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1 This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research education use, including for instruction at the authors institution sharing with colleagues. Other uses, including reproduction distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving manuscript policies are encouraged to visit:
2 Theoretical Population Biology 74 (008) 6 3 Contents lists available at ScienceDirect Theoretical Population Biology journal homepage: Joint stationary moments of a two-isl diffusion model of population subdivision Amir R.R. Kermany a,, Xiaowen Zhou a, Donal A. Hickey b a Department of Mathematics Statistics, Concordia University, Montreal, QC H3G M8, Canada b Department of Biology, Concordia University, Montreal, QC H4B R6, Canada a r t i c l e i n f o a b s t r a c t Article history: Received 7 May 008 Available online 6 July 008 Keywords: Population subdivision Diffusion process Joint stationary moments Itô formula Reversibility An expression for joint stationary moments of a diffusion approximation to a generalized Wright Fisher model, corresponding to two finite populations of equal sizes, with migration mutation, is derived. This gives a complete description of the stationary distribution of allele frequencies in the balance between migration, mutation genetic drift. We derive the sampling formula in terms of the joint stationary moments, we also prove that the diffusion process corresponding to this model of population division is not reversible. 008 Elsevier Inc. All rights reserved.. Introduction Many studies have been carried out to characterize the stationary distribution of allele frequencies in a single panmictic population (Wright, 93, 969; Ethier Kurtz, 99; Ethier, 98; Watterson, 976; Ewens, 004). However, studies on the stationary states of alleles in a substructured population have been more limited. For many species geographic barriers, as well as the spatial distribution of individuals, impose population subdivisions that must be taken into account in mathematical models of the evolution of a population. Models of population subdivision include isolation by distance (Wright, 943), the isl model (Wright, 93) the stepping-stone model (Kimura Weiss, 964; Weiss Kimura, 965; Malécot, 966). Shiga Uchiyama (986) proved that the diffusion approximation of the stepping stone model has a unique stationary distribution (see also Itatsu (989, 987) for the results on discrete time Markov model). Fu et al. (003) presents the exact form of the first second moments of the stationary distribution of the Markov chain corresponding to a subdivided population; (see also Nagylaki (980, 000) Nagylaki Lou (007) for the results on the strong weak migration limits of a diffusion model of subdivided populations.) Corresponding author. addresses: arrajike@alcor.concordia.ca (A.R.R. Kermany), xzhou@mathstat.concordia.ca (X. Zhou), donal.hickey@concordia.ca (D.A. Hickey). URL: akermany (A.R.R. Kermany). While most of the studies on the stationary distribution of a subdivided population are focused on characterization of some statistics of this distribution, there is no complete description of the stationary distribution of alleles for arbitrary values of mutation migration rates. Our aim in this paper is to give a detailed description of the joint stationary distribution in terms of the joint moments of the allele frequencies in a two isl model. Throughout this paper we assume that there are only two alleles on the locus under study. We assume that there is a gene flow between the two sub-populations, that the alleles are selectively neutral. Denoting K τ (i) with 0 K τ (i) N for the number of genes of the allele A in isl i at generation τ, where τ is a non-negative integer, the state of the two-isl system is uniquely identified by the vector X (N) τ ( K () τ N, K () τ N ). We approximate the generalized Wright Fisher Markov model of the allele frequencies in two isls, by a two-dimensional diffusion process, X(t) (X (t), X (t)) which is the solution to a system of two stochastic differential equations. In order to characterize the stationary distribution of the diffusion process, we proceed to identify the joint moments of X(t) (X (t), X (t)) under the stationary distribution µ, i.e. we want to find an expression for E µ X i (t)x j (t), i, j Z+. As an application, we find the sampling formula for the number of genes of the allele A in a sample taken from both isls. We also prove that the diffusion process corresponding to the two-isl model is not reversible /$ see front matter 008 Elsevier Inc. All rights reserved. doi:0.06/j.tpb
3 A.R.R. Kermany et al. / Theoretical Population Biology 74 (008) The diffusion model Consider a haploid population that is divided into two isls of equal size N. We assume that the rate of mutation is independent of the isl, that the mutation rate from the allele A to the allele A is u (N) with the reverse mutation rate u (N). In addition, we assume that there is a gene flow (migration) between the two isls with equal migration rates denoted by θ (N) > 0. Assuming that mutation takes effect prior to the stage at which adults migrate from one population to the other, the expected frequency of adults carrying allele A in isl i, prior to the sampling stage, is given by θ (N) u (N) + K (j) t N ( u(n) ) ( θ (N) ) u (N) + K (i) t N ( u(n) ), i j {, }, where u (N) denotes the total rate of mutation u (N) + u (N). Let us assume that all the rates are of order of /N, i.e. u (N) u N + O(/N), θ (N) θ N + O(/N), u(n) u N + O(/N) where u, u, θ O(). If we measure time in units of N generations, then as N, the evolution of X τ (N) can be approximated by a two dimensional diffusion process X(t) (X (t), X (t)), taking values in 0, : 0, 0, (Shiga Uchiyama, 986). The generator of the limiting diffusion process can be written as G : a (x) + a (x) + b (x) + b (x) x x x x in which b i (x) u ux i + θ ( x j x i ) a i (x) x i ( x i ); i j {, }, where u : u + u.. The joint stationary moments.. The system of equations for the joint stationary moments The process X is the solution to the system of stochastic differential equations given by dx (t) u ux (t) + θ (X (t) X (t)) dt + X (t)( X (t))db (t), () dx (t) u ux (t) + θ (X (t) X (t)) dt + X (t)( X (t))db (t), () where B (t) B (t) are independent Brownian motions. By Itô s formula for any f C ( 0, ), i.e. all twice continuous functions on 0,, we have df (X (t), X (t)) i + f (X (t), X (t)) dx i (t) X i (t) i,j f (X (t), X (t)) d X i, X j t, (3) X i (t) X j (t) where X i, X j t denotes the quadratic variation between X i X j, such that d X i, X j t X i (t)x j (t) ( X i (t)) ( X j (t) ) δ dt, where δ denotes Dirac s delta function. Let us denote by M mk the joint moments of the stationary distribution M mk : E µ X m (t)x k m (t). (4) Applying Itô s formula to the function f (x, y) x m y k m, taking expectation with respect to the stationary measure µ, we get the following system of equations for the joint moments of the stationary distribution mm m,k + (k m)m m+,k C mk M mk + a m M m,k + a k m M m,k 0, (5) where C mk : m + (k m) θ a m : m (m + u ), θ ( ) u + k + θ for all integers m k. For more details on deriving (5) see Appendix A. Our goal is to solve the system of equations in (5) for all m, k Z +, which provides the joint moments of the stationary distribution µ. But we first recover the following result on the support of X i (t). Proposition. For u i, θ > 0, i, we have P µ {X i (t) 0 or } 0, i,. Proof. Take k m, then by (5), 0 a m M m,m C mm M mm + mm m,m a m M m,m C mm M mm + mm m,m. It follows that M mm a m + m C mm M m,m (θ + u ) + m (θ + u) + m M m,m. By induction, m (θ + u ) + i M mm (θ + u) + i M. i Clearly, M mm 0 as m. So, P µ {X (t) } 0. The desired result follows from symmetry... Joint stationary moments of the two-isl model In this section we proceed to find the expression for M mk by solving the system of equations in (5). Define the (k + )-dimensional column vectors M k : (M 0k, M k,..., M kk ) T ; k Z. In addition let the (k + ) (k + ) matrix B k the (k + ) k matrix A k be B k : iδ i,j + δ (k i)δ i+,j ; i, j 0,,... k (7) (6)
4 8 A.R.R. Kermany et al. / Theoretical Population Biology 74 (008) 6 3 A k : a k i δ + a i δ i,j ; i 0,,..., k; j 0,,..., k. For any non-negative integer k let k : {0,,..., k} for any C k let us define A C : {A : A C, x, y A, x y : x y }, Ψ k (C) : A A C ( ) A i A (k i)(i + ) C i+,k with the convention Ψ k ( ) :, where is defined in (6) A denotes the number of elements in A. ( ) Proposition. Write B k β (k), for the inverse matrix of B k. For 0 i, j k, β (k) is given by (k i)! Ψ k (k \ {i, i,..., j}), i < j; (k j)! j Ψ k (k) C lk β (k) li : i! Ψ k (k \ {j, j,..., i}), i j. j! i Ψ k (k) C lk lj Proof. See Appendix B for the proof. Now, let us define the (k + ) k matrix Φ k : B k A k (9) where B k is given by Proposition. The following proposition gives the solution to the system of equations in (5). Proposition 3. For any k > 0, we have M k Φ k Φ k Φ where Φ k is defined by (9) Proof. It is easy to see that the system of equations in (5) can be written in the matrix form as B k M k A k M k, where the matrices B k A k are defined by (7) (8). Then M k Φ k M k. Since M 0, we have M k Φ k Φ k... Φ. Clearly knowing matrix Φ k for all integers k, we will be able to express all the moments of measure µ in terms of mutation migration rates. See Appendix C for some examples. The mean, variance covariance of this distribution are E µ X E µ X u u, Var µ X Var µ X (θ + u)(u u )u u ( u + 4θu + u + θ ) Cov µ X, X θ(u u )u u ( u + 4θu + u + θ ). It is worth noticing that the variance covariance of this distribution are equal to the limit of the variance covariance of the stationary distribution of the Markov chain model as described in Fu et al. (003) for the special case of two subpopulations as N. (8) 3. Some applications 3.. Sampling formula Suppose that the process X(t) (X (t), X (t)) has reached its stationary distribution µ. Take a simple rom sample of size m from isl one, an independent simple rom sample of size n from isl two. Denote the respective number of genes of allelic type A in the samples from isls by ξ η. Let Y ki if the ith sample from isl k has allele A Y ki 0 otherwise. Then m E µ ξ E µ Y i mm 0, i ( ) m E µ ξ E µ Y i mm 0 + m(m )M 0 i m,n E µ ξη E µ Y i Y j mnm. Therefore, E µ ξ mu u, i,j Var µ ξ mu (u u )m(u + θ) + u(u + θ) u ( u + 4θu + u + θ ) Cov µ ξ, η mnθu (u u ) u ( u + 4θu + u + θ ). Where we used values of M 0, M 0 M given in Eqs. (C.) (C.). Moreover, for 0 i m, 0 j n, we have Pr µ {ξ i, η j} ( m ) ( ) n E µ X i i j (t)( X (t)) m i X j (t)( X (t)) n j. Exping the terms in the above equation, we have Pr µ {ξ i, η j} ( m ) ( ) n m i n j ( ) ( ) m i n j E µ i j p q p0 q0 ( ) m i+n j (p+q) X m p X n q ( ) m i+n j ( m i ) ( n j ) m i n j ( ) ( ) m i n j p0 q0 ( ) p+q M m p,m+n (p+q). (0) This equation describes the probability distribution of genes of allele A, in a sample taken from both isls, in terms of the joint stationary moments. Unfortunately because of the complex form of the joint moments, an explicit form of the sampling formula in terms of the mutation migration rates is very difficult to obtain. Fig. shows the probability distribution of the number of genes of allelic type A in a sample of size 30 taken from one of the two isls, where we used Eq. (0) to calculate the probabilities. The results are compared to a sample that is drawn from a population with Beta(u, u ) distribution, corresponding to the stationary distribution of a population with no migration, Beta(θu /u, θ( u /u)) distribution corresponding to the Wright s isl model (Wright, 93). p q
5 A.R.R. Kermany et al. / Theoretical Population Biology 74 (008) GX (t) (θ + u )X (t) + (u + ) X (t) + θx (t)x (t), then by () we have E µ θx 3 (t) + u X (t) + (θ + u + )X (t)x (t) θx (t)x (t) (u + ) X (t)x (t) 0 or equivalently, θm 33 + u M + (θ + u + )M 3 θm 3 (u + ) M 0. (3) Applying the same argument on f (x, x ) x x, g(x, x ) x it follows that Fig.. Plot of probabilities of observing i genes of allelic type A in a sample of size 30 taken from an isl with N 0 5 u (N) u (N) Each plot corresponds to a different stationary distribution: (- -): samples are drawn from a population under stationary distribution corresponding to the two-isl model. The probabilities are calculated using Eq. (0) with migration rate equal (- -) The distribution of alleles in the sample assuming a Beta distribution of the Wright s Isl model with θ (- -) The distribution of alleles in the sample assuming a Beta distribution, corresponding to a single population without migration. 3.. Reversibility of the process In population genetics it is important to have a retrospective view of the process, i.e. knowing the current distribution we want to get information about the past states of the system. If a process is reversible, then the backward forward processes have the same law. For example the one dimensional limit diffusion of a panmictic population is a reversible process (Karlin Taylor, 98, Ch. 5, Sec. 3). However as we show below, the diffusion approximation to the generalized Wright Fisher model for a twoisl model is not reversible. The process X(t) (X (t), X (t)), with the stationary distribution π is reversible if f, g C ( 0, ) E π (f (X(t))Gg(X(t))) E π (g(x(t))gf (X(t))), () where G is the generator for the process (Jiang et al., 004, ch. 3, sec. 3). From () () we have f C (0, ), Gf (x, x ) u (θ + u)x + θx x f (x, x ) + u (θ + u)x + θx x f (x, x ) + x ( x ) + x ( x ) x x f (x, x ) f (x, x ). Now we state our result regarding the reversibility of the twoisl model. Theorem 4. Suppose that θ, u, u > 0, then the two-isl model is not reversible. Proof. The proof is by contradiction. Assume that X(t) is reversible. Then from () we have E µ X (t)gx (t) X (t)gx (t) 0, t 0. () Since GX (t) u + θx (t) (θ + u)x (t), θm 33 + u M (θ + u)m 3 0. (4) Since M 3 M 3 from (3) (4), then (u + )M 3 (u + )M 0. Now by replacing for M 3, M after simplification, the above equation reduces to θ (u u ) (u u ) u u ( u + 4θu + u + θ ) ( 3u + 4u + θ(4u + ) + ) 0. Since θ, u > u > 0, we must have u u/. On the other h, if we let f (x, x ) x 3 g(x, x ) x, direct calculation of Ef Gg ggf for u u/ results in E µ θx 4 (t) + ux 3(t) + (θ + u + 3)X 3 (t)x (t) ( u ) 3θX (t)x (t) 3 + X (t)x (t) θm 44 + u M 33 + (θ + u + 3)M 34 ( u ) 3θM M 3 3θu(θ + u) 0, D where D > 0 is given by D 4 ( u + 4θu + u + θ ) ( 8(4u + 3)θ + 4(u(6u + ) + 9)θ + (u + )(u + 3)(4u + 3)). This contradicts the assumption of reversibility. 4. Discussion We have derived the expression for all the moments of the stationary distribution of a two isl model of population subdivision. This was done through applying the Itô formula corresponding to the two dimensional diffusion process to the function f (x, y) x m y n solving the resulting system of equations for all the moments. Using expressions for the moments of the stationary distribution, we derived a formula for the joint probability of the number of genes of allele A in a rom sample taken from the two isls. We also proved that for all u, u, θ > 0 this process is not reversible. In this paper, we only considered the model with equal population sizes. However, under certain conditions on the ratio of population sizes we can find the diffusion approximation to the system of two populations of different size with migration between the two. Applying the same method we can find the joint stationary moments of the more general model.
6 30 A.R.R. Kermany et al. / Theoretical Population Biology 74 (008) 6 3 Acknowledgments We would like to thank Sabin Lessard anonymous reviewers for their constructive comments on an earlier version of this paper. This work was supported by an NSERC Discovery Grant. Appendix A. Derivation of Eq. (5) Let us consider f (x, y) x m y n. Then by (3) ( d X m (t)x n (t)) mx m (t)x n (t)dx (t) + nx m n (t)x (t)dx (t) m(m ) + X m X n (t)x (t) ( X (t)) dt By () () we have n(n ) + d ( X m (t)x n (t)) mx m (t)x n (t)b (X(t)) dt X m (t)x n (t)x (t) ( X (t)) dt. + nx m n (t)x (t)b (X(t)) dt m(m ) + X m (t)x n X (t) ( X (t)) dt n(n ) + X m n (t)x (t)x (t) ( X (t)) + dm t, where M t is a martingale. After some algebra we further have ( { d X m (t)x n (t)) m(m ) mu + X m (t)x n (t) n(n ) + nu + X m n (t)x (t) m(m ) + n(n ) (m + n)(θ + u) + X m (t)x n (t) } + mθx m (t)x n+ (t) + nθx m+ (t)x n (t) dt + dm t. (A.) Note that since µ is the stationary distribution of the two-isl model, E µ X m (t)x n (t) is independent of t, therefore de µ X m (t)x n (t) 0. If we take expectation with respect to µ from both sides of Eq. (A.), divide both sides of the equation by θ > 0 we obtain the following equation mm m,k + (k m)m m+,k C mk M mk + a m M m,k + a k m M m,k 0, where M mk is as defined in (4) ( ) C : i + (j i) (u + u ) + j +, θ θ a i : i (i + u ) θ for all integers i j. Appendix B. Proof for Proposition Now we want to show that B k β (k). Let us define A k to be the matrix with components β (k), i.e. A k β (k). We proceed by showing that B k A k I k+ A k B k where I k+ is the (k + ) (k + ) identity matrix. Let k : B k A k. Then k pq k B pj A jq j0 k ( pδ p,j + C pk δ pj (k p)δ p+,j )β jq j0 pβ p,q + C pk β pq (k p)β p+,q. Our goal is to show that pq δ pq. We first prove the following lemma. Lemma 5. For any nonnegative integer k any q C k, we have A C\{q} A C \ A (q) C\{q,q,q+}, where A (q) C : {A : A B {q}, B A C }. (B.) Proof. For any A A C, if q A, then {q, q + } A A A (q) C\{q,q,q+} ; if q A, then A A C\{q}. Clearly, A C A (q) C\{q,q,q+} A C\{q}. Since A (q) C\{q,q,q+} A C\{q} are disjoint, (B.) follows readily. The next result follows from Lemma 5 the definition of Ψ k. Corollary 6. For any non-negative integer k any q C such that q C k, we have Ψ k (C \ {q}) Ψ k (C) + or equivalently Ψ k (C \ {q}) Ψ k (C) (k q)(q + ) C qk C q+,k Ψ k (C \ {q, q, q + }), (k q)(q + ) C qk C q+,k Ψ k (C \ {q, q, q + }). (B.) In order to prove Proposition we are going to consider three different cases: p 0, 0 < p < k p k. Case one, p 0: For p 0 we further consider four cases for q: i.e. q 0, q, < q < k, q k. If q 0, then we have k 00 C 0k β 00 kβ 0 Ψ (k \ {0}) Ψ (k) Ψ (k), k Ψ (k \ {0, }) C 0k C k Ψ (k) ( Ψ (k \ {0}) k C 0k C k Ψ (k \ {0, }) where we have used (B.) for the second to the last equation. If q, then k 0 C 0k β 0 kβ kψ (k \ {0, }) C k Ψ (k) 0. kψ (k \ {0, }) C k Ψ (k) )
7 A.R.R. Kermany et al. / Theoretical Population Biology 74 (008) If < q < k, then k 0q C 0k β 0q kβ q k! (k q)! q i k! (k q)! q 0. If q k, we then have i k 0k C 0k β 0k kβ k k! k Ψ (k) i 0. Ψ (k \ {0,..., q}) Ψ (k) Ψ (k \ {0,..., q}) Ψ (k) k! k Ψ (k) i Case two, 0 < p < k: Let us first assume that q < p < k then, q p we have p(p )! pβ p,q q! p C pkp! C pk β pq q! p Ψ (k \ {q,..., p }), Ψ (k) Ψ (k \ {q,..., p}) Ψ (k) (k p)(p + )! (k p)β p+,q q! p+ Ψ (k \ {q,..., p + }). Ψ (k) For simplicity in notation we denote C : k \ {q,..., p }. Then k pq Ψ (k) p! Ψ (C) + q! p p! q! p (k p)(p + )! Ψ (C \ {p, p + }) q! p+ p! Ψ (k)q! p But by (B.) we have Ψ (C \ {p}) Ψ (C) Ψ (C \ {p}) Ψ (C) (k p)(p + ) C pk C p+,k Ψ (C \ {p, p + }) Ψ (C \ {p}). (k p)(p + ) C pk C p+,k Ψ (C \ {p, p, p + }). Note that p C : k \ {q,..., p }. Then C \ {p, p, p + } C \ {p, p + } k pq 0. Case three, p q: Then we have k pp pβ p,p + C pk β pp (k p)β p+,p p(k p + ) Ψ (k \ {p, p, p}) Ψ (k) C p,k C pk + Ψ (k \ {p, p}) (k p)(p + ) Ψ (k \ {p, p, p + }). C p,k C p+,k By (B.) we have (k p + )p Ψ (k \ {p, p, p}) C p,k C p,k Ψ (k \ {p }) Ψ (k). In addition, since k\{p, p} (k\{p })\{p}, by Corollary 6 we have (k p)(p + ) Ψ (k \ {p, p}) Ψ (k \ {p }) + C p,k C p+,k Ψ ((k \ {p }) \ {p, p, p + }) (k p)(p + ) Ψ (k \ {p }) + Ψ (k \ {p, p, p + }). C pk C p+,k Putting these together we have k pp. For the case 0 < p < q < k, the same argument as the case 0 < q < p < k gives k pq 0. For p k, similar to the case p 0 we can show that k kq if only if q k. Therefore k pq 0 for p q k pq 0 for p q; i.e. k pq δ pq. Hence, the matrix k B k A k is the identity matrix. The proof for A k B k I k+ is similar. Appendix C. Some examples In this section we give some particular values of the vector M k for k,, 3, for the general case in which the backward forward mutations have different values. Since the expressions become more complicated for higher values of k, we just consider the special case of u u for the case of k 4. Define u : u +u. Then M M ( u u, u ) T, (C.) u u uu(u + ) + θ(4u + ) (M 0, M, M ) T, where M 0 M M 0 u(u + ) + θ(4u + ), M u (u + ) + θ(4u + ). For k 3 we have M 3 u um 3 (M 03, M 3, M 3, M 33 ) T where M 03 M 33, M 3 M 3 m 3 θ (u + )(4u + ) + θ ( 0u 3 + 7u + 0u + ) + u(u + )(u + )(3u + ), M 03 θ (u + ) (4u + ) + θ 6u + 6u + + u (4u + ) (5u + 4) + u(3u + ) (u + ) (u + ), M 3 θ (u + ) (4u + ) + θ 4u + 8u + + u (4u + ) (5u + ) + u (u + )(3u + ) (u + ). (C.)
8 3 A.R.R. Kermany et al. / Theoretical Population Biology 74 (008) 6 3 Now we assume that u u/ then for k 4 we have M 4 4m 4 (M 04, M 4, M 4, M 34, M 44 ) T, where M 04 M 44, M 4 M 34 m 4 8θ 3 ( 6u + 6u + 3 ) + 4θ ( 64u 3 + 0u + 64u + 9 ) + θ ( 60u u u + 08u + 9 ) + u(u + ) ( 8u + 8u + 9 ), M 04 6θ 3 ( u + 5u + 3 ) + θ ( 64u u + 50u + 7 ) + θ ( 40u u u + 50u + 8 ) + u ( 8u u 3 + 0u + 75u + 8 ), M 4 3θ 3 ( u + 5u + 3 ) + 4θ ( 3u u + 3u + 36 ) + θ ( 80u u u + 34u + 36 ) + u ( 6u u 3 + 4u + 8u + 8 ) M 4 6 ( u + 5u + 3 ) θ 3 + ( 64u u + 8u + 7 ) θ + θ ( 40u 4 + 5u u + 08u + 8 ) + u(u + ) ( 8u + 8u + 9 ). A Mathematica v.6 notebook for all the calculations that have been carried out in this paper is available upon request. References Ethier, S.N., 98. A class of infinite-dimensional diffusions occurring in population genetics. Indiana University Mathematics Journal 30 (6), Ethier, S.N., Kurtz, T.G., 99. On the stationary distribution of the neutral diffusion model in population genetics. The Annals of Applied Probability (), Ewens, W.J., 004. Mathematical Population Genetics I. Theoretical Introduction, nd edition. In: Interdisciplinary Applied Mathematics, Springer, Berlin. Fu, R., E Gelf, A., Holsinger, K.E., 003. Exact moment calculations for genetic models with migration, mutation, drift. Theoretical Population Biology 63 (3), Itatsu, S., 987. Equilibrium measures of the stepping stone model with selection in population genetics. In: Proceedings of a Workshop on Stochastic Methods in Biology. Springer-Verlag New York, Inc., New York, NY, USA, pp Itatsu, S., 989. Ergodic properties of the stepping stone model. Nagoya Mathematical Journal 4, Jiang, D.Q., Qian, M., Qian, M.P., 004. Mathematical Theory of Nonequilibrium Steady States: On the Frontier of Probability Dynamical Systems. In: Lecture Notes in Mathematics, Springer. Karlin, S., Taylor, H., 98. A Second Course in Stochastic Processes. Academic Press, New York. Kimura, M., Weiss, G.H., 964. The stepping stone model of population structure the decrease of genetic correlation with distance. Genetics 49 (4), Malécot, G., 966. Identical loci relationship. In: Proceedings of the Fifth Berkeley Symposium on Mathematics. Statistics Probability 4, Nagylaki, T., 980. The strong-migration limit in geographically structured populations. Journal of Mathematical Biology 9 (), 0 4. Nagylaki, T., 000. Geographical invariance the strong-migration limit in subdivided populations. Journal of Mathematical Biology 4 (), 3 4. Nagylaki, T., Lou, Y., 007. Evolution under multiallelic migration selection models. Theoretical Population Biology 7 (), 40. Shiga, T., Uchiyama, K., 986. Stationary states their stability of the stepping stone model involving mutation selection. Probability Theory Related Fields 73 (), Watterson, G.A., 976. The stationary distribution of the infinitely-many neutral alleles diffusion model. Journal of Applied Probability 3 (4), Weiss, G.H., Kimura, M., 965. A mathematical analysis of the stepping stone model of genetic correlation. Journal of Applied Probability (), Wright, S., 93. Evolution in mendelian populations. Genetics 6, Wright, S., 943. Isolation by distance. Genetics 8 (), Wright, S., 969. Evolution the Genetics of Populations: The Theory of Gene Frequencies, vol.. University of Chicago Press, Chicago.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research education use, including for instruction at the authors institution
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
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