text: 12 TABLE I. Mean exit times T− and T to leave the left and right wells, respectively, as derived from numerical simulations + and theory. We compare the results obtained using diﬀerent start and end points. → → T− (left right) T (right left) + x x Eq. (23) SSA x x Eq. (23) SSA 0 1 0 1 ρ−(x ) (x∗ + x )/2 11.1190 11.1275 ρ (x ) (x− + x∗)/2 11.5366 11.5573 0 + + 0 x− (x∗ + x )/2 11.4443 11.4610 x (x− + x∗)/2 11.8714 11.9482 + + 0 (x∗ + x )/2 11.6121 11.3253 1000a (x− + x∗)/2 11.9853 12.1354 + 0 x∗b 13.7263 13.7463 1000a x∗b 11.2982 11.5974 (cid:16) (cid:17) 260 (a) 240 k s (t) = − 1 − e−(k−+k+)t . (25) + k + k 220 − + 200 The stationary value s = k− = 1 − θ, represents + k−+k+ X180 the proportion of time that X spends in the right-hand 160 well on average. For S to be a good approximation to X this should be equal to the integral of ρ(x) over the 140 right-hand well. This gives us one relationship between 120 k and k , which determines the ratio between the mean + − switching times T . It can be shown that choosing k 100 ± + 0 1 2 3 4 5 6 and k which satisfy this constraint will ensure that the time t − 0.2 reduced process accurately captures the mean behaviour (b) of Y . 0 To accurately capture the variance in Y we need to capture the rate of approach to this stationary solution /b) −0.2 )∞ accurately, i.e. we need to determine the time constant X −0.4 ψ = 1/(k +k ). We can avoid the diﬃculty of determin- − ing when−a sw+itch in X has occurred by considering the X(t) −0.6 time dependent mean of both X and the reduced process g(( o −0.8 l S. If we can match the decay rates of these means, then we will have determined the rate constant ψ accurately. −1 The mean of X under the reduced model is given by −1.2 0 1 2 3 4 5 6 time t X(t) = θX +(cid:82)(1−θ)X −(1−θ)(X −X )e−t/ψ, (26) − + + − FIG. 14. Estimation of ψ from (26). (a) Time-evolution of the mean X(t) with X(0) = X− (solid blue line). The stationary value X∞ is shown as a dash red line. The curve has been where X = xρ (x) dx are the mean values of X ± Ω± ± obtained from 105 SSA−realisations of the full−model (6−). (b) restricted to the right and left well respectively. The equi- Transformation log((X X∞)/b) with b = (θ 1)(X X−) + librium value is X∞ = θX− + (1 − θ)X+. By comparing using X(t) in (a). The parameter ψ is estimated from the (26) to an ensemble of short-time stochastic simulations slope of this curve, using (26). The black dashed line shows of the full process X(t) we are able to get a good estimate the section we use for the linear ﬁt (shown in as a dot-dashed ψ, which is then enough to determine k and k . red line). − + In Figure ??(a) we show the computed time evolution of X, the mean of the full X process (obtained as the average of 105 runs). If from this we subtract the large time behaviour X and then take a logarithm we should ∞ We see that these are in the same range as those in obtain a straight line with gradient −1/ψ, as shown in Fig. ??(b). We ﬁt a straight line to the part of the curve Table I. These are the values of T± that have been used in all simulations for RM2 and RM3 presented in Sub- in which X −X lies between 80% and 40% of its initial ∞ sections and . value, in order to avoid any initial transients. The ﬁt is IV. CALCULATING EXTINCTION TIMES IN A good, and leads to the following values for θ, ψ and T : ± PREDATOR–PREY SYSTEM θ = 0.4729, ψ = 5.6602, In this section we apply the model reduction methods (27) T = 10.7379, T = 11.9697. developed above to an ecological model. In particular, we − + ##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##

text: TABLE

text: I.

text: Mean

text: exit

text: times

text: T−

text: and

text: T

text: to

text: leave

text: the

text: left

text: and

text: right

text: wells,

text: respectively,

text: as

text: derived

text: from

text: numerical

text: simulations

text: and

text: theory.

text: We

text: compare

text: the

text: results

text: obtained

text: using

text: diﬀerent

text: start

text: and

text: end

text: points.

label: table

text: T−

text: (left

text: right)

text: T

text: (right

text: left)

text: x

text: x

text: Eq.

text: SSA

text: x

text: x

text: Eq.

text: SSA

text: ρ−(x

text: (x∗

text: x

text: ρ

text: (x

text: (x−

text: x−

text: (x∗

text: x

text: x

text: (x−

text: (x∗

text: x

text: (x−

text: x∗b

text: x∗b

text: (a)

text: k

text: s

text: (t)

text: e−(k−+k+)t

text: k

text: k

text: The

text: stationary

text: value

text: s

text: k−

text: θ,

text: represents

text: k−+k+

text: the

text: proportion

text: of

text: time

text: that

text: X

text: spends

text: in

text: the

text: right-hand

text: well

text: on

text: average.

text: For

text: S

text: to

text: be

text: a

text: good

text: approximation

text: to

text: X

text: this

text: should

text: be

text: equal

text: to

text: the

text: integral

text: of

text: ρ(x)

text: over

text: the

text: right-hand

text: well.

text: This

text: gives

text: us

text: one

text: relationship

text: between

text: k

text: and

text: k

text: which

text: determines

text: the

text: ratio

text: between

text: the

text: mean

text: switching

text: times

text: T

text: It

text: can

text: be

text: shown

text: that

text: choosing

text: k

text: and

text: k

text: which

text: satisfy

text: this

text: constraint

text: will

text: ensure

text: that

text: the

text: time

text: t

text: reduced

text: process

text: accurately

text: captures

text: the

text: mean

text: behaviour

text: (b)

text: of

text: Y

text: To

text: accurately

text: capture

text: the

text: variance

text: in

text: Y

text: we

text: need

text: to

text: capture

text: the

text: rate

text: of

text: approach

text: to

text: this

text: stationary

text: solution

text: /b)

text: accurately,

text: i.e.

text: we

text: need

text: to

text: determine

text: the

text: time

text: constant

text: X

text: ψ

text: +k

text: We

text: can

text: avoid

text: the

text: diﬃculty

text: of

text: determin-

text: ing

text: when−a

text: sw+itch

text: in

text: X

text: has

text: occurred

text: by

text: considering

text: the

text: X(t)

text: time

text: dependent

text: mean

text: of

text: both

text: X

text: and

text: the

text: reduced

text: process

text: g((

text: o

text: l

text: S.

text: If

text: we

text: can

text: match

text: the

text: decay

text: rates

text: of

text: these

text: means,

text: then

text: we

text: will

text: have

text: determined

text: the

text: rate

text: constant

text: ψ

text: accurately.

text: −1

text: The

text: mean

text: of

text: X

text: under

text: the

text: reduced

text: model

text: is

text: given

text: by

text: time

text: t

text: X(t)

text: θX

text: −X

text: )e−t/ψ,

text: FIG.

text: Estimation

text: of

text: ψ

text: from

text: (a)

text: Time-evolution

text: of

text: the

text: mean

text: X(t)

text: with

text: X−

text: (solid

text: blue

text: line).

text: The

text: stationary

text: value

text: X∞

text: is

text: shown

text: as

text: a

text: dash

text: red

text: line.

text: The

text: curve

text: has

text: been

text: where

text: X

text: xρ

text: (x)

text: dx

text: are

text: the

text: mean

text: values

text: of

text: X

text: Ω±

text: obtained

text: from

text: SSA−realisations

text: of

text: the

text: full−model

text: (b)

text: restricted

text: to

text: the

text: right

text: and

text: left

text: well

text: respectively.

text: The

text: equi-

text: Transformation

text: log((X

text: X∞)/b)

text: with

text: b

text: (θ

text: X−)

text: librium

text: value

text: is

text: X∞

text: θX−

text: θ)X+.

text: By

text: comparing

text: using

text: X(t)

text: in

text: (a).

text: The

text: parameter

text: ψ

text: is

text: estimated

text: from

text: the

text: to

text: an

text: ensemble

text: of

text: short-time

text: stochastic

text: simulations

text: slope

text: of

text: this

text: curve,

text: using

text: The

text: black

text: dashed

text: line

text: shows

text: of

text: the

text: full

text: process

text: X(t)

text: we

text: are

text: able

text: to

text: get

text: a

text: good

text: estimate

text: the

text: section

text: we

text: use

text: for

text: the

text: linear

text: ﬁt

text: (shown

text: in

text: as

text: a

text: dot-dashed

text: ψ,

text: which

text: is

text: then

text: enough

text: to

text: determine

text: k

text: and

text: k

text: red

text: line).

text: In

text: Figure

text: ??(a)

text: we

text: show

text: the

text: computed

text: time

text: evolution

text: of

text: X,

text: the

text: mean

text: of

text: the

text: full

text: X

text: process

text: (obtained

text: as

text: the

text: average

text: of

text: runs).

text: If

text: from

text: this

text: we

text: subtract

text: the

text: large

text: time

text: behaviour

text: X

text: and

text: then

text: take

text: a

text: logarithm

text: we

text: should

text: We

text: see

text: that

text: these

text: are

text: in

text: the

text: same

text: range

text: as

text: those

text: in

text: obtain

text: a

text: straight

text: line

text: with

text: gradient

text: as

text: shown

text: in

text: Fig.

text: ??(b).

text: We

text: ﬁt

text: a

text: straight

text: line

text: to

text: the

text: part

text: of

text: the

text: curve

text: Table

text: I.

text: These

text: are

text: the

text: values

text: of

text: T±

text: that

text: have

text: been

text: used

text: in

text: all

text: simulations

text: for

text: and

text: presented

text: in

text: Sub-

text: in

text: which

text: X

text: −X

text: lies

text: between

text: and

text: of

text: its

text: initial

text: sections

text: and

text: value,

text: in

text: order

text: to

text: avoid

text: any

text: initial

text: transients.

text: The

text: ﬁt

text: is

text: IV.

text: CALCULATING

text: EXTINCTION

text: TIMES

text: IN

text: A

text: good,

text: and

text: leads

text: to

text: the

text: following

text: values

text: for

text: θ,

text: ψ

text: and

text: T

text: PREDATOR–PREY

text: SYSTEM

text: θ

text: ψ

text: In

text: this

text: section

text: we

text: apply

text: the

text: model

text: reduction

text: methods

text: T

text: T

text: developed

text: above

text: to

text: an

text: ecological

text: model.

text: In

text: particular,

text: we

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text: 17 at n = N such that the results are insensitive to N; we In Table ?? we compare the theoretical results of ﬁnd that N = 150 is a good choice. The region of va- Fig. ?? with those obtained from multiple repetitions lidity of each reduced model is set by the switching scale of the SSA. We do this to have an idea of the compu- τ ∼ 102 (e.g., when τ ∼ 102 the system is in Regime tational time reduction that each reduced model gives s y 2). relative to the original system. We choose three values of the timescale τ such that they are representative of y 107 each regime: τy = 104 for Regime 1, τy = 100 for Regime 2, and τ = 1 for Regime 3. To compute T numerically, y n 106 we run the SSA of the full system (??) and the three reduced models RM1 (??), RM2 (??), and RM3 (??). 105 We initialise the system with Y (0) = 30 and X(0) ∼ ρ(x) [equivalently, for RM2 and RM3, S(0) = S with prob- 104 − T ability θ] and run the simulation until Y (t) = 0. We n 103 record the ﬁnal time and repeat the process until the standard error in T is below 1%. The twelve simulation n 102 values of T30 in Table ?? required O(104) runs to achieve Full such relative error. 101 RRMM21 Finally, in Fig. ?? we show the histograms of the time RM3 to extinction T obtained from the simulations. We com- 100 30 10−1 100 101 102 103 104 105 pare the distribution of T30 of the full system with those τy from the reduced models. We see that the Regime 2 model does a good job of capturing the distribution for T FIG. 17. Theoretical mean time to extinction n of Y as a the whole range of values of τy. In contrast the reduced function of τy. Curves obtained from the reduced models RM1 models 1 and 3 fail to capture the distribution of T30 (??), RM2 (??) and RM3 (??), and the exact full system MTE near the origin when used outside of their regime of va- (??). We use n = 30, N = 150 and the parameter values (??) lidity. This would be important if, for instance, we were and (??). The system (??) is in Regime 2 for τ around 100, y interested in the probability that the predator population and moves towards Regime 1 (3) for larger (smaller) values of became extinct within one year. τ . y V. SUMMARY AND DISCUSSION We ﬁnd that the MTE computed from the full model increases with τ roughly linearly (as expected) from y τ ∼ O(102), but that for faster timescales it becomes In this paper we developed a model reduction tech- y less sensitive to τ (see black line with circles in Fig. ??). nique for slow–fast stochastic systems with metastability. y This is because, when the system is in Regime 3 and Y (t) This is a generalisation of previous approaches based on is fast, the main contribution to the T is the time it takes stochastic averaging principles,? when the average of the for a switch in X from the right to the left well to occur fast process switches between diﬀerent quasi-stationary (i.e. T = 1/k on average) multiplied by the proba- densities instead of relaxing to a unique stationary mea- + + bility that X started in the right well, which is 1 − θ. sure. Interestingly, we found that we can still accurately Once the switch to the low-level prey has occurred, the describe the behaviour of metastable slow–fast systems extinction of Y is almost instantaneous for τ → 0. This while improving numerical eﬃciency substantially by in- y is because this extinction occurs at a rate r [see (??)] troducing a switch variable to describe the metastable − which scales like 1/τ . If we do this simple calculation, process and using stochastic averaging separately in each y we ﬁnd that T should tend to T (1 − θ) = 178.94 as of the basins of attraction. n + τ → 0, using (??). This value is consistent with the We focused on a simple class of metastable slow–fast y results plotted in Fig. ??. We see that the approxima- stochastic systems consisting of two species, one of which tion of the MTE given by RM2 agrees very well with the (X) is fast with exponentially long bistable switches. The exact MTE from the full system throughout the whole second species (Y ), which we assume is the species of in- range of timescales τ except for very small scales where terest, has dynamics that are coupled to the bistable y a small error can be perceived. As expected, the approxi- species. We identiﬁed three dynamical regimes that led mation to the MTE of RM1 is good for slow timescales τ to diﬀerent model reductions in both of our two-species y such that the system is in Regime 1 (τ → ∞), but very metastable systems. When the dynamics of Y are much y poor for fast timescales in Regime 3 when the MTE is slower than the switches in X (and thus not aﬀected by highly dependent on the bistable prey behaviour. Con- the bistable behaviour) then the reduced model RM1 versely, the approximation of RM3 is good in Regime is equivalent to the standard fast-variables reduction 3 (τ small), but underestimates the MTE of Y when used in monostable systems.? However, in the other two y the predator is very slow (and is thus not as sensible to regimes, when the evolution of Y is of the same order switches to the low-level in the prey population). or faster than the switches in X, then the switches in ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: at

text: n

text: N

text: such

text: that

text: the

text: results

text: are

text: insensitive

text: to

text: N;

text: we

text: In

text: Table

text: we

text: compare

text: the

text: theoretical

text: results

text: of

text: ﬁnd

text: that

text: N

text: is

text: a

text: good

text: choice.

text: The

text: region

text: of

text: va-

text: Fig.

text: with

text: those

text: obtained

text: from

text: multiple

text: repetitions

text: lidity

text: of

text: each

text: reduced

text: model

text: is

text: set

text: by

text: the

text: switching

text: scale

text: of

text: the

text: SSA.

text: We

text: do

text: this

text: to

text: have

text: an

text: idea

text: of

text: the

text: compu-

text: τ

text: (e.g.,

text: when

text: τ

text: the

text: system

text: is

text: in

text: Regime

text: tational

text: time

text: reduction

text: that

text: each

text: reduced

text: model

text: gives

text: s

text: y

text: relative

text: to

text: the

text: original

text: system.

text: We

text: choose

text: three

text: values

text: of

text: the

text: timescale

text: τ

text: such

text: that

text: they

text: are

text: representative

text: of

text: y

text: each

text: regime:

text: τy

text: for

text: Regime

text: τy

text: for

text: Regime

text: and

text: τ

text: for

text: Regime

text: To

text: compute

text: T

text: numerically,

text: y

text: n

text: we

text: run

text: the

text: SSA

text: of

text: the

text: full

text: system

text: and

text: the

text: three

text: reduced

text: models

text: and

text: We

text: initialise

text: the

text: system

text: with

text: Y

text: and

text: ρ(x)

text: [equivalently,

text: for

text: and

text: S

text: with

text: prob-

text: T

text: ability

text: θ]

text: and

text: run

text: the

text: simulation

text: until

text: Y

text: (t)

text: We

text: n

text: record

text: the

text: ﬁnal

text: time

text: and

text: repeat

text: the

text: process

text: until

text: the

text: standard

text: error

text: in

text: T

text: is

text: below

text: The

text: twelve

text: simulation

text: n

text: values

text: of

text: in

text: Table

text: required

text: runs

text: to

text: achieve

text: Full

text: such

text: relative

text: error.

text: Finally,

text: in

text: Fig.

text: we

text: show

text: the

text: histograms

text: of

text: the

text: time

text: to

text: extinction

text: T

text: obtained

text: from

text: the

text: simulations.

text: We

text: com-

text: pare

text: the

text: distribution

text: of

text: of

text: the

text: full

text: system

text: with

text: those

text: τy

text: from

text: the

text: reduced

text: models.

text: We

text: see

text: that

text: the

text: Regime

text: model

text: does

text: a

text: good

text: job

text: of

text: capturing

text: the

text: distribution

text: for

text: T

text: FIG.

text: Theoretical

text: mean

text: time

text: to

text: extinction

text: n

text: of

text: Y

text: as

text: a

text: the

text: whole

text: range

text: of

text: values

text: of

text: τy.

text: In

text: contrast

text: the

text: reduced

text: function

text: of

text: τy.

text: Curves

text: obtained

text: from

text: the

text: reduced

text: models

text: models

text: and

text: fail

text: to

text: capture

text: the

text: distribution

text: of

text: and

text: and

text: the

text: exact

text: full

text: system

text: MTE

text: near

text: the

text: origin

text: when

text: used

text: outside

text: of

text: their

text: regime

text: of

text: va-

text: We

text: use

text: n

text: N

text: and

text: the

text: parameter

text: values

text: lidity.

text: This

text: would

text: be

text: important

text: if,

text: for

text: instance,

text: we

text: were

text: and

text: The

text: system

text: is

text: in

text: Regime

text: for

text: τ

text: around

text: y

text: interested

text: in

text: the

text: probability

text: that

text: the

text: predator

text: population

text: and

text: moves

text: towards

text: Regime

text: for

text: larger

text: (smaller)

text: values

text: of

text: became

text: extinct

text: within

text: one

text: year.

text: τ

text: y

text: V.

text: SUMMARY

text: AND

text: DISCUSSION

text: We

text: ﬁnd

text: that

text: the

text: MTE

text: computed

text: from

text: the

text: full

text: model

text: increases

text: with

text: τ

text: roughly

text: linearly

text: (as

text: expected)

text: from

text: y

text: τ

text: but

text: that

text: for

text: faster

text: timescales

text: it

text: becomes

text: In

text: this

text: paper

text: we

text: developed

text: a

text: model

text: reduction

text: tech-

text: y

text: less

text: sensitive

text: to

text: τ

text: (see

text: black

text: line

text: with

text: circles

text: in

text: Fig.

text: nique

text: for

text: slow–fast

text: stochastic

text: systems

text: with

text: metastability.

text: y

text: This

text: is

text: because,

text: when

text: the

text: system

text: is

text: in

text: Regime

text: and

text: Y

text: (t)

text: This

text: is

text: a

text: generalisation

text: of

text: previous

text: approaches

text: based

text: on

text: is

text: fast,

text: the

text: main

text: contribution

text: to

text: the

text: T

text: is

text: the

text: time

text: it

text: takes

text: stochastic

text: averaging

text: principles,?

text: when

text: the

text: average

text: of

text: the

text: for

text: a

text: switch

text: in

text: X

text: from

text: the

text: right

text: to

text: the

text: left

text: well

text: to

text: occur

text: fast

text: process

text: switches

text: between

text: diﬀerent

text: quasi-stationary

text: (i.e.

text: T

text: on

text: average)

text: multiplied

text: by

text: the

text: proba-

text: densities

text: instead

text: of

text: relaxing

text: to

text: a

text: unique

text: stationary

text: mea-

text: bility

text: that

text: X

text: started

text: in

text: the

text: right

text: well,

text: which

text: is

text: θ.

text: sure.

text: Interestingly,

text: we

text: found

text: that

text: we

text: can

text: still

text: accurately

text: Once

text: the

text: switch

text: to

text: the

text: low-level

text: prey

text: has

text: occurred,

text: the

text: describe

text: the

text: behaviour

text: of

text: metastable

text: slow–fast

text: systems

text: extinction

text: of

text: Y

text: is

text: almost

text: instantaneous

text: for

text: τ

text: This

text: while

text: improving

text: numerical

text: eﬃciency

text: substantially

text: by

text: in-

text: y

text: is

text: because

text: this

text: extinction

text: occurs

text: at

text: a

text: rate

text: r

text: [see

text: troducing

text: a

text: switch

text: variable

text: to

text: describe

text: the

text: metastable

text: which

text: scales

text: like

text: If

text: we

text: do

text: this

text: simple

text: calculation,

text: process

text: and

text: using

text: stochastic

text: averaging

text: separately

text: in

text: each

text: y

text: we

text: ﬁnd

text: that

text: T

text: should

text: tend

text: to

text: T

text: θ)

text: as

text: of

text: the

text: basins

text: of

text: attraction.

text: n

text: τ

text: using

text: This

text: value

text: is

text: consistent

text: with

text: the

text: We

text: focused

text: on

text: a

text: simple

text: class

text: of

text: metastable

text: slow–fast

text: y

text: results

text: plotted

text: in

text: Fig.

text: We

text: see

text: that

text: the

text: approxima-

text: stochastic

text: systems

text: consisting

text: of

text: two

text: species,

text: one

text: of

text: which

text: tion

text: of

text: the

text: MTE

text: given

text: by

text: agrees

text: very

text: well

text: with

text: the

text: (X)

text: is

text: fast

text: with

text: exponentially

text: long

text: bistable

text: switches.

text: The

text: exact

text: MTE

text: from

text: the

text: full

text: system

text: throughout

text: the

text: whole

text: second

text: species

text: (Y

text: which

text: we

text: assume

text: is

text: the

text: species

text: of

text: in-

text: range

text: of

text: timescales

text: τ

text: except

text: for

text: very

text: small

text: scales

text: where

text: terest,

text: has

text: dynamics

text: that

text: are

text: coupled

text: to

text: the

text: bistable

text: y

text: a

text: small

text: error

text: can

text: be

text: perceived.

text: As

text: expected,

text: the

text: approxi-

text: species.

text: We

text: identiﬁed

text: three

text: dynamical

text: regimes

text: that

text: led

text: mation

text: to

text: the

text: MTE

text: of

text: is

text: good

text: for

text: slow

text: timescales

text: τ

text: to

text: diﬀerent

text: model

text: reductions

text: in

text: both

text: of

text: our

text: two-species

text: y

text: such

text: that

text: the

text: system

text: is

text: in

text: Regime

text: (τ

text: but

text: very

text: metastable

text: systems.

text: When

text: the

text: dynamics

text: of

text: Y

text: are

text: much

text: y

text: poor

text: for

text: fast

text: timescales

text: in

text: Regime

text: when

text: the

text: MTE

text: is

text: slower

text: than

text: the

text: switches

text: in

text: X

text: (and

text: thus

text: not

text: aﬀected

text: by

text: highly

text: dependent

text: on

text: the

text: bistable

text: prey

text: behaviour.

text: Con-

text: the

text: bistable

text: behaviour)

text: then

text: the

text: reduced

text: model

text: versely,

text: the

text: approximation

text: of

text: is

text: good

text: in

text: Regime

text: is

text: equivalent

text: to

text: the

text: standard

text: fast-variables

text: reduction

text: (τ

text: small),

text: but

text: underestimates

text: the

text: MTE

text: of

text: Y

text: when

text: used

text: in

text: monostable

text: systems.?

text: However,

text: in

text: the

text: other

text: two

text: y

text: the

text: predator

text: is

text: very

text: slow

text: (and

text: is

text: thus

text: not

text: as

text: sensible

text: to

text: regimes,

text: when

text: the

text: evolution

text: of

text: Y

text: is

text: of

text: the

text: same

text: order

text: switches

text: to

text: the

text: low-level

text: in

text: the

text: prey

text: population).

text: or

text: faster

text: than

text: the

text: switches

text: in

text: X,

text: then

text: the

text: switches

text: in

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text: 20 where ρ(x) is the no(cid:90)rmalised steady solution of (??) Equation (??) is the Fokker–Planck equation associ- [given by (13)], and ated with the reduced stochastic model (16) appropriate for Regime 1 when Y is approximated by a continuous ∞ random variable. This asymptotic reduction relies on the q(y) = P(x, y, 0) dx. fact that the stochastic process for X reaches steady state 0 on a timescale which is faster than the timescale for the Now we move back to O(1) times and consider equation evolution of Y . (cid:28) ∼ (??). Expanding P ∼ P(0) + (cid:15)/τy P(1) + · · · , gives, at 2. Regime 2: (cid:15) τs τy leading-order,(cid:26) (cid:27) To analyse Regime 2 asymptotically we need to ensure that X is metastable, and quantify the switching time. ∂ ∂ [d(x)P(0)] − v(x)P(0) = 0. (A4) We suppose then that 1/δ, the typical equilibrium value ∂x ∂x of X, is large, and scale the rate constants as in (7). We set xˆ = δx, where xˆ is O(1) as δ → 0. We will see that Thus τ ∼ er/δ for some constant r > 0. In terms of the new s scaled variables eq(cid:26)uation (??) becomes (cid:27) P(0)(x, y, t) = C(y, t)ρ(x), (A5) (cid:26) (cid:27) ∂P τ ∂ ∂ = y [δdˆ(xˆ)P] − vˆ(xˆ)P ∂t (cid:15) ∂xˆ ∂xˆ where C(y, t) is arbitrary at this stage. Matching this (A9) ∂ ∂ solution for long times with the short times solution (??) + [D(xˆ, y)P] − V (xˆ, y)P , gives C(y, 0) = q(y). At the next order equation (??) ∂y ∂y gives (cid:26) (cid:27) where dˆ(xˆ) = δd(x), etc. With this scaling we see that the diﬀusion δdˆ(xˆ) of X is weaker than the drift ∂P(0) ∂ ∂(cid:26) (cid:27) = [d(x)P(1)] − v(x)P(1) vˆ(xˆ) when δ (cid:28) 1, which is the reason that switches do ∂t ∂x ∂x not occur frequently (τ (cid:29) (cid:15)); the parameter δ makes (A6) s ∂ ∂ explicit this separation of timescales between drift of X + ∂y ∂y [D(x, y)P(0)] − V (x, y)P(0) . and switches in X (this is the weak noise limit? ). It is convenient to write (??) in the form Integrating this equation with respect to x (using no- ﬂux boundary conditions at x = 0, ∞) gives the following ∂P ∂J solvability condi(cid:26)tion for C: (cid:27) (cid:15)ˆ + (cid:15)ˆ y = Lδ, (A10) (cid:2) (cid:3) ∂t ∂y ∂C ∂ ∂ where (cid:15)ˆ = (cid:15)/τ , J = − ∂ [D(xˆ, y)P] + V (xˆ, y)P and (y, t) = D(y)C(y, t) − V (y)C(y, t) (A7) y y ∂y ∂t ∂y ∂y Lδ is a linear opera(cid:18)tor acting on the vari(cid:19)able xˆ only: (cid:90) where ∂ ∂ L P ≡ [δdˆ(xˆ)P] − vˆ(xˆ)P . (A11) δ ∂xˆ ∂xˆ F(y) = F(x, y)ρ(x) dx, for F = D, V. (cid:90) The marginal stationary density in xˆ [ρ(x) in (13)] cor- Finally, since responds to the zero-eigenvalue eigenfunction L , that is δ ∞ Lδρ(xˆ) = 0, giving, in the(cid:34)sca(cid:90)led variabl(cid:35)es, C(y, t) = P(x, y, t) dx 0 A 1 xˆ vˆ(s) we see that C(y, t) is the marginal density for Y (t), so ρ(xˆ) = exp ds , (A12) that (??) gives the evolution of the probability density dˆ(xˆ) δ 0 dˆ(s) function of a reduced process for Y (t) where the fast variable X(t) has been averaged out. Going back to the where A is the normalisation constant. original time variable w(cid:26)e can then write (cid:27) The novelty in this bistable problem is the coupling of (cid:2) (cid:3) X with Y , that is, in the term J in (??). We sketch y the following calculation since it follows closely that of ∂p 1 ∂ ∂ (y, t) = D(y)p − V (y) (A8) Refs. ? and ? . By using a WKB approximation for ∂t τy ∂y ∂y small δ (with (cid:15)ˆ (cid:28) δ) we ﬁnd solutions to (??) of the form ##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: where

text: ρ(x)

text: is

text: the

text: steady

text: solution

text: of

text: Equation

text: is

text: the

text: Fokker–Planck

text: equation

text: associ-

text: [given

text: by

text: and

text: ated

text: with

text: the

text: reduced

text: stochastic

text: model

text: appropriate

text: for

text: Regime

text: when

text: Y

text: is

text: approximated

text: by

text: a

text: continuous

text: random

text: variable.

text: This

text: asymptotic

text: reduction

text: relies

text: on

text: the

text: q(y)

text: P(x,

text: y,

text: dx.

text: fact

text: that

text: the

text: stochastic

text: process

text: for

text: X

text: reaches

text: steady

text: state

text: on

text: a

text: timescale

text: which

text: is

text: faster

text: than

text: the

text: timescale

text: for

text: the

text: Now

text: we

text: move

text: back

text: to

text: times

text: and

text: consider

text: equation

text: evolution

text: of

text: Y

label: section

text: Expanding

text: P

text: gives,

text: at

text: Regime

text: τs

text: τy

text: To

text: analyse

text: Regime

text: asymptotically

text: we

text: need

text: to

text: ensure

text: that

text: X

text: is

text: metastable,

text: and

text: quantify

text: the

text: switching

text: time.

text: We

text: suppose

text: then

text: that

text: the

text: typical

text: equilibrium

text: value

text: ∂x

text: ∂x

text: of

text: X,

text: is

text: large,

text: and

text: scale

text: the

text: rate

text: constants

text: as

text: in

text: We

text: set

text: xˆ

text: δx,

text: where

text: xˆ

text: is

text: as

text: δ

text: We

text: will

text: see

text: that

text: Thus

text: τ

text: er/δ

text: for

text: some

text: constant

text: r

text: In

text: terms

text: of

text: the

text: new

text: s

text: scaled

text: variables

text: becomes

text: y,

text: t)

text: C(y,

text: t)ρ(x),

text: ∂P

text: τ

text: y

text: [δdˆ(xˆ)P]

text: vˆ(xˆ)P

text: ∂t

text: ∂xˆ

text: ∂xˆ

text: where

text: C(y,

text: t)

text: is

text: arbitrary

text: at

text: this

text: stage.

text: Matching

text: this

text: solution

text: for

text: long

text: times

text: with

text: the

text: short

text: times

text: solution

text: [D(xˆ,

text: y)P]

text: V

text: (xˆ,

text: y)P

text: gives

text: C(y,

text: q(y).

text: At

text: the

text: next

text: order

text: equation

text: ∂y

text: ∂y

text: gives

text: where

text: dˆ(xˆ)

text: δd(x),

text: etc.

text: With

text: this

text: scaling

text: we

text: see

text: that

text: the

text: diﬀusion

text: δdˆ(xˆ)

text: of

text: X

text: is

text: weaker

text: than

text: the

text: drift

text: vˆ(xˆ)

text: when

text: δ

text: which

text: is

text: the

text: reason

text: that

text: switches

text: do

text: ∂t

text: ∂x

text: ∂x

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text: occur

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text: (τ

text: the

text: parameter

text: δ

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text: s

text: explicit

text: this

text: separation

text: of

text: timescales

text: between

text: drift

text: of

text: X

text: ∂y

text: ∂y

text: [D(x,

text: V

text: (x,

text: and

text: switches

text: in

text: X

text: (this

text: is

text: the

text: weak

text: noise

text: limit?

text: It

text: is

text: convenient

text: to

text: write

text: in

text: the

text: form

text: Integrating

text: this

text: equation

text: with

text: respect

text: to

text: x

text: (using

text: no-

text: ﬂux

text: boundary

text: conditions

text: at

text: x

text: gives

text: the

text: following

text: ∂P

text: ∂J

text: solvability

text: for

text: C:

text: y

text: Lδ,

text: ∂t

text: ∂y

text: ∂C

text: where

text: J

text: [D(xˆ,

text: y)P]

text: V

text: (xˆ,

text: y)P

text: and

text: (y,

text: t)

text: D(y)C(y,

text: t)

text: V

text: (y)C(y,

text: t)

text: y

text: y

text: ∂y

text: ∂t

text: ∂y

text: ∂y

text: Lδ

text: is

text: a

text: linear

text: acting

text: on

text: the

text: xˆ

text: only:

text: where

text: L

text: P

text: [δdˆ(xˆ)P]

text: vˆ(xˆ)P

text: δ

text: ∂xˆ

text: ∂xˆ

text: F(y)

text: F(x,

text: y)ρ(x)

text: dx,

text: for

text: F

text: D,

text: V.

text: The

text: marginal

text: stationary

text: density

text: in

text: xˆ

text: [ρ(x)

text: in

text: cor-

text: Finally,

text: since

text: responds

text: to

text: the

text: zero-eigenvalue

text: eigenfunction

text: L

text: that

text: is

text: δ

text: Lδρ(xˆ)

text: giving,

text: in

text: C(y,

text: t)

text: P(x,

text: y,

text: t)

text: dx

text: A

text: xˆ

text: vˆ(s)

text: we

text: see

text: that

text: C(y,

text: t)

text: is

text: the

text: marginal

text: density

text: for

text: Y

text: (t),

text: so

text: ρ(xˆ)

text: exp

text: ds

text: that

text: gives

text: the

text: evolution

text: of

text: the

text: probability

text: density

text: dˆ(xˆ)

text: δ

text: dˆ(s)

text: function

text: of

text: a

text: reduced

text: process

text: for

text: Y

text: (t)

text: where

text: the

text: fast

text: variable

text: X(t)

text: has

text: been

text: averaged

text: out.

text: Going

text: back

text: to

text: the

text: where

text: A

text: is

text: the

text: normalisation

text: constant.

text: original

text: time

text: variable

text: can

text: then

text: write

text: The

text: novelty

text: in

text: this

text: bistable

text: problem

text: is

text: the

text: coupling

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text: X

text: with

text: Y

text: that

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text: the

text: term

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text: sketch

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text: following

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text: it

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text: of

text: ∂p

text: (y,

text: t)

text: D(y)p

text: V

text: (y)

text: Refs.

text: and

text: By

text: using

text: a

text: WKB

text: approximation

text: for

text: ∂t

text: τy

text: ∂y

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text: small

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text: δ)

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text: ﬁnd

text: solutions

text: to

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text: 22 (cid:28) (cid:29) with the orthogonality relationship (cid:104)ρ , ξ (cid:105) = δ where i j ij δij is the Kronecker delta. The adjoint eigenfunction cor- ∂P ∂A− ∂A+ responding to λ0 = 0 is simply ξ0 = 1. The ﬁrst adjoint 1, ∂t = Φ− ∂t + Φ+ ∂t , (A29) eigenfunction ξ (xˆ) is approximately (see e.g. Ref. ? ) 1  (cid:18)(cid:113) (cid:19) (cid:28)and (cid:29) ξ1(xˆ)∼ΦΦ−−((ΦΦΦΦ++++++ΦΦ−−)) − 2Φ1− erf |γ2∗δ|(xˆ − xˆv∗) xxˆˆ ≈< xxˆˆv∗v∗,, 1, ∂∂(cid:90)Jyy (cid:18) (cid:19) (Φ+−+1Φ−) xˆ > xˆv∗, = (cid:90) ∂∂y V(cid:18)(xˆ, y)A− − ∂∂y [D(xˆ, y)A−] (cid:19)φ(xˆ) dxˆ (A23) Ω− ∂ ∂ + V (xˆ, y)A − [D(xˆ, y)A ] φ(xˆ) dxˆ. + + The ﬁrst eigenvalue can now be computed by taking Ω+ ∂y ∂y the inner product of (??) with a suitable test function ϕ: (A30) Taking the integrals over Ω inside the y derivatives ± (cid:28) (cid:29) (cid:18) (cid:19) (cid:104)ϕ, L ρ (cid:105) = λ (cid:104)ϕ, ρ (cid:105). (A24) and using that ρ (xˆ) = φ(xˆ)/Φ , we ﬁnd δ 1 1 1 ± ± ∂J ∂ (cid:18) ∂[D (y)A ] (cid:19) Here we u(cid:90)se ϕ = 1Ω−. T(cid:18)he left hand side gives(cid:19)(cid:12) 1, y = Φ− V −(y)A− − − − (cid:12) ∂y ∂y ∂y (cid:12) (cid:12) ∂ ∂[D (y)A ] xˆv∗ ∂ + Φ+ V +(y)A+ − + + , (cid:104)ϕ, L ρ (cid:105) = L ρ dxˆ = [δdˆ(xˆ)ρ ] − vˆ(xˆ)ρ ∂y ∂y δ 1 δ 1 1 1 0 (cid:18) ∂xˆ (cid:19) (cid:114) xˆ=xˆv∗ (A31) √ = δdˆ(xˆv)ρ(cid:48) (z)| (cid:90) ∗ 1 z=0 where √ Φ |γ | = − δeu(xˆv∗)/δ 1 + − ∗ . Φ 2π + F (y) := F(xˆ, y)ρ (xˆ) dxˆ, for F = D, V. (A32) ± ± (A25) Ω± Thus The right-hand side is (cid:115) (cid:90) (cid:18) (cid:19) ∂A ∂A − + 0 = Φ + Φ xˆv∗ 2πδ − ∂t + ∂t (cid:104)ϕ, ρ1(cid:105) = 0 ρ1 dxˆ = Φ− ∼ γ− φ(xˆv−), (A26) + Φ− ∂ (cid:18)V −(y)A− − ∂[D−(y)A−](cid:19) (A33) ∂y ∂y using (??). Combining (??) and (??) gives the expo- + Φ ∂ V (y)A − ∂[D+(y)A+] . nentially small eigenvalue (cid:115) + ∂y + + ∂y (cid:18) (cid:19) (cid:18) (cid:19) Next we take the inner product of (??) with the ﬁrst (cid:28) (cid:29) dˆ(xˆv ) Φ |γ | u(xˆv) − u(xˆv ) adjoint eigenfunction: λ ∼ − − 1 + − ∗ exp ∗ − . 1 2π Φ γ δ + − ∂P ∂J (A27) (cid:15)ˆ ξ , + y = (cid:104)ξ , L P(cid:105) = (cid:104)L∗ξ , P(cid:105) = λ (cid:104)ξ , P(cid:105). 1 ∂t ∂y 1 δ δ 1 1 1 Note that the argument of the exponential is negative (A34) since u(xˆv) < u(xˆv ). ∗ − Finally, we seek the diﬀerential equations describing Using (??) and (??) gives another PDE for A− and (cid:18) (cid:19) the evolution of A+ and A−. First we integrate the equa- A+: tion (??) with respect to xˆ (equivalent to taking the inner pro(cid:28)duct with the (cid:29)adjoint eigenfunction ξ0 = 1): ∂A− − ∂(cid:18)A+ − ∂ ∂[D−(y)A−] −(cid:19)V (y)A − − ∂t ∂t ∂y ∂y ∂P ∂J ∂ ∂[D (y)A ] (cid:15)ˆ 1, + y = (cid:104)ξ , L P(cid:105) = (cid:104)L∗ξ , P(cid:105) = 0. (A28) + + + − V (y)A (A35) ∂t ∂y 0 δ δ 0 ∂y ∂y + + λ = 1 (A − A ). − + From the left-hand side we have (cid:15)ˆ ##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##

label: equation

text: with

text: the

text: orthogonality

text: relationship

text: ξ

text: δ

text: where

text: i

text: j

text: ij

text: δij

text: is

text: the

text: Kronecker

text: delta.

text: The

text: adjoint

text: eigenfunction

text: cor-

text: ∂P

text: ∂A−

text: ∂A+

text: responding

text: to

text: λ0

text: is

text: simply

text: ξ0

text: The

text: ﬁrst

text: adjoint

text: ∂t

text: Φ−

text: ∂t

text: Φ+

text: ∂t

text: eigenfunction

text: ξ

text: (xˆ)

text: is

text: approximately

text: (see

text: e.g.

text: Ref.

text: ξ1(xˆ)∼ΦΦ−−((ΦΦΦΦ++++++ΦΦ−−))

text: 2Φ1−

text: erf

text: xˆv∗)

text: xxˆˆ

text: xxˆˆv∗v∗,,

text: xˆ

text: xˆv∗,

text: ∂∂y

text: y)A−

text: ∂∂y

text: [D(xˆ,

text: y)A−]

text: dxˆ

text: Ω−

text: V

text: (xˆ,

text: y)A

text: [D(xˆ,

text: y)A

text: φ(xˆ)

text: dxˆ.

text: The

text: ﬁrst

text: eigenvalue

text: can

text: now

text: be

text: computed

text: by

text: taking

text: Ω+

text: ∂y

text: ∂y

text: the

text: inner

text: product

text: of

text: with

text: a

text: suitable

text: test

text: function

text: ϕ:

text: Taking

text: the

text: integrals

text: over

text: Ω

text: inside

text: the

text: y

text: derivatives

text: L

text: ρ

text: λ

text: ρ

text: and

text: using

text: that

text: ρ

text: (xˆ)

text: φ(xˆ)/Φ

text: we

text: ﬁnd

text: δ

text: ∂J

text: ∂[D

text: (y)A

text: Here

text: we

text: ϕ

text: 1Ω−.

text: left

text: hand

text: side

text: y

text: Φ−

text: V

text: −(y)A−

text: ∂y

text: ∂y

text: ∂y

text: ∂[D

text: (y)A

text: xˆv∗

text: Φ+

text: V

text: +(y)A+

text: L

text: ρ

text: L

text: ρ

text: dxˆ

text: [δdˆ(xˆ)ρ

text: vˆ(xˆ)ρ

text: ∂y

text: ∂y

text: δ

text: δ

text: ∂xˆ

text: xˆ=xˆv∗

text: (z)|

text: where

text: Φ

text: |γ

text: δeu(xˆv∗)/δ

text: Φ

text: 2π

text: F

text: (y)

text: F(xˆ,

text: y)ρ

text: (xˆ)

text: dxˆ,

text: for

text: F

text: D,

text: V.

text: Ω±

text: Thus

text: The

text: right-hand

text: side

text: is

text: ∂A

text: ∂A

text: Φ

text: Φ

text: xˆv∗

text: 2πδ

text: ∂t

text: ∂t

text: ρ1

text: dxˆ

text: Φ−

text: γ−

text: φ(xˆv−),

text: Φ−

text: −(y)A−

text: ∂y

text: ∂y

text: using

text: Combining

text: and

text: gives

text: the

text: expo-

text: Φ

text: V

text: (y)A

text: ∂[D+(y)A+]

text: nentially

text: small

text: eigenvalue

text: ∂y

text: ∂y

text: Next

text: we

text: take

text: the

text: inner

text: product

text: of

text: with

text: the

text: ﬁrst

text: dˆ(xˆv

text: Φ

text: |γ

text: u(xˆv)

text: u(xˆv

text: adjoint

text: eigenfunction:

text: λ

text: exp

text: 2π

text: Φ

text: γ

text: δ

text: ∂P

text: ∂J

text: ξ

text: y

text: L

text: λ

text: ∂t

text: ∂y

text: δ

text: δ

text: Note

text: that

text: the

text: argument

text: of

text: the

text: exponential

text: is

text: negative

text: since

text: u(xˆv)

text: u(xˆv

text: Finally,

text: we

text: seek

text: the

text: diﬀerential

text: equations

text: describing

text: Using

text: and

text: gives

text: another

text: PDE

text: for

text: A−

text: and

text: the

text: evolution

text: of

text: A+

text: and

text: A−.

text: First

text: we

text: integrate

text: the

text: equa-

text: A+:

text: tion

text: with

text: respect

text: to

text: xˆ

text: (equivalent

text: to

text: taking

text: the

text: inner

text: with

text: the

text: eigenfunction

text: ξ0

text: ∂A−

text: ∂[D−(y)A−]

text: (y)A

text: ∂t

text: ∂t

text: ∂y

text: ∂y

text: ∂P

text: ∂J

text: ∂[D

text: (y)A

text: y

text: L

text: V

text: (y)A

text: ∂t

text: ∂y

text: δ

text: δ

text: ∂y

text: ∂y

text: λ

text: (A

text: A

text: From

text: the

text: left-hand

text: side

text: we

text: have

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 4 (cid:114) is set by the decay rate k /τ . Thus if (cid:15) and τ are 6 y y to represent these timescales we should choose k and 2 (cid:115) k to be of order unity. The other rates determine the v(X) 2d(X) 6 dX = dt + dW (t), (8a) equilibrium values of X and Y . If we suppose a typical (cid:15) (cid:15) X equilibrium value of X is O(δ−1), this corresponds to V (X, Y ) 2D(X, Y ) k1 = O(δ−1), k3 = O(δ), k4 = O(δ2). In Fig. 1(a) we plot dY = dt + dWY (t), (8b) τ τ one realisation of the SSA of (6) obtained for X(0) = 400, y y Y (0) = 200, τ = 25, and y which are of the same form as the slow–fast system of stochastic diﬀerential equations (3). The drift coef- ﬁcients v(x), V (x, y) and the diﬀusion coeﬃcients d(x) k = 0.6 δ−1, k = 1, k = 0.48 δ, 1 2 3 and D(x, y) are given by k = 0.0666 δ2, k (x) = 0.833 x, k = 1, (7) 4 5 6 (cid:15) = 0.025, δ = 0.01. v(x) = k − k x + k x(x − 1) − k x(x − 1)(x − 2), 1 2 3 4 1 d(x) = [k + k x + k x(x−1) + k x(x − 1)(x − 2)] , 1 2 3 4 2 The parameters values are chosen such that system (6) is bistable in X(t) (as in Ref. ? ) and Y (t) evolves in V (x, y) = k5(x) − k6y, the same timescale as switches in X. The rate k (x) 1 5 D(x, y) = [k (x) + k y]. (9) (which we will also use in the simulations that follow) 2 5 6 is such that the production reaction for Y can be writ- ten as X −→ X + Y with rate 0.833/τ . We also plot y Let P(x, y, t)dxdy be the probability that X(t) ∈ the stationary distribution of X, obtained by a long time [x, x + dx) and Y (t) ∈ [y, y + dy) at time t. Then simulation of the SSA with ﬁnal time t = 106 and record- the chemical Fokker–Planck (FP) equation for the joint ing the value of X every ∆t = 0.1 in Fig. 1(b). We see probability distribution function corresponding to (8) is that for these parameter values the system (6) has two given by (cid:26) (cid:27) favourable states for X. We see that X(t) spends quite a long time ﬂuctuating around one of these states be- fore switching to the other. In section A 2 we will derive ∂P 1 ∂ ∂(cid:26) (cid:27) an analytical approximation of the mean switching times (x, y, t) = [d(x)P] − v(x)P ∂t (cid:15) ∂x ∂x between states and ﬁnd that is exponentially large in the 1 ∂ ∂ parameter δ. We want to understand how the oscillation + [D(x, y)P] − V (x, y)P . between these two states inﬂuences the dynamics of Y , τy ∂y ∂y and obtain an eﬃcient and accurate way to take this ef- (10) fect into account without having to simulate the full X dynamics. This equation is complemented with no-ﬂux boundary conditions on [0, ∞)2 since the probability P must remain A. Continuous approximation: the chemical Langevin and normalised for all times. the Fokker–Planck equations The stationary distributions of X and Y , ρ(x) and q(y) respectively, correspond to taking the limit t → ∞ in the For ease of exposition we will apply our model reduc- marginal pro(cid:90)bability densities (cid:90) tion techniques not to (6) directly, but to the continu- ous approximation of this system given by the chemical Langevin equation. This approximation is similar to the ρ(x) = lim P(x, y, t) dy, q(y) = lim P(x, y, t) dx. numerical method of τ-leaping.? The idea of the approxi- t→∞ t→∞ mation is that for large X and Y the propensity functions (11) do not change signiﬁcantly after each individual reaction event, which enables us to jump forward many reaction Since the dynamics of X do not depend on Y , it is events without updating the propensities. The change in straightforward to solve for ρ(x) by integrating (10) with the number of molecules between timesteps is then Pois- respect to y to gi(cid:18)ve (cid:19) son distributed. If we now take a limit in which many reactions are considered between timesteps, but still with ∂ ∂ a small relative change in the number of molecules, X and [d(x)ρ] − v(x)ρ = 0, (12) ∂x ∂x Y may be approximated by continuous random variables and the change in molecular numbers between timesteps is approximately normally distributed (see Refs. ? ? for where we have used no-ﬂux boundary conditions at more details of this approximation). In that case we ar- y = 0 and y = ∞. Thus the stationary distribution of X rive at the chemical Langevin equations is given by ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: is

text: set

text: by

text: the

text: decay

text: rate

text: k

text: /τ

text: Thus

text: if

text: and

text: τ

text: are

text: y

text: y

text: to

text: represent

text: these

text: timescales

text: we

text: should

text: choose

text: k

text: and

text: k

text: to

text: be

text: of

text: order

text: unity.

text: The

text: other

text: rates

text: determine

text: the

text: v(X)

text: dX

text: dt

text: dW

text: (t),

text: equilibrium

text: values

text: of

text: X

text: and

text: Y

text: If

text: we

text: suppose

text: a

text: typical

text: X

text: equilibrium

text: value

text: of

text: X

text: is

text: this

text: corresponds

text: to

text: V

text: (X,

text: Y

text: Y

text: O(δ),

text: In

text: Fig.

text: we

text: plot

text: dY

text: dt

text: dWY

text: (t),

text: τ

text: τ

text: one

text: realisation

text: of

text: the

text: SSA

text: of

text: obtained

text: for

text: y

text: y

text: Y

text: τ

text: and

text: y

text: which

text: are

text: of

text: the

text: same

text: form

text: as

text: the

text: slow–fast

text: system

text: of

text: stochastic

text: diﬀerential

text: equations

text: The

text: drift

text: coef-

text: ﬁcients

text: v(x),

text: V

text: (x,

text: y)

text: and

text: the

text: diﬀusion

text: coeﬃcients

text: d(x)

text: k

text: δ−1,

text: k

text: k

text: δ,

text: and

text: D(x,

text: y)

text: are

text: given

text: by

text: k

text: k

text: (x)

text: x,

text: k

text: δ

text: v(x)

text: k

text: k

text: x

text: k

text: x(x

text: k

text: x(x

text: d(x)

text: [k

text: k

text: x

text: k

text: k

text: x(x

text: The

text: parameters

text: values

text: are

text: chosen

text: such

text: that

text: system

text: is

text: bistable

text: in

text: X(t)

text: (as

text: in

text: Ref.

text: and

text: Y

text: (t)

text: evolves

text: in

text: V

text: (x,

text: y)

text: the

text: same

text: timescale

text: as

text: switches

text: in

text: X.

text: The

text: rate

text: k

text: (x)

text: D(x,

text: y)

text: [k

text: (x)

text: k

text: y].

text: (which

text: we

text: will

text: also

text: use

text: in

text: the

text: simulations

text: that

text: follow)

text: is

text: such

text: that

text: the

text: production

text: reaction

text: for

text: Y

text: can

text: be

text: writ-

text: ten

text: as

text: X

text: X

text: Y

text: with

text: rate

text: We

text: also

text: plot

text: y

text: Let

text: P(x,

text: y,

text: t)dxdy

text: be

text: the

text: probability

text: that

text: X(t)

text: the

text: stationary

text: distribution

text: of

text: X,

text: obtained

text: by

text: a

text: long

text: time

text: [x,

text: x

text: dx)

text: and

text: Y

text: (t)

text: [y,

text: y

text: dy)

text: at

text: time

text: t.

text: Then

text: simulation

text: of

text: the

text: SSA

text: with

text: ﬁnal

text: time

text: t

text: and

text: record-

text: the

text: chemical

text: Fokker–Planck

text: (FP)

text: equation

text: for

text: the

text: joint

text: ing

text: the

text: value

text: of

text: X

text: every

text: ∆t

text: in

text: Fig.

text: We

text: see

text: probability

text: distribution

text: function

text: corresponding

text: to

text: is

text: that

text: for

text: these

text: parameter

text: values

text: the

text: system

text: has

text: two

text: given

text: by

text: favourable

text: states

text: for

text: X.

text: We

text: see

text: that

text: X(t)

text: spends

text: quite

text: a

text: long

text: time

text: ﬂuctuating

text: around

text: one

text: of

text: these

text: states

text: be-

text: fore

text: switching

text: to

text: the

text: other.

text: In

text: section

text: A

text: we

text: will

text: derive

text: ∂P

text: an

text: analytical

text: approximation

text: of

text: the

text: mean

text: switching

text: times

text: (x,

text: y,

text: t)

text: [d(x)P]

text: v(x)P

text: ∂t

text: ∂x

text: ∂x

text: between

text: states

text: and

text: ﬁnd

text: that

text: is

text: exponentially

text: large

text: in

text: the

text: parameter

text: δ.

text: We

text: want

text: to

text: understand

text: how

text: the

text: oscillation

text: [D(x,

text: y)P]

text: V

text: (x,

text: y)P

text: between

text: these

text: two

text: states

text: inﬂuences

text: the

text: dynamics

text: of

text: Y

text: τy

text: ∂y

text: ∂y

text: and

text: obtain

text: an

text: eﬃcient

text: and

text: accurate

text: way

text: to

text: take

text: this

text: ef-

text: fect

text: into

text: account

text: without

text: having

text: to

text: simulate

text: the

text: full

text: X

text: dynamics.

text: This

text: equation

text: is

text: complemented

text: with

text: no-ﬂux

text: boundary

text: conditions

text: on

text: since

text: the

text: probability

text: P

text: must

text: remain

text: A.

text: Continuous

text: approximation:

text: the

text: chemical

text: Langevin

text: and

text: normalised

text: for

text: all

text: times.

text: the

text: Fokker–Planck

text: equations

text: The

text: stationary

text: distributions

text: of

text: X

text: and

text: Y

text: ρ(x)

text: and

text: q(y)

text: respectively,

text: correspond

text: to

text: taking

text: the

text: limit

text: t

text: in

text: the

text: For

text: ease

text: of

text: exposition

text: we

text: will

text: apply

text: our

text: model

text: reduc-

text: marginal

text: densities

text: tion

text: techniques

text: not

text: to

text: directly,

text: but

text: to

text: the

text: continu-

text: ous

text: approximation

text: of

text: this

text: system

text: given

text: by

text: the

text: chemical

text: Langevin

text: equation.

text: This

text: approximation

text: is

text: similar

text: to

text: the

text: ρ(x)

text: lim

text: P(x,

text: y,

text: t)

text: dy,

text: q(y)

text: lim

text: P(x,

text: y,

text: t)

text: dx.

text: numerical

text: method

text: of

text: τ-leaping.?

text: The

text: idea

text: of

text: the

text: approxi-

text: t→∞

text: t→∞

text: mation

text: is

text: that

text: for

text: large

text: X

text: and

text: Y

text: the

text: propensity

text: functions

text: do

text: not

text: change

text: signiﬁcantly

text: after

text: each

text: individual

text: reaction

text: event,

text: which

text: enables

text: us

text: to

text: jump

text: forward

text: many

text: reaction

text: Since

text: the

text: dynamics

text: of

text: X

text: do

text: not

text: depend

text: on

text: Y

text: it

text: is

text: events

text: without

text: updating

text: the

text: propensities.

text: The

text: change

text: in

text: straightforward

text: to

text: solve

text: for

text: ρ(x)

text: by

text: integrating

text: with

text: the

text: number

text: of

text: molecules

text: between

text: timesteps

text: is

text: then

text: Pois-

text: respect

text: to

text: y

text: to

text: son

text: distributed.

text: If

text: we

text: now

text: take

text: a

text: limit

text: in

text: which

text: many

text: reactions

text: are

text: considered

text: between

text: timesteps,

text: but

text: still

text: with

text: a

text: small

text: relative

text: change

text: in

text: the

text: number

text: of

text: molecules,

text: X

text: and

text: [d(x)ρ]

text: v(x)ρ

text: ∂x

text: ∂x

text: Y

text: may

text: be

text: approximated

text: by

text: continuous

text: random

text: variables

text: and

text: the

text: change

text: in

text: molecular

text: numbers

text: between

text: timesteps

text: is

text: approximately

text: normally

text: distributed

text: (see

text: Refs.

text: for

text: where

text: we

text: have

text: used

text: no-ﬂux

text: boundary

text: conditions

text: at

text: more

text: details

text: of

text: this

text: approximation).

text: In

text: that

text: case

text: we

text: ar-

text: y

text: and

text: y

text: Thus

text: the

text: stationary

text: distribution

text: of

text: X

text: rive

text: at

text: the

text: chemical

text: Langevin

text: equations

text: is

text: given

text: by

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: ∗ (a) Equilibrium polymer volume fraction, θ p (b) Donnan swelling pressure, z Ψ m e 15 (c) crosslink fraction, α Figure 4: Variables in equilibrium versus calcium bath concentration, C (in mol/lt) for WT-mucus b at pH = 7.2 (solid curve), CF-infected mucus at pH = 7.2 (dashed curve), WT-mucus at pH = 5.0 (dash-dot curve), CF-infected mucus at pH = 5.0 (dotted curve). The bath concentration of sodium ◦ and the temperature are ﬁxed at N = 0 and T = 37 C, respectively. b ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

text: Equilibrium

text: polymer

text: volume

text: fraction,

text: θ

text: p

text: (b)

text: Donnan

text: swelling

text: pressure,

text: z

text: Ψ

text: m

text: e

text: (c)

text: crosslink

text: fraction,

text: α

text: Figure

text: Variables

label: caption

text: in

text: equilibrium

text: versus

text: calcium

text: bath

text: concentration,

text: C

text: (in

text: mol/lt)

text: for

text: WT-mucus

text: b

text: at

text: pH

text: (solid

text: curve),

text: CF-infected

text: mucus

text: at

text: pH

text: (dashed

text: curve),

text: WT-mucus

text: at

text: pH

text: (dash-dot

text: curve),

text: CF-infected

text: mucus

text: at

text: pH

text: (dotted

text: curve).

text: The

text: bath

text: concentration

text: of

text: sodium

text: and

text: the

text: temperature

text: are

text: ﬁxed

text: at

text: N

text: and

text: T

text: C,

text: respectively.

text: b

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##

text: ##LTFigure##


text: − with symbol C = c, h, n, c , z = z = 1, z = 2 and z = 1. The subscript ‘b’ denotes l n h c c l the corresponding bath concentrations [35]. The electrostatic potential, Ψ , is determined by the e electroneutrality constraint inside the gel, namely, − (2c + n + h c )θ + z m = 0, (19) l s m T where z is the average residual charge of the unbound monomers which depends on the amount m of binding with ions, − z m = w m. (20) m T Since both the electrostatic potential and polymer particle fraction are assumed to be zero outside the gel, electroneutrality in the bath requires that − 2c + n + h c = 0. (21) b b b lb Finally, the crosslink fraction (or the fraction of monomers bound with calcium) is α = x , where Eqns. (15, 17) deﬁne the concentrations, m , x, respectively. mT T In summary, the dynamical motion of a freely swelling mucus-gel is modeled by the system of equations including the mass conservation P D E (2), total volume conservation P D E (3) together with force balance P D E S (4-5) and interface condition Eqn. (13), subject to the constraints Eqn. (15) (monomer conservation), Eqn. (18) (ion motion) and Eqn. (19) (electroneutrality). 2.3 Linearized analysis of spherically symmetric swelling gels We now consider the swelling kinetics of a radially symmetric mucus gel using a linearized analysis of the governing equations, described in the previous two sections. The volume fraction ≤ θ is non-zero on the domain 0 r < R(t), with R (maximum radius of the sphere) a function of p time. Since only the ﬁnite solutions of the governing equations are of interest (in particular, ﬁnite solutions of the force balance, P D E S (4, 5)) inside this domain, we assume that v = v = 0 at p s r = 0. The volume conservation, P D E (3), implies the constraint θ v + θ v = 0 throughout the p p s s domain. Using this constraint, we reduce the two force balance equations into a single equation (by multiplying(cid:20)P D(cid:18)E (4) by volume fract(cid:19)io(cid:21)n θ and P(cid:26)D E (cid:20)(5) by v(cid:18)olume(cid:19)fraction θ (cid:18)and subt(cid:19)ra(cid:21)c(cid:27)ting) s p ∇ · (cid:18) ∂v (cid:19)λ ∂ ∇ · ∂ θ v λ ∂ r2θ v θ θ η p + p r2v + θ θ η p p + s p p s p p ∂r r2 ∂r p p s s ∂r θ r2 ∂r θ s s φ φ − ∇ + ξv θ p + s θ θ (Σ ) = 0, (22) p p p s net ν ν s p (cid:18) (cid:19) (cid:18) (cid:19) whereas the interface condition (Eqn. (13)) reduces to ∂v ∂ θ v λ ∂ λ ∂ r2θ v η p + η p p + p r2v + s p p = Σ at r = R(t). (23) p ∂r s ∂r θ r2 ∂r p r2 ∂r θ net s s ∇ − ∇ Note that ( µp µs ) = (Σnet), where Σnet is the interface swelling pressure deﬁned in Eqn. (14). νm νs − The velocities inside the gel are v = v and outside the gel are v+ = 0. Next, we assume that p/s p/s ∗ p/s ∗ the linearized velocity v is small (the equilibrium velocity v = 0) and that θ = θ + δθ , where ∗ p p p p p θ is the equilibrium polymer volume fraction. Sircar et al. [35] gives details on the equilibrium p 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: symbol

text: C

text: c,

text: h,

text: n,

text: c

text: z

text: z

text: z

text: and

text: z

text: The

text: subscript

text: ‘b’

text: denotes

text: l

text: n

text: h

text: c

text: c

text: l

text: the

text: corresponding

text: bath

text: concentrations

text: The

text: electrostatic

text: potential,

text: Ψ

text: is

text: determined

text: by

text: the

text: e

text: electroneutrality

text: constraint

text: inside

text: the

text: gel,

text: namely,

text: n

text: h

text: c

text: )θ

text: z

text: m

text: l

text: s

text: m

text: T

text: where

text: z

text: is

text: the

text: average

text: residual

text: charge

text: of

text: the

text: unbound

text: monomers

text: which

text: depends

text: on

text: the

text: amount

text: m

text: of

text: binding

text: with

text: ions,

text: z

text: m

text: w

text: m.

text: m

text: T

text: Since

text: both

text: the

text: electrostatic

text: potential

text: and

text: polymer

text: particle

text: fraction

text: are

text: assumed

text: to

text: be

text: zero

text: outside

text: the

text: gel,

text: electroneutrality

text: in

text: the

text: bath

text: requires

text: that

text: n

text: h

text: c

text: b

text: b

text: b

text: lb

text: Finally,

text: the

text: crosslink

text: fraction

text: (or

text: the

text: fraction

text: of

text: monomers

text: bound

text: with

text: calcium)

text: is

text: α

text: x

text: where

text: Eqns.

text: deﬁne

text: the

text: concentrations,

text: m

text: x,

text: respectively.

text: mT

text: T

text: In

text: summary,

text: the

text: dynamical

text: motion

text: of

text: a

text: freely

text: swelling

text: mucus-gel

text: is

text: modeled

text: by

text: the

text: system

text: of

text: equations

text: including

text: the

text: mass

text: conservation

text: P

text: D

text: E

text: total

text: volume

text: conservation

text: P

text: D

text: E

text: together

text: with

text: force

text: balance

text: P

text: D

text: E

text: S

text: and

text: interface

text: condition

text: Eqn.

text: subject

text: to

text: the

text: constraints

text: Eqn.

text: (monomer

text: conservation),

text: Eqn.

text: (ion

text: motion)

text: and

text: Eqn.

text: (electroneutrality).

text: Linearized

text: analysis

text: of

text: spherically

text: symmetric

text: swelling

text: gels

text: We

text: now

text: consider

text: the

text: swelling

text: kinetics

text: of

text: a

text: radially

text: symmetric

text: mucus

text: gel

text: using

text: a

text: linearized

text: analysis

text: of

text: the

text: governing

text: equations,

text: described

text: in

text: the

text: previous

text: two

text: sections.

text: The

text: volume

text: fraction

text: θ

text: is

text: non-zero

text: on

text: the

text: domain

text: r

text: R(t),

text: with

text: R

text: (maximum

text: radius

text: of

text: the

text: sphere)

text: a

text: function

text: of

text: p

text: time.

text: Since

text: only

text: the

text: ﬁnite

text: solutions

text: of

text: the

text: governing

text: equations

text: are

text: of

text: interest

text: (in

text: particular,

text: ﬁnite

text: solutions

text: of

text: the

text: force

text: balance,

text: P

text: D

text: E

text: S

text: inside

text: this

text: domain,

text: we

text: assume

text: that

text: v

text: v

text: at

text: p

text: s

text: r

text: The

text: volume

text: conservation,

text: P

text: D

text: E

text: implies

text: the

text: constraint

text: θ

text: v

text: θ

text: v

text: throughout

text: the

text: p

text: p

text: s

text: s

text: domain.

text: Using

text: this

text: constraint,

text: we

text: reduce

text: the

text: two

text: force

text: balance

text: equations

text: into

text: a

text: single

text: equation

text: (by

text: by

text: volume

text: θ

text: and

text: E

text: by

text: θ

text: s

text: p

text: ∂v

text: θ

text: v

text: λ

text: v

text: θ

text: θ

text: η

text: p

text: p

text: θ

text: θ

text: η

text: p

text: p

text: s

text: p

text: p

text: s

text: p

text: p

text: ∂r

text: ∂r

text: p

text: p

text: s

text: s

text: ∂r

text: θ

text: ∂r

text: θ

text: s

text: s

text: φ

text: φ

text: ξv

text: θ

text: p

text: s

text: θ

text: θ

text: (Σ

text: p

text: p

text: p

text: s

text: net

text: ν

text: ν

text: s

text: p

text: whereas

text: the

text: interface

text: condition

text: (Eqn.

text: reduces

text: to

text: ∂v

text: θ

text: v

text: λ

text: λ

text: v

text: η

text: p

text: η

text: p

text: p

text: p

text: s

text: p

text: p

text: Σ

text: at

text: r

text: R(t).

text: p

text: ∂r

text: s

text: ∂r

text: θ

text: ∂r

text: p

text: ∂r

text: θ

text: net

text: s

text: s

text: Note

text: that

text: µp

text: µs

text: (Σnet),

text: where

text: Σnet

text: is

text: the

text: interface

text: swelling

text: pressure

text: deﬁned

text: in

text: Eqn.

text: νm

text: νs

text: The

text: velocities

text: inside

text: the

text: gel

text: are

text: v

text: v

text: and

text: outside

text: the

text: gel

text: are

text: v+

text: Next,

text: we

text: assume

text: that

text: p/s

text: p/s

text: p/s

text: the

text: linearized

text: velocity

text: v

text: is

text: small

text: (the

text: equilibrium

text: velocity

text: v

text: and

text: that

text: θ

text: θ

text: δθ

text: where

text: p

text: p

text: p

text: p

text: p

text: θ

text: is

text: the

text: equilibrium

text: polymer

text: volume

text: fraction.

text: Sircar

text: et

text: al.

text: gives

text: details

text: on

text: the

text: equilibrium

text: p

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##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 E and covariance operators such as | and Cov | will be equivalent to E | and Cov | , reUspYectively. WeUwYill prefer where r =∆ min{p, p} and s =∆ min{q, q}. Thus, instead of X Y X Y ≥ ≥ ≤ to use expectation operators in terms of the primary variables proving f(p, q) 0, it sufﬁces to prove f(r, s) 0 for 0 ≤ ≥ X and Y rather than the secondary (derived) variables such as r, s 1/2. In fact, using (18), it sufﬁces to prove f(r, s) 0 ≤ ≤ ≤ ≤ ≤ ≤ U, A, B, to emphasize that the underlying space is (X, Y). for 0 r s 1/2. Assuming 0 r s 1/2, it is We note that, due to the independence of Y and Y , A and straightforward to show that 1 2 1 A are independent; likewise, B and B are independent. 2 1 2 A. Covariance decomposition step ∗ ≥ ∗ rs ≤ r s ≤ 1 r s r s and ∗ ∗ . (20) As the ﬁrst step of the proof of Theorem ??(cid:48), we use the r s r s 2 covariance decomposition formula (13) to write Thus, if we write out the expression for f(r, s), as in (16) with (r, s) in place of (p, q), we can see easily that each of E Cov(hout,1, hout,2) = Y CovX|Y(hout,1, hout,2) the four factors on the right hand side of that expression are E E + CovY( X|Yhout,1, X|Yhout,2). (15) non-negative. More speciﬁcally, the logarithmic term is non- negative due to the ﬁrst inequality in (20) and the bracketed term is non-negative due to the second inequality in (20). This For brevity, we will use the notation ≥ ∈ completes the proof that f(p, q) 0 for all p, q [0, 1]. Cov1 =∆ EY CovX|Y(hout,1, hout,2) Next, we identify the nec≤essary a≤nd sufﬁcient conditions for f(p, q) to be zero over 0 p, q 1. Clearly, f(p, q) = 0 Cov2 =∆ CovY(EX|Yhout,1, EX|Yhout,2) iff one of the four factors on the right hand side of (16) equals zero. By straightforward algebra, one can verify the to denote the two terms on the right hand side of (15). Our ∗ proof of Theorem ??(cid:48) will consist in proving the following follow∈ing{ statements.}The ﬁrst factor p q∗ equals zero iff (p, q) (0, 1), (1, 0) . The second factor p q equals zero iff two statements. ∈ { } Proposition 5. We have Cov ≥ 0, with equality iff either (p, q) (0, 0), (1, 1) . The log term equals zero iff p = 1/2 1 or q = 1/2. Finally the difference of the entropy terms equals (X , Y ) or (X , Y ) is an erasing BDE. ∗ ∗ ∗ − ∗ 1 1 2 2 zero iff pq/p q = pq/p q or pq/p q = 1 pq/p q which ≥ ∈ { } ∈ { } Proposition 6. We have Cov 0. in turn is true iff p 0, 1/2, 1 or q 0, 1/2, 1 . Taking 2 the logical combination of these conditions we conclude that Remark 2. We note that Cov = 0 iff, of the two BDEs ∈ { } ∈ { } 2 f(p, q) = 0 iff p 0, 1/2, 1 or q 0, 1/2, 1 . (X , Y ) and (X , Y ), either one is extreme or both are pure. 1 1 2 2 We note this only for completeness but do not use it in the Lemma 5. We have paper. E E The rest of the section is devoted to the proof of the above Cov = f(A) = f(B). (21) 1 propositions. B. Proof of Proposition 5 (cid:18) (cid:19) Proof. Fix a(cid:8)s(cid:2)ample y = (y1, y2). N(cid:3)ote that (cid:9) ∈ For p,fq(p,[q0), 1=∆],(dpe∗ﬁnq(cid:20)e)(p(cid:18)∗ q) lo(cid:19)g p ∗∗ (cid:18)q (cid:19)(cid:21) Co=vXE(cid:88)|yX(|hyout,1h,(hUo1u|t,y2)) −(cid:2)=HCo(UvX1||yy)(hh(U(U12|y|U),1h(cid:3), y(U)2|U1, y)) p q | | − | | = p | (u y) h(u (cid:2)y) H(U y) H(U(cid:3) u , y). × H p q − H p q U1 Y 1 1 1 2 1 ∗ ∗ . (16) u1 p q p q | − | After some algebra, the term h(u y) H(U y) simpliﬁes We will soon give a formula for Cov1 in terms of this function. to 1− 1| First, a number of properties of f(p, q) will be listed. The − | 1 p | (u y) following symmetry properties are immediate: (1 p | (u y)) log U1 Y |1 . U1 Y 1 p | (u y) U1 Y 1 f(p, q) = f(p, q) = f(p, q) = f(p, q), (17) Substituting this in the preceding equation and writing out the sum over U explicitly, we obtain 1 (cid:2) | | (cid:3) f(p, q) = f(q, p). (18) Co·vX|yp(hou|t,1(,0h|you)t,2) = pU|1|Y(0 y)pU−1|Y(1 y)| log U1 Y | H(U U = 1, y) H(U U = 0, y) . p | (1 y) 2 1 2 1 ≥ ∈ U1 Y Lemma 4. We have f(p, q) 0 for all p, q [0, 1] with ∈ { } ∈ { } Expressing each factor on the right side of the above equation equality iff p 0, 1/2, 1 or q 0, 1/2, 1 . in terms of a = α(y ), i = 1, 2, we see that it equals i i E Proof. We use (17) to write f(a , a ). Taking expectations, we obtain Cov = f(A). 1 2 E 1 The alternative formula Cov = f(B) follows from the fact 1 f(p, q) = f(r, s) (19) that f(B) = f(A) due to the symmetries (17). ##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: E

text: and

text: covariance

text: operators

text: such

text: as

text: and

text: Cov

text: will

text: be

text: equivalent

text: to

text: E

text: and

text: Cov

text: reUspYectively.

text: WeUwYill

text: prefer

text: where

text: r

text: min{p,

text: p}

text: and

text: s

text: min{q,

text: q}.

text: Thus,

text: instead

text: of

text: X

text: Y

text: X

text: Y

text: to

text: use

text: expectation

text: operators

text: in

text: terms

text: of

text: the

text: primary

text: variables

text: proving

text: f(p,

text: q)

text: it

text: sufﬁces

text: to

text: prove

text: f(r,

text: s)

text: for

text: X

text: and

text: Y

text: rather

text: than

text: the

text: secondary

text: (derived)

text: variables

text: such

text: as

text: r,

text: s

text: In

text: fact,

text: using

text: it

text: sufﬁces

text: to

text: prove

text: f(r,

text: s)

text: U,

text: A,

text: B,

text: to

text: emphasize

text: that

text: the

text: underlying

text: space

text: is

text: (X,

text: Y).

text: for

text: r

text: s

text: Assuming

text: r

text: s

text: it

text: is

text: We

text: note

text: that,

text: due

text: to

text: the

text: independence

text: of

text: Y

text: and

text: Y

text: A

text: and

text: straightforward

text: to

text: show

text: that

text: A

text: are

text: independent;

text: likewise,

text: B

text: and

text: B

text: are

text: independent.

text: A.

text: Covariance

text: decomposition

text: step

text: rs

text: r

text: s

text: r

text: s

text: r

text: s

text: and

text: As

text: the

text: ﬁrst

text: step

text: of

text: the

text: proof

text: of

text: Theorem

text: we

text: use

text: the

text: r

text: s

text: r

text: s

text: covariance

text: decomposition

text: formula

text: to

text: write

text: Thus,

text: if

text: we

text: write

text: out

text: the

text: expression

text: for

text: f(r,

text: s),

text: as

text: in

text: with

text: (r,

text: s)

text: in

text: place

text: of

text: (p,

text: q),

text: we

text: can

text: see

text: easily

text: that

text: each

text: of

text: E

text: Y

text: the

text: four

text: factors

text: on

text: the

text: right

text: hand

text: side

text: of

text: that

text: expression

text: are

text: E

text: E

text: CovY(

text: non-negative.

text: More

text: speciﬁcally,

text: the

text: logarithmic

text: term

text: is

text: non-

text: negative

text: due

text: to

text: the

text: ﬁrst

text: inequality

text: in

text: and

text: the

text: bracketed

text: term

text: is

text: non-negative

text: due

text: to

text: the

text: second

text: inequality

text: in

text: This

text: For

text: brevity,

text: we

text: will

text: use

text: the

text: notation

text: completes

text: the

text: proof

text: that

text: f(p,

text: q)

text: for

text: all

text: p,

text: q

text: EY

text: Next,

text: we

text: identify

text: the

text: nec≤essary

text: a≤nd

text: sufﬁcient

text: conditions

text: for

text: f(p,

text: q)

text: to

text: be

text: zero

text: over

text: p,

text: q

text: Clearly,

text: f(p,

text: q)

text: iff

text: one

text: of

text: the

text: four

text: factors

text: on

text: the

text: right

text: hand

text: side

text: of

text: equals

text: zero.

text: By

text: straightforward

text: algebra,

text: one

text: can

text: verify

text: the

text: to

text: denote

text: the

text: two

text: terms

text: on

text: the

text: right

text: hand

text: side

text: of

text: Our

text: proof

text: of

text: Theorem

text: will

text: consist

text: in

text: proving

text: the

text: following

text: follow∈ing{

text: statements.}The

text: ﬁrst

text: factor

text: p

text: q∗

text: equals

text: zero

text: iff

text: (p,

text: q)

text: The

text: second

text: factor

text: p

text: q

text: equals

text: zero

text: iff

text: two

text: statements.

text: Proposition

text: We

text: have

text: Cov

text: with

text: equality

text: iff

text: either

text: (p,

text: q)

text: The

text: log

text: term

text: equals

text: zero

text: iff

text: p

text: or

text: q

text: Finally

text: the

text: difference

text: of

text: the

text: entropy

text: terms

text: equals

text: (X

text: Y

text: or

text: (X

text: Y

text: is

text: an

text: erasing

text: BDE.

text: zero

text: iff

text: pq/p

text: q

text: pq/p

text: q

text: or

text: pq/p

text: q

text: pq/p

text: q

text: which

text: Proposition

text: We

text: have

text: Cov

text: in

text: turn

text: is

text: true

text: iff

text: p

text: or

text: q

text: Taking

text: the

text: logical

text: combination

text: of

text: these

text: conditions

text: we

text: conclude

text: that

text: Remark

text: We

text: note

text: that

text: Cov

text: iff,

text: of

text: the

text: two

text: BDEs

text: f(p,

text: q)

text: iff

text: p

text: or

text: q

text: (X

text: Y

text: and

text: (X

text: Y

text: either

text: one

text: is

text: extreme

text: or

text: both

text: are

text: pure.

text: We

text: note

text: this

text: only

text: for

text: completeness

text: but

text: do

text: not

text: use

text: it

text: in

text: the

text: Lemma

text: We

text: have

text: paper.

text: E

text: E

text: The

text: rest

text: of

text: the

text: section

text: is

text: devoted

text: to

text: the

text: proof

text: of

text: the

text: above

text: Cov

text: f(A)

text: f(B).

text: propositions.

text: B.

text: Proof

text: of

text: Proposition

text: Proof.

text: Fix

text: y

text: that

text: For

text: q)

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