Grasman, J. (Johan)

English

Grasman, Johan

NARCIS 

On a class of elliptic singular perturbations with applications in population genetics : (preprint)

CWI's Institutional Repository

stichting mathematisch centrum AFDELING TOEGEPASTE WISKUNDE (DEPARTMENT OF APPLIED MATHEMATICS) TW 184/78 ON A CLASS OF ELLIPTIC SINGULAR PERTURBATIONS WITH APPLICATIONS IN POPULATION GENETICS J. GRASMAN Preprint ~ MC OKTOBER 2e boerhaavestraat 49 amsterdam PJun:ted at ;the Ma.thema.ti.c.ai. Cer,;t,r,e, 49, 2e BoeJr.ha.a.vu.ttuuit, Am6;te.Julam. The Ma.thema.ti.cai. Cenbz.e, 6oun.ded ;the 11-;th 06 F-ebJc.WVLy 1946, .l6 a non-pll.o 6U .ln1>:ti:tu:Uo n a,i.mbig a.;t ;the pll.omoUo n o 6 pUILe ma.thema.ti.c.6 and w appUc.a:Uon6. 1;t .l6 .6pon1>01C.ed by ;the Ne:the1Lf.and6 Gove/lnmen:t fuou.gh ;the Ne:thelli.a.ndt, OJr.gaYU.za.ti.on 6oJc. ;the Advanc.emen:t on PU/le Rue.a1tc.h (Z.W.O}. AMS(MOS) subject classitication scheme (1970): 35B25,35J25 On a class of elliptic singular perturbations with applications • . • *) in population genetics by J. Grasman ABSTRACT With the maximum principle for differential equations asymptotic esti-mates are made for a class of linear elliptic singular perturbation problems with resonant turning point behaviour in some of the independent variables. The method is applied to stationary solutions of the Kolmogorov backward equation from population genetics. KEY WORDS & PHRASES: mazimum prinaipZe, eZZiptia singula!' pePturobation, population genetias. *) This report will be submitted for publication elsewhere. I. INTRODUCTION In this paper we consider elliptic singular perturbations of first or-der differential operators vanishing at an interior surface of a domain. For Dirichlet problems of this type we construct asymptotic solutions and prove their validity by using the maximum principle. DE JAGER [4] considered a similar class of problems in which a para-bolic boundary layer occurs at the interior surface. We will investigate the case where the first order operator has the opposite sign giving arise to ordinary boundary layers along the boundaries of the domain. For this problem standard singular perturbation techniques do not lead to a uniquely determined outer solution. Similar to the method for elliptic singular per-turbation problems with turning points of GRASMAN & MATKOWSKY [3], we pose an additional condition, so that a unique outer solution can be derived. Adding boundary layer corrections we obtain a uniform asymptotic approxima-tion; its validity is proved by estimating asymptotically the remainder term. This proof, based on the maximum principle for elliptic differential equations, differs from the ones given by DE JAGER [4] and ECKHAUS & DE JAGER [I], as near the surface where the first order operator vanishes, the approx-imate solution varies in the normal direction in a way unsuitable for apply-ing the maximum principle. In this paper we construct barrier functions that also take into account the ~ehaviour of the asymptotic solution along the surface, so that the maximum principle will lead to meaningful results. This method requires a higher order accuracy in a neighbourhood of the sur-face. The type of elliptic singular perturbations.we deal with occur in prob-lems from population genetics. The elliptic perturbation models the effect of random mating, while the parameter£ denotes the inverse of the popula-tion size. We will not attempt to give a complete description of the class of genetic problems to which our method applies, but confine ourselves to two examples: a one-locus model with migration and a two-locus model. Our asymptotic results hold for a subdomain of the continuous state space of possible genetic distributions; the elliptic equations for these problems degenerate at the boundaries of the full domain. In general existence of solutions of this last type of Dirichlet problems is not guaranteed; see FRIEDMAN [2,p.308]. 2 2. FORMULATION OF THE MATHEMATICAL PROBLEM We consider the Dirichlet problem for a function cj> (x 1, ••• ,~ ,y 1, ••• ,ym; £) satisfying the linear uniformly elliptic differential equation (2. 1) inn with boundary values (2.2) cj> = h(x,y;e) on an, where£ is a small positive parameter. The domain n is a bounded domain in n lR , n = k+m, of a form such that (2.3) (x,y) E n implies (x,O) En. The first and second order differential operators L1 and L2 have coefficients that are Holder continuous inn, (2.4) (2.5) m LI - l j=I a b. -"-, J oy, J k L2 = l i,j=I Furthermore, it is assumed that (2.6) b(x,y) = 0 iff lyl (2. 7) S(x,y)•b(x,y) :,; 0 on (2.8) m 2 I b. (x,y)y. :,; -Llyl j=I J J = o, an, in n, where S(x,y) is the outward normal to an, La positive constant independent of£ and lyl the Euclidean length of y. The behaviour of the solution depends strongly upon the first term of L2 near the surface lyl = 0. We define the bounded domain r c lR.k by 3 (2. 9) r = {x I (x,O) c: Q} and state the following lennna, which is easily proved from the definition of ellipticity; see for example [9,p.56]. LEMMA 2.1. Let the differential operator 1 2 be unifoY1171ly elliptic in rl; then the operator (2.10) k 32 I a .. (x, O) 3 . . l iJ dX. X. i,J= i J is unifoY1171Z.y eUiptic in r. 3. THE MAXIMUM PRINCIPLE For the elliptic operator L given by (2.1) we formulate the maximum E principle as follows: a twice continuously differentiable function¢ satis-fying L ¢ > 0 in a domain Q cannot have a maximum in Q; see PROTTER & E WEINBERGER [9,p.61]. The following lennna is a direct consequence of the maximum principle. LEMMA 3.1. If the -twice continuously differentiable functions¢ and iµ satisfy (3. 1 ) IL ¢ I < -L 1jJ E E in r1 -and if 1¢1 s 1jJ on 3Q~ then 1¢1 s 1/J in Q. PROOF. From the maximum principle and (3.1) we deduce that ¢-iµ cannot have a maximum in Q and since ¢-iµ s O on 3Q, we conclude that ¢-iµ s O in Q. Simi-larly, -¢-~1 does not have a maximum in Q and -¢-iµ s O at 3Q, so that -¢-iµ s O in~- Combining these results we obtain 1¢1 s 1jJ in rl. D In the next step we give an asymptotic estimate for the solution of (2.1), (2.2). For that purpose use will be made of so-called barrier func-tions: Lemma 3.1 is applied with a given function 1jJ as barrier function. THEOREM 3.1. Let the -twice continuously differentiable function ¢ satisfy 4 (3.2) and I <I> I ~ N on an with M and N positive constants independBnt of e:. Then a constant K indBpendBnt of e: exists such that (3.3) l<l>I ~ K in Q. PROOF. We introduce the barrier function (3.4) ~(x,y) = -U(x) + Rlyl 2 + S, in which we choose R > M/L with L given by (2.8) and U(x) such that (3.5) k a2u l a. .. (x, o) a a •• I 1J x. x. 1,J= 1 J k = 2M + 2R l i,j=l y .. (x,O) 1J in r. Since the coefficients a. .. and y .. are Holder continuous, there exists a 1J 1J positive constant F, such that (3. 6) k a2u k . J. a.iJ0 (x,y) ax.ax. - 2R . J. 1,J=l 1 J 1,J=l 2 For ly I ~ (I+ F/M)e: we have (3. 7) y •• (x,y) > -F 1J in Q. -L ~ e: { k a2u m } m = e: .. I a.iJ0 ax.ax. - 2R . J. I yiJ" - 2R .l 1,J=l 1 J 1,J= J=l b.y. > J J 2 2 > -e:F + 2RL ly I ~ M( ly I + e: ). Because of the Holder continuity of a. .. and y .. a-t ly I = 0 the following 2 1] 1] estimate can be made for lyl < e:(I +F/M) and e: sufficiently small; see (3.5). (3. 8) k a2u I a. .. cx,y) a a • • I 1] x. X, 1,J= 1 J m - 2R l i,j=l Thus, for lyl 2 < e:(I + F/M) we have (3. 9) 2 2 -L ~ > e:M + 2RLlyl > M( lyl + e:). e: y .. (x,y) > M. 1J 5 Finally S of (3.4) is taken sufficiently large such that (3. IO) ip 2".: N on an. From (3.7) and (3.9) we conclude IL ~I ~-Lip inn, while from (3.10) it E: E: follows that 1~1 ~ ip on an. Using Lennna 3.1 we obtain the estimate l~I ~ ip in ri. Sinc1e the function U(x) as well as the domain n is bounded, a positive constant K can be found such that~~ Kin Q, which completes the proof of the theorem. D COROLLARY 3.1. Let the twice continuously differentiable function ~(x,y;e) satisfy (3.11) IL ~ I E: inn and l~I ~ Noh(e) on an with of and oh continuous positive functions for O < e < e0 (e 0 sufficiently small) and with Mand N independent of e. Then a constant K independent of e exists such that in Q (3.12a) or (3.12b) if oh/of is bounded fore+ o, PROOF. As a barrier function we take and proceed as in the proof of Theorem 3.1. D Thus, Corollary 3.1 produces an asymptotic estimate for the solution of (2.1), (2.:2) from the asymptotic estimates of the data f and h. 4. ASYMPTOTIC APPROXIMATION Let us assume that by some matched asymptotic expansion procedure we have found a formal uniformly valid asymptotic approximation, say~ , of as ~ satisfying (I.I) and (1.2). Its validity is proved as follo-ws. Substitu-tion of ~ := ~ + R into (I. I) and (I .2) yields as 6 (4.la) (4.lb) L R = f - L ~ £ £o/as R = h - cj> as inn, on an. If we are able to show that the right-hand sides of (4.1) and (4.2) have the appropriate asymptotic behaviour, then by application of Corollary 3.1 the smallness of the remainder term R is established. It is to be expected that the solution of (I.I), (1.2) has a boundary layer structure, which ~ay com-plicate the construction of a suitable function cj> as its derivatives may as be of a larger order of magnitude in the boundary layer. this difficulty is surmounted by including (small) boundary layer corrections to the asymptotic approximation. Depending on the shape of the domain different types of boundary layers may arise. In the sequel we restrict ourselves to the case m= I for convex domains with nowhere characteristic boundaries, so that inequality (2.7) is strict-ly satisfied. These domains have the form (4.3) + + with p-(x) > 0 in r, p-(x) = 0 on ar and because of our method of approxima-tion p± E c3 (r). We consider the Dirichlet problem for the function cj>(x 1, ••• ,xk,y;£) satisfying (4.4) 00 -with a .. ,S.,y EC (Q). This problem is assumed to have continuous boundary 1J 1 values -(4.5) for XE f with h± E c2 (r). The asymptotic approximation of cj> has the form (4.6) with u0 (x) satisfying (4.7a) k a2u l a .. (x, O) 0 0 in r = i,j=I l.J ax. ax. l. J (4.7b) u0(x) = h(x) on ar, and with (4.8a) [ + + + = h±(x)exp p-(x){~-(x)+y}], v0(x,y;E) Eq-(x) k ± ± k + ± a - + ± ~+ + (4. 8b) l ap ap ± l q = ----a + _£_f3- + y ' h- = h-- uo, ax. ax. ij ax. i i,j=I l. J i=l l. + + + + w:iere a:.,s:,y- = a .. ,f3.,y(x,±p-(x)). l.J l. l.J l. THEOREM 4.1. Let the function $(x1, ••• ,~,y;E) satisfy (4.4) in the domain Q defined by (4.3) with boundary values (4.5). Then there exists a positive constant K independent of E auch that (4. 9) I$-$ I ~ KE in Q as with $ given b.f ( 4 • 6 ) - ( 4 • 8) • as PROOF. We introduce the local coordinate n respect to E, -1 + + + LE - E MO+ Mi+ M2+ ••• , + + a2 + a MO - q-(x) -2 - p-(x) arj" , an + while }r, m > o, is of the form m + k + a2 k }1 - l r:. (x) + l m i,j=l l.Jm ax.ax. i=I l. J + = (p-(x)+y)/E and expand L with E + a2 s: (x) im ax. an l. + a2 + t-(x) --+ m an2 (r .. 1 =0). l.J We introduce additional boundary layer terms (4.10) + - + -$as (x,y;E) = U(x) + V 0(x, n) + v0(x, n) + dV 1 (x, n) + V 1 (x, n)} + 2 + -+ E {v2(x,n)+ v2(x,n)} 7 8 with V. satisfying ]. + + + ~+ (4.lla) MoVo = o, v0(x,O) = h-(x), + + + + + (4.llb) M°oVt = -M1v0, v1(x,O) = o, + + + + + + + ·(4.llc) M-v- = -Mj"Vj" - ~VO' v2(x,O) = o, 0 2 + i (4.lld) v:(x,n) + 0 as n + co, = 0,1,2. ]. + The expression for v0 we gave in (4.8); + v: ]. with i > 0 is of the type (4.12) Let R = ~-~ • By straightforward calculation one finds that a constant M as exists such that IL RI s ME 2 inn, while also IRI s NE on an for some N > O. . E Fro~ Corollary 3.1 we conclude that IRI s KE inn for some K. Finally, the • + proof is completed by checking the additional boundary layer terms El.vi: i = 1,2 which are O(E) in Q. D REMARKS. When making higher order approximations, one has to take into ac-count corner layer contributions in an E-neighbourhood of (x,O), x far. The higher order terms for the outer- and boundary layer expansions follow from the fundamental iteration process (see [1]) with an additional equation of the type (4.7a) for the terms of the outer expansion. 5. APPLICATION TO PROBLEMS IN POPULATION GENETICS A population consisting of different genotypes with random mating can be described by stochastic as well as by deterministic mathematical models. We will deal with a deterministic model, a diffusion equation known as the ' Kolmogorov backward equation, being the limit of a stochastic model as the population size increases indefinitely; see MARUYAMA [6,p.221]. Our asymp-totic analysis applies to the stationary solution of the Kolmogorov back-ward equation of a certain class of genetic problems. We will give two illustrating examples. EXAMPLE 5.1. We consider a diploid population with two alleles a and A at one locus divided into two colonies of each N individuals. Let p. denote l. the fraction of allele A at colony i. Assuming random mating without selec-tion or mutation and with nett migration proportional to the difference in pi, we obtain the Kolmogorov backward equation (5. I) where ~(p 1,p2,t) denotes the probability density of the fractions pi at time t. This equation holds in the square S = { (p 1 ,p2) I O <Pi, p2 < I}. Substitution of (5.2) transforms the stationary equation of (5.1) into (5. 3) f 2 2 (a 2~ a2~) a2~ ] a~ e: lo -x -y ) - + - - sxy -- - y - = o, ax2 ay2 axay 3y e: = I / ( 4 µN), in a domain Q = {(x,y) I lx±yl < I}. We consider the Dirichlet problem of (5.3) with O < e: << I for a subdomain Q0 c Q of the form (5.4) Qo = { (x,y) I lyl· < I - ✓x2-o 2+2o, lxl < I - o} -with boundary values hon an0 . Equation (5.2) relates a point (x,y) E Q to the distribution of alleles at some time. Let p0,e:(x,y;x0 ,y0) denote the probability density of leaving Q0 the first time at (x,y) E an0 if starting at (x0 ,y0) E Q0• The following relation between Po,e: and~, is kno-wn to be valid (5.5) Ian Po,e:(x,y;xo,Yo)h(x,y;e:)dcr = ~(xo,Yo;e:) 0 where dcr denotes a positive measure on an0 ; see MATKOWSKY & SCHUSS [7]. If (x0 ,y0) is chosen in the outer region of Q0 the system leaves Q0 at either a point of an0 with x < -1+2o or with x > 1-20 with probabilities that tend to 9 (5.6ab) 1-o-x0 pr(left exit)= - 2-_-2-0-, pr(right exit)= 1-o+xo 2-20 10 as E + 0. This result is derived from (5.5) by choosing appropriate boundary values h. As o + 0 this asymptotic result tends to the exact solution of the problem for the full domain with arbitrary E > 0. EXAMPLE 5.2. A population of N diploid individuals, each characterized by its genotype with respect to two loci and with two alleles at each locus, is described by the fractions of gametes of types AB, Ab, aB and ab. Let these fractions be denoted by p., i = 1,2,3,4. In case of random mating such I. system is modeled by the Kolmogorov backward equation (5. 7) ~= at 3 I i=l 2 3 I I i=l j=i+l + (5.8) transforms the equation for the stationary problem into (5.9) E = l/(2+4Nµ), while the domain S transforms into a domain n satisfying (2.3), (2.7) and (2.8). Again we consider the Dirichlet problem of (5.9) with O < E << 1 for a subdomain n0 c n with an0 bounded away from an and with an0 + an as o+O. In the limit E + 0 the probability of leaving n0 at some point of an0, if starting at the outer region of n0, depends according to formula (5.5) entirely on the function U(x 1,x2) satisfying 11 (5. I Oa) a2u a2u 0 in r O = { (x l' x2) I (x1 ,x2,0) e: n0 } XI (I-xi )-2 + x20-x2)-2 = ax 1 ax2 (5. 1 Ob) u = h on ar0 , where his some appropriately chosen boundary value. Using this result one can prove that for£+ 0 the two-locus system, if starting in the outer region of n, tends to linkage equilibrium (y= 0) along the subcharacteristic of L1 by choosing an appropriate domain nn with n arbitrary small but in-dependent of£; see Figure I. For a more extensive discussion of this prob-lem we refer to LITTLER [5]. y n n x2 (x0 ,0) + n + xi Fig. 1 The path towards linkage equilibrium as£+ 0. REMARKS. The asymptotic solution (5.10) for the outer region tends to a regular limit as o + 0. From this limit expression one may derive the probability of first fixation of a specified allele in a same manner as we find the probability of loosing either one of the two alleles in Example 5.1 from (5.6) by letting o + O. Finally it is mentioned that for both examples more accurate approximations can be obtained by computing the next terms of the asymptotic expansion in£ as we remarked in Section 4. In Example 5.2 this would lead to new quantitative results for linkage dis-equilibrium when£ is small; see also OHTA & KIMURA [8]. 12 ACKNOWLEDGEMENTS. The author is gratefull to Dr. E.D. Fackerell (Univ. of Sydney) for suggest-ing the biological problems of Section 5 and to J.A.J. Metz (Inst. Theor. Biol., Leyden) for some valuable remarks. REFERENCES [1] ECKHAUS, W. & E.M. DE JAGER, Asymptotic solutions of singular perturba-tion problems for linear partial differential equations of eUip-tic type, Arch. Rat. Mech. & Anal. Q (1966), p.26-86. [2] FRIEDMAN, A., Stochastic differential equations and applications, vol. 2, Acad. Press, New York (1976). [3] GRASMAN, J. & B.J. MATKOWSKY, A variational approach to singularly perturbed bounda:ry value problems for ordina:ry and partial differential equations with turning points, SIAM J. Appl. Math. ~ (1977), p.588-597. [4] JAGER, E.M. DE, Singular elliptic perturbations of vanishing first order operators, Conference on the Theory of Ordinary and Partial Differential Equations, Lecture Notes in Math. 280, Springer-Verlag, Berlin (1972). [5] LITTLER, R.A., Linkage disequilibrium in -two-locus, finite, random mating models without selection or mutation, Theor. Pop. Biol. 4 (1973), p.259-275. [6] MARUYAMA, T., Stochastic problems in population genetics, Lecture Notes in Biomath • .!2_, Springer·-Verlag, Berlin (1977). [7] MATKOWSKY, B.J. & Z. SCHUSS, The exit problem for randomly perturbed ·dynamical systems, SIAM J. Appl. Math. 33 (1977), p.365-382. [8] OHTA, T. & M. KIMURA, Linkage disequilibrium due to random genetic d:Pift, Genet. Res. Camb. _!l (1969), p.47-55. [9] PROTTER, M.H. & H.F. lvEINBERGER, Ma:x:imum principles in differential equations, Prentice Hall, Englewood Cliffs, New Jersey (1967). 

On a class of elliptic singular perturbations with applications in population genetics : (preprint)

Abstract

Similar works

Full text

Available Versions

NARCIS

CWI's Institutional Repository