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text: S.

text: Kaune,

text: K.

text: Pussep,

text: C.

text: Leng,

text: A.

text: Kovacevic,

text: G.

text: Tyson,

text: and

text: R.

text: Stein-

text: metz.

text: Modelling

text: the

text: internet

text: delay

text: space

text: based

text: on

text: geographical

text: loca-

text: tions.

text: In

text: Proc.

text: Conf.

text: Parallel,

text: Distributed

text: and

text: Network-based

text: Process-

text: ing,

text: pages

text: Feb

text: R.

text: Kohavi

text: and

text: R.

text: Longbotham.

text: Online

text: experiments:

text: Lessons

text: learned.

text: IEEE

text: Computer

text: Magazine,

text: F.

text: Li,

text: M.

text: Li,

text: R.

text: Lu,

text: H.

text: Wu,

text: M.

text: Claypool,

text: and

text: R.

text: Kinicki.

text: Measur-

text: ing

text: queue

text: capacities

text: of

text: IEEE

text: wireless

text: access

text: points.

text: In

text: Proc.

text: BroadNets,

text: pages

text: Sep

text: F.

text: F.-H.

text: Nah.

text: A

text: study

text: on

text: tolerable

text: waiting

text: time:

text: how

text: long

text: are

text: web

text: users

text: willing

text: to

text: wait?

text: Behaviour

text: Information

text: Technology,

text: J.

text: Padhye

text: and

text: H.

text: F.

text: Nielsen.

text: A

text: comparison

text: of

text: SPDY

text: and

text: HTTP

text: per-

text: formance.

text: Microsoft

text: Technical

text: Report

text: G.

text: Podjarny.

text: Not

text: as

text: spdy

text: as

text: you

text: thought.

text: technical/not-as-spdy-as-you-thought/,

text: C.

text: Severance.

text: JavaScript:

text: Designing

text: a

text: language

text: in

text: days.

text: IEEE

text: Computer

text: Magazine,

text: A.

text: Sheth,

text: S.

text: Nedevschi,

text: R.

text: Patra,

text: S.

text: Surana,

text: E.

text: Brewer,

text: and

text: L.

text: Sub-

text: ramanian.

text: Packet

text: loss

text: characterization

text: in

text: WiFi-based

text: long

text: distance

text: networks.

text: In

text: Proc.

text: INFOCOM,

text: pages

text: Y.

text: X.

text: Skadberg

text: and

text: J.

text: R.

text: Kimmel.

text: Visitors’

text: ﬂow

text: experience

text: while

text: brows-

text: ing

text: a

text: web

text: site:

text: its

text: measurement,

text: contributing

text: factors

text: and

text: consequences.

text: Computers

text: in

text: human

text: behavior,

text: P.

text: Sun,

text: M.

text: Yu,

text: M.

text: J.

text: Freedman,

text: and

text: J.

text: Rexford.

text: Identifying

text: perfor-

text: mance

text: bottlenecks

text: in

text: CDNs

text: through

text: TCP-level

text: monitoring.

text: In

text: SIG-

text: COMM

text: workshop

text: on

text: Measurements

text: Up

text: the

text: Stack,

text: pages

text: ACM,

text: S.

text: Sundaresan,

text: W.

text: de

text: Donato,

text: N.

text: Feamster,

text: R.

text: Teixeira,

text: S.

text: Crawford,

text: and

text: A.

text: Pescap`e.

text: Broadband

text: internet

text: performance:

text: a

text: view

text: from

text: the

text: gateway.

text: In

text: Proc.

text: SIGCOMM,

text: pages

text: ACM,

text: B.

text: Thomas,

text: R.

text: Jurdak,

text: and

text: I.

text: Atkinson.

text: SPDYing

text: up

text: the

text: web.

text: Com-

text: mun.

text: of

text: ACM,

text: Dec

text: R.

text: Trace,

text: A.

text: Foresti,

text: S.

text: Singhal,

text: O.

text: Mazahir,

text: H.

text: F.

text: Nielsen,

text: B.

text: Raymor,

text: R.

text: Rao,

text: and

text: G.

text: Montenegro.

text: HTTP

text: Speed+Mobility.

text: Jun

text: IETF

text: Network

text: Working

text: Group.


text: – 17 – where f is the symmetric normalized velocity distribution with respect to the center of the galaxy, β is the asymmetry parameter and X is given, ?), by: X2 (cid:16)= −1 (cid:112) (cid:17) (1 β)s2 c √ (cid:16) (cid:112) − − 2 − 3δ cos α cos Φ 2 1 ξ2 sin φ + 2δ sin α sin Φ − − (cid:0) (cid:1)(cid:1) β 2 1 ξ2 cos φ (δ cos α + 2) sin Θ+ √ − (cid:0) (cid:0) 2 (cid:1)(cid:1) δ cos Θ 2 cos Φ sin α 3 cos α sin Φ + √ (cid:16) (cid:112) − 2 2ξy + (δ cos α + 2) cos Θ + δ sin Θ 2 cos Φ sin α 3 cos α sin Φ + − − (cid:0) (cid:1)(cid:1) 2 1 ξ2 cos φ + (δ cos α + 2) sin Θ+ √ − 2 δ cos Θ 3 cos α sin Φ 2 cos Φ sin α .(35)The direction of the WIMP velocity is speciﬁed by ξ = cos θ and φ. The direction of observation is speciﬁed by the angles Θ and Φ. 6. Discussion In the present work we studied how one can construct the velocity distribution in the Eddington approach starting from dark matter density proﬁles. By modifying the distribution function by suitable angular momentum functions one can obtain asymmetric velocity distributions as well. We clariﬁed some of the issues involved in this approach by considering a simple model which can yield analytic solutions. Results of realistic calculations for dark matter searches, employing the present technique and using realistic density proﬁles ?), ?), ?), will appear elsewhere ?). Acknowledgments: This research has been partially supported by the European Union Social Fund (ESF) and the Greek national funds through the program THALIS of the Hrllenic Open University: Development and Applications of Novel Instrumentation and ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: where

label: equation

text: f

text: is

text: the

text: symmetric

text: normalized

text: velocity

text: distribution

text: with

text: respect

text: to

text: the

text: center

text: of

text: the

text: galaxy,

text: β

text: is

text: the

text: asymmetry

text: parameter

text: and

text: X

text: is

text: given,

text: by:

text: −1

text: c

text: 3δ

text: cos

text: α

text: cos

text: Φ

text: ξ2

text: sin

text: φ

text: 2δ

text: sin

text: α

text: sin

text: Φ

text: β

text: ξ2

text: cos

text: φ

text: (δ

text: cos

text: α

text: sin

text: Θ+

text: δ

text: cos

text: Θ

text: cos

text: Φ

text: sin

text: α

text: cos

text: α

text: sin

text: Φ

text: (δ

text: cos

text: α

text: cos

text: Θ

text: δ

text: sin

text: Θ

text: cos

text: Φ

text: sin

text: α

text: cos

text: α

text: sin

text: Φ

text: ξ2

text: cos

text: φ

text: (δ

text: cos

text: α

text: sin

text: Θ+

text: δ

text: cos

text: Θ

text: cos

text: α

text: sin

text: Φ

text: cos

text: Φ

text: sin

text: α

text: direction

text: of

text: the

text: WIMP

text: velocity

text: is

text: speciﬁed

text: by

text: ξ

text: cos

text: θ

text: and

text: φ.

text: The

text: direction

text: of

text: observation

text: is

text: speciﬁed

text: by

text: the

text: angles

text: Θ

text: and

text: Φ.

text: Discussion

text: In

text: the

text: present

text: work

text: we

text: studied

text: how

text: one

text: can

text: construct

text: the

text: velocity

text: distribution

text: in

text: the

text: Eddington

text: approach

text: starting

text: from

text: dark

text: matter

text: density

text: proﬁles.

text: By

text: modifying

text: the

text: distribution

text: function

text: by

text: suitable

text: angular

text: momentum

text: functions

text: one

text: can

text: obtain

text: asymmetric

text: velocity

text: distributions

text: as

text: well.

text: We

text: clariﬁed

text: some

text: of

text: the

text: issues

text: involved

text: in

text: this

text: approach

text: by

text: considering

text: a

text: simple

text: model

text: which

text: can

text: yield

text: analytic

text: solutions.

text: Results

text: of

text: realistic

text: calculations

text: for

text: dark

text: matter

text: searches,

text: employing

text: the

text: present

text: technique

text: and

text: using

text: realistic

text: density

text: proﬁles

text: will

text: appear

text: elsewhere

text: Acknowledgments:

text: This

text: research

text: has

text: been

text: partially

text: supported

text: by

text: the

text: European

text: Union

text: Social

text: Fund

text: (ESF)

text: and

text: the

text: Greek

text: national

text: funds

text: through

text: the

text: program

text: THALIS

text: of

text: the

text: Hrllenic

text: Open

text: University:

text: Development

text: and

text: Applications

text: of

text: Novel

text: Instrumentation

text: and

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: – 19 – 0.6 0.5 0.4 → 0.3 ) (cid:48)y 0.2 , β ( f 0.1 2 (cid:48)y π 4 1 2 3 4 (cid:48) −→ y Fig. 6.—: We show the angular average of the properly normalized velocity distribution for values of the asymmetry parameter β = (0.0.1, 0.2, 0.3, 0.4, 0.5). In the plots β is increasing from right to left). The results depend on the value of the potential in our vicinity. Here the value of ξ(xs) = 10 was adopted. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: β

text: f

text: π

text: y

text: Fig.

text: We

text: show

text: the

text: angular

text: average

text: of

text: the

text: properly

text: normalized

text: velocity

text: distribution

text: for

text: values

text: of

text: the

text: asymmetry

text: parameter

text: β

text: In

text: the

text: plots

text: β

text: is

text: increasing

text: from

text: right

text: to

text: left).

text: The

text: results

text: depend

text: on

text: the

text: value

text: of

text: the

text: potential

text: in

text: our

text: vicinity.

text: Here

text: the

text: value

text: of

text: ξ(xs)

text: was

text: adopted.

text: ##LTFigure##

label: figure

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: 12 (a) (b) Re=100 0 0.2 0.4 x 0.6 0.8 1 1 0.4 0.8 0.2 y 0.6 0 v(x,0.5) 0.4 -0.2 0.2 -0.4 Ghia Present 0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 u(0.5,y) (c) (d) Fig. 3. Lid-driven cavity ﬂow with Re = 100 : (a) stream function; (b) pressure; (c) vorticity; (d) centerline velocities u and v. Results from Ghia et al. [?] are compared with current numerical solutions. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: (b)

text: x

text: y

text: Ghia

text: Present

text: (c)

text: (d)

text: Fig.

text: Lid-driven

label: caption

text: cavity

text: ﬂow

text: with

text: Re

text: (a)

text: stream

text: function;

text: (b)

text: pressure;

text: (c)

text: vorticity;

text: (d)

text: centerline

text: velocities

text: u

text: and

text: v.

text: Results

text: from

text: Ghia

text: et

text: al.

text: are

text: compared

text: with

text: current

text: numerical

text: solutions.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 where ∆ the Laplace operat(cid:16)or in ﬁn(cid:17)ite diﬀerence type, the I, J component of the h matrix, Ψ[α,β] = Ψ[α,β] (x ) M×N J I IJ We introduce the virtual interpolation point for matrix D∗ corresponding to the discrete divergence operator. The virtual interpolation points for virtual staggered ∈ grid points for velocity ﬁeld and pressure at virtual interpolation point of node x T I are: − z+ = x + (h/2, 0), z− = x (h/2, 0), I,1 I I,1 I − z+ = x + (0, h/2), z− = x (0, h/2), I,2 I I,2 I and deﬁne the discrete divergence (cid:88) (cid:104) (cid:105) (cid:88) (cid:104) (cid:105) (D∗U) = I 1 N − 1 N − Ψ[0,0](z+ ) Ψ[0,0](z− ) u + Ψ[0,0](z+ ) Ψ[0,0](z− ) v . h J I,1 J I,1 J h J I,2 J I,2 J J=1 J=1 So we can write the numerical du(cid:32)al operator (cid:33)of divergence by numerical derivative matrix D, D Ψ[0,0] D = h M×N , D Ψ[0,0] h M×N 2M×N where D means ﬁnite diﬀerence operator. h Remark 2.2. By adopting periodic boundary condition, we can extend to whole plane. We employ the discrete divergence operator D∗ to deﬁne the discrete Laplace operator AU = D∗(DU) instead of stiﬀness matrix A. Instead of the gradient matrix GP of the pressure, we formulate the velocity equations by DP. But the replacement is simply for the convenience of analysis and the existence proof will hold for GP after considering projection error, too. Now we prove that the virtual point collocation scheme is stable for the Stokes ﬂow − (3.1) AU + DP = F, D∗U = 0, { } when the approximation node set x = R is suﬃciently dense. To be more I speciﬁc we introduce a deﬁnition. { } Definition 3.1. (Realization) The node set R = x , I = 1, ..., N realizes the { } I ∈ R set of virtual interpolation point T = y , J = 1, ..., M if for each U M there is ∈ R (cid:88)J u N such that N U = Ψ (y )u . J I J I I=1 ##LTLine## ##LTLine##

text: where

text: the

text: Laplace

text: in

text: diﬀerence

text: type,

text: the

text: I,

text: J

text: component

text: of

text: the

text: h

text: matrix,

text: Ψ[α,β]

text: Ψ[α,β]

text: (x

text: M×N

text: J

text: I

text: IJ

text: We

text: introduce

text: the

text: virtual

text: interpolation

text: point

text: for

text: matrix

text: D∗

text: corresponding

text: to

text: the

text: discrete

text: divergence

text: operator.

text: The

text: virtual

text: interpolation

text: points

text: for

text: virtual

text: staggered

text: grid

text: points

text: for

text: velocity

text: ﬁeld

text: and

text: pressure

text: at

text: virtual

text: interpolation

text: point

text: of

text: node

text: x

text: T

text: I

text: are:

text: z+

text: x

text: z−

text: x

text: I

text: I

text: z+

text: x

text: z−

text: x

text: I

text: I

text: and

text: deﬁne

text: the

text: discrete

text: divergence

text: (D∗U)

text: I

text: N

text: N

text: u

text: v

text: h

text: J

text: J

text: J

text: h

text: J

text: J

text: J

text: So

text: we

text: can

text: write

text: the

text: numerical

text: operator

text: divergence

text: by

text: numerical

text: derivative

text: matrix

text: D,

text: D

text: D

text: h

text: M×N

text: D

text: h

text: M×N

text: where

text: D

text: means

text: ﬁnite

text: diﬀerence

text: operator.

text: h

text: Remark

text: By

text: adopting

text: periodic

text: boundary

text: condition,

text: we

text: can

text: extend

text: to

text: whole

text: plane.

text: We

text: employ

text: the

text: discrete

text: divergence

text: operator

text: D∗

text: to

text: deﬁne

text: the

text: discrete

text: Laplace

text: operator

text: AU

text: D∗(DU)

text: instead

text: of

text: stiﬀness

text: matrix

text: A.

text: Instead

text: of

text: the

text: gradient

text: matrix

text: GP

text: of

text: the

text: pressure,

text: we

text: formulate

text: the

text: velocity

text: equations

text: by

text: DP.

text: But

text: the

text: replacement

text: is

text: simply

text: for

text: the

text: convenience

text: of

text: analysis

text: and

text: the

text: existence

text: proof

text: will

text: hold

text: for

text: GP

text: after

text: considering

text: projection

text: error,

text: too.

text: Now

text: we

text: prove

text: that

text: the

text: virtual

text: point

text: collocation

text: scheme

text: is

text: stable

text: for

text: the

text: Stokes

text: ﬂow

text: AU

text: DP

text: F,

text: D∗U

text: when

text: the

text: approximation

text: node

text: set

text: x

text: R

text: is

text: suﬃciently

text: dense.

text: To

text: be

text: more

text: I

text: speciﬁc

text: we

text: introduce

text: a

text: deﬁnition.

text: Definition

text: (Realization)

text: The

text: node

text: set

text: R

text: x

text: I

text: N

text: realizes

text: the

text: I

text: R

text: set

text: of

text: virtual

text: interpolation

text: point

text: T

text: y

text: J

text: M

text: if

text: for

text: each

text: U

text: M

text: there

text: is

text: R

text: u

text: N

text: such

text: that

text: N

text: U

text: Ψ

text: (y

text: )u

text: J

text: I

text: J

text: I

text: ##LTLine##

text: ##LTLine##


text: 9 We have that − − (cid:107)∇ (cid:107) Γv1,x1(x) = v1,x1(xI) + v1,x1x1(xI)(x1 − xI,1) + v1,x1x2(xI)(x2 − xI,2) + C(cid:107)∇2v(cid:107)CαO(h1+α) Γv2,x2(x) = v2,x2(xI) + v2,x2x2(xI)(x2 xI,2) + v2,x1x2(xI)(x1 xI,1) + C 2v CαO(h1+α), and we also have that, from divergence free condition, − − v (x )(x x ) + v (x )(x x ) = 0 1,x1x2 I 2 − I,2 2,x2x2 I 2 − I,2 v (x )(x x ) + v (x )(x x ) = 0. 2,x1x2 I 1 I,1 1,x1x1 I 1 I,1 ∇ Taking divergence of Γv and noting that (divv)(x ) = 0, we also have I (cid:107)∇ (cid:107) divΓv(xI) = 2v CαO(h1+α), and similarly from mean value theorem (cid:107)∇ (cid:107) (D∗Γv)I = 2v CαO(h1+α). ∈ Similarly, we suppose q C1,α and we have ∇ ∇ (cid:107)∇ (cid:107) ( Γq)(xI) = q(xI) + q CαO(hα), and from mean value theorem (cid:107) − ∇ (cid:107) ≤ (cid:107)∇ (cid:107) (DΓq)I Γq(xI) C q Cαhα. − ∇ ∈ Since ∆v(x ) + q(x ) = f(x ) and f Cα we have the error equation for virtual I I I interpolation method, − − − − (cid:107)∇ (cid:107) (cid:107)∇ (cid:107) (3.3) A(U Γv) + D(P Γq) = (F Γf) + ( 2v Cα + q Cα)O(hα), − (cid:107)∇ (cid:107) (cid:107) (cid:107) D∗(U Γv) = 2v CαO(h1+α) = f CαO(h1+α). where (U, P) are solution of the discrete Stokes equations (??). Theorem 3.4. We suppose that all the node sets satisfy the conditions in The- ≥ ∈ orem ??, and the reproducing degree m 2. We also assume that f Cα. We let × (v, q) the true solution in C2,α(Ω) C1,α(Ω) and (U, P) discrete solution. Then, there per per is an absolute constant C such that (cid:107) − (cid:107) (cid:107) − (cid:107) ≤ (cid:107) (cid:107) D(U Γv) + P Γq Chα−1 f . l2 l2 Cα(Ω) ∈ Proof. If we have f Cα, from Calderon-Zygmund theory of Stokes equations we ∈ ∈ have v C2,α and q C1,α. From Sobolev embedding we have (cid:107)∇ (cid:107) (cid:107)∇ (cid:107) ≤ (cid:107) (cid:107) 2v Cα + q Cα C f Cα, ∈ and in case f Cα, we have (cid:107) − (cid:107) ≤ (cid:107) (cid:107) F Γf ∞ Chα f . l C1,α

text: We

text: have

text: that

text: C

text: and

text: we

text: also

text: have

text: that,

text: from

text: divergence

text: free

text: condition,

text: v

text: (x

text: )(x

text: x

text: v

text: (x

text: )(x

text: x

text: I

text: I

text: v

text: (x

text: )(x

text: x

text: v

text: (x

text: )(x

text: x

text: I

text: I

text: Taking

text: divergence

text: of

text: Γv

text: and

text: noting

text: that

text: (divv)(x

text: we

text: also

text: have

text: I

text: divΓv(xI)

text: and

text: similarly

text: from

text: mean

text: value

text: theorem

text: (D∗Γv)I

text: Similarly,

text: we

text: suppose

text: q

text: and

text: we

text: have

text: Γq)(xI)

text: q(xI)

text: q

text: CαO(hα),

text: and

text: from

text: mean

text: value

text: theorem

text: (DΓq)I

text: Γq(xI)

text: C

text: q

text: Cαhα.

text: Since

text: ∆v(x

text: q(x

text: f(x

text: and

text: f

text: Cα

text: we

text: have

text: the

text: error

text: equation

text: for

text: virtual

text: I

text: I

text: I

text: interpolation

text: method,

text: A(U

text: Γv)

text: D(P

text: Γq)

text: (F

text: Γf)

text: Cα

text: q

text: Cα)O(hα),

text: D∗(U

text: Γv)

text: f

text: where

text: (U,

text: P)

text: are

text: solution

text: of

text: the

text: discrete

text: Stokes

text: equations

text: Theorem

text: We

text: suppose

text: that

text: all

text: the

text: node

text: sets

text: satisfy

text: the

text: conditions

text: in

text: The-

text: orem

text: and

text: the

text: reproducing

text: degree

text: m

text: We

text: also

text: assume

text: that

text: f

text: Cα.

text: We

text: let

text: (v,

text: q)

text: the

text: true

text: solution

text: in

text: and

text: (U,

text: P)

text: discrete

text: solution.

text: Then,

text: there

text: per

text: per

text: is

text: an

text: absolute

text: constant

text: C

text: such

text: that

text: D(U

text: Γv)

text: P

text: Γq

text: f

text: Cα(Ω)

text: Proof.

text: If

text: we

text: have

text: f

text: Cα,

text: from

text: Calderon-Zygmund

text: theory

text: of

text: Stokes

text: equations

text: we

text: have

text: v

text: and

text: q

text: From

text: Sobolev

text: embedding

text: we

text: have

text: Cα

text: q

text: Cα

text: C

text: f

text: Cα,

text: and

text: in

text: case

text: f

text: Cα,

text: we

text: have

text: F

text: Γf

text: Chα

text: f

text: l


text: 2 INTRODUCTION The increasing rate of habitat destruction and fragmentation poses a major threat to the persistence of many species [1, 2]. Connectivity among populations has been long recognised to play a key role in preventing extinction [3, 4]: immigration can lead to the recolonisation of previously extinct patches, thus promoting the long-term persistence of the network of populations, i.e. the metapopulation [3]. Also in the absence of population turnover, immigration can reduce the probability of local extinction by boosting numbers of inhabitants under adverse conditions, thus buﬀering local populations against demographic and environmental stochasticity. This reduction in local extinction probability was termed the “rescue eﬀect” by Brown & Kodric-Brown [5], and ecologists have accumulated a wealth of empirical evidence of its potential importance [e.g. 1, 2, 6, 7]. While it is intuitively obvious that the rescue eﬀect can play a pivotal role in determining the persistence of local populations as well as of the entire metapopulation, quantifying the link between local dynamics and the strength of the rescue eﬀect is a challenging endeavour. The classical Levins’ metapopulation model [8] considers patches as either occupied or empty, thus ignoring local population dynamics within patches. In an attempt to integrate the rescue eﬀect in Levins’ model, several authors have assumed that the extinction probability of a given patch is a function of the number of occupied patches [9, 10]. However, the shape of this function cannot be derived from ﬁrst principles, and has to be ﬁtted to the data for each species of interest. Thus, the lessons learnt from this approach are speciﬁc to the dataset in question. In order to investigate the factors that aﬀect the strength of the rescue eﬀect it is necessary to explicitly model the local stochastic population dynamics. Agent-based models can be used for this purpose [e.g. 11], but it is often diﬃcult to understand the mechanisms that determine the global metapopulation dynamics, and to quantify the inﬂuence of the diﬀerent factors characterising the local population demography. An analytical treatment of the metapopulation is much more desirable and informative, but it is not trivial. Lande et al. [12] were the ﬁrst to develop such an analytical model based on a diﬀusion approximation for the local population dynamics. Their approach characterises the metapopulation dynamics as a sequence of isolated, local extinction and recolonisation events, and relies on the assumption of very low migration rates. This assumption is appropriate when a large fraction of suitable patches is not occupied near demographic equilibrium. This scenario is at odds with the observation that the majority of suitable patches tend to remain occupied in many metapopulations [e.g. 13–17]. Keeling [18] took a diﬀerent approach, relying on a moment-closure technique, to investigate the persistence of stochastic metapopulations. This approach relies on knowing the distribution of population sizes, a problematic restriction if we are interested in investigating the dynamics of a metapopulation close to extinction, for which the shape of the distribution is likely to diﬀer from the corresponding shape near demographic equilibrium. In this paper, we develop an analytical framework, based on the identiﬁcation of the slow and fast modes underlying the metapopulation dynamics. Our approach allows us to investigate the link between local stochastic dynamics and the strength of the rescue eﬀect. We compare our results to spatially explicit simulations to explore the spatial scales at which our analytical results are relevant. While for this paper we focus on a model that describes the local dynamics in terms of a Ricker map [19], the approach can be applied to other models for population dynamics, formulated in terms of discrete non-overlapping generations or in terms of continuously occurring birth and death processes. METAPOPULATION MODEL Our model considers N patches connected by global dispersal, and we speciﬁcally focus on metapopulations consist- ing of a large number of patches. Each patch can be inhabited by a population that evolves in discrete, non-overlapping generations. Each generation has two phases, the ﬁrst corresponding to local recruitment and the second to dispersal to other populations. For biological realism, we use the model by Melbourne & Hastings [20] to represent local recruit- ment and density-dependent survival. This model is a stochastic version of the classical deterministic Ricker model Ricker [19] that incorporates the eﬀect of both demographic and environmental stochasticity within populations. In our model each individual has a Poisson-distributed number of progeny, with mean given by the recruitment rate R i for individual i. The individual recruitment rate R is itself random (independent between generations), giving rise to i demographic and environmental stochasticity. The environmental stochasticity determines a mean (environmental) recruitment rate, R , for the local population (R is gamma distributed with mean R and shape parameter k ). Given E E E the environmental state of the local population, individual recruitment rates R are independent gamma distributed i stochastic variables with mean R and shape parameter k (giving rise to demographic recruitment stochasticity). E D − Finally, each progeny survives (independently) to adulthood with density-dependent probability exp( αn), where n is the number of adults in the population and α regulates the strength of the density dependence. As shown by Melbourne & Hastings [20], the expected number of progeny surviving to adulthood is the same as in the classical Ricker dynamics (with recruitment R and density dependence α), but the variance of the distribution is strongly aﬀected by the amount of demographic and environmental recruitment stochasticity (deﬁned by the shape parameters k and k ). Putting it all together (following [20]) the probability of having, after the ﬁrst phase, j individuals in a E D

text: INTRODUCTION

label: section

text: The

text: increasing

text: rate

text: of

text: habitat

text: destruction

text: and

text: fragmentation

text: poses

text: a

text: major

text: threat

text: to

text: the

text: persistence

text: of

text: many

text: species

text: Connectivity

text: among

text: populations

text: has

text: been

text: long

text: recognised

text: to

text: play

text: a

text: key

text: role

text: in

text: preventing

text: extinction

text: immigration

text: can

text: lead

text: to

text: the

text: recolonisation

text: of

text: previously

text: extinct

text: patches,

text: thus

text: promoting

text: the

text: long-term

text: persistence

text: of

text: the

text: network

text: of

text: populations,

text: i.e.

text: the

text: metapopulation

text: Also

text: in

text: the

text: absence

text: of

text: population

text: turnover,

text: immigration

text: can

text: reduce

text: the

text: probability

text: of

text: local

text: extinction

text: by

text: boosting

text: numbers

text: of

text: inhabitants

text: under

text: adverse

text: conditions,

text: thus

text: buﬀering

text: local

text: populations

text: against

text: demographic

text: and

text: environmental

text: stochasticity.

text: This

text: reduction

text: in

text: local

text: extinction

text: probability

text: was

text: termed

text: the

text: “rescue

text: eﬀect”

text: by

text: Brown

text: Kodric-Brown

text: and

text: ecologists

text: have

text: accumulated

text: a

text: wealth

text: of

text: empirical

text: evidence

text: of

text: its

text: potential

text: importance

text: [e.g.

text: While

text: it

text: is

text: intuitively

text: obvious

text: that

text: the

text: rescue

text: eﬀect

text: can

text: play

text: a

text: pivotal

text: role

text: in

text: determining

text: the

text: persistence

text: of

text: local

text: populations

text: as

text: well

text: as

text: of

text: the

text: entire

text: metapopulation,

text: quantifying

text: the

text: link

text: between

text: local

text: dynamics

text: and

text: the

text: strength

text: of

text: the

text: rescue

text: eﬀect

text: is

text: a

text: challenging

text: endeavour.

text: The

text: classical

text: Levins’

text: metapopulation

text: model

text: considers

text: patches

text: as

text: either

text: occupied

text: or

text: empty,

text: thus

text: ignoring

text: local

text: population

text: dynamics

text: within

text: patches.

text: In

text: an

text: attempt

text: to

text: integrate

text: the

text: rescue

text: eﬀect

text: in

text: Levins’

text: model,

text: several

text: authors

text: have

text: assumed

text: that

text: the

text: extinction

text: probability

text: of

text: a

text: given

text: patch

text: is

text: a

text: function

text: of

text: the

text: number

text: of

text: occupied

text: patches

text: However,

text: the

text: shape

text: of

text: this

text: function

text: cannot

text: be

text: derived

text: from

text: ﬁrst

text: principles,

text: and

text: has

text: to

text: be

text: ﬁtted

text: to

text: the

text: data

text: for

text: each

text: species

text: of

text: interest.

text: Thus,

text: the

text: lessons

text: learnt

text: from

text: this

text: approach

text: are

text: speciﬁc

text: to

text: the

text: dataset

text: in

text: question.

text: In

text: order

text: to

text: investigate

text: the

text: factors

text: that

text: aﬀect

text: the

text: strength

text: of

text: the

text: rescue

text: eﬀect

text: it

text: is

text: necessary

text: to

text: explicitly

text: model

text: the

text: local

text: stochastic

text: population

text: dynamics.

text: Agent-based

text: models

text: can

text: be

text: used

text: for

text: this

text: purpose

text: [e.g.

text: but

text: it

text: is

text: often

text: diﬃcult

text: to

text: understand

text: the

text: mechanisms

text: that

text: determine

text: the

text: global

text: metapopulation

text: dynamics,

text: and

text: to

text: quantify

text: the

text: inﬂuence

text: of

text: the

text: diﬀerent

text: factors

text: characterising

text: the

text: local

text: population

text: demography.

text: An

text: analytical

text: treatment

text: of

text: the

text: metapopulation

text: is

text: much

text: more

text: desirable

text: and

text: informative,

text: but

text: it

text: is

text: not

text: trivial.

text: Lande

text: et

text: al.

text: were

text: the

text: ﬁrst

text: to

text: develop

text: such

text: an

text: analytical

text: model

text: based

text: on

text: a

text: diﬀusion

text: approximation

text: for

text: the

text: local

text: population

text: dynamics.

text: Their

text: approach

text: characterises

text: the

text: metapopulation

text: dynamics

text: as

text: a

text: sequence

text: of

text: isolated,

text: local

text: extinction

text: and

text: recolonisation

text: events,

text: and

text: relies

text: on

text: the

text: assumption

text: of

text: very

text: low

text: migration

text: rates.

text: This

text: assumption

text: is

text: appropriate

text: when

text: a

text: large

text: fraction

text: of

text: suitable

text: patches

text: is

text: not

text: occupied

text: near

text: demographic

text: equilibrium.

text: This

text: scenario

text: is

text: at

text: odds

text: with

text: the

text: observation

text: that

text: the

text: majority

text: of

text: suitable

text: patches

text: tend

text: to

text: remain

text: occupied

text: in

text: many

text: metapopulations

text: [e.g.

text: Keeling

text: took

text: a

text: diﬀerent

text: approach,

text: relying

text: on

text: a

text: moment-closure

text: technique,

text: to

text: investigate

text: the

text: persistence

text: of

text: stochastic

text: metapopulations.

text: This

text: approach

text: relies

text: on

text: knowing

text: the

text: distribution

text: of

text: population

text: sizes,

text: a

text: problematic

text: restriction

text: if

text: we

text: are

text: interested

text: in

text: investigating

text: the

text: dynamics

text: of

text: a

text: metapopulation

text: close

text: to

text: extinction,

text: for

text: which

text: the

text: shape

text: of

text: the

text: distribution

text: is

text: likely

text: to

text: diﬀer

text: from

text: the

text: corresponding

text: shape

text: near

text: demographic

text: equilibrium.

text: In

text: this

text: paper,

text: we

text: develop

text: an

text: analytical

text: framework,

text: based

text: on

text: the

text: identiﬁcation

text: of

text: the

text: slow

text: and

text: fast

text: modes

text: underlying

text: the

text: metapopulation

text: dynamics.

text: Our

text: approach

text: allows

text: us

text: to

text: investigate

text: the

text: link

text: between

text: local

text: stochastic

text: dynamics

text: and

text: the

text: strength

text: of

text: the

text: rescue

text: eﬀect.

text: We

text: compare

text: our

text: results

text: to

text: spatially

text: explicit

text: simulations

text: to

text: explore

text: the

text: spatial

text: scales

text: at

text: which

text: our

text: analytical

text: results

text: are

text: relevant.

text: While

text: for

text: this

text: paper

text: we

text: focus

text: on

text: a

text: model

text: that

text: describes

text: the

text: local

text: dynamics

text: in

text: terms

text: of

text: a

text: Ricker

text: map

text: the

text: approach

text: can

text: be

text: applied

text: to

text: other

text: models

text: for

text: population

text: dynamics,

text: formulated

text: in

text: terms

text: of

text: discrete

text: non-overlapping

text: generations

text: or

text: in

text: terms

text: of

text: continuously

text: occurring

text: birth

text: and

text: death

text: processes.

text: METAPOPULATION

text: MODEL

text: Our

text: model

text: considers

text: N

text: patches

text: connected

text: by

text: global

text: dispersal,

text: and

text: we

text: speciﬁcally

text: focus

text: on

text: metapopulations

text: consist-

text: ing

text: of

text: a

text: large

text: number

text: of

text: patches.

text: Each

text: patch

text: can

text: be

text: inhabited

text: by

text: a

text: population

text: that

text: evolves

text: in

text: discrete,

text: non-overlapping

text: generations.

text: Each

text: generation

text: has

text: two

text: phases,

text: the

text: ﬁrst

text: corresponding

text: to

text: local

text: recruitment

text: and

text: the

text: second

text: to

text: dispersal

text: to

text: other

text: populations.

text: For

text: biological

text: realism,

text: we

text: use

text: the

text: model

text: by

text: Melbourne

text: Hastings

text: to

text: represent

text: local

text: recruit-

text: ment

text: and

text: density-dependent

text: survival.

text: This

text: model

text: is

text: a

text: stochastic

text: version

text: of

text: the

text: classical

text: deterministic

text: Ricker

text: model

text: Ricker

text: that

text: incorporates

text: the

text: eﬀect

text: of

text: both

text: demographic

text: and

text: environmental

text: stochasticity

text: within

text: populations.

text: In

text: our

text: model

text: each

text: individual

text: has

text: a

text: Poisson-distributed

text: number

text: of

text: progeny,

text: with

text: mean

text: given

text: by

text: the

text: recruitment

text: rate

text: R

text: i

text: for

text: individual

text: i.

text: The

text: individual

text: recruitment

text: rate

text: R

text: is

text: itself

text: random

text: (independent

text: between

text: generations),

text: giving

text: rise

text: to

text: i

text: demographic

text: and

text: environmental

text: stochasticity.

text: The

text: environmental

text: stochasticity

text: determines

text: a

text: mean

text: (environmental)

text: recruitment

text: rate,

text: R

text: for

text: the

text: local

text: population

text: (R

text: is

text: gamma

text: distributed

text: with

text: mean

text: R

text: and

text: shape

text: parameter

text: k

text: Given

text: E

text: E

text: E

text: the

text: environmental

text: state

text: of

text: the

text: local

text: population,

text: individual

text: recruitment

text: rates

text: R

text: are

text: independent

text: gamma

text: distributed

text: i

text: stochastic

text: variables

text: with

text: mean

text: R

text: and

text: shape

text: parameter

text: k

text: (giving

text: rise

text: to

text: demographic

text: recruitment

text: stochasticity).

text: E

text: D

text: Finally,

text: each

text: progeny

text: survives

text: (independently)

text: to

text: adulthood

text: with

text: density-dependent

text: probability

text: exp(

text: αn),

text: where

text: n

text: is

text: the

text: number

text: of

text: adults

text: in

text: the

text: population

text: and

text: α

text: regulates

text: the

text: strength

text: of

text: the

text: density

text: dependence.

text: As

text: shown

text: by

text: Melbourne

text: Hastings

text: the

text: expected

text: number

text: of

text: progeny

text: surviving

text: to

text: adulthood

text: is

text: the

text: same

text: as

text: in

text: the

text: classical

text: Ricker

text: dynamics

text: (with

text: recruitment

text: R

text: and

text: density

text: dependence

text: α),

text: but

text: the

text: variance

text: of

text: the

text: distribution

text: is

text: strongly

text: aﬀected

text: by

text: the

text: amount

text: of

text: demographic

text: and

text: environmental

text: recruitment

text: stochasticity

text: (deﬁned

text: by

text: the

text: shape

text: parameters

text: k

text: and

text: k

text: Putting

text: it

text: all

text: together

text: (following

text: the

text: probability

text: of

text: having,

text: after

text: the

text: ﬁrst

text: phase,

text: j

text: individuals

text: in

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text: 11 #"0.025 9.:*('+%,-# Theory ;S<i0m%u2l4a#tion 0.02 # " " 0.015 ! !!(t) I! 0.01 0.005 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 I(t) !!""# 0.4 !" 9T.:he*o(r'y+%,-# 0.35 ;S<i0m%u2la4t#ion 0.3 "t#=51##0#67# 4# "t#=55##0#87# &y 0.25 t=100 ,nc "#5#677# 0e *qu 0.2 3e 0Fr 0.15 2 1 0.1 0.05 0 0 5 10 15 20 Patch pop size i $%&'(#)%)*('+%,#-./0# FIG. 3. Comparison ofFitgh e3o: rCyo(m)ptaorissiomnu olaft tihoneosroyf tEo qsi(m6u).laPtiaonnesl. APancoeml Ap acroems pthareecsh tahneg cehianndgeis pine rdsiaslperarstae,l r∆atIe,, aI,s a function of I, from the simulations (red line) to the theory (black line). Panel B compares the distribution of local population sizes as a function of I, from the simulations (red line) to the theory (black line). Panel B compares from the simulations (points) to the theoretical predictions (lines), at three diﬀerent points in time (t =10, 50, 100). Parameter the distribution of local population sizes from the simulations (points) to the theoretical values: R = 1.2, α = 0.01, m = 0.1, k = 1, k = 10 (environmental recruitment stochasticity). predictions (linDes), at tEhree different points in time (t=10, 50, 100). Parameter values: R=1.2, α=0.01, m=0.1, kD=1, kE=10 (environmental recruitment stochasticity). 18 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Panel

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text: Panel

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text: distribution

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text: the

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text: (points)

text: to

text: the

text: theoretical

text: values:

text: R

text: α

text: m

text: k

text: k

text: (environmental

text: recruitment

text: stochasticity).

text: predictions

text: (linDes),

text: at

text: tEhree

text: different

text: points

text: in

text: time

text: Parameter

text: values:

text: (environmental

text: recruitment

text: stochasticity).

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text: 5 &/88- <1=082-(00'+>?- !" !"') :2;1,6- &/88-(00'+>?- %" !"!'$ !"!'# %%" !"'# ))"n !"!' "#$###.+*,-*-**//!! !!"'"'' ((*-nn+0!123 !!""!!!!%& $ / ! n !"!!$ # ! ! !"!( !"!!# !"!& ! ! !"# !"$ !"% !"& ' ! # $ % & '! '# &'()*+,-+.-+))/*0123-0(4)526-#" @2(,-001+230?-61A2-&" #" !"'$ $" + !"'# '"# $"9!""!%#--4/ #$%%!" !!"!"'& +'- !"'& !"%!!"-!!"5%+!4-$",4-!/!6!%#-!46/$.",!!-!6%6"-%/!6-/#.!$6!/%/%- ""!!+4! !"!% ()4 !"% ! & !! !"!$ !"$ !"!# !"# ! ! + ! !"# !"$ !"% !"& ' '"# ! !"# !"$ !"% !"& ' '"# 71602'6(8-4'(42-!" 71602'6(48-'(42-!" FIG. S3. Comparison of our model to Levins’ model. A: In Levins’ model the change in fraction of occupied patches (∆p) Fig. S3:∗ C−omparison o∗f our model to Levins’ model. A: In Levins’ model the change in fraction is proportional to p(p p) where p is the value of p in the stationary state of the metapopulation (c.f. Eq. (8) in the main text). Accordinogf toocLceuvpiniesd’ sipmaptcleheeqs u(aΔtipo)n ,isi nptrhoepaobrsteionncealo ftoa rpe(spc*ue‐pe)ﬀ ewcht,e∆rep /pp*( p∗is− tph)e isvcaolunest aonft p(b lianc kthlien e). Deviations ∗ − of ∆p/p(p p)stfarotimontahrey csotnasttea notfv athluee mreevetaalphoopwultahtieorne s(ccu.fe. eEﬀqe.c t7e mine rtghees fmroamint hteexlto)c. aAl cpcooprudlaintigo ntod yLneavminisc’s in our model. We show the full deterministic dynamics (Eq. (6), blue line) for a trajectory starting from mean population size 0.1, and the simple equation, in the absence of a rescue effect, Δp/p(p*‐p) is constant (black line). corresponding curves for the slow-mode theory (, green line) and the simple approximation (Eq. (7), red line). We observe a Deviations of Δp/p(p*‐p) from the constant value reveal how the rescue effect emerges strong rescue eﬀect and excellent agreement between the analytical approximations and the full dynamics. In contrast, the corresponding farnoamly stihsef o rlotchael pmoepaunlaptoiopunl adtyionnamsiizces (in¯n, opuarn meloBd)eal.n Wd ed isspheorwsa lthreat feu(llI d, eptaenreml iCn)issthico wdysntahmatictsh e dynamics of these observabl(ebsluisev elirnyec) lofoser tao tLreavjeincst’omryo dsteal r(taifntger farosmho rmt teraann spieonptu).laDti:oTnh seiztew o0f.a1c, taonrsdd tehteer mcoinrrinegsptohendchinagn ge in dispersal rate, ∆I, in thecusirmvepsle foapr pthroex aimpaptriooxni,maastaiofnusn c(trieodn aonfdI .gAresecna nlinbees,s eseene ilnegpeannedl).C Waen dobfrsoemrvteh ea rsetdrolninge ,re∆sIcuise approximately ∗ − ∗ proportional toeIff(eIct aIn),dw heexrceeIllenist thaegsrteeeamdye-nstta tbeedtiwspeeernsa ltrhaete .aTnahliystwicoaul ldaspepermoxtiomiamtpiolynsa naanbds entchee offurells cue eﬀect, but plotting the two factors in the simple approximation of ∆I [Eq. (6)] separately reveals that the eﬀect of an increasing trend in the excess emigdryantiaomn ircast.e In(D co(In)t−rasIt,, gtrheee ncolirnree)spisoonﬀdsientgi nan∆aIlysbiys afocro trhrees pmoenadnin pgodpeucrlaetaisoinng sitzreen (d i,n ptahneeell aBs)t icity (−λ (I), 2 blue line). Paraamndet derisvpaelruseasl: rRat=e (1I,. 2p,aαne=l 0C.)0 s1h, omw=s t0h.1a,t kthe= d1y,nkam=ics1 0of( ethnveisreo nombesentravlabrelecrsu iist mveenryt cstloocshea tsoti city). D E Levin’s model (after a short transient). D: The two factors determining the change in dispersal rate, ΔI, in the simple approximation, as a function of I. As can be seen in panel C [22] Drechsler, M. & Wissel, C., 1997 Separability of local and regional dynamics in metapopulations. Theor. Popul. Biol. 51, and from the red line, ΔI is approximately proportional to I(I*‐I), where I* is the steady‐state 9–21. dispersal rate. This would seem to imply an absence of rescue effect, but plotting the two [23] Eriksson, A., Elías-Wolﬀ, F. & Mehlig, B., 2013 Metapopulation dynamics on the brink of extinction. Theor. Popul. Biol. 83, 101–12f2a.ctors in the simple approximation of ΔI (Eq. 6) separately reveals that the effect of an [24] Chesson, Pin. cLr.e, a1s9i8n2g Tthreensdta binil iztihneg eeﬀxeccetsosf aemraingdraotmionen vriartoen m(Ee(nIt).‐I,J .gMreaetnh. lBinioel)o giys 1o5ff,s1e–t3 6in. ΔI by a [25] Ovaskainen, O. & Hanski, I., 2002 Transient dynamics in metapopulation response to perturbation. Theor. Popul. Biol. 61, 285–29c5o.rresponding decreasing trend in the elasticity (–λ2(I), blue line). Parameter values: R=1.2, [26] Higgins, Kα.,=200.0091,M me=ta0p.1o,p kuDla=t1io, nkEe=x1t0in (cetniovnirorinskm: eDnitsaple rresaclr’usitdmupelnicti tsyt.ocThhaesotri.ciPtyo)p. u l. Biol. 76, 146–155. [27] Sæther, B. E., Engen, S. & Lande, R., 1999 Finite metapopulation models with density–dependent migration and stochastic local dynamics. Proc. R. Soc. London B 266, 113–118. [28] Casagrandi, R. & Gatto, M., 1999 A mesoscale approach to extinction risk in fragmented habitats. Nature 400, 560–562. 23 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ))"n

text: n

text: FIG.

text: Comparison

text: of

text: our

text: model

text: to

text: Levins’

text: model.

text: A:

text: In

text: Levins’

text: model

text: the

text: change

text: in

text: fraction

text: of

text: occupied

text: patches

text: (∆p)

text: Fig.

text: C−omparison

text: o∗f

text: our

text: model

text: to

text: Levins’

text: model.

text: A:

text: In

text: Levins’

text: model

text: the

text: change

text: in

text: fraction

text: is

text: proportional

text: to

text: p(p

text: p)

text: where

text: p

text: is

text: the

text: value

text: of

text: p

text: in

text: the

text: stationary

text: state

text: of

text: the

text: metapopulation

text: (c.f.

text: Eq.

text: in

text: the

text: main

text: text).

text: Accordinogf

text: toocLceuvpiniesd’

text: sipmaptcleheeqs

text: u(aΔtipo)n

text: ,isi

text: nptrhoepaobrsteionncealo

text: ftoa

text: rpe(spc*ue‐pe)ﬀ

text: ewcht,e∆rep

text: /pp*(

text: p∗is−

text: tph)e

text: isvcaolunest

text: aonft

text: p(b

text: lianc

text: kthlien

text: e).

text: Deviations

text: of

text: ∆p/p(p

text: p)stfarotimontahrey

text: csotnasttea

text: notfv

text: athluee

text: mreevetaalphoopwultahtieorne

text: s(ccu.fe.

text: eEﬀqe.c

text: mine

text: rtghees

text: fmroamint

text: hteexlto)c.

text: aAl

text: cpcooprudlaintigo

text: ntod

text: yLneavminisc’s

text: in

text: our

text: model.

text: We

text: show

text: the

text: full

text: deterministic

text: dynamics

text: (Eq.

text: blue

text: line)

text: for

text: a

text: trajectory

text: starting

text: from

text: mean

text: population

text: size

text: and

text: the

text: simple

text: equation,

text: in

text: the

text: absence

text: of

text: a

text: rescue

text: effect,

text: Δp/p(p*‐p)

text: is

text: constant

text: (black

text: line).

text: corresponding

text: curves

text: for

text: the

text: slow-mode

text: theory

text: green

text: line)

text: and

text: the

text: simple

text: approximation

text: (Eq.

text: red

text: line).

text: We

text: observe

text: a

text: Deviations

text: of

text: Δp/p(p*‐p)

text: from

text: the

text: constant

text: value

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text: how

text: the

text: rescue

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text: emerges

text: strong

text: rescue

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text: and

text: excellent

text: agreement

text: between

text: the

text: analytical

text: approximations

text: and

text: the

text: full

text: dynamics.

text: In

text: contrast,

text: the

text: corresponding

text: farnoamly

text: stihsef

text: o

text: rlotchael

text: pmoepaunlaptoiopunl

text: adtyionnamsiizces

text: (in¯n,

text: opuarn

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text: Wd

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text: these

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text: tbeedtiwspeeernsa

text: ltrhaete

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text: naanbds

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text: D:

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text: change

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text: ΔI,

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text: as

text: a

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text: I.

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text: panel

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text: M.

text: Wissel,

text: C.,

text: Separability

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text: local

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text: dynamics

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text: metapopulations.

text: Theor.

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text: and

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text: the

text: red

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text: ΔI

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text: I(I*‐I),

text: where

text: I*

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text: steady‐state

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text: but

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text: Eriksson,

text: A.,

text: Elías-Wolﬀ,

text: F.

text: Mehlig,

text: B.,

text: Metapopulation

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text: the

text: brink

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text: extinction.

text: Theor.

text: Popul.

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text: I.,

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text: Parameter

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text: Higgins,

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text: u

text: l.

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text: Sæther,

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text: E.,

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text: S.

text: Lande,

text: R.,

text: Finite

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text: Proc.

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text: 10 20 e z i 15 s n o 6 a l u 10 p Theory o p Panmic6c dispersal e Dispersal dist. = 5 g a 5 Dispersal dist. = 2 r e v Dispersal dist. = 1 A 0 0.001 0.01 0.1 1 Probability of dispersal (m) FIG. 2. The eﬀect of the dispersal distance and dispersal rate on the mean stationary population size in a spatially explicit model (20 by 20 patches in a square lattice). Simulations of panmictic dispersal (squares, all patches equally likely to receive a dispersing individual) are in excellent agreement with the theory for inﬁnite number of patches (Eq. 6, line). Simulations of dispersal with random direction and exponentially distributed distance (circles and triangles) show that as soon as the mean dispersal distance exceeds the distance between demes, the theory is in very good agreement with the simulations. Shown for R = 1.5, α = 0.02, k = 1, and k = 10 (environmental recruitment stochasticity). Each simulation data point was estimated D E from 10,000 independent measurements. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: e

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text: square

text: lattice).

text: Simulations

text: of

text: panmictic

text: dispersal

text: (squares,

text: all

text: patches

text: equally

text: likely

text: to

text: receive

text: a

text: dispersing

text: individual)

text: are

text: in

text: excellent

text: agreement

text: with

text: the

text: theory

text: for

text: inﬁnite

text: number

text: of

text: patches

text: (Eq.

text: line).

text: Simulations

text: of

text: dispersal

text: with

text: random

text: direction

text: and

text: exponentially

text: distributed

text: distance

text: (circles

text: and

text: triangles)

text: show

text: that

text: as

text: soon

text: as

text: the

text: mean

text: dispersal

text: distance

text: exceeds

text: the

text: distance

text: between

text: demes,

text: the

text: theory

text: is

text: in

text: very

text: good

text: agreement

text: with

text: the

text: simulations.

text: Shown

text: for

text: R

text: α

text: k

text: and

text: k

text: (environmental

text: recruitment

text: stochasticity).

text: Each

text: simulation

text: data

text: point

text: was

text: estimated

text: D

text: E

text: from

text: independent

text: measurements.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: (a) (b) k 10 subcritical 8 supercritical 6 4 2 0 T 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (c) (d) (cid:104) (cid:105) FIG. 6. Fixing ω = 3, ∆ = 1, τ˜ + τ = T + τ = 3, Var(τ˜ + τ ) = T2 = 3, the ﬁrst bifurcation 0 0 0 0 n value and its direction is drawn in (a), and all bifurcation curves are illustrated in (b). The dark region stands for local stable region of incoherent state. Numbers indicates the quantity of coherent | | attractors. When (c) T = 0.5(k¯ = 2.6992) and (d) T = 2.8(k¯ = 7.0388), r is shown in black dots by simulating system (??) with N = 128. As k increases, two even more Hopf bifurcations yield the coexistence of coherent states which makes the dynamical behavior more delicate. As we have seen in FIG. ?? (c), when | | k > 5, the standard variance of r among 100 times of simulations is rather large, which reveals that several stable attractors coexist. They may be originated from the second branch of Hopf bifurcation (black circles). Another reason makes this standard variance large may be the unstable hysteresis loop near k = 6, when a subcritical bifurcation occur. When k is suﬃciently near this critical point, hysteresis loop no longer contains stable states, because after the ﬁrst Hopf bifurcation the trivial solution of (??) is certainly unstable. Notice that 15 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: (b)

text: k

text: subcritical

text: supercritical

text: T

text: (c)

text: (d)

text: FIG.

text: Fixing

text: ω

text: τ˜

text: τ

text: T

text: τ

text: Var(τ˜

text: τ

text: the

text: ﬁrst

text: bifurcation

text: n

text: value

text: and

text: its

text: direction

text: is

text: drawn

text: in

text: (a),

text: and

text: all

text: bifurcation

text: curves

text: are

text: illustrated

text: in

text: (b).

text: The

text: dark

text: region

text: stands

text: for

text: local

text: stable

text: region

text: of

text: incoherent

text: state.

text: Numbers

text: indicates

text: the

text: quantity

text: of

text: coherent

text: attractors.

text: When

text: (c)

text: T

text: and

text: (d)

text: T

text: r

text: is

text: shown

text: in

text: black

text: dots

text: by

text: simulating

text: system

text: with

text: N

text: As

text: k

text: increases,

text: two

text: even

text: more

text: Hopf

text: bifurcations

text: yield

text: the

text: coexistence

text: of

text: coherent

text: states

text: which

text: makes

text: the

text: dynamical

text: behavior

text: more

text: delicate.

text: As

text: we

text: have

text: seen

text: in

text: FIG.

text: (c),

text: when

text: k

text: the

text: standard

text: variance

text: of

text: r

text: among

text: times

text: of

text: simulations

text: is

text: rather

text: large,

text: which

text: reveals

text: that

text: several

text: stable

text: attractors

text: coexist.

text: They

text: may

text: be

text: originated

text: from

text: the

text: second

text: branch

text: of

text: Hopf

text: bifurcation

text: (black

text: circles).

text: Another

text: reason

text: makes

text: this

text: standard

text: variance

text: large

text: may

text: be

text: the

text: unstable

text: hysteresis

text: loop

text: near

text: k

text: when

text: a

text: subcritical

text: bifurcation

text: occur.

text: When

text: k

text: is

text: suﬃciently

text: near

text: this

text: critical

text: point,

text: hysteresis

text: loop

text: no

text: longer

text: contains

text: stable

text: states,

text: because

text: after

text: the

text: ﬁrst

text: Hopf

text: bifurcation

text: the

text: trivial

text: solution

text: of

text: is

text: certainly

text: unstable.

text: Notice

text: that

text: ##LTFigure##

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text: (cid:90) ∞ (cid:104) − (cid:105) + − − r(t) = ξ(t τ) = ξ(t τ)h(τ τ )dτ (3) 0 τ0 with (cid:90) (cid:90) ∞ + 2π i ξ(t) = f(ω, θ, t)e θdθdω (4) −∞ 0 The distribution density f(ω, θ, t) characterizes the state of the oscillators’ system at time t in frequency ω and phase θ. Now we are about to restate some results about the Ott-Antonsen’s reduction of a system with distributed delay ﬁrst derived(cid:40)b(cid:34)y Lee et al[? ]. Rewr(cid:35)itin(cid:41)g system (??) as ∂ ∂ k −i − i ∗ f + ω + (e θr e θr ) f = 0 (5) ∂t ∂θ 2i (cid:40) (cid:40) (cid:34) (cid:35)(cid:41)(cid:41) and restricting this partial diﬀerential equa(cid:88)tion on the Ott-Antonsen manifold ∞ g(ω) i f : f = 1 + αm(ω, t)e mθ + c.c. 2π m=1 with c.c. the complex conjugate of the formal terms, we substitute the Fourier series of f into (??). After comparing the coeﬃcient of the same harmonic terms, a reduced equation is obtained − k ∗ − k α˙ (ω, t) = iωα(ω, t) + r rα2(ω, t) (6) 2 2 (cid:82) ∞ ∗ Obviously, from (??) we have ξ(t) = −+∞ g(ω)α (ω, t)dω, then Eq.(??) yields (cid:90) (cid:90) ∞ ∞ + + ∗ − − r(t) = g(ω)α (ω, t τ)dωh(τ τ )dτ (7) −∞ 0 τ0 For the sake of theoretical analysis, the distribution density g(ω) is usually chosen as Lorentzian distribution, that is ∆ − ∞ ∞ g(ω) = − , < ω < + (8) π[(ω ω )2 + ∆2] 0 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: r(t)

text: ξ(t

text: τ)

text: ξ(t

text: τ)h(τ

text: τ

text: )dτ

text: τ0

text: with

text: 2π

text: i

text: ξ(t)

text: f(ω,

text: θ,

text: t)e

text: θdθdω

text: The

text: distribution

text: density

text: f(ω,

text: θ,

text: t)

text: characterizes

text: the

text: state

text: of

text: the

text: oscillators’

text: system

text: at

text: time

text: t

text: in

text: frequency

text: ω

text: and

text: phase

text: θ.

text: Now

text: we

text: are

text: about

text: to

text: restate

text: some

text: results

text: about

text: the

text: Ott-Antonsen’s

text: reduction

text: of

text: a

text: system

text: with

text: distributed

text: delay

text: ﬁrst

text: Lee

text: et

text: al[?

text: system

text: as

text: k

text: −i

text: i

text: f

text: ω

text: (e

text: θr

text: e

text: θr

text: f

text: ∂t

text: ∂θ

text: and

text: restricting

text: this

text: partial

text: diﬀerential

text: on

text: the

text: Ott-Antonsen

text: manifold

text: g(ω)

text: i

text: f

text: f

text: αm(ω,

text: t)e

text: mθ

text: c.c.

text: 2π

text: with

text: c.c.

text: the

text: complex

text: conjugate

text: of

text: the

text: formal

text: terms,

text: we

text: substitute

text: the

text: Fourier

text: series

text: of

text: f

text: into

text: After

text: comparing

text: the

text: coeﬃcient

text: of

text: the

text: same

text: harmonic

text: terms,

text: a

text: reduced

text: equation

text: is

text: obtained

text: k

text: k

text: α˙

text: (ω,

text: t)

text: iωα(ω,

text: t)

text: r

text: t)

text: Obviously,

text: from

text: we

text: have

text: ξ(t)

text: g(ω)α

text: (ω,

text: t)dω,

text: then

text: Eq.(??)

text: yields

text: r(t)

text: g(ω)α

text: (ω,

text: t

text: τ)dωh(τ

text: τ

text: )dτ

text: τ0

text: For

text: the

text: sake

text: of

text: theoretical

text: analysis,

text: the

text: distribution

text: density

text: g(ω)

text: is

text: usually

text: chosen

text: as

text: Lorentzian

text: distribution,

text: that

text: is

text: g(ω)

text: ω

text: π[(ω

text: ω

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: of the pulsar in orbital motion. If one takes the time, t and red shift will change its position. And thus, the k (where k = 1, 2), as observer’s time, then it implies that distribution of the projected velocity is no longer uniform the two pulses, sent at position 1 and 2, can reach the with respect to the observer. observer instantaneously. Consequently, net Doppler shift to the frequency of (cid:90) In such case, the two signals, which sent with a time pulse is given(Gong 2005), − discrepancy say, t t = 1.0s, are thought to be detected 2 1 as ν(t ) and ν(t + 1.0) respectively. 1 1 1 Pb z xK − e2 Whereas, the true situation is that the second signal ∆ν = ∆ν(t)d(t + ) = νπ(1 ) (4) is act−ually detected as, ν(t1 + 1.0 + δz/c), where δz/c = Pb 0 c Pb 4 z /c z /c. 2 1 It is this additional time that changes the distribution Notice that it is this additional time delay dz/c that of the projected velocity with respect to the observer. makes the shift of pulse frequency of Eq.4 non-zero. This E.g., for a pulsar in circular motion around its companion result means that in every orbital period the signal fre- star, it will have 1/4 orbital period of blue shift (close quency of one object in the binary system is shifted by to), and 1/4 orbital period of red shift (away) at the a value described by Eq.4, which is a Newtonian second two sides of the point of closest distance to the observer, Doppler eﬀect. without considering the propagation time. But when the Such a residual Doppler shift aﬀects three typical bi- adDdritaifotnavlertismioen iMs ainyc2lu6,d2e0d2,0the turning point of the blue nary pulsars, PSR B1259-63, 4U1627-67 and PSR J2051- Preprint typeset using LATEX style emulateapj v. 12/16/11 0827 diﬀerently. In the following sections, they will be analysed one by one. EVIDENCE OF RESIDUAL DOPPLER SHIFT ON THREE PULSARS, PSR B1259-63, 4U1627-67 AND PSR J2051-0827 Bi-Ping Gong1, Z.Q. Yan1 0 1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China Subject headings: stars: neutron—X-rays: binaries—X-rays: individual (PSR B1259-63, 4U1627-67, PSR J2051-0827, PSR 0823+26, PSR J0631+1036, Her X-1, Cen X-3, GX 1+4, OAO 1657-415, Vela X-1, 4U 1907+09) 2. PSR B1259-63 and hence frequency shift of pulsar at moment t, from which the variation of ∆ν(t) versus time is obtained as PSR B1259-63 was discovered in a large-scale high- shown in panel a of Fig. 2. frequency survey of the Galactic plane (Johnston 1992). As analysed above, the time t is actually the local time It is a radio pulsar in orbit about a massive, main- at rest to the center mass of the pulsar. At this very sequence B2e star. With an orbital period of 1237 d and companion mass of > 10M(cid:12), the 23yr data(Shannon et time, the orbital phase of the pulsar is f(t), whereas, the pulse signal sent at this phase is measured as ∆ν(t+z/c) al. 2013) indicates a signiﬁcant orbital period decay, × − instead of ∆ν(t). 1.4(7) 10 8, while the timing residual versus orbital The discrepancy between the two shift of pulse fre- phase has similar shape as ten years ago(Wang et al. − quency, ∆ν(t + z/c) ∆ν(t), is equivalent to O-C 2004). (observation-calculation). Expand ∆ν(t + z/c) in Taylor The huge derivative of orbital period corresponds to a · ≈ series, ∆ν(t + z/c) = ∆ν(t) + ν˙(t) (z/c) + ... Then the time residual in one orbit, δTo = P˙bPb 1(s). inte(cid:90)gration of the s(cid:90)econd term at rig(cid:90)ht hand side gives, Can residual Doppler shift reproduce such a signiﬁ- cant timing residual ? This can be tested easily by − putting x = 1296.3s, P = 1236.7d, ν = 20.9s 1 and Pb z Pb z − Pb z˙ b ν˙(t) dt = ∆ν(t) dt ∆ν(t) dt (6) e = 0.87(Wang et al. 2004; Shannon et al. 2013), into c c c 0 0 0 the residual Doppler shift in one orbit(Gong 2005), Apparently, the ﬁrst term at right hand side of Eq.6 xK − e2 × − − equals zero, considering the trigonometric function of ∆ν = νπ(1 ) = 1.0 10 7(s 1) (5) Pb 4 Eq.1 and Eq.2, and the second−term is equivalent to Eq.4, which is actually ∆ν(t + z/c) ∆ν(t). The variation of Therefore, the predicted additional time residual in ev- such a discrepancy versus time is as shown in panel b of ≈ ≈ ery orbital period is δT ∆νP /ν 0.6s, which explains Fig. 2. ≈ b the residual, δT 1(s), corresponding to the signiﬁcant The accumulated change of pulse frequency corre- o derivative of orbital period(Shannon et al. 2013), with sponds to a timing residual as shown in the bottom a ﬂying color. panel of Fig. 3, which is obtained by, Further, the timing residual versus orbital phase can (cid:48) ∆ν − also be tested. Input t into Eq.1, obtains phase, velocity, Res(t ) = Res(t − ) + (t t − ) (7) k k 1 ν k k 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: the

text: pulsar

text: in

text: orbital

text: motion.

text: If

text: one

text: takes

text: the

text: time,

text: t

text: and

text: red

text: shift

text: will

text: change

text: its

text: position.

text: And

text: thus,

text: the

text: k

text: (where

text: k

text: as

text: observer’s

text: time,

text: then

text: it

text: implies

text: that

text: distribution

text: of

text: the

text: projected

text: velocity

text: is

text: no

text: longer

text: uniform

text: the

text: two

text: pulses,

text: sent

text: at

text: position

text: and

text: can

text: reach

text: the

text: with

text: respect

text: to

text: the

text: observer.

text: observer

text: instantaneously.

text: Consequently,

text: net

text: Doppler

text: shift

text: to

text: the

text: frequency

text: of

text: In

text: such

text: case,

text: the

text: two

text: signals,

text: which

text: sent

text: with

text: a

text: time

text: pulse

text: is

text: given(Gong

text: discrepancy

text: say,

text: t

text: t

text: are

text: thought

text: to

text: be

text: detected

text: as

text: ν(t

text: and

text: ν(t

text: respectively.

text: Pb

text: z

text: xK

text: Whereas,

text: the

text: true

text: situation

text: is

text: that

text: the

text: second

text: signal

text: ∆ν

text: ∆ν(t)d(t

text: νπ(1

text: is

text: act−ually

text: detected

text: as,

text: δz/c),

text: where

text: δz/c

text: Pb

text: c

text: Pb

text: z

text: /c

text: z

text: /c.

text: It

text: is

text: this

text: additional

text: time

text: that

text: changes

text: the

text: distribution

text: Notice

text: that

text: it

text: is

text: this

text: additional

text: time

text: delay

text: dz/c

text: that

text: of

text: the

text: projected

text: velocity

text: with

text: respect

text: to

text: the

text: observer.

text: makes

text: the

text: shift

text: of

text: pulse

text: frequency

text: of

text: non-zero.

text: This

text: E.g.,

text: for

text: a

text: pulsar

text: in

text: circular

text: motion

text: around

text: its

text: companion

text: result

text: means

text: that

text: in

text: every

text: orbital

text: period

text: the

text: signal

text: fre-

text: star,

text: it

text: will

text: have

text: orbital

text: period

text: of

text: blue

text: shift

text: (close

text: quency

text: of

text: one

text: object

text: in

text: the

text: binary

text: system

text: is

text: shifted

text: by

text: to),

text: and

text: orbital

text: period

text: of

text: red

text: shift

text: (away)

text: at

text: the

text: a

text: value

text: described

text: by

text: which

text: is

text: a

text: Newtonian

text: second

text: two

text: sides

text: of

text: the

text: point

text: of

text: closest

text: distance

text: to

text: the

text: observer,

text: Doppler

text: eﬀect.

text: without

text: considering

text: the

text: propagation

text: time.

text: But

text: when

text: the

text: Such

text: a

text: residual

text: Doppler

text: shift

text: aﬀects

text: three

text: typical

text: bi-

text: adDdritaifotnavlertismioen

text: iMs

text: turning

text: point

text: of

text: the

text: blue

text: nary

text: pulsars,

text: PSR

text: and

text: PSR

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: diﬀerently.

text: In

text: the

text: following

text: sections,

text: they

text: will

text: be

text: analysed

text: one

text: by

text: one.

text: EVIDENCE

text: OF

text: RESIDUAL

text: DOPPLER

text: SHIFT

text: ON

text: THREE

text: PULSARS,

text: PSR

text: AND

text: PSR

text: Bi-Ping

text: Z.Q.

text: of

text: Physics,

text: Huazhong

text: University

text: of

text: Science

text: and

text: Technology,

text: Wuhan

text: China

text: Subject

text: headings:

text: stars:

text: neutron—X-rays:

text: binaries—X-rays:

text: individual

text: (PSR

text: PSR

text: PSR

text: PSR

text: Her

text: Cen

text: GX

text: OAO

text: Vela

text: PSR

text: and

text: hence

text: frequency

text: shift

text: of

text: pulsar

text: at

text: moment

text: t,

text: from

text: which

text: the

text: variation

text: of

text: ∆ν(t)

text: versus

text: time

text: is

text: obtained

text: as

text: PSR

text: was

text: discovered

text: in

text: a

text: large-scale

text: high-

text: shown

text: in

text: panel

text: a

text: of

text: Fig.

text: frequency

text: survey

text: of

text: the

text: Galactic

text: plane

text: (Johnston

text: As

text: analysed

text: above,

text: the

text: time

text: t

text: is

text: actually

text: the

text: local

text: time

text: It

text: is

text: a

text: radio

text: pulsar

text: in

text: orbit

text: about

text: a

text: massive,

text: main-

text: at

text: rest

text: to

text: the

text: center

text: mass

text: of

text: the

text: pulsar.

text: At

text: this

text: very

text: sequence

text: star.

text: With

text: an

text: orbital

text: period

text: of

text: d

text: and

text: companion

text: mass

text: of

text: the

text: data(Shannon

text: et

text: time,

text: the

text: orbital

text: phase

text: of

text: the

text: pulsar

text: is

text: f(t),

text: whereas,

text: the

text: pulse

text: signal

text: sent

text: at

text: this

text: phase

text: is

text: measured

text: as

text: ∆ν(t+z/c)

text: al.

text: indicates

text: a

text: signiﬁcant

text: orbital

text: period

text: decay,

text: instead

text: of

text: ∆ν(t).

text: while

text: the

text: timing

text: residual

text: versus

text: orbital

text: The

text: discrepancy

text: between

text: the

text: two

text: shift

text: of

text: pulse

text: fre-

text: phase

text: has

text: similar

text: shape

text: as

text: ten

text: years

text: ago(Wang

text: et

text: al.

text: quency,

text: ∆ν(t

text: z/c)

text: ∆ν(t),

text: is

text: equivalent

text: to

text: O-C

text: (observation-calculation).

text: Expand

text: ∆ν(t

text: z/c)

text: in

text: Taylor

text: The

text: huge

text: derivative

text: of

text: orbital

text: period

text: corresponds

text: to

text: a

text: series,

text: ∆ν(t

text: z/c)

text: ∆ν(t)

text: ν˙(t)

text: (z/c)

text: Then

text: the

text: time

text: residual

text: in

text: one

text: orbit,

text: δTo

text: P˙bPb

text: of

text: the

text: term

text: at

text: hand

text: side

text: gives,

text: Can

text: residual

text: Doppler

text: shift

text: reproduce

text: such

text: a

text: signiﬁ-

text: cant

text: timing

text: residual

text: This

text: can

text: be

text: tested

text: easily

text: by

text: putting

text: x

text: P

text: ν

text: and

text: Pb

text: z

text: Pb

text: z

text: Pb

text: z˙

text: b

text: ν˙(t)

text: dt

text: ∆ν(t)

text: dt

text: ∆ν(t)

text: dt

text: e

text: et

text: al.

text: Shannon

text: et

text: al.

text: into

text: c

text: c

text: c

text: the

text: residual

text: Doppler

text: shift

text: in

text: one

text: orbit(Gong

text: Apparently,

text: the

text: ﬁrst

text: term

text: at

text: right

text: hand

text: side

text: of

text: xK

text: equals

text: zero,

text: considering

text: the

text: trigonometric

text: function

text: of

text: ∆ν

text: νπ(1

text: Pb

text: and

text: and

text: the

text: second−term

text: is

text: equivalent

text: to

text: which

text: is

text: actually

text: ∆ν(t

text: z/c)

text: ∆ν(t).

text: The

text: variation

text: of

text: Therefore,

text: the

text: predicted

text: additional

text: time

text: residual

text: in

text: ev-

text: such

text: a

text: discrepancy

text: versus

text: time

text: is

text: as

text: shown

text: in

text: panel

text: b

text: of

text: ery

text: orbital

text: period

text: is

text: δT

text: ∆νP

text: /ν

text: which

text: explains

text: Fig.

text: b

text: the

text: residual,

text: δT

text: corresponding

text: to

text: the

text: signiﬁcant

text: The

text: accumulated

text: change

text: of

text: pulse

text: frequency

text: corre-

text: o

text: derivative

text: of

text: orbital

text: period(Shannon

text: et

text: al.

text: with

text: sponds

text: to

text: a

text: timing

text: residual

text: as

text: shown

text: in

text: the

text: bottom

text: a

text: ﬂying

text: color.

text: panel

text: of

text: Fig.

text: which

text: is

text: obtained

text: by,

text: Further,

text: the

text: timing

text: residual

text: versus

text: orbital

text: phase

text: can

text: ∆ν

text: also

text: be

text: tested.

text: Input

text: t

text: into

text: obtains

text: phase,

text: velocity,

text: Res(t

text: Res(t

text: (t

text: t

text: k

text: k

text: ν

text: k

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 SHAHRAM MOHSENIPOUR (cid:48) (cid:48) for the player II in the game G(S , e ). Suppose the player I plays according to the following strategy: µ , µ , . . . , µ (cid:48). 1 2 e Since S is e-superlarge, then the player II has a winning strategy: γ , . . . , γ , . . . , γ , . . . i < θ 1 e e+i − − for the game G(S, e). Put i 1 = p (if there is no m such that i > u, then put p = i 1 and ≥ m+1 m+1 ≥ 1 note that i > u 1). There are several cases to be considered. Case I: e p. Case II: e < p. 1 ≥ Case I: (e p) (cid:104) (cid:105) First recall the deﬁnition of the elementary end extension chain of initial submodels M ; i < θ | | i from the previous section. For simplicity we denote M by M∗ and set M∗ = χ. Assume that (cid:63) is a γp new symbol diﬀerent from any element of M. In this case we face with three subcases: Subcase (Ia): (cid:48) (cid:48) e = p. Subcase (Ib): p < e = e . Subcase (Ic): p < e < e . Subcase (Ia): (e = p) (cid:48) − Let e p = d where d > 0. Suppose the following are the ordinals given by the wining strategy of the player II against the above mentioned strategy of player I in the game G(S, e): γ (µ ), . . . , γ (µ , . . . , µ ), . . . , γ (µ , . . . , µ ), . . . i < θ 1 1 e 1 e e+i 1 e · · · This implies that γ < γ < < γ < . . . for i < θ and there exist sets: 1 2 i ∈ − ∈ − X [F 1(γ )]µ1, . . . , X [F 1(γ )]µe 1 1 e e as well as the following sets: X ⊂ F(cid:8)−1(γ ) for(cid:9)1 ≤ i < θ e+i e+i such that | | (cid:89)(cid:2) (cid:3)sup (cid:2)Xe+i(cid:91); i < θ =(cid:3)θ and e × | ⊂ r r,f X F ( X ) S, i e+i ≤ i=1 1 i<θ − where f = s e. Now we move towards deﬁning β’s which guarantee the winning of the player II in (cid:48) (cid:48) the game G(S , e ). Let ≤ ≤ β (µ , . . . , µ ) = γ (µ , . . . , µ ) for 1 j p. j 1 j j 1 j (cid:105) Suppose µ is given. Put λ = max(µ , 2χ). Let κ be a cardinal with θ > κ > − (λ ) and δ p+1 1| | ≥ p+1 1 1 r 1 1 1 is the least ordinal such that X κ . Now set e+δ1 1 β (µ , . . . , µ ) = γ (µ , . . . , µ ). p+1 1 p+1 e+δ1 1 e (cid:48) (cid:48) If d = 1, then this completes the description of the strategy of the player II in the game G(S , e ). ≤ If d > 1, then for 1 < i d suppose we have deﬁned β , . . . , β − and µ is given. Set (cid:105) p+1 p+(i 1) p+i λi = max(2κi−1, µp+i) and|let κi |b≥e any cardinal > r−1(λi) and < θ. Suppose δi is the least ordinal < θ and > δ − such that X κ . Now we deﬁne i 1 e+δi i β (µ , . . . , µ ) = γ (µ , . . . , µ ). p+i 1 p+i e+δi 1 e So far we have deﬁned β , . . . , β (cid:48). For 1 < i < θ let 1 e β (cid:48) (µ , . . . , µ (cid:48)) = γ (µ , . . . , µ ). e +i 1 e e+δd+i 1 e (cid:48) (cid:48) This completes the description of the strategy of the player II in the game G(S , e ). It remains to ∈ − ≤ ≤ (cid:48) show that it i⊂s a w−inning strategy. We should ﬁnd{s|ubset|s Yi }[F 1(βi)]µi for 1 i e as well as subsets Y (cid:48) F 1(β (cid:48) ) for i < θ such that sup Y (cid:48) ; i < θ = θ and e +i e +i e +i ##LTLine## ##LTLine## ##LTLine##

text: SHAHRAM

text: MOHSENIPOUR

text: for

text: the

text: player

text: II

text: in

text: the

text: game

text: G(S

text: e

text: Suppose

text: the

text: player

text: I

text: plays

text: according

text: to

text: the

text: following

text: strategy:

text: µ

text: µ

text: µ

text: e

text: Since

text: S

text: is

text: e-superlarge,

text: then

text: the

text: player

text: II

text: has

text: a

text: winning

text: strategy:

text: γ

text: γ

text: γ

text: i

text: θ

text: e

text: e+i

text: for

text: the

text: game

text: G(S,

text: e).

text: Put

text: i

text: p

text: (if

text: there

text: is

text: no

text: m

text: such

text: that

text: i

text: u,

text: then

text: put

text: p

text: i

text: and

text: note

text: that

text: i

text: u

text: There

text: are

text: several

text: cases

text: to

text: be

text: considered.

text: Case

text: I:

text: e

text: p.

text: Case

text: II:

text: e

text: p.

text: Case

text: I:

text: (e

text: p)

text: First

text: recall

text: the

text: deﬁnition

text: of

text: the

text: elementary

text: end

text: extension

text: chain

text: of

text: initial

text: submodels

text: M

text: i

text: θ

text: i

text: from

text: the

text: previous

text: section.

text: For

text: simplicity

text: we

text: denote

text: M

text: by

text: M∗

text: and

text: set

text: M∗

text: χ.

text: Assume

text: that

text: is

text: a

text: γp

text: new

text: symbol

text: diﬀerent

text: from

text: any

text: element

text: of

text: M.

text: In

text: this

text: case

text: we

text: face

text: with

text: three

text: subcases:

text: Subcase

text: (Ia):

text: e

text: p.

text: Subcase

text: (Ib):

text: p

text: e

text: e

text: Subcase

text: (Ic):

text: p

text: e

text: e

text: Subcase

text: (Ia):

text: (e

text: p)

text: Let

text: e

text: p

text: d

text: where

text: d

text: Suppose

text: the

text: following

text: are

text: the

text: ordinals

text: given

text: by

text: the

text: wining

text: strategy

text: of

text: the

text: player

text: II

text: against

text: the

text: above

text: mentioned

text: strategy

text: of

text: player

text: I

text: in

text: the

text: game

text: G(S,

text: e):

text: γ

text: (µ

text: γ

text: (µ

text: µ

text: γ

text: (µ

text: µ

text: i

text: θ

text: e

text: e

text: e+i

text: e

text: This

text: implies

text: that

text: γ

text: γ

text: γ

text: for

text: i

text: θ

text: and

text: there

text: exist

text: sets:

text: i

text: X

text: [F

text: X

text: [F

text: )]µe

text: e

text: e

text: as

text: well

text: as

text: the

text: following

text: sets:

text: X

text: i

text: θ

text: e+i

text: e+i

text: such

text: that

text: i

text: θ

text: and

text: e

text: r

text: r,f

text: X

text: F

text: X

text: S,

text: i

text: e+i

text: i<θ

text: where

text: f

text: s

text: e.

text: Now

text: we

text: move

text: towards

text: deﬁning

text: β’s

text: which

text: guarantee

text: the

text: winning

text: of

text: the

text: player

text: II

text: in

text: the

text: game

text: G(S

text: e

text: Let

text: β

text: (µ

text: µ

text: γ

text: (µ

text: µ

text: for

text: j

text: p.

text: j

text: j

text: j

text: j

text: Suppose

text: µ

text: is

text: given.

text: Put

text: λ

text: max(µ

text: Let

text: κ

text: be

text: a

text: cardinal

text: with

text: θ

text: κ

text: (λ

text: and

text: δ

text: r

text: is

text: the

text: least

text: ordinal

text: such

text: that

text: X

text: κ

text: Now

text: set

text: β

text: (µ

text: µ

text: γ

text: (µ

text: µ

text: e

text: If

text: d

text: then

text: this

text: completes

text: the

text: description

text: of

text: the

text: strategy

text: of

text: the

text: player

text: II

text: in

text: the

text: game

text: G(S

text: e

text: If

text: d

text: then

text: for

text: i

text: d

text: suppose

text: we

text: have

text: deﬁned

text: β

text: β

text: and

text: µ

text: is

text: given.

text: Set

text: p+(i

text: p+i

text: λi

text: µp+i)

text: and|let

text: κi

text: |b≥e

text: any

text: cardinal

text: and

text: θ.

text: Suppose

text: δi

text: is

text: the

text: least

text: ordinal

text: θ

text: and

text: δ

text: such

text: that

text: X

text: κ

text: Now

text: we

text: deﬁne

text: i

text: e+δi

text: i

text: β

text: (µ

text: µ

text: γ

text: (µ

text: µ

text: p+i

text: p+i

text: e+δi

text: e

text: So

text: far

text: we

text: have

text: deﬁned

text: β

text: β

text: For

text: i

text: θ

text: let

text: e

text: β

text: (µ

text: µ

text: γ

text: (µ

text: µ

text: e

text: +i

text: e

text: e+δd+i

text: e

text: This

text: completes

text: the

text: description

text: of

text: the

text: strategy

text: of

text: the

text: player

text: II

text: in

text: the

text: game

text: G(S

text: e

text: It

text: remains

text: to

text: show

text: that

text: it

text: i⊂s

text: a

text: w−inning

text: strategy.

text: We

text: should

text: ﬁnd{s|ubset|s

text: Yi

text: }[F

text: for

text: i

text: e

text: as

text: well

text: as

text: subsets

text: Y

text: F

text: for

text: i

text: θ

text: such

text: that

text: sup

text: Y

text: i

text: θ

text: θ

text: and

text: e

text: +i

text: e

text: +i

text: e

text: +i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##