewrite (6.9) by decomposing AN into a martingale term A and a deter ministic drift term: <{t) = ^(0)+M^(t)+ f [^(s-)mx(TTN(s-))^(s-) y=-LJo -TT?(S- my(iTN(s-))^(s-)] ds + p J2 / K ( s - ) -Kx(s-) . . _ T JO ds u=-L TT»(0) + M?(t) + ( *Z(s-)[mx(wN(s-)) - m x J 0 +fi[l-(2L+l)nx(s-)] ds (6.11) 71 where we define AN = AN — \ N to be the martingale part of AN and n»(3-)mx(irN{s-))*»(8-y 1 N i U < K2 ' ^(s-)my(7rN(s-))^(s-y K2 -r/. + jj j J l^<^(s-))kN(ds,x,dy,d^2) - f j\^<^.{s-))AN(ds,dy,x,d^,2) . j AN(ds,x,dy,d£,l) AN(ds,dy,x,d£, 1) We estimate the quadratic variation of the martingale term M% (t) and show that it con-verges to 0 as ./V —> co: (MXN) ^(s-)mx(irN(s-))ir»(s-)\ m - 2 K2 NK ds KtM<--i:f ir»(s-)myfrN(s-))n»(8-) K2 NK2 ds < < y=-L 2K2 L '* N ds y=-L 2{K2 + n)(2L + l)t N (6.12) which —> 0 as JV —> oo. Since the maximum jump size in M£ is jj, by Burkholder's inequality (see for example Theorem 21.1 in [Burkholder 1973]), we have E ^ s u p M f ( i ) ) ^ <C(E(M?)T + ^)^0 (6.13) as N - * 0. The other two terms in (6.11), i.e. TT^(0) and £\ix%{s-)[mx{jrN [s-)) -rn„.jv( s _)] + /j[l — (2L + l)7rx(s—)] ds, are both C-tight, (0) being constant and the integrand in the integral bounded uniformly between constants. Therefore TT^ is C-tight for each x. If there exists a sequence such that TTNTI converges to ir weakly and IT is contin-uous, then by bounded convergence, / 0 '7n^"(s—)\m x(irN n(s—)) — m 7 r w„( s _)] + — (2L + l)7Ta;(s—)] ds converges to / irx(s)[mx(ir(s)) - fn^^] + n[l - (2L + l)nx{s)] ds, Jo 72 and since TTN has representation (6.11), TT satisfies the fol lowing determinis t ic integral equa-t ion : irx(t) = TTx(0) 4- f 7Tx(s)[mx(7T(s)) - Mv{s)] + - (2L + 1)TTX(S)\ ds. (6.14) Jo A continuous (7r(£),0 < t < oo) solves the integral equat ion (6.14) i f and on ly i f i t satisfies the O D E system (6.10). B y wel l -known results from O D E theory (e.g. T h e o r e m 1.1.1 from [Wiggins 1988]), the solut ion to (6.10) is unique because its right-hand-side is C°° i n TT. Therefore solut ion to (6.14) is unique as wel l , and the proof is complete. • 6.2.2 The Weak Selection Model For our mode l w i t h weak selection, the part icle system w i t h N part icles undergoes the fol lowing: • Selection: A part ic le at x gets replaced by a part icle at y at rate my(7TN)iTy ; • M u t a t i o n : A part ic le at x gets replaced by a part icle at y at rate N/J,TT^ ; • Replacement sampl ing: A part icle at x gets replaced by a par t ic le at y at rate Nl nN N 2 "x "y Just as i n the s t rong selection model , this process can be constructed using a Po i s son point process AN(dt, dx, dy, d£, de) on K + x {(x,y) e E2 : x ^ y} x [0,1] x { 1 , 2 , 3 } , w i t h intensi ty measure XN(A xBxCxD) = l(A)(#B)(#C)k(D), where I denotes the Lebesgue measure on K + , # denotes the count ing measure, D C [0,1] x { 1 , 2 , 3 } , and k — I x (NK25i + N + ^j-63). T h e weak selection mode l can then be expressed i n terms of the following formula for ir^(t): 1 + N 1 + N TT"(s-)mx(irN(s-))iT"(s-)\ „ K2 7rf ( S - )m y (7^(s - ) )<(s - ) ' K2 + N J Jl(Z<TTy*(s-))AN(ds,x,dy,dt;,2) J* j l(Z<TT£(s-))AN(ds,dy,x,dZ,2) f j 1(£ < 7T»(s-)7r»(s-))AN(dS, x, dy, d£, 3) 'j' J 1(£ < ^(s-)^(s-))AN(ds,dy,x,d^) AN(ds,dy,x,d£, 1) (6.15) 73 The two integrals inside the first set of brackets correspond to selection, those inside the second set of brackets correspond to mutation, and those inside the third set of brackets correspond to replacement sampling. By carrying out computations similar to those done in the proof of Proposition 6.2.1 (and the tightness proof in [Perkins 2002]), one can conclude the processes {irN : iV G N} is C-tight, and that each weak limit point TT satisfies the following martingale problem: Kx(t) = TTx{0) + f 7 r f f i ( s ) [ m E ( 7 r ( s ) ) - m 7 r ( s ) ] + / i [ l - ( 2 L + l)7ra!(s)]-ds + M a ! ( £ ) , Jo where Mx is a continuous (F t 7 r)-martingale such that Mx(0) = 0, and {Mx,My)t— / 8xy7Tx{s) - TTx(s)TTy(s) ds. (6.16) Jo Here 5xy = l ( x = y). Ito's formula shows that for F G C2{E), F(ir{t)) - F(TT(0)) - / QF{TT{S)) ds Jo is a bounded continuous martingale, where L \ir fm„(ir\ — rn _\ Mid] — (9 T. 4- 1 W..V, Ja7T. x , Qp QF(TT) = 2_j [nx(mx(7T) — mv) + /J(1 — (2L + 1)TT X)]-^-x= — L L L d2F + 0 E E ^x(5Xy-TTy) dTTxdlTv In particular, if F takes on the form F(TT) = /((TT, (f>)) where (TT, cj>) = J2Z nz<Pz and (f> : E is a function, then gF{TT) = ]T f\{TT,(j)))(l)x[TTx{mx{n)-m^ + fi{l-(2L+l)TTx)} X = — L 1 L L +2 E E nM))Mv**x&*v - (6-i7) X=—Ly~~L This is a special (finite E and symmetric mutation) case of generator for the Fleming-Viot process with selection. The martingale problem associated with Q has a unique solution (see Chapter 10.1.1 from [Dawson 1993]), so TTN converges weakly to TT on J D ( R + , M\{E)). For the process described by the martingale problem (6.16), Lemma 4.1 from [Ethier and Kurtz 1994] says that u{dTT) = C ^ n ^ d7T~L • • • dlTL is the unique stationary distribution. Here C is the normalizing constant such that v is a probability measure on V{E). Notice that if mx{Tr) = 0 for all x and TT, then mv = 0 for 74 all TT, and in that case, v(dir) = C \Jlx=_L TTXJ dTr_/_, • • • OITTL is the unique stationary distribution of the Fleming-Viot process on E with symmetric mutation [Ethier and Kurtz 1981]. Adding selection to the model has the effect of putting weight e771' (where is the mean fitness of TT) on the phenotypical distribution TT. But even the least fit TT has weight at least 1, since exp(min7rm7r) = e° = 1, and the fittest TT has weight at most e. The effect of fitness on the density at a particular phenotypical distribution TT is thus only marginal. In contrast, as we shall see later, fitness will have a much more pronounced effect in the strong selection model. For the rest of this part of the thesis, we deal with the strong selection model outlined in Chapter 6.2.1. 75 C h a p t e r 7 The Selection-Mutation Equation As discussed in Chapter 6.2.1, the particle model with strong selection converges weakly to the selection-mutation equation as the number of particles N tends to infinity: dtTTx = TTx{rnx - m f f) + - (2L + 1)TTX), (7.1) where, as before, mx = mx(n) is the fitness of site x in population TT, and is the mean fitness of the population: mx = KXY,BX-ZKZTTZ (7.2) z m First, we state a few assumptions on the parameters involved: 1. x G E = [-L,L]CiZ = {-L, . . . , - 1 , 0 , 1 , . . . , L}, 2. K:E^ (0,1], 3. B : Z^r [0,1]. We establish a few basic facts about the system (7.1). First of all, mx (uniformly in x) and both lie in [0,1], therefore mx — mv > — 1. Thus dTTTX = TTx{mx - fn„) + fj.(l - (2L + 1)TTX) > -TTX + n(l - (2L + 1)TTX) = fi- (2LU.+ U. + \)TTX. (7.4) ^ ' K x < 2Lfj,+n+i' then dfKx > 0 regardless of TT. This means that there can be no stationary points of the system (7.1) with any 7TX < 2 L M + M + 1 ' a n ( ^ furthermore, for those x where 7 V x < 2(2LM+M+I) hiitially, ^x will increase at a positive (bounded away from 0) speed and 76 eventually irx > 2 (2L^+M+I) ^ o r a u x ' Secondly, since the sum over all x of the right hand side of (7.1) is ^ T j ( m j - r r v ) +/x(l - (2L + l)irx) X = ^T] TXXTXIx - Wiir TTX + (2L + l)/j - /j(2£ + 1) nx x x x = mn-m.K + (2L+l)ij,-(2L + l)fj, = 0, (7.5) we have hence the total mass J2X TTX remains constant. This, together with the first observation we just made, imply that if at t = 0, 7r(0) is a probability measure, i.e. J2xwx(0) = 1, then ir(t) remains a probability measure for all t. Since TTX will become instantly nonzero at x where wx = 0 initially, we restrict our attention to polymorphic initial conditions, i.e. 7^(0) > 0 for all x, which is equivalent o to saying that TT(0) lies in the interior A of the set A . For polymorphic initial conditions, irx(t) > 0 for all x and t, and therefore (7.1) can be written as: dTTTX = 7rx ^mx - mv + - n(2L + l)j . (7.6) o Furthermore, there is a Lyapunov function V„ : A —> R for the dynamical, system (7.6): VTT = ^ T T V + fl ^ lo§ nv (7-7) X Notice that• 9 T l m T = 2 ^ z KxBx-zKzirz = 2mx. The assertion that is a Lyapunov function for (7.6) can then be verified by the following simple calculus exercise: a: = ^ ^mjc + 7rK + ^ - - JU(2L +1)^ = ] P + 7rx ^ + - - n(2L + l)j - (mn + fj,(2L + 1)) Y V ( m I + - - m I - f i ( 2 L + l ) ) . Notice that J2X irx (jnx + — mv — p(2L + 1)^ = 0 by (7.5), therefore AV* = E n* (m* + ~" " m* ~ ^ 2 L + X)) - 0 ' 77 hence VT is a Lyapunov function for (7.6) as claimed. In fact, according to Theorem A.9 of [Burger 2000], (7.6) is a so-called Svirezhev-Shahshahani gradient system with potential K- as defined in (7.7), i.e. dtTT = WV(TT), where X7V(TT) = G 7 r VV r (7r) and GV is the matrix formed by entries gxy = TTx(5xy — iry). Any gradient system, such as (7.6), has the property that all orbits, regardless of initial condition, converge to some point in the w-limit set Dw = {p : p is an accumulation point of irx(t) as t —» oo}. Al l points in Du are stationary points of (7.6). Since TT is a stationary point of (7.6) if and only if mx - ma + p (J^~ - (2L + 1)^ = 0 for all x G E, (7.8) where we write mx = mx(ir), all points in satisfies (7.8). We observe that if TT is a stationary distribution and rhx > rhy, then condition (7.8) means that 4 — (2L 4- 1) < j (2L + 1), therefore TTX > Tty, i.e. rhx > my => T\X> TTV. (7.9) In words, fitter sites have more mass. We will try to characterize the stationary points of the dynamical system (7.6) for K and B satisfying the following conditions: K is symmetric and unimodal with Ko = 1, and B is of the form Bx = b+{1- 6)1{| X |>M} with b G [0,1] and L < M < 2L. (7.10) Define I = M — L, then for x G [—1 + 1,1 — 1], the cooperation intensity between x and any other site z G [—L,L], Bx-Z, is equal to b, which means that mx = Kx Y BX-ZKZTTz = bKx ^ KZTTZ. (7-11) z z 7.1 Mild Competition: b close to 1 If b = 1, then BX = b for all x, hence there is equal competition between all sites. This actually means that competition plays no part in how fit site x is and mx is proportional to Kx. Therefore, since Kx is unimodal (hence Kx is strictly increasing in [—L,0] and strictly decreasing in [0, L\), the fitness should be unimodal, too. Recall from (7.9) that stationary distributions of (7.6) has the property of fitter sites having more mass, thus we expect the stationary distribution TT to be unimodal as well. In particular, it should attain its maximum at x = 0. As p —> 0, we expect the "peak" of n concentrated around 0 to become sharper and sharper, approaching So, the 5-measure concentrated at 0. In fact, as we shall see, b only needs to be somewhat close to 1 for this behaviour to occur. We now show that for any stationary distribution TT of (7.6), site 0 is fitter than any other site for b G (| , 1] sufficiently close to 1, i.e. r%o > mx if x ^ 0. This will mean that as 78 fx —» 0, any TT approaches the <5-measure. Recall from (7.11) that for x £ [—I + 1,1 — 1], we have mx = bKxy^Kznz. z We recall that K Q = 1 and Kx is assumed to be strictly increasing in [-L, 0] and strictly decreasing in [0, L], therefore Kx attains its maximum at x = 0, and thus for x £ [—1 + 1, -1] U [1,1 - 1], K x - K 0 < K i - K 0 < 0. Therefore for x £ [-1 + 1, -1] U [1, l - l ] , m x - m 0 = b(Kx - K0) ^ Kznz < -b(K0 - K{) ]T] KZTTZ. z z We now apply the bound ^ 2 Kzirz > i n f T £ A J2Z KZ~KZ = K L to the above inequality to obtain m x - m Q < -b(K0 - K{)KL. Since b is assumed to be > | , we have for x £ [—1 + 1, —1] U [1,1 — 1], m x - m 0 < ~ ( K 0 - K 1 ) K L . (7.12) We also bound the fitness for sites x in [l,L]: mx = bKx ^2 KZTTZ + (1 - b)Kx ^ K ^ z z \x-z\>M = bKx ^ Kzwz + (1 — b)Kx ^2 KZTTZ since a; > I z s—z>M x —M < bKx ^2 K z i t z + (1 - 6)JfE ]P if* since TTZ < 1 z z=-L - i < M G ^ T ^ T T * + (1 - b)Kx J2 K * Z z= — L < bKx^2Kznx + (l-b)(L-l + l)KxKi, (7.13) z where in the last line, we use the following: for z £ [—L,x — M] C [-L, —l],.Kz < K-i = K[. Similarly, for x £ [-L, —I], we have the same bound (7.13). Therefore for x G [-L, —l]L)[l, L], we have mx < b K ^ K ^ + i l - b ^ L - l + l ^ K i z < bKx^2Kzirz + (l-b)(L-l + l)Kf, (7.14) z where again we use the bound Kx < Ki for x £ [-L, —I] U [l,L]. We use (7.11) and (7.14) to estimate mx — mo for x £ [-L, —I] U [I, L]\ m x - m 0 < bKxJ2Kznz + (l-b)(L-l + l)Kf-bK0J2KziTz z z = b(Kx-K0)J2Kzirz + (l-b)(L-l + l)K?. 79 Since Kx — Ko < 0 for x 6 [-L, -1} U [I, L], we have mx-mo < -b inf {KQ - Kx) inf V KZTTZ + (1 - 6)(L - I + 1)K? x€[-L,-l]U[l,L] 7 r 6 A ^ - / = -b{K0-Ki)KL + (l-b)(L-l + l)Kf. Since 6 is assumed to be > \ , the above bound can be simplified to: mx-mo<0--b)(L-l + l)Kf-^(KQ-Ki)KL. (7.15) Thus if b is so close to 1 that ^ ^ ( L - l + ' ( 7 - 1 6 ) then let <5i = |(iv"o — K{)KL be a positive constant and we have (l-b)(L-l + l)K? <^(Ko-Kl)KL = ±(Ko-Kl)KL-61. (7.17) Thus if condition (7.16) holds, then (7.15) and (7.17) imply that for x G [-L, -I] U [l,L], mx - mo < —Si. (7-18) Define 5 = min (Si, \{KQ — KI)KL), then the estimates in (7.12) and (7.18) mean that for all x G F\{0} and any TT, mx — mo < —S. (7-19) This shows that for b satisfying condition (7.16) and for any TT, site 0 is fitter than any other site. We will use this bound to establish the following. Theorem 7.1.1. IfK is symmetric and unimodal with Ko = 1, and Bx = b+(l — b)l{\x\>M) with L < M <2L, I — M - L, and b G i _ {K0-KI)KL i 1 4{L-l+l)K'{ '  l , then as p —> 0, sup{||7rM — <5o||oo : TT^ is an stationary distribution for mutation parameter p] —> 0. Proof. Recall condition (7.8): TT^ is a stationary distribution of (7.6) if and only if for all xe[-L,L], m£ - m*,' + p (J^ - (2L + 1)^ = 0, (7.20) where we write m% = mx(Tt^). We make the following observations using condition (7.20): if m£ > m V , then + (2L + 1) > 0, which implies that TT£ > 1/(2L + 1); similarly, if mx < m>/« then fig < 1/(2L + 1). Since l/fig > 1, the following bound holds for all x: m% - m V + p{\ - (2L + 1)) < 0, 80 which implies that m£ - m*,. < 2Lp. (7.21) We consider p, small enough such that 2L\i < | , where 5 is defined right after (7.18). Then the estimate (7.21) applied to x = 0 means that TOQ — m*(» < 2Lp < -, which implies that - u _ 6 m% < m*,. + - . Applying the above to estimate (7.19), we get, for all x 0, ( m£<m%-5 <m** (7.22) In particular, the only x where m£ > m^v is a; = 0. Using the bound (7.22), condition (7.20) implies that for x ^ 0, ^ - ( 2 L + 1 ) ) 4 Therefore which —> oo as /tt —> 0. Hence 7r£ —> 0 as fi —>"0 for all x -fi 0. Notice that the proof does not depend on which 7rM we pick, therefore we are done. • Remark 7.1.2. The crucial estimate for the above proof is (7.19). In the case of equal competition, i.e. 6=1, it is very easy to derive (7.19): . m x - m 0 = b ( K X - K 0 ) Y K z i r z < -b inf ( K 0 - K X ) inf Y KZTTZ z—' X^0 7r€Az—' Z 2 = - H K O - K ^ K L , which is a positive constant independent of p. 7.2 Intense Competition: b close to 0 Results from Chapter 7.1 show that if the competition between pairs of sites that are far away from each other is not intense, i.e. b is close to 1, then as p, —> 0, the stationary distribution(s) converge to 5Q, and therefore, there is no speciation. In this section, we show that if there is the most intense competition between pairs of sites that are far away from each other, i.e. 6 = 0, then we do see speciation in the stationary distribution(s). More 81 interesting behaviour arises in the case of positive but small 6. We will show that, in this case, if p is small enough, then there are at least two vastly different stationary distributions, one resembling the 5-measure, the other bimodal and having almost zero mass in the middle; on the other hand, if p is sufficiently large, then all stationary distributions are bimodal and have little mass in the middle. Thus for small p, whether speciation occurs eventually in the dynamical system depends on the initial state of the system. But for large enough p, speciation will occur eventually if one waits long enough. We first illustrate this behaviour in a system with 3 phenotypes { — 1 , 0 , 1 } , whose stationary points we can calculate explicitly. 7.2.1 Study of A One-dimensional System Here we will take the simplest possible scenario, and show that the dynamical system (7 .6) has exactly two stable stationary points. Let L = 1, E = {—1,0 ,1} , KQ — 1, and K\ = K-i = \. Let BX = b + (1 — b)l{|x|>2}> i-e. phenotype —1 cooperates with phenotype 0 and itself at level b (i.e. some competition), and cooperates with phenotype 1 at level 1 (i.e. no competition). We only consider symmetric distributions, i.e. T T _ I = TT\. Taking into account that 7r_i + TTQ + m = 1, the dynamical system (7 .6) has only one variable, say TTQ. The fitness of the 3 sites in E and the mean fitness are: m0 = K0b(K0TT0 + 2 i \ i ? T i ) = b(ir0 + TTI) mx = K^bKoiro + (1 + b)Knn) = ^ ( 6 T T 0 + ^y^i) TOTT = &7ro (7T 0 + T T i ) + 7ri(&7T0 + ^—~-TTl) Thus the dynamical system (7 .6) with variable TTQ can be written as a single ordinary dif-ferential equation: dtn0 = 7To(m 0 - mV) + p(l - 37r 0 ) = 7To fo(7To + 7Ti) - &7To(7T0 + 7Ti) - 7Ti(o7T0 - 4^~~7rl)>) + ~ 37I"°) Substituting in TX\ = i ( l — TTQ), we get b + 1 3 1 - 6 o 36 - 1 - 24/x _ n o , dtiro = g — T r g + — T r g + TTQ + p. . (7 .23) Now we take b = \ , then (7 .23) can be simplified: 3 3 1 2 / I 3 ( 3 4 , fl _ \ 2 0 = -2o( 7 r o-3 7 r o + U + 2 0^ 7 r o-y^ We define the polynomial p(x) — xz — | x 2 + (5 + 20^)a: — ^ p, then p has roots x\ = \ , X2 = 5(1 + A /1 ~~ 8 0 / i ) , and x3 — | (1 — ^ 1 — 80/a). Notice that-the root x\ does not depend on p, although this is only true for 6 = jr. For other 6's, all roots depend on p. Two plots of — -^p(x) are shown in figure 7 .1 . 82 Figure 7.1: One dimensional case: 6 = | For fi < -gQ, there are three real roots, as in figure 7.1(a). In this case, if 7To(0) < | , then 7To(£) —> x3 « 0.16 as t —> oo; since 7r_i = TTI and 7 r_ i + TTO + TTI = 1, 7r_i = 7Ti —> 0.42 as t —> oo. This distribution has much larger mass on sites —1 and 1 than on site 0, thus we can say that speciation occurs eventually. But if 7To(0) > | , then wo(t) —> x2 « 0.84 as t —> oo, then 7r_i = 7Ti —+ 0.08. This distribution has much larger mass on site 0 than on sites —1 and 1, and we say that speciation never occurs. On the other hand, if \x > g^, there is only one real root, as in figure (7.1(b). In this case, regardless of initial condition, ffo(t) —* g a s t —>'oo. 83 Two plots for b = ^ are shown below. For sufficiently large p., e.g. p — in figure 7.2(b), regardless of initial condition, TT converges to a configuration with little mass in the middle (approximately 0.037), i.e. speciation occurs. But for \x sufficiently small, e.g. p = jog in figure 7.2(a), there may or may not be speciation depending on the initial condition. The time evolution of TTO with different initial conditions for both M = 300 a n d / i = gig are also shown. Q | | 1 1 1 1 gl ; 1 1 | , | 1 L_ 0 50 100 , 150 200 250 0 50 100 150 , 200 250 300 350 (C) M = 500 (d) M = 300 Figure 7.2: One dimensional case: 6 = A 84 7". 2.2 Large enough /J Analysis of the dynamical system (7.6) in its simplest form in the last subsection shows that if fi is small, there may be two vastly different types of stationary points for (7.6). In subsections 7.2.3 and 7.2.4, we establish this for (7.6) in its general form. But in this section, we examine the behaviour of (7.6) when fx > 4Ki ^L_i-) and establish that all stationary points of (7.6) are bimodal. We maintain the assumption in (7.10) that K is symmetric and unimodal with K0 = 1, and B is of the form Bx = b + (1 - 6)1{|x|>M} with b G [0,1] and L < M < 2L. We will need a uniform (in fi) lower bound on m^,.. We first establish a crude lower bound on m%n that does depend on fi. Recall from (7.8) that T T m is a stationary distribution if and only if for all x G [-L, L], fn)t — m-jtn • A « Q * - ( 2 L + 1)) =0. (7.24) Therefore /•* I 7^* — + 1)) = rn-a - < m% <. sup mx(it). \ 7 T X / x€E,-weA Since K and B lie in [0,1], mx = Kx J2Z Bx-zKznz < J2Z irz = 1, and therefore ^ G H 2 ^ ) - 1 ' which means that r£ > r > fi. (7.25) For x G [I, L], we have (recall the first three steps of (7.13)), x-M mx{ir) = bKx ^ Kznz + (1 - b)Kx ^ Kzirz z=-L x — M > bKL KZTTZ + (1 - b)KL ^ K zirz since Kx is decreasing in [0, L] 2 2 = — L > bKl + (l-b)K2L = K \ . (7.26) Since K and B are both symmetric, the same bound applies if x G [-L, —I}. Therefore (7.25) and (7.26) imply the following estimate of the mean fitness m, : L -i x x=l x=—L L > iY^Klfi = 2Klfi(L-l). (7.27) X—l 85 This crude lower bound on m#;. depends on p, but in Lemma 7.2.1 we will improve it such that it does not depend on p for sufficiently small 6. For now, we use it to establish an estimate on 7rM([—I + 1,1 — 1]) that will be needed for the proof of Lemma 7.2.1 below. Condition (7.24) implies that for x £ [—1 + 1,1 — 1], / i ( - £ - ( 2 L + l ) ) =m#, -m%. (7.28) For x G [—1 + 1,1 — 1], J2Z Bx-zKzirz = b^2z KZTTZ is constant, thus the maximum fitness is attained at x = 0, with m0(7r) = bK0 Y K**z < b, (7-29) z since Kx is increasing in [—L, 0] and decreasing in [0, L]. If b < 2K\p{L — I), then we can apply the estimate (7.27) on m x , and bound the right hand side of (7.28): m*/» — rhx > m#e — mg > 2K2p(L — I) — b, which is > 0 if b < 2K2Lp[L-l). Thus for b < 2K\p[L-l), (7.28) implies that ^ - - ( 2 i + l ) > 0, i.e. nH < 1 2L + 1 Hence we can bound the mass in [—1 + 1,1 — 1}: 2/ + 1 + | _ i ] ) < _ ± _ (7.30) Before we state the theorem of this section, we establish the following lemma, which improves upon the bound (7.27) such that it does not depend on p: Lemma 7.2.1. For p and b in the region Rx = {(p-,b) : 0 < b < mm(4:pK2(L — I), 4(2L+T)^)}' taere *s a positive constant c\ that depends on L, I, and K but not on p, such that rti%tL > c\ for any stationary distribution . Proof. We notationally suppress the dependence of 7rM, m£, and m>/« on p. Suppose that TTXTXv < S for all x G [-L, -1} and y G [x + M,L], (7-31) where K\(L -1) 6 = 2(2L + 1)3 - b ^ is a positive constant if (p, b) G Ri- The pairs (x,y), with x G [—L, —I] and y G [x + M,L], are exactly those that contribute weight 1 to the calculation of the mean fitness m> as defined 86 in (7.3), while the pairs (x1, y') with x' G [-L, -1} and y' G [-L, x + M - 1] contribute only weight b. Condition (7.31) implies that 771*= ^ KxKzTfxnz + b ^ KXKZTTXTTZ \x-z\>M \x-z\<M < KI 5 + B E Ko \x-z\>M \x-z\<M < 2(L-l + l)25 + 2b(2L + l)2 < 2(2L + l)2(5 + b). (7.33) Recall from (7.30) that fc([-l +1,1-1}) < which is equivalent to ir([—L, —I] U [I, L\) > Therefore either it([—L, —I}) > o r Hil,L]) > 2X+T- Suppose n([l,L}) > then we can bound the fitness of site — L: L L m-L = K^L Y, B-L-zKzirz>K„LJ2KzTrz z= — L z=l > KlJ2*z=Kl*{[l,L])>Kl±=±. (7.34) z=l Since 5 + b = , (7.33) and (7.34) together imply that m _ L - m* > K2 - 2(2L 4 1)2(6 + b) = 0. As a result of the inequality above, condition (7.24) implies that 1 (2L + 1)<0, hence • > ^ 1 - (7-35) Combining the two bounds (7.34) and (7.35) on and r n _ L , we have L - l '(2L + 1) 2 ' m* = E^ x ^ - K-Lm-L > K2L (7.36) By similar reasoning, if Tt(\—L, —I}) > -£t+h> then we have ifiL > -^L2T+I a n ^ ^L > 2L+i ' which also implies that ma > K\ (2L+[)i • Therefore condition (7.31) implies that m% > (2i+i)2 • This would actually contradict (7.33) for sufficiently small 5 and b, but it does not matter since in that case, it just says that condition (7.31) is impossible and the analysis below applies. On the other hand, if condition (7.31) is not satisfied, i.e. there is at least one pair of phenotypes, say (x,y), with x G [—L, —I] and y G [x 4- M,L], such that nx^y > 5. Then it is easy to see that m*> ]T KxKznxnz > K2L7txTty > K\& > f f t i ' i i , (7-37) \x-z\>M ^ ' 87 / n the last inequality due to (7.32) and the requirement 6 < ^L+I)*- Therefore combin-ing (7.36) and (7.37), we conclude that . (K\(L-l) KHL-l)\ n This bound is uniform for any positive (/J, b) satisfying the condition b < 2K\\x[L — I) and b < 4 ^ L + I ) 3 a n < ^ t n e c o n c l u s i ° n °f the lemma follows. • Theorem 7.2.2. IfK is symmetric and unimodal with Ko = 1, and Bx = 6+(l — b)l{\X\>M} with L < M < 2L, then with the constant c\ = c\(K, L, I) defined in Lemma 7.2.1, we have TTx < ^ for x G [—I + 1, I — 1] and (/x, 6) lying in R = ((/a, b) : 0 < b < min UnK\{L - I), % ^ " l ) 2 '4(2L + 1) 3, Proof. Recall from (7.29) that for a; € [—1 + 1,1 — 1], the maximum fitness is attained at x = 0, and rno(7r) < b. If b < with C\ from Lemma 7.2.1, then mn(7r) < ^ . Hence for \x and 6 lying in i? C where R\ is defined in Lemma 7.2.1, Lemma 7.2.1 implies that rn*» - TOQ1 > y • (7.38) Condition (7.24) and (7.38) together imply that for x G [-1 + 1,1-1] and (/x, b) G R, k1 ^ ~ ( 2 L + = m * " ~ ™x > £ l 2 ' Therefore for i e [—1 + 1,1 — 1] and (/J, 6) G R, - M 1 2/x and the proof is complete. • Remark 7.2.3. 7/6 = 0, iften Theorem 7.2.2 works for any fi, no matter how small. Remark 7.2.4. The constant c\ in Lemma 7.2.1 is small. 7.2.3 Small fi: Existence of 5-like Stationary Measure Results from Chapter 7.2.1 for one-dimensional systems indicate that when b and fi are sufficiently small, there should be at least two stationary distributions, one resembling the (5-measure, the other bimodal and having little mass in the middle. In this section, we show the existence of <5-like stationary distributions. More specifically, we establish the following: Proposit ion 7.2.5. Define k — min x \KX — Kx-i\. If \x < ^ e i , then the set A\ is an invariant set for the dynamical system (7.6), where we define A\ = {ir G A : irx < e\ for all x ^ 0}, with ei < min( 3L, 2 K - i $ _ l + 1 ) , j ^ ) -88 Proof. Let /J < ^ e i be fixed. For TT G A\, since TTx < e\ < j^, we have 7ro > 1 — 2Lei > \ . Recall that l — M-L and that for x G [-Z + 1,1 - 1], m x = bKxYKznz. z Since iv"x is increasing in [—L,0] and decreasing in [0,L] with K0 = 1, sites in [0, Z — 1] have decreasing fitness, and sites in [—Z 4-1,0] have increasing fitness, and we also have the following estimates: mo - mi — b(K0 - K\) Y^^z^z > bkK0TT0 > — since 7r0 > \, (7.39) z mi < bKxKo <bKi. (7.40) For 7r G Ai and x G [I, L], X — M mi - m x = KtbYKz^z - Kx 6 ^ 1 v V r 2 + ( ! -&) Y z=-L X—M ( K T - Kx)b ] T KZTTz -Kx(l-b) ] T KZTTZ z z= — L -I > kbKoTTQ ~ Kt Y Kz-*z z=-L kh > '^-Kf(L-l + l)e1 by (7.39) > 0, 1 (7.41) since we assume t\ < 2 R- 2 (L - i+ i ) 1 ^ t n e s a m e calculation as in (7.41), m_; — mx > 0 for x G [-L, —Z] as well. Therefore sites in [-L, —Z] are less fit than site —I, and sites in [l,L] are less fit than site Z; furthermore, sites in [—Z, 0] have increasing fitness, while sites in [0, Z] have decreasing fitness. In particular, among sites in \—L, —1] U [1,L], sites 1 and —1 have maximum fitness. Any measure TT G A\ looks like the ^-measure. For such measures, the mean fitness rnn is close to but less than mo. We estimate the difference between m^ and mi: mV. — mi = ^ ^ m x 7 r x — mi > mo7To — mi > mo(l — 2Lt{) — m.i, x since 7ro > 1 — 2Lei from the beginning of the proof. Now using estimates (7.39) and (7.40), we continue estimating —m\\ bk m 0 ( l - 2Lei) - mx = (1 - 2Lei)(m 0 - mi) - 2Leimi > (1 - 2Lei)— - 2Le1bK1. Recalling from the beginning of the proof that 1 — 2Le\ > \, we use the assumption ei < 1 6 £ K to estimate the right hand side of the above inequality: » .bk n r , r , bk k bk ( l - 2 L C 0 T - 2 L e 1 W f 1 > T - 2 i 6 i f 1 — > T . _ 89 Therefore " i x - mi > (7-42) Since among sites in [—L, —1] U [1, L], sites 1 and —1 have maximum fitness, (7.42) implies that for all x ^  0, _ bk mx - mv < - —. o • We write dA± = Bi U B2 U B3, where B\ = {TT G C M I : 7rx = 0 for some x and 7TX ^ ei for all x}, B2 = {TT G 9Ai : irx = e\ for some x and 7rx ^ 0 for all x}, B3 = {IT G <9AI : 7rx — t\ for some a; and % = 0 for some y}. For 7r G B i , we have shown following (7.4) that 9 t7r x > 0 at x where 7rx = 0. Therefore 9t7r points toward the interior of A. For 7r G B2, we have for x where 7rx = ei, <9t7rx = 7 r x ( m x - m x ) + / j ( l - ( 2 L + l )7r x ) bk < - e i y + M < 0, (7.43) since p < ^ e i - Thus 9t7r also points toward the interior of A . For 7r G B3, we can apply (7.4) to sites x where 7rx = 0 and (7.43) to sites y where iry — €I, and conclude that c>t7r also points toward the interior of A. Therefore the set A\ is invariant for the dynamical system (7.6), as required. • 7.2.4 Small fi: Existence of Bimodal Stationary Measure Results from Chapter 7.2.3 show that the set A\, members of which resemble the <5-measure, is invariant for the dynamical system (7.6). In this section, we assume that < and show that there is another invariant set A2 = JTT : 7r x < e2 for all x <£ {p, -p}, and | log ^ | < e3j , (7.44) where p = M/2 and L < M <2L, 1 . / ^Kp(Kp-Kp+1) u K$\ c2 = - m i n i ( 1 - & ) - * — ^ — * >b,j£\, (7.45) . ( 1 KP(KP-KP+1) K2p e2 = min 1 — - — 8 ( 2 L + 1)' 8K2+1{L -l + l)' 16Kt(L -l + l)' c2 16(2L+l)KpJ ' (7.46) e3 = min( log2, log( l + ^ ^ . (7.47) 90 Apparently, M must be even; this is such that i? p_(_ p) = 1 but 5( p_i)_(_ p) — 6. Notice that since M < 2L, I = M - L < M - f = f = p, and also p = f < L. Thus l<p<L. Members of A2 are bimodal distributions with very sharp peaks at sites p and —p. For 7T £ A2, ftp + 7 T _ P = 1 - Ysx^p-p^x > 1 - (2 i - 1)8(21+1) b v t h e condition TTX < 8(22+i) for x {p, -p}, therefore 7r p + T T _ p > | . Now since | log ^ | < log 2, we have < 1. This means that min(7rp, 7 T _ p ) > (7.48) for otherwise, say 7r p < \, then 7r_p > | , which means that < § < 3 -We use the same idea that we used to establish the invariance property of A\ to show that A2 is also an invariant set for the dynamical system, for p < %2-e2. Lemma 7.2.6. For any TT £ A2, the following estimate holds: c2 min(m p, m_ p) -mx> — forxe [-L, L]\{p, -p}, where A2 and c2 are defined in (7.44) and (7-45), respectively. (7.49) Proof. As in the proof of Proposition 7.2.5, we first establish a few bounds on fitness of various sites for ir £ A2. For x = p, nir, > Kp(l-b)K-pn-p> z=-L (l-b)K% (7.50) where we use (7.48) in the last inequality. For x £ \p + 1, L], bj2K^z + (i-b) z=-L x-M bj2Kzvz + (i-b) Y K**z - p z=-L x — M (Kp - Kx)b Y Kzftz + (Kp - Kx)(l -b)Y Kzitz - Kx(l - b) £ Kznz. z=-L 2 = - p + l Since x £ [p 4- 1, L] and Kx is decreasing in [0, L], we have Kp — Kx > Kp — Kp+y > 0. We apply these facts, along with (7.48) and the requirements on TTX for 7r £ A2, to the right hand side and obtain m p - m x > 0 + (Kp-Kp+1)(l-b)K-p^-Kp+i{l-b)(L-l + l)K_p+1e2 (1-6) KP(KP - Kp+1) Kp+1Kp-i(L - I + l)e2 > (i b)Kp(KP-Kp+1) (7.51) 91 > / X A A A A / N J A / y X A A A A A A A A A A A A A A A / X A ^ A A A A / ^ —L —p —I I p L Figure 7.3: Illustration of A2 for x£\p+l,L] since e2 < ^ [ ^ l - i + i y F ° r xe[0,l- 1], mx = z therefore using (7.50) and the inequality above, we have <e2 mp — mx > (1 - b)K2 - -—p--b>b (7.52) K. since < For x £ [l,p— 1], x — M 6 ^ ^ 7 r z + ( l - 6 ) Y K ^ z=-L < KtibKoY,** + Ko E e 2 ) < Ki(b+(L-l + l)e2), therefore using (7.50) and the inequality above, we have m p - m x > ( 1 ~ ^ K p -Ki(b+{L-l + l)e2) K2 (K2 \ = -f-b{-f+Klj-Kl(L-l + l)e2. K2 K2 {K2 \ K2 Since b < 2Kn+xlKo, we have -f-—b l-f - + KA > -f-. Furthermore, since t2 < we have K2 m p - mx > —r-16 (7.53) The estimates (7.51), (7.52), and (7.53) compare m p with mx for a; s [0, L]. Similar cal-culations comparing m _ p with mx for x £ [—L, 0] yield similar results. Then recalling the 92 definition of c 2 in (7.45), we have m p - m x > c 2 for x G [0, L]\{p), and m-p — mx>C2 for x G [—L, 0]\{—p}. To establish the lemma, it suffices to compare mp and m_ p : -p = Kp(l-b) 6 ^ i v V r 2 + ( l - 6 ) K ^ z z=-L L b ^ i v V r z + ( 1 - 6 ) ^ 7 ^ z r, z=p z=p L < Kp(l-b)Y,Kz\n-z-7xz\. (7.54) Since Kz < Kp for z G [p, L], Kz < K_p — Kp for z G -p], and 7rz < e2 for z ^ -p,p, , we have K - m _ p | < i v - 2 ( l - 6 ) ( | 7 r _ p - 7 r p | + ( £ - p ) 2 e 2 ) < K*\ir-p - TTP\ + 2K2p(L - p)e2. (7.55) We treat the two terms in the above sum separately. We first deal with the second term 2K2(L - p)e2. Since L - p < 2L + 1 and Kp > K2, we have IQ^2L+I)KP — 8(L—%)K^ • Then the definition of e2 in (7.46) means that e2 < 1 6 ( 2 L + I ) ^ P — 8 ( L - p ) K 2 ' which implies 2Kl(L-p)e2<4- (7.56) For the first term in (7.55), we can divide into twocases: TTP > 7r_ p and irp < 7r_ p . If TTp > 7 T - p ; then 0 < 7T p — 7 T _ p = 7T p — e e 3 7 T _ p + e £ 37r_ p - 7 T _ p = (7T p — e£37T_p) + (e'3 — l ) 7 T _ p . The definition of e3 in (7.47) implies that 7TP < e£37r_p and e6 3 — 1 < jg?, so continuing the calculation in the line above, we obtain i I ^ n , ° 2 ° 2 F P - TT-PI = % - TT_ P < 0 + J ^ 7 1 " - ? < ^ • (7.57) If 7Tp < 7r_p, we get the same bound. Therefore applying (7.56) and (7.57) to (7.55), we have I | , c 2 c 2 c 2 K - T O _ p | < - + j = y -93 This result means that the estimate in (7.54) can be generalized to min(m p, m_ p) — mx > — for x £ [-L, L]\{p, —p}, as required by the lemma. • Proposition 7.2.7. If p < ^-e2, then the set A2 defined in (7.44) *s a n invariant set for the dynamical system (7.6). Proof. As in the proof of Proposition 7.2.5, it suffices to observe that dtirx > 0 where TTX = 0 as shown by the argument following (7.43), and check that the following inequalities hold: ( f t T » ) l ^ = £ a , w 6 A 2 < 0 for x ^ p, -p, (7.58) < 0, (7.59). dt log 7T_ and f dt log '^—E- J log •=-£-=€3iirGA2 K —p < 0. (7.60) log -PJL=€3,TTGA2 The estimate (7.49) means that for 7r £ A2 and x £ {p, —p}, mx is significantly smaller than min(m p, m_ p). But the mean fitness m , cannot be much smaller than min(m p, m_ p): .. min(m p, m_ p) — m^ = min(rop, m_ p) — mxTTx x < min(mp, m_ p) — m p7r p — m_ p 7r_ p < min(mp, m_ p) — min(m p, m-p)(wp + 7r_ p) = (1 — 7rp — 7r_ p) min(m p, m_ p) < (2L — l)e2 min(m p, m_ p) by the requirement that T r s < e2 for x p,—p and TT £ Ai. Since TOp = -Kp ^ z BP-ZKZTTZ < KPKQ and similarly m _ p < KPKQ, we have min(m p, m_ p) — mY < (2L — 1)C2KPKQ = (2L — l)e2KP. We use the definition of e2 in (7.46) to obtain e2< 16(2L+I)K < &(2L-I)KP > which, when applied to the estimate in the line above, implies that c2 min(mp,m_p) — < —. (7-61) 8 Now (7.49) and (7.61) imply that for TT £ A2 and x £ {p, -p}, m x - m„. < mx + — - mm(rap, ra_p) < - — + — = ——. Thus for TT £ dA2 fl {TTZ = t2 for some z ^  {p, — p}}, we have for a: where 7rx = e2, (dtnx)^^ = irx(mx -m\) + p(l - (2L + 1)^)1^=^ . 3c2 < -e2—+p < 0, 94 since p < 3f2-e2. This verifies (7.58). Now we deal with TT G dA2 fl {log — £3}. By (7.6), we have dt\og—^- = (m p - m f f 4- — - p(2L + 1)) - (m_ p — + p(2L + l)) m - p + LI (7.62) If Jhu = eC3 > 1, then 7r„ > 7 r _ D , which means that — — < 0. Therefore it remains to check the sign of mp — m_ p : m p Ul — p = K , bY,Ksitz + {\-b)Y,Kz-Kz If Kp(l - b) (KP{TT-P - 7TP) + ]T ^ ( 7 r _ z - 7 r z ) . V Z=p+1 / e63, then (7.63) T T _ P — 7Tp = T T _ P ( 1 — et3) = 7r_pmax(—1, _ £ 2 _ A K 2 ) < C 2 16/Cp2 (7.64) by the definition of e3 in (7.47) and the fact T T _ p > | established in (7.48). For w G A2 and z > p 4 - 1 , TT-Z — TTZ < 2e2, therefore KZ{TT.Z - TTi) < ( L -p)Kp2e2. (7.65) Applying (7.64) and (7.65) to (7.63), and using the requirement e2 < ie(2L+i)Kp m (7-46),. which implies e2 < 3 2 K £ ( L _ p ) , we conclude that mp — m _ p < 0 if — e6 3. Therefore (7.62) implies that dt log — <0, -=e'3 which verifies (7.59). The verification of (7.60) is similar, and the proof is complete. • Propositions 7.2.5 and 7.2.7 imply that if b is sufficiently small and p, < c3b2, where c 3 is a small constant dependent on K and L, then (7.6) has at least two stationary distri-butions, one resembling a 5-measure and the other bimodal and having little mass in the middle. But Theorem 7.2.2 imply that if 6 is sufficiently small and LI > C 4 6 , where C 4 is a large constant dependent on K and L, then all stationary distributions are bimodal and have very little mass in the middle. This phenomenon is illustrated in figure 7.4. We conjecture that there is a phase transition between a unique stationary distribution and two stationary distributions in the behaviour of (7.6) for small p and b. But since we cannot come up with a straightforward comparison argument in either p or b, this remains a conjecture. 95 6 Figure 7.4: Conjecture of phase transition in (7.6) for small fj, and 6 (Shaded region is where Theorem 7.2.2, Propositions 7.2.5 and 7.2.7 work) 96 C h a p t e r 8 Stationary Distributions In Chapter 7, we examined the large-time behaviour of the deterministic dynamical sys-tem (7.1), i.e. linit^oo limjv-foo irN(t). In this chapter, however, we will take the limit t —> oo first and examine the behaviour of , the stationary distribution of irN; more specifically, we will do this by taking the limits N —> oo then LI —> 0 and examine l i m ^ o limjv-»oo i ' M ' J v . For this, we consider the case of symmetric, strictly positive, and unimodal K, with Kx strictly decreasing for x E [0,L] and Ko = 1, and Bx = 1{|X|>M} with L < M < 2L. Define I = —L + M, then mx = 0 for x E [-1 + 1,1-1]. Define C3 — sup^m^. For the strong selection model with N particles and mutation rate LL described in Chapter 6.2.1, we observe that this continuous-time finite-state Markov process has the property that all states communicate, and therefore it has a unique stationary distribution v11'1* [Durrett 1991]. Let (QN,V(QN)) denote the generator associated with this Markov process, then for all F G C°°(V(E)) C V{QN), we have J GN F{TT)^'n (dn) =0. (8.1) Let (Q,V(G)) denote the generator associated with the deterministic process de-scribed by the ODE (6.10): dtirx = irx{m(x,ir) - m x ) + LI(1 - (2L + l)irx). We calculate the effect of the generator QN on a C°°-function F(irN)=F(*»L,...,w?,. gN F(TTN) = E E X yjtx = ^ E E x y ^ x x^(m(y,nN)^ + p). 77 I N N 1 - „ N , I „jv\ T ? / N \ F I TT_L,...,TTX -Jj,---,KY + — , . . . , 7 T L I - t (TT ) 97 Performing a Taylor expansion on F, we continue the above computation: gNF(xN) = " E E X yjtx - E E X y^kx 1 9 f ( o 4 S ( ^ ) + o ( n Ndir>? 5 < dir? ^ ( m ( y , T T ^ X + a) + H i ( i V , F ) ( T T W ) . (8.2) Since supx^ ir?(m(y, nN) + /i)0{N-2) = 0(N~2), we have C\\F\\ Rii^F)^) < N ' where we suppress the dependence on L. Therefore Ri(N,F)(w) is 0(N 1) uniformly for all 7T G V(E). Here we use the norm associated with the Sobolev space H2(RE) for F, where F denotes the Fourier transform of F. The first term in (8.2) is in fact equal to QF(irN) by the following computation: E E x y^x dF OF 9% 07TX E E x y OF dF W - ~ (Tf. d7T^ G»7Ta: irx(m(y,ir)TTy + fj.) •Kx{m(y,Tv)Tvy + ji) ^ ^ QP ^ ^ Qp x 2/ Y g- i r( 7 r )( m ( a ; ' 7 r ) 7 r x - + (2L + l ) / i ) ^ 7 r x — ( 7 r ) Therefore (8.2) can be written in the following much-simplified form gNF(TTN) = gF(7TN) + R1(N,F)(7rN). Then (8.1) implies J QF(ir)u^'N (d,7r) = J R^N, F)(n)^N (dn) < J ^ » (dw C\\F\\ N ' J gF(7T)^N(d7T)=0{N-1). (8.3) 98 Since E is compact, so is V(E) and V(V(E)), therefore for each p, we can take a sequence Nk{p) such that u"'N"^ converges weakly to some v*1 G V(V(E)). By (8.3), v^1 satisfies: for all.F € C°°(V(E)), gF{ir)iS(dn) = / ^ [ 7 r x ( m ( x , 7 r ) - m x ) + / j ( l - ( 2 L + l)7r : c)] — ( 7 r ) ^ ( d 7 r ) = 0. (8.4) Therefore is an stationary distribution for the deterministic flow Q. Now we take a sequence p\. —> 0, such that vllk converges weakly to some v° G V(V(E)), and by (8.4) and the following estimate: as p —> 0, dF J ^ / x ( l - ( 2 L + l ) 7 r I ) ^ ( 7 r K ( d 7 r ) <pC(F,L) J v^dn) pC(F,L)->0, i/° satisfies: QJ? ^7r x (m(x , 7r ) -mlx)^-(-K)v0(dTx) = 0. (8.5) After establishing several lemmas, we will use the above characterization of a n d VQ ^Q prove the following: Theorem 8.0.8. Suppose K is symmetric, unimodal, and strictly decreasing for x £ [0,L], with K0 = 1, and Bx = 1{|x|>M} with L < M < 2L. Define I = — L + M. If is a weak limit point of i ^ , A r , then for any z G [—1 + 1,1 — 1], we have {/'{TT : TTZ > 6} < 5%+p(2L + iy where 52 = min ( ^ J r y , 4(2L+\)K*> 2(2? | i )») • Corollary 8.0.9. Under the same assumption on K and B as in Theorem 8.0.8, we have v°{ir : TTX = 0 Vx G [-I + 1,1 - 1]} = 1 is a weak limit point of , and consequently, for any 5 > 0, ^ . ^ ( M , ) ^ . 7Xx < 5 Vx G [-1 + 1,1 - 1]} > 1 - 5 for some sufficiently large i and j = ji. Proof of Corollary 8.0.9. We recall that if a sequence of random variables Xn =£> X Q O , then liminfn^oo P(Xn G A) > P(Xco G A) for any open set A. Thus Theorem 8.0.8 implies that for any z G [—/ + 1,1 — 1] and sequence pi such that =>• v°, U°{TT : 7T 2 > 5'} < lim inf ^ { T T : TT 2 > 5'} < lim inf 4 * = 0. 1 i—*oo J - i-oo 5 ' ^ + A t i ( 2 L + l) 99 This holds for any positive 5', so U°{TT : irz > 0} = 0 and therefore, i / °{7r : TT X = 0 Va; G [-1 + 1, J - 1]} = 1. • Lemma 8.0.10. If fj, <1, then Proof. Let z G [ -L, L] be an arbitrary site. Let 5 < 2(2L+2) ^ e a s m a u positive constant and / G C°°(R) be a function that satisfies the following requirements: (a) / '(x) = 1 for x < f; (b) / '(x) = 0 for x > 5; (c) f'{x) G [0,1] for x G [§,5]; and (d) / ( l ) = 0 . Define F ( T T _ L , . . . ,TTL) = f{irz). Then (8.4) implies: \irz(mt-m\)+ii(l-(2L + l)wx)]f\nz)^(dirj=0. (8.6) Since m z > 0 and m,r is bounded above by 1 uniformly in TT, we have mz — > — 1. The integrand in the above integral is nonzero only for TTZ G [0,5], therefore (8.6) can be rewritten as: / [TTz(mz - m\) + M(l - (2L + l)TTz)}f'(TTz)^(d7T) = 0. (8.7) Furthermore, the integrand is bounded below: TTz{mz-mir)+H{1-{2L + 1)TTZ) > -irz + - (2L + l)irz) •= -(1 + fi(2L + 1))TTz + p. (8.8) Since \x < 1, we have 5 < 2 ( 1 + 2 L + 1 ) < 2(1+ (^2^ +1))' a n c * therefore if TTZ G [0,5] C [0, 2 ( 1 + M f 2 L + 1 ) ) ] , then (1 + IM{2L + 1))TTZ < §. Thus (8.8) implies that 7Tz(mz-m7r) + p(l-(2L + l)TTZ) > | . (8.9) Applying the estimate (8.9) to (8.7), we obtain a 2 0 > £ / / ' ( ^ ( o V ) > / fM^idir) since /'(a;) > 0 for x G [5/2,5] -'{7r:7r,<(5/2} i/"(d7r) since /'(x) = 1 for x G [0,5/2] z<5/2} - ^ITT:TTZ<6-100 Therefore which implies that x^ * | T T : TT, < 11 == 0 Z 1 | T T : wx < ^ for some x G [-L,L] j < ] T ^ | T T : irx < ^ j = 0. Thus the lemma follows. Lemma 8.0.10 implies that ^M-a.s., 16(2L + 2) 2 ' ( K2 K1 K2 \ 2 (2L+i) ' 4(2L+i)K?' 2(2L+i) 2 ) i s independent of /z. We define <5i • (8.10) 16(2L+2)2 a n ( l and the function i/> : V(E) —>. 1 2L + 1 (8.11) (8.12) We recall from a formula following (7.7) that = 2mx, and observe that Qmv — Y [ T r x ( m x - m x ) + /j(l - ( 2 i + 1)^)]—— = Ekg("ta! ~ ^ x) + M(1 - (2L + l)7r x)]2m x = 2 ^ irxmx(mx - m T) + /x(2L + 1) ^  rox f ^ - 7TX x a; * Adding — 2 ^ x Kxm-„(mx — mv) = 0 to the right hand side, we continue as follows: Qmn = 2 . X X + M(2i + 1) ^  rox ( giVl ~ Y n x ( m x ~m*)2 + K2L + i)Em x (2ZTT _ 7 r a : ) Therefore ^m,r = 2ip(n). (8.13) 101 We will establish in the following two lemmas that on the set A defined in (8.11), ib(n) is bounded below by 4 ( ~ ^ + 1 ) m 2 r . We write A as a disjoint union of two sets A\ and A2, where A\ = j 7T G A : TTX < 1 for all x with mx ^ 01, ( 2L +1 J A2 — In £ A : irx > 1 for some x with mx ^ 0 i , ( 2L 4- 1 J and prove a lemma for each case. Lemma 8.0.11. For any n £ A\, we have > 2^r[1fn2T. Proof. For any n £ A\, there are two cases: Case 1 TTX < 2i~fT f°r a n X e [ — L , —I] U [I, L } ; Case 2 irx > and mx = 0 for some (possibly more than one) a; G [-L, —I] U [/,L], but for any x with mx ^ 0, 7r x is still < 2£,+i. For Case 1, (8.12) implies l-i , 1 4>(TT)> Y ^x(mx - m^)2 + LI(2L + 1) Y m x ( 2L + 1 ~ ? x = - i + l x-.m^O Since m x = 0 for x £ [-1 + 1,1-1} and irx < 1^-^ for all x £ [—1,1}, the second sum on the right hand side is nonnegative, and therefore 1-1 1-1 ip{ft)> Y nx{mx -m-*)2 = Y nxml' (8-14) X=-l + \ x=-l+l Using 1—1 —^ ^ 2^  ^ E *•* =1 - ** - ^ 2 T T 1 ' x = —i+1 x=—L x=l (8.14) implies that We now turn to Case 2. Suppose y is a site with iry > 2 L 1 + 1 and = 0. We bound IP(TC) in (8.12) a bit differently from Case 1. Since nx < JE+H ^ o r a n y X w ^ t n m : E ^ 0, T N E second sum in (8.12) is nonnegative, and therefore X Therefore (8.15) and (8.16) imply the lemma. • 102 Lemma 8.0.12. If LL < 8 ( 2 L + ^ 4 g 2 , then for any TT £ A2, we have V ,(71") — 4(2L+i) • Proof. For 7r € A 2 , the main inconvenience is that the second term in (8.12), i.e. the term involving 2L+I ~ ^x, c a n b e negative. We divide into three cases and show that in each case, we have ™2 4(2L+ 1)' Case 1. For all x £ [-L,-l], irx < 22^1 > a n d i G [hL] is the rightmost site with n. > J X ^ J , i.e. there is no x to the right of i with irx > 2L+I '•> Case 2. For all x £ [l,L], irx < oxrr, and i £ [-L, —I] is the leftmost site with n. > 2E+i^ i - e -there is no x to the left of i with irx > 1 2L+1> Case 3. i\ £ [-L, —1} is the leftmost site with rr. > 2L+i> and i2 £ [l,L] is the rightmost site W i t h 7T. > 2L+1 • /VAAAAAAAAAAAAAAAAAAAAAAA/VAAAJAAA^  > 1 2L+1 < 1 2L+1 -L i-M Figure 8.1: Illustration of Case 1 We first deal with Case 1. The terms in the first sum of the definition of ip in (8.12) are squares and therefore nonnegative, hence we can throw some of them away and obtain the following: i-M ip{ft) > Y nx(mx - mv)2 + LI(2L + 1) Y m x 1 ~ +Tvi(mi-m7T)2. (8.17) For x £ [-L, i — M], Bx-i = 1, therefore K2 mX=KX Y BX-ZKZTTZ > K2LTTi > 2 L ^ X Z=X+M 103 Applying the above and the requirement < 2 ( 2 L + i ) f°V7T€:A to (8.17), we obtain: x=—L x=—L x „ 4 i - M This deals with the first term in (8.17). For the second term in (8.17), we observe that mx = 0 for x £ [-1,1], and j^pr - irx > 0 for x £ [-L, -l] U [i + 1, L], therefore Applying the universal bound 2Z+1 — TT^ > — 1 to right hand side above, we obtain Y m x ( 2 X T T _7rx) - ~Ymx- (8-19) Now applying (8.18), (8.19), and the requirement TTJ > J E + I to (8.17), we obtain ^ ( 7 r ) - 4 (2L + 1)2 E ^ ~ ^ 2 L + 1) E m * + 2 T + l ( m i ~ ^ ) 2 ' ( 8 " 2 0 ) ^ ' x=—L x=l We observe that i i x—M i x—M i i—M E% = YK* E ^%<^2E E % ^ 2 E E ^  x=i x=i y=—L x=l y=—L x=l y=—L i-M < K?(L- l + l) Y Kx < K?(2L + l)ir([-L,i-M]). " (8.21) x= — L Therefore the computation in (8.20) can be continued as follows: i>M > 4 ( 2 L + 1 ) 2 ^ - ^ » K i ( 2 L + VMl-L,i- M)) + 2 i T T ( T O i - ^ ) 2 * 8 ^ ^ ^ - L ^ - M ] ) + 2 x T T ( T O l - ^ ) 2 • ( 8 ' 2 2 ) K since < 8 ( 2 L + i ) 4 A ' 2 • ^ n e right hand side of the above is a sum of two nonnegative terms, both of which cannot be small at the same time. Indeed, notice that i-M x=—L = Kt Y K x 7 r x ^ Ki<[-L,i-M}). 104 .Therefore, if n([—L,i - M)) < 2-^m7r, then rm < \m^, and we have (m* - mv)2 > \m\, 1 —2 hence the second term in (8.22) alone implies i/>(7r) > A^L+I) • Otherwise, ir([—L, i — M]) > 2 ^ r m T , then the first term in (8.22) implies ip(ir) > I6^L+I)2K2 • ^° w e n a v e t n e f°U°wing estimate on ip(n): n) > min .. „ , „ , , ml Vl6(2L + l ) 2 ^ 2 ' 4 ( 2 L + l ) (8.23) Kim-, Since 52 < A { 2 L + \ ) K i ' o n t h e s e t A = {IT : < <52}, we have 16{2LL+1"2Kf > 4(2L+i) = therefore (8.23) implies —2 V>(TT) > 4(2L+l ) ' -L i 2 - M A -/ \ A A A A A A A / \ A A A A A A A A A A A A A A A / \ A A A | A A ^ > i 2L+1 < 1 2L+1 «2 H + M Figure 8.2: Illustration of Case 1 Case 2 follows by exactly the same argument. For Case 3, we first observe that if K i2 — ii > M then Bi2-i1 = 1 and on A = {w : m x < 52}, where by definition S2 < 2(2L+i) we have 52 > m^ > -K^K^Ki^^ > ^ L + I ) * ' which is impossible. Thus i2 — ii < M, and i2-M mx-m^, VK71") > E nx(mx - mv)2 + TTi2(mi2 -ra,) 2+ ^ nx( x=—L x = i i + M +wh (mh - mv)2 + n(2L + 1) ^ mx (^2^ + 1 - ^xj • (8.24) We can use techniques similar to those used for Case 1 (leading to (8.18) and (8.21)) to obtain the following bounds: i-2 — M L E nx(mx - m , ) 2 + Y Kx(mx - m f f ) 2 ~ 4(2L + l ) 2 { 7 T { [ ~ L ' k ~ M ] ) + 7 r ( [ ' 1 + M ' 105 and L E m*(21V1 ~**) - E ^ ^ - ^ ) + i>*(^ri-^) x=-L N ' x=l x ' x=ii x 1 2 — 1 > - mx -x=l x=ii > -K?(2L+l)ir([-L,i2-M}U[i1 + M,L}). Since fi < 8 ( 2 L + I ) 4 ^ > ^ e a D O V e t w 0 estimates applied to (8.24) implies that M*) > 4(2f| 1)2<[~L, t 2 - M ] U [h + M,L\) -H(2L + l)Kf(2L + 1)TT([-L, i2 - M] U [ii + M , L] ) ' + 2 L T T ( m i 2 _ M J 2 + 2 Z T T ( m i ] " ™ w ) 2 KAL „ , 1 > iif 4 1 +mLTW7r{[il +M'L]) + 2 X T T K I - ^ ) 2 -We can now apply the technique leading to (8.23) to the sum of the first two terms above, and then to the sum of the last two terms, to obtain: V>(7r)>2min' L 16(2L+1) 2 X2 '4(2L + 1)7 ' which by virtue of m„ < S2 < 4 ( 2 L+\)K 2 ° N A implies V>(TT) > ml 2(2L+1)' Thus we have established the necessary bound on ^(n) for 7r G A2 in all three cases. . • Lemma 8.0.13. If Si = iggr+V a n d 6 2 = min(27jil)> 4(2L+\)K'?' 2 ( 2 L | I ) * ) » I F T E N ^ J T T : ^ < y | =0. Proof. Recall from (8.10) that z/'-a.s., > Si. • (8.25) Let / G C°°(M) to be a function that satisfies the following requirements: (a) f'(x) increases from 0 to -j? for a; G [0,5i]; 106 (b) f'(x) = £ for 6, < x < f ; (c) f'(x) decreases from to 0 for x G [4f, (d) / '(x) = 0 for x > 2§*. Define F(TT-L, • • •, TTL) = f{mv). Then (8.4) implies: y / ' ( m I ) g m I / ( ( J ? r ) = 0 . Substituting (8.13) into the above equation, we obtain J f(mMTr)^(dir) = 0. (8.26) Lemmas 8.0.11 and 8.0.12 imply that IP(TT) is bounded below by jr^^fn2 on the set A = {IT : mv < 62}, defined in (8.11). Applying this fact and (8.25) to (8.26), we obtain 0 = 2 / ^(TT)/'(fn^v^(dir) since f'(x) only nonzero for x in [0, J {Tr:0<m„<^-} = 2 / V ' M / ' ^ T T K W since V»{TT :mn < Sx} = 0 by (8.10) J{Tr:Si<m„<^-} > 2 / ^(7r)/'(mw)i/"(dir) since VM/'(™>r) > 0 if G [ f , 2|a] 2 > 2 [ - 7 — r 2 - — r = o ^ ( d 7 r ) by the bound on ^(m,) for 7r G A • / { x ^ m , , ^ } 4 ( 2 i + 1) m% 2(2^+1) J { T : 5 ! < m „ < ^ } Therefore j IT : " V < y J = J>" {TT : 0 < m\ < 5t} + i / M | T T : <5i < < y J = 0 + 0 = 0, as required. • Proof of Theorem 8.0.8. For an arbitrary site z £ [-1 + 1,1- 1], we have mz = 0. If we take F(n) = TTZ, then by (8.4), we have 0 = J TTz(m(z,ir) -m*.) + n(l - {2L + \)-Kz)v^{d-K) = J n-(mn + fi(2L + l))irzvii(diT), so H = j(mv +n{2L + l ) )7r 2 ^(d7r) = / (mff + /i(2L+l))7r zi/"(d7r) + / {fnv + fx{2L + l ) )7r 2 ^(d7r) J { 7 r : m „ < i 5 2 / 2 } 107 where 52 is as defined in Lemma 8.0.13. The same lemma shows that {TT : < o~2/2} has -measure 0, so the second integral in the above equation is 0, thus (m„ + /i(2L + l))7r 2 i /"(d7r)> / (^ + p(2L + 1)) n^dir), . i.e. J K 1 ~ f +/i(2L + l) The observation V^{TT : TTZ > 5} < | / ^^(dir) completes the proof. • 108 A p p e n d i x A A Result on the Conditioned Dieckmann-Doebeli Model In this section, we deal with a special case of the conditioned Dieckmann-Doebeli model described in (6.2) and (6.3), and show that in this special case, there exist symmetric bimodal stationary distributions. Let E = [-L, L] n Z, A be a Markov transition matrix associated with mutation, ix(t) G V(E) for all t G Z + , and M be an even constant such that L — 1 < M < 2(L — 1), then the equation of the discrete-time dynamical system is as follows: y where Vx(ir) = V^\TT) = ZzKZ(t)vz(K(t))-Kr Cx-z^z Kx = 1{M<£-1}> and Cx = 1{|X|<M}- (A.l) Every step of (A.l) can be divided into three sub-steps: Resampling ir'x(t) = nx(t)Vx(ir(t)); Mutation Txx(t) = Y/A(y,x)'Klx(t); y Normalization wx{t + 1) = „ x J. . . (A.2) 2^yKy(t) Notice that performing the normalization step before the mutation step does not change the model, but for this section, we will use the step order in (A.2). If K, C, and A(y, x) = A(y—x) are symmetric about 0, then the map Tr(t) i—»7r(£+l) maps the set of symmetric probability measures on [-L, L] to itself. Therefore by Brouwer's fixed point theorem, there exists symmetric stationary distribution(s). We first derive a few simple facts about symmetric stationary distributions in the no-mutation case, i.e. when 109 Figure A . l : Relative locations of various sites of interest A = I. In this case, if v° is a stationary distribution of (A.l), we must have if i/° ^ 0 , then Vx(u°) = K-r i-iz z z l^z=x-M uz is a constant. (A.3) This is the same condition as (6.5). Since KX = 0 outside the interval [—L + l, L — 1], Vx — 0 outside that interval, i.e. at x = +L, therefore the support of any stationary distribution v° must lie in [-L + 1,L — 1]. We restrict our attention to sites in [—L + 1,L — 1}. If the competition intensity function C is rectangular, as in (A.l), two sites either compete at intensity 1 or they do not compete against each other at all. If M = L — 1, then 0 is the only site that competes with every other site in [-L + 1,L — 1]; but if M = 2{L — 1) — 1, then for any site x £ [-L + 2, L — 2], x competes with every site in \—L + 1, L — 1]. Define I = — (L — l—M), then [—1,1] contains the sites that compete with every site in [—L+l, L — l], therefore T T- / o\ KX _ r I vx\y ) - X+M o - 1 o z=x-M "z X £ [-1,1] 0 x £ (oo, —L] U [L,oo) (A.4) We also observe that f > / because M — 21 = M + 2(L — 1 — M) = 2(L - 1) — M > 0. li there is some mass in [-L + 1, -4^ - 1] U [4f + 1, L - 1], then since no site in [4f + 1, L - 1] competes with any site in [-L + 1, —4^  — 1], we have x+M z=x — M for x £ [-L + l , -the following: M " 2 l ] U [ f + 1,L- 1], which by the definition of Vx(ir) in (A.l) implies Vx{u°) > 1 for x£ ' T ,  M - L + L - y 1 U (A.5) Combining the results on fitness Vx(ft) in (A.4) and (A.5) and condition (A.3), we conclude that stationary distributions with rectangular K and C as defined in (A.l) must have all the mass falling in either [-L + 1, -f - l] U [4/ + 1, L - l] or [-4/, 4/]. Now we turn to the case with small mutations. We take A11 to be an operator that corresponds to a small 1-step mutation, i.e. convolution with /J<5_I + (1 — 2/J)<5O + LI8\, and define to be a stationary distribution of (A.l) with mutation kernel A11. Any stationary distribution of (A.l) satisfies the following condition: Vx £ [—L,L], / u / ^ V x - i + (1 - 2riv£Vx + ^ + 1 V X + 1 = Vv%, (A.6) 110 where Vy = Vy(v^) and V = V{v») = vzVz{^) Z-M'^O is the normalization constant. Condition (A.6) implies that is nowhere zero in [—L,L]; otherwise, say v% = 0, then then v^_x = vz+1 = 0 as well, which by induction means that vg = 0 for all x, a clear contradiction. Since the support of has expanded on both sides each by 1 site compared to u°, the sites where V^(^M) is constant 1 should correspondingly contract by 1 site oh each side: ( 1 xG [-1 + 1,1-1] Yxi"") = ^ X + M " p. = { >1 xe[-L + l,-l]U[l,L-l) . . (A.7) l^z=xrMv" [ o x G (oo, —L] U [L, oo) Therefore, for sites in the middle, i.e. x £ [—1 + 2,1 — 2], v*1 satisfies: K - i + (i - 2 ^ K + K + i = v(>?K-Since every site x in [-L + 1, L — 1] competes with all sites lying on the same side (with respect to the origin) as x, and the stationary distributions we consider are symmetric, YZtx-M vx is at least \ for 1 6 [—L + 1,L — 1], therefore Vx(^) < 2 for x G [-L+ 1,L- 1], (A.8) i.e. the normalization constant V^) is bounded above by 2. We also need a nontrivial lower bound of V(^ M ) for symmetric z/M that is uniform for small fi for the proof of the upcoming theorem. We rewrite condition (A.6) for x = L and x = L — 1, taking into account the fact VL(v^) = 0 from (A.7): K - r V i - i K ) =7KK, K - 2 v i - 2 K ) + (i-2M)^_1vi_1K) = F K K _ r Dividing both sides of the above two equations, we get K K - I V L - I ( ^ ) K _ 2 V L - 2 ( ^ ) + (1 - 2 / x K _ 1 V L _ i ( ^ ) K _ I V L - I K ) /x < ( l - 2 / i ) i ^ _ 1 y L _ i ( i / M ) l - 2 / x ' which is < 1 if /x < | . This means that v£ — < \, for otherwise, f£ = u^_L > | implies that v^-i = — \ - y L < I» bence > 1, a contradiction. Since Vx(u^) > 1 for x G [-L + 1, L - 1] by (A.7), we have L - l L - l FK)> J2 "JW)> E 2=-L+l 2=-L+l 111 The fact v£ = vtL < \ then implies V(v»)>\-v»_L-vl>\. (A.9) In particular, is bounded between | and 2. We take a sequence fxn —» 0, such that i / n = converges to some u°, then since V(^) is a continuous function of u, Vn = V(un) also converges to a positive constant V. We will prove the following: Theorem A.0.14. If vn = p^n is a convergent sequence of symmetric stationary distribu-tions for the conditioned Dieckmann-Doebeli model in (A.l), then vV M M , - » 0 asn->oo. [ 2 ' 2 J Proof. Since un —> i/° and V{yn) —> V as n —> oo, condition (A.6) converges to the following: \/xe[-L,L],V°Vx^°) = Vu°. Therefore u° is in fact an stationary distribution for the no-mutation case, i.e. (A.l) with A = I. If some mass of i/° lies in [-L + 1, -f - l] U [4f + 1, L - l ] , then ^ ^ must be zero, which means that M M 1 —> 0 when n —> oo as required. Therefore it suffices to show that _ l ] u [ f + 1 L _ l ] > 0. We assume, toward a contradiction, that v? M . = u?M , 1 = 0 . Then [ -L+1,——-lj [-^-+1,L—1\ for any positive 6, we have v? M , = i/fV, . < <5 for sufficiently large n. We [-L + l , - - 5 - - l ] [-2- + l , L - l j first derive more refined (than (A.9) and (A.8) respectively) lower and upper bounds for Vn- Because of the supposition v?M -. < <5, (A.6) and (A.7) with x = L and the bound v2-i < <S imply that < W V L - I K ) < 2 5 ^ ^ n * n by (A.8). Applying the estimate (A.9) to the right hand side, we have ..n „ 2 5 < 45Hn- (A.11) Therefore L - l Y s i n c e = Vt(" n ) = 0 by (A.7) z = - L + l L - l > YI V" s i n c e ^ ( ^ n ) > 1 for x G [-£ + 1, £ - 1] z = - L + l > 1 - 85pn, (A.12) 112 by (A.11). For are [ - M M ] , Vx{vn) = +Af - < — M = i _ 9,.n - 1 _ 2(5 4- 4 5 „ ) ( ' A ' 1 3 ) by (A.11) and the supposition ^ ? M + 1 < <5- Therefore L - l v>n) = E T O O = E TOO+ 2 £ TO"") < 7 7 ^ — r - ^ + 4<5 (A.14) - l - 2 ( 5 + 45/in) v ' using (A.13) for the first sum and (A.8) for the second. Let r = • ' J I M _,_,] = % , , * ] < L H E N E L — l + M „ v*1^ n 1 _ 7/n — 7/™ — 7/™ > (A.15) 1 — r Condition (A.6) applied with x = L — 1 implies (1 - 2^)ul_1VL-1(un) < F ( « / > L - I , hence Then (A.14) and (A.15) imply i.e. 1 - r ~ l - 2 / i „ Vl-2(<5 4-45/j„) 1-2/in (1 -2 /%)( l - 2 5 -85/j n ) 1-35 ' 1 - V > : — = — — —; ——z > ———: = 1 75 l-2(5+4(5/i„) for sufficiently small Lin. Therefore + 45 1 + 45 - 852 - 3262Lin " 1 + 45 1 + 45 l - f H - r ^ f ] - i + 4 5 - 7 d -The above inequality and the supposition v™_L+1 _ M _ X J — ^ [ M + I L - I ] < ^ i m P ^ t n a t ^ - L + I , - « - I ] = I / [ J + I , L - I ] < 8 < J - ( A . 1 6 ) 113 Let p = v™. We bound Vi(vn). Since site -L is the only site in the support of vn that does not compete with site I, we have 1 < Vi(un) = —?—— = K l < (A.17) by (A. 11). We will establish the following lemma after the proof of the present theorem: Lemma A.0.15. Let vn = be as in Theorem A.0.14, and suppose = v[i+i L - i ] < 8^> ^en 1. and vf is bounded away from 0 as n —> 00; 8- vf_l < uJ1 + 5 for sufficiently large n. Since Vy_L+x < 88 by (A.16), we have from (A. l l ) , £ ( A - 1 8 ) because sites — L and — L + 1 are the only sites in [-L, L] that do not compete with site I + 1. We estimate J2y A(y, l)vyVy{vn) - i f : = ^"P-i^-it"") + (1 - 2 , ^ ) ^ ( 0 + /xni/r+iVi+i(i/n) - v? £ M p + ^) + T ^ T + i - 8 5 - 4 ^ using i f . , < p + <5, Vi_i ( i / n ) = 1, (A.17), i f + 1 < 5, and (A.18). Simplifying the right hand side of the above, we get E A, , \ r i T T - / n\ n ^ ~ 4 / i 2 <5p + p - / J n p / i n o " . -4 / j 2 5p - pnp + Apn5p pn8 + Lln8 + 1 - A5pn ^ 1-88- A8pn pnp{l-48) pn8 < ; 777.— + M " 1 - A8pn ^ 1 - 88 - A8pn 2 for sufficiently small 5 and pn. This estimate means that after resampling and mutation (and before normalization), i f decreases by at least pnp/2. On the other hand, vf+l can only increase: let q = vf+1, then q < 8 and £ > ( y , i + i)Wvn)-v?+1 v = wWiv") + (1 - 2pn)v?+1Vl+1{vn) + pnv?+2Vl+2(vn) - i f + 1 > UnP + (1 - 2pn)q - q, 114 since Vi(vn) > 1 and Vi+l(un) > 1. Therefore Y A(y, I + \)^Vy{vn) - v?+1 > Lin(p - 2q) > pn(p -28)>0 • y if S is small enough. After normalization, i.e. dividing by V(yn), vj1 and u^+1 cannot possibly return to their original values. This contradicts the assumption that-z/1 is an stationary distribution for (A.l) with mutation kernel A^n, and the proof is complete. • Proof of Lemma A.0.15. Define Cc = TOO (A-19) Notice that Cx depends on n, but notationally we suppress this dependence. For x £ [-1 + 1,1 - 1], Cx = since Vx{vn) = 1 from (A.7). For x £ [-1 + 1,1- 1], we rewrite condition (A.6) as follows: A*„Cx-l + (1 " 2pn - Vn)C,x + MnCx+l = 0. (A.20) This is a recurrence relation with general solution £ x = Afiif + Bfi^, where \3\ and #2 are the two roots of the quadratic polynomial pn + (1 — 2pn — Vn)r + /x„r2; or £x = (A + Bx)(3f, where [3y is the double root of the polynomial. Elementary calculation shows that for the solution Cx = (A + Bx)6f to satisfy the symmetry requirement for L > 1, either B = 0 or Pi = 0; Pi = 0 leads to the solution of Cx =0, and B — 0 leads to the conclusion Pi = 1 and Cx = A; both these two scenarios will be included in Case 2 below. For the solution Cx = Apx + Bpx, simple calculation leads to: Pl,p2 = ~ (2pn + Vn - 1 ± y / ( F r l - l ) 2 + 4 / X n ( F „ - l ) ) . We divide into three cases: 1. If pi and P2 are two real roots, then since C is symmetric, we must have Pi = I/P2 with Pi > 0, and the solution is Cx = A(Pf + P~x) for x £ [-1,1]. 2. If pi = P2, then the solution is Cx = A for x £ [—1,1]. 3. If Pi and P2 are complex roots, then we write Pi = -yel6 and p2 = 7e~*e, and the solution is Cx = A-ycosfaO) for x £ [—1,1]. Define an = Vn-l, (A.21) then for Pi and P2 to be complex, a 2 4- Apnan = an(an + Apn) < 0, which means that either an < 0 and an + 4pn > 0 . (A.22) or an > 0 and an + Ap,n < 0. (A.23) 115 Now (A.23) is clearly impossible since \xn > 0, and (A.22) implies that an < 0. Furthermore, tan# -a 2 - 4fj,nan _ ( al + 4 / i n a n + 4/u2 x 1 / 2 (an + 2/x„)2 \ al + A/j,nan 4/4 ^ " 1 / 2 al + 4finan / ' + < A ' 2 4 ) Hence (A.12) and (A.21) imply that an > —85fin, and since an < 0,we have 0 < -an/pn < 85. We conclude from (A.24) that for sufficiently small 5, tan# is also very small, Note that —I and I are the boundary sites for (A.20), therefore statements about Cx in the three cases above all hold for x G [—1,1], even though (A.20) holds for only x£ [-1 + 1 J-l]. In case 1, C is a linear combination of two convex functions, therefore C is convex for x € [—1,1]. In case 2, C is flat for x £ [—1,1]. And in case 3, C is concave for a; G [—/, I], but since # is small for small 5, it is close to being flat for small 5. Therefore recalling the definition of Cx in (A.19) and using (A.17), we have (1-4<W)0 <«>P < Ci-In summary, for x £ [—Z,Z], C is convex, or flat, or nearly flat for small enough 5; = Cx for x £ [—2 + 1,2 — 1] and vf = v™l is smaller than but very close to Cz = C-f> furthermore, by (A. l l ) and (A.16), we have « f _ M ] = 1 " v"Lh - l f - L + 1 , - 1 - 1 ] - i t f + i . L - ! ] - *l > 1 - 8<5Mn - 165, i.e. ^ n has almost all its mass on [—1,1]. We can then use the symmetry assumption on vn to arrive at the conclusion of the lemma. • 116 B i b l i o g r a p h y [Burkholder 1973] Distribution function inequalities for martingales. D. L. Burkholder. Ann. Probability 1 (1973), 19-42. [Burger 2000] The mathematical theory of selection, recombination, and mutation. Reinhard Burger. John Wiley, 2000. [Dawson 1993] Measure-valued Markov Processes, Lecture Notes in Mathematics, 1541. D. A. Dawson. Springer-Verlag, Berlin, 1993. [Dawson and Perkins 1998] Long-time behavior and coexistence in a mutually catalytic branching model. D. A. Dawson and E. A. Perkins. Ann. Probab. 26 (1998), no. 3, 1088-1138. [Del Moral 1998] Measure-valued processes and interacting particle systems: Application to nonlinear filtering problems. P. Del Moral. Ann. Appl. Probab. 8 (1998), no. 2, 438-495. [Dieckmann and Doebeli 1999] On the origin of species by sympatric speciation. U . Dieck-. mann and M . Doebeli. Nature v. 400 no. 6742 (July 22 1999) p. 354-7. [Durrett 1991] Probability : theory and examples. R. Durrett. Wadsworth & Brooks, 1991. [Durrett 1995] Ten lectures on particle systems, Lecture Notes in Mathematics 1608 R. Durrett. Springer, Berlin, 1995. [Durrett and Neuhauser 1994] Particle systems and reaction-diffusion equations. R. Durrett and C. Neuhauser. Ann. Probab. 22 (1994), no. 1, 289-333. [Ethier and Kurtz 1981] The infinitely-many-neutral-alleles diffusion model. S. N . Ethier and T. G. Kurtz. Adv. in Appl. Probab. 13 (1981), no. 3, 429-452. [Ethier and Kurtz 1994] Convergence to Fleming-Viot processes in the weak atomic topology. S. N . Ethier and T. G. Kurtz. Stochastic Process. Appl. 54 (1994), no. 1, 1-27. [Liggett 1985] Interacting particle systems. Thomas M . Liggett. Springer-Verlag, 1985. [Liggett 1999] Stochastic interacting systems : contact, voter, and exclusion processes. Thomas M . Liggett. Springer, 1999. [Mayr 1963] Animal species and evolution. Enst Mayr. Harvard University Press, 1963. [Perkins 2002] Dawson-Watanabe superprocesses and measure-valued diffusions, Lectures on probability theory and statistics (Saint-Flour, 1999), 125-324, Lecture Notes in Mathematics 1781 E. A. Perkins. Springer, Berlin, 2002. [Taylor 1996] Partial differential equations. M . E. Taylor. Springer, 1996. 117 [Wiggins 1988] Global bifurcations and chaos : analytical methods. Stephen Wiggins. Springer-Verlag, 1988. 118 

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