Moment Closure for the Stochastic Logistic Model 1

Transcription

1 Momet Closure for the Stochastic Logistic Model 1 Abhyudai Sigh ad João Pedro Hespaha Departmet of Electrical ad Computer Egieerig, Uiversity of Califoria, Sata Barbara, CA Abstract Cotiuous-time birth-death Markov processes serve as useful models i populatio biology. Whe the birth-death rates are oliear, the time evolutio of the first order momets of the populatio is ot closed, i the sese that it depeds o momets of order higher tha. For aalysis purpose, the time evolutio of the first order momets is ofte made to be closed by approximatig these higher order momets as a oliear fuctio of momets up to order, which we refer to as the momet closure fuctio. I this paper, a systematic procedure for costructig momet closure fuctios of arbitrary order is preseted for the stochastic logistic model. We obtai the momet closure fuctio by first assumig a certai separable form for it, ad the matchig time derivatives of the exact (ot closed) momet equatios with that of the approximate (closed) equatios for some iitial time ad set of iitial coditios. The separable structure esures that the steady-state solutios for the approximate equatios are uique, positive ad real, while the derivative matchig guaratees a good approximatio, at-least locally i time. Moreover, the accuracy of the approximatio ca be improved by icreasig the order of the approximate model.to the best of our kowledge, this paper is the first to propose a systematic procedure to costruct momet closure fuctios of arbitrary order that guaratee biologically meaigful equilibria. A host of other momet closure fuctios previously proposed i the literature are also ivestigated. Amog these we show that oly the oes that achieve derivative matchig provide a close approximatio to the exact solutio. Moreover, we improve the accuracy of several previously proposed momet closure fuctios by forcig derivative matchig. However, for certai rages of parameter models, momet closure fuctios that lack the separability property lead to biologically meaigless scearios of imagiary ad eve stable egative steady-states of the closed momet equatios. Key words: Momet Closure, Stochastic Logistic Model, Stochastic Hybrid Systems 1 This material is based upo work supported by the Istitute for Collaborative Biotechologies through grat DAAD19-03-D-0004 from the U.S. Army Research Office ad by the Natioal Sciece Foudatio uder Grat No. CCR abhi@egieerig.ucsb.edu, hespaha@ece.ucsb.edu Preprit submitted to Elsevier Sciece 16 Jauary 006

2 1 INTRODUCTION Cotiuous-time birth-death Markov processes have bee extesively used for modelig stochasticity i populatio biology (Matis ad Kiffe, 00, 1996; Matis et al., 1998). The time evolutio of these processes is typically described by a sigle equatio for a grad probability fuctio, where time ad species populatios appear as idepedet variables, called the Master or Kolmogorov equatio (Bailey, 1964). However, this equatio ca oly be solved for relatively few, highly idealized cases ad a more reasoable goal is to determie the time evolutio of a few low-order statistical momets. I this paper, a method for estimatig low-order statistical momets is itroduced for the stochastic logistic model. This model was first itroduced by Barlett et al. (1960) ad is a cotiuous-time birth-death Markov process ivolvig a sigle specie, with birth ad death rates beig polyomials of order. Although, oe ca directly use the Kolmogorov equatio to derive differetial equatios for the time evolutio of momets of the process, i this paper we use a alterative method. We model the stochastic logistic model as a Stochastic Hybrid System (SHS) whose state x is the populatio of the specie. The, the time evolutio of the momets is obtaied usig results from the SHS literature (e.g., Hespaha, 004). Details of the stochastic logistic model ad its modelig as a SHS are preseted i Sectio. Table 1 Separable derivative-matchig momet closure fuctio ϕ+1 s (µ) for {,3,4}. = = 3 = 4 ϕ+1 s (µ) µ 3 µ 1 4µ4 3 µ 10µ5 4 µ 1 3 µ 6 µ 1 5µ10 3 Let µ m be the m th order ucetered momet of x, i.e. µ m (t) = E[x(t) m ] where x(t) deotes the populatio of the specie at time t. We will show i Sectio 3 that for the stochastic logistic model the time derivative of µ m is a liear combiatio of momets up to order m + 1. Hece, if oe stacks all momets i a ifiite vector µ = [µ 1, µ, ] T, its dyamics ca be writte as µ = A µ, (1) for some ifiite matrix A. As the above ifiite dimesioal system caot be solved aalytically, we trucate (1) by creatig a vector µ = [µ 1,..., µ ] T, where is the order of the trucatio. Its dyamics, give by µ = Aµ + Bµ +1 () for some matrices A ad B is ot closed, i the sese that the time evolutio of the vector µ depeds o the + 1 th order momet µ +1. For aalysis purposes, we close the above system by approximatig µ +1 as a oliear fuctio ϕ +1 (µ) of

3 momets up to order. This procedure is commoly referred to as momet closure. We call ϕ +1 (µ) as the momet closure fuctio for µ +1. Let ν = [ν 1,...,ν ] T deote the state of the ew closed system. The, its dyamics is give by ν = Aν + Bϕ +1 (ν) (3) ad is referred to as the trucated momet dyamics. We deote the states of () ad (3) usig differet symbols because µ refers to the actual momet dyamics whereas ν to a approximated momet dyamics. I Sectio 4, we cosider momet closure fuctios which have the followig separable form: ϕ+1 s (µ) = µγ 1 1 µγ... µγ for appropriately chose costats γ m R. These costats are obtaied by matchig time derivatives of µ +1 ad ϕ+1 s (µ) i () ad (3) respectively, at some iitial time t 0, for a determiistic iitial coditio x(t 0 ) = x 0 with probability oe. The reaso for this lies i the fact that the class of determiistic distributios forms a atural basis for the ifiite dimesioal space cotaiig the vector µ. We refer to this momet closure as the separable derivative-matchig momet closure. We show that for all, this determies the fuctio ϕ+1 s uiquely, which is idepedet of the birth ad death rates. Table 1 shows the fuctios ϕ+1 s that we obtaied for trucatios of order =, 3 ad 4. The strikig feature of the separable derivative-matchig momet closure is that the accuracy of the approximate momet dyamics improves by icreasig order of trucatio ad the depedece of higher order momets o lower order oes is cosistet with x(t) beig logormally distributed, i spite of the fact that the derivative matchig procedure used to costruct ϕ+1 s did ot make ay assumptio o the distributio of the populatio. Alterative momet closure methods which have appeared i literature typically costruct the momet closure fuctios ϕ by directly assumig the probability distributio to be ormal (Whittle, 1957), logormal (Keelig, 000), poisso or biomial (Nasell, 003a). We refer to them as ormal, logormal, poisso ad biomial momet closures respectively ad review them i Sectio 5. I Sectio 6, they are compared with the separable derivative-matchig momet closure. I Sectio 6.1 the comparisos are doe based o how well the momet closure fuctio ϕ +1 (µ) approximates µ +1. Towards that ed, we itroduce the error e +1 (t) := µ +1 (t) ϕ +1 (µ(t)) = i=0 (t t 0 ) i ε+1 i i! (x 0) where we expaded the error as a Taylor series with ε i +1 (x 0) defied to be ε+1 i (x 0) := di µ +1 (t) t=t0 dt i di ϕ +1 (µ(t)) t=t0 dt i. 3

4 We call ε i +1 (x 0) the i th order derivative matchig error. Ideally oe would like this error to be zero but this is geerally ot possible. Whe x(t 0 ) = x 0 with probability oe, the derivative matchig error is typically a polyomial i x 0. The lesser the order of this polyomial, the lesser is the error e +1 (t), ad hece the better is ϕ +1 (µ) i approximatig µ +1. We show that for =, all the above momet closure fuctios perform derivative matchig except the poisso momet closure fuctio by Nasell (003a). This is because, it has a 0 th order derivative matchig error ε 0 3 (x 0) which grows liearly with x 0 while for separable derivative-matchig, logormal, biomial ad ormal momet closure fuctios the 0 th order error is always zero. Hece, Nasell s poisso momet closure fuctio exhibits a larger iitial error tha the others. We propose a alterative poisso momet closure fuctio, for which ε 0 3 (x 0) = 0, ad show that it performs better tha the oe proposed by Nasell (003a). Although the above momet closures provide good estimates for a secod order of trucatio ( = ), it is typically beeficial to cosider 3 because they lead to better momet approximatios ad reduce the errors by a few orders of magitude. However, for distributios like the ormal ad the logormal, which are characterized by less tha parameters, 3 typically leads to multiple ormal ad logormal momet closure fuctios. I Sectio 6.1. we illustrate how derivative matchig ca be used as a tool for gaugig performaces of these multiple momet closure fuctios ad choosig the oes amog them that yield the least derivative matchig errors. I particular, we show that amog the momet closure fuctios cosistet with the logormal distributio, the separable derivative-matchig momet closure fuctio yields the least order polyomial for the derivative matchig error, ad hece, exhibits the best performace. Based o derivative matchig we also propose families of ormal momet closure fuctios which provide good approximatios for µ +1. Towards the ed, for = 3, we propose a ew momet closure fuctio, for which, ulike the logormal or ormal momet closure fuctios both the 0 th ad 1 st order derivative matchig error are zero, ad hece, provides the best estimates for µ 4 as compared to them, at least locally i time. I Sectio 6. comparisos are doe based o the steady-state solutios of the trucated momet dyamics (3). We show that the separable derivative matchig momet closure, always yields a uique o-trivial positive real steady-state N. Thus, it is preferable to the other momet closures techiques which lack the separable structure ad exhibit spurious, imagiary ad eve stable egative steadystates, which would be biologically meaigless. 4

5 Stochastic Logistic Model.1 Model formulatio The stochastic logistic model is the stochastic birth-death aalogous model of the well-kow determiistic Verhulst-Pearl equatios (Pielou, 1977) ad has bee extesively used for modelig stochasticity i populatio biology (Matis ad Kiffe, 00, 1996; Matis et al., 1998). For this cotiuous-time birth-death Markov process, the coditioal probabilities of a uit icrease ad decrease, respectively, i a ifiitesimal time iterval (t,t + dt] is give by P{x(t + dt) = x + 1 x(t) = x} = { η(x)dt, P{x(t + dt) = x 1 x(t) = x} = χ(x)dt, x U 0, otherwise where x(t) N represets the populatio size at time t, η(x) := a 1 x b 1 x > 0, χ(x) := a x + b x > 0, x (0,U) (4) ad U := a 1 /b 1 N, a 1 > 0, a > 0, b 1 > 0, b 0. We assume that the iitial coditio satisfies x(t 0 ) {1,,...,U}, ad hece, x(t) {0, 1,...,U}, t [0, ) with probability oe. We call U the populatio limit.. Statioary ad Quasi-Statioary Distributios Sice the birth ad death rates are zero for x = 0 (η(0) = χ(0) = 0) we have that x = 0 is a absorbig state ad evetual covergece to the origi is certai. Thus, the statioary distributio is degeerate with probability oe at the origi. However, though out this paper we assume the mea time to extictio to be very large, i which case there exists a quasi-statioary distributio (Nasell, 001). Barlett et al. (1960) shows that a good approximate for P x (the probability that x = x at quasi-equilibrium ) ca be umerically obtaied usig the followig recurrece relatioship χ(x) P x = η(x 1) P x 1, x {,3,...,U}. (5) 5

6 .3 Trasiet Distributios Ideally oe would like to determie the exact probability distributio of x(t) at ay time t. It ca be show that the probability P x (t) that x(t) = x satisfies the followig differetial equatios Ṗ 0 (t) = χ(1)p 1 (t) Ṗ 1 (t) = χ()p (t) [η(1) + χ(1)]p 1 (t) Ṗ x (t) = χ(x + 1)P x+1 (t) [η(x) + χ(x)]p x (t) + η(x 1)P x 1 (t),. Ṗ U (t) = χ(u)p U (t) + η(u 1)P U 1 (t) (6a) (6b) (6c) (6d) (Bailey, 1964). These differetial equatios are commoly referred to as the Master or Kolmogorov equatios. If the populatio limit U is small, the above system of equatios ca be solved umerically. However, for large U, a more reasoable goal (ad oe that is of primary iterest i applicatios) is to determie the evolutio of the some lower-order momets of x(t). 3 Time Evolutio of Momets 3.1 Modelig the Stochastic Logistic Model To model the time evolutio of x(t), we cosider a special class of systems kow as Stochastic Hybrid Systems (SHS). These systems were itroduced by Hespaha ad Sigh (005) to model the stochastic time evolutio of the populatios of differet species ivolved i a chemical reactio. More specifically, to fit the framework of our problem, these system are characterized by two reset maps: x φ 1 (x) := x + 1, x φ (x) := x 1 (7) oe correspodig to a birth ad the other to a death, with associated trasitio itesities give by λ 1 (x) := η(x), λ (x) := χ(x). (8) Betwee births ad deaths the populatio remais costat ad thus ẋ = 0. I essece, wheever a birth evet or a death evet takes place, the correspodig reset φ i (x) is activated ad x is reset accordigly, furthermore, the probability of the activatio takig place i a ifiitesimal time iterval (t,t +dt] is determied by the associated trasitio itesities λ i (x)dt. 6

7 3. Momet Dyamics Give m {1,,...}, we defie the m th order (ucetered) momet of x to be µ m (t) = x m P x (t) := E[x(t) m ], t 0. (9) x=1 The time evolutio of momets is give by the followig result, which is a straightforward applicatio of Theorem 1 i Hespaha (004) to the above SHS. Theorem 1: The time evolutio of µ m is give by dµ m dt = E [ [(φ i (x)) m x m ]λ i (x) i=1 ]. (10) Usig the above Theorem, we show i Appedix A that oe ca coclude µ m = m+1 C m m+p r f (m + p r, p)µ r, (11) r=1 where we defie C m j ad f ( j, p) as follows 3 j, m, p N. C m j := f ( j, p) := { m! (m j)! j! m j 0 0 m < j 0 j = 0 a 1 + ( 1) j a j > 0, p = 1 b 1 + ( 1) j b j > 0, p =. (1) (13) Oe ca see from the right-had-side of (11), that the time derivative of µ m is a liear combiatio of the momets µ r, up to order r = m + 1. Hece, if oe stacks all momets i a ifiite vector µ = [µ 1, µ, ] T, its dyamics ca be writte as µ = A µ, (14) for some ifiite matrix A. Let µ = [µ 1, µ,..., µ ] T R cotais the top elemets of µ. The, usig (11) the evolutio of µ is give by µ = Aµ + Bµ +1, (15) 3! deotes the factorial of. 7

8 for some ad 1 matrices A ad B which have the followig structure A = , B =., (16) 0 where the deotes o-zero etries. Our goal is to approximate (15) by a fiitedimesioal oliear ODE of the form ν = Aν + Bϕ +1 (ν), ν = [ν 1,ν,...,ν ] T (17) where the map ϕ +1 : R R should be chose so as to keep ν(t) close to µ(t). This procedure is commoly referred to as momet closure. We call (17) the trucated momet dyamics ad ϕ +1 (µ) the momet closure fuctio for µ +1. Whe a sufficietly large but fiite umber of derivatives of µ(t) ad ν(t) match poit-wise, the, the differece betwee solutios to (15) ad (17) remais close o a give compact time iterval. This follows from a Taylor series approximatio argumet. To be more precise, for each δ > 0 ad iteger N, there exists T R, for which the followig result holds: Assume that for every t 0 0, µ(t 0 ) = ν(t 0 ) ad d i µ(t) t=t0 dt i = di ν(t) t=t0 dt i, i {1,...,N} (18) where di µ(t) ad di ν(t) represet the i th time derivative of µ(t) ad ν(t) alog the dt i dt i trajectories of system (14) ad (17) respectively. The, µ(t) ν(t) δ, t [t 0,T ], (19) alog solutios of (14) ad (17), where µ deote the first elemets of µ. I the ext sectio we use (18) to costruct momet closure fuctios ϕ +1 (µ). 4 Separable Derivative-Matchig Momet Closures I this sectio we costruct trucated momet dyamics (17) for the stochastic logistic model usig (18). After replacig (15) ad (17) i (18), equality (18) becomes a PDE o ϕ +1. We will seek for solutios ϕ +1 to this PDE that have the followig separable form ϕ s +1 (µ) = γ µ m m (0) m=1 8

9 for appropriately chose costats γ m R. As we will see i Sectio 6., this separable structure esures that the steady-state solutios for the trucated momet dyamics (17) are uique, positive real, ad hece, biologically meaigful. I the sequel we refer to such ϕ+1 s (µ) as a separable derivative-matchig momet closure fuctio for µ +1. Oe ca see that the ifiite vector µ Ω ca be expressed as E[x(t)] x E[x(t) µ = x E[x(t) 3 = ] x=1 x 3 P x (t)... Hece, the ifiite vectors µ = [x,x,x 3,...] T, which correspods to a determiistic distributio, i.e. x(t) = x with probability oe, form a atural basis for Ω. I particular, we will fid costats γ m which satisfy (18) for each vector µ (t 0 ) belogig to this basis, i.e. for the class of determiistic iitial coditios. However, ofte it is ot possible to fid γ m for which (18) holds exactly. We will therefore relax this coditio ad simply demad the followig µ(t 0 ) = ν(t 0 ) ad d i µ(t) t=t0 dt i = di ν(t) t=t0 dt i + E[ε i (x(t 0 ))], (1) i {1,}, where each elemet of the vector ε i (x(t 0 )) is a polyomial i x(t 0 ). Oe ca thik of (1) as a approximatio to (18) that is valid as log as di µ (t) dt i t=t0 domiates E[ε i (x(t 0 ))]. The followig theorem summarizes the mai result, the proof of which is give i Appedix B. Theorem : Let γ m, m {1,...,} be chose as γ m = ( 1) m C +1 m. () The, for every determiistic iitial coditio ν(t 0 ) = µ(t 0 ) = [x 0,...,x 0 ]T which correspods to x(t 0 ) = x 0 with probability oe, we have dµ(t) dt t=t0 = dν(t) dt t=t0 (3a) d µ(t) t=t0 dt = d ν(t) t=t0 dt + ε (x 0 ), (3b) where di µ(t) represet the i th time derivative of µ(t) ad ν(t) alog the dt i dt i trajectories of the systems (14) ad (17), respectively, ad the th elemet of the vector ε (x 0 ) is a polyomial i x 0 of order with all other elemets beig zero. ad di ν(t) 9

10 Remark 1. Usig (11) it ca be show that d µ (t) is a liear combiatio of momets of x up to order +. Thus, d µ (t) dt dt t=t0 is a polyomial i x 0 of order +, ad hece, for 4 x 0 >> 1, ε (x 0)/ d µ (t) dt elemet of ε (x 0 ). Hece, the term d µ (t) dt t=t0 = O(x0 ), where ε (x 0) is the th t=t0 domiates ε (x 0) by x0. Remark. It ca be verified that the separable derivative-matchig momet closures also matches derivatives of order higher tha i (3) with small errors. For example for {,3,4} ad i {,...,9}, we have dµ(t) dt t=t0 = dν(t) dt t=t0 (4a) d i µ(t) t=t0 dt i = di ν(t) t=t0 dt i + ε i (x 0 ), (4b) where the m th elemet of ε i (x 0 ) is a polyomial i x 0 of order m + i, for m + i ad equal to zero otherwise. We cojecture that the above equality holds N ad i N but we oly verified it for up to 4 ad i up to 9. As above, the elemets of vector di µ(t) dt i t=t0 domiate the correspodig elemets of ε i (x 0 ) by x0, ad hece, with icreasig, the trucated momet dyamics ν(t) should provide a more accurate approximatios to the lower order momets µ(t). 5 Distributio based Momet Closures Most momet closure techiques that appeared i the literature start by assumig a specific class of distributios D for the populatio, ad use this assumptio to express higher order momets as a fuctio of the lower order oes. We refer to such a momet closure fuctio as beig cosistet with the distributio D ad defie them as follows. Defiitio: Let D be a class of distributios parameterized by m parameters (q 1,...,q m ) Q, with the k th order momet µ k give i terms of the q 1,...,q m as follows µ k = f k (q 1,...,q m ), k {1,,...}. (5) The momet closure fuctio ϕ D +1 (µ) for µ +1 is said to be cosistet with the distributio D if, for every (q 1,...,q m ) Q, oe has that µ +1 = f +1 (q 1,...,q m ) = ϕ D +1(µ) (6) 4 O(.) deotes order of. 10

11 Table Uique Momet Closure Fuctios for the m th order trucatio ( = m) ad differet distributios D. where D m Uique momet closure fuctio Normal ϕ g 3 (µ) = 3µ µ 1 µ 3 1 Logormal ϕ l 3 (µ) = µ3 µ 3 1 Poisso 1 ϕ p (µ) = µ 1 + µ 1 Biomial ϕ b 3 (µ) = (µ µ 1 ) µ 1 (µ µ 1 ) + 3µ 1µ µ 3 1 µ 1 f 1 (q 1,...,q m ) µ :=. =.. (7) f (q 1,...,q m ) µ For well kow classes of distributios such as logormal, ormal, poisso, or biomial we simply say that ϕ+1 D is the logormal, ormal, poisso, or biomial momet closure fuctio. 5.1 Techiques for obtaiig ϕ D +1 Geerically, whe the dimesio m of the parameter space Q is the same as the dimesio of the domai of the momet closure fuctio ϕ+1 D (µ), the fuctioal equatio (6) (7) i the ukow ϕ+1 D ( ) has a uique solutio 5. I fact, to determie ϕ+1 D (µ) oe ca start by solvig (7) for q 1,...,q m i terms of µ 1,..., µ m, ad the substitutig these back i (6) to obtai a uique momet closure fuctio ϕ+1 D ( ). As we choose D to be ormal (Whittle, 1957), log-ormal (Keelig, 000), poisso, or biomial (Nasell, 003a), this procedure results i the differet momet closure fuctios show i Table. Difficulties arise whe the dimesio m of the parameter space Q is strictly smaller tha the dimesio of the domai of the momet closure fuctio ϕ+1 D (µ), because i this case the fuctioal equatio (6) (7) does ot have a uique solutio 6 ad oe ca fid ifiitely may momet closure fuctios cosistet with 5 The reader is ivited to covice herself of this by imagiig that all fuctios are locally liear, i which case ϕ+1 D (µ) is represeted by a vector with ukows. Sice (6) must hold for a (local) basis of Q, we have exactly m equatios to determie the = m ukows. 6 I cotrasts to the previous case, we ow have ukows to represet the local li- 11

12 the same family of distributios. However, there is a strog icetive to cosider this case because, as well shall see shortly, large values for geerally lead to sigificatly more accurate momet closures. I the sequel, we illustrate some optios for momet closures with > m, Example 1: Cosider the class of poisso distributios characterized by their expected value θ (m = 1). Their momets are give by µ 1 = f 1 (θ) := θ µ = f (θ) := θ(1 + θ) µ 3 = f 3 (θ) := θ(1 + 3θ + θ ),... Nasell (003a) proposed the followig poisso momet closure fuctio for = : ϕ p1 3 (µ) = µ 1 + 3µ 1 µ µ 3 1, (8) for which it is straightforward to verify that (6) (7) holds because µ 3 = θ(1 + 3θ + θ ) = ϕ p1 3 (µ), µ = [ θ, θ(1 + θ) ] T. However, a alterative choice for the poisso momet closure fuctio that also satisfies (6) (7) is give by ϕ p 3 (µ) = µ µ 1 + 3µ 1 µ µ 3 1, (9) which, as we will see i the ext sectio, performs better that (8). The explaatio for this lies i the fact that (9) has better derivative matchig properties tha (8), i the sese of (1). I the sequel we refer to (8) ad (9) as the Nasell-poisso ad ew-poisso momet closure fuctio, respectively. Example : Cosider ow the class of ormal distributios parameterized by their mea ω ad variace σ (m = ). Their momets are give by µ 1 = f 1 (θ,σ) := ω µ = f (θ,σ) := ω + σ µ 3 = f 3 (θ,σ) := ω(ω + 3σ ) µ 4 = f 4 (θ,σ) := ω 4 + 6ω σ + 3σ 4,... For = 3, ay fuctio of the followig form is a ormal momet closure fuctio for µ 4 ϕ 4 (µ) := 4µ 1 µ 3 + 3µ 1µ µ 1 + 6µ h(µ 1, µ µ 1,µ 3 3µ µ 1 + µ 3 1 ) earizatio of ϕ+1 D (µ) but a (local) basis of Q oly provides m < equatios. 1

13 where h(x,y,z) is ay fuctio with the property that h(x,y,0) = 0. To verify that this is so, we ote that (6) (7) holds because for µ = [ ω, ω + σ, ω(ω + 3σ ) ] T, we obtai h(µ 1, µ µ 1, µ 3 3µ µ 1 + µ 3 1 ) = h(ω,σ,0) = 0 ad therefore ϕ 4 (µ) = ω 4 + 6ω σ + 3σ 4 = µ 4. Example 3: Fially cosider the class of logormal distributios characterized by the parameters α > 0, β > 0 (m = ), whose momets are give by µ k = f k (α,β) := α k β k, k {1,, }. (30) For every, the separable derivative-matchig momet closure fuctios defied by (0), with coefficiet give by () i Theorem are logormal momet closure fuctios for µ +1. We ca verify this by otig that (6) (7) holds because, from (30) ad (B.1) used i the proof of Theorem (Appedix B) we coclude that ϕ s +1 (µ) = α( m=1 γ mm) β ( m=1 γ mm ) = α ( m=1 γ mc m 1 ) β ( m=1 γ m{c m +Cm 1 }), = α +1 β C+1 +C +1 1 = α +1 β (+1) = µ +1, where we used the facts that k = C k 1, k = C k + Ck 1, k N. 5. Cumulat closure fuctios Most of the literature o momet closure prefers to work with a vector κ = [κ 1,...,κ ] where κ (t) is the th order cumulat 7, istead of the previously itroduced vector µ of ucetered momets i (15). The, istead of doig momet closure oe performs cumulat closure by approximatig κ +1 by a oliear fuctio φ +1 (κ) of κ 1,...,κ, which we refer to as the cumulat closure fuctio. The disadvatage of workig with κ istead of µ is that the dyamics of κ is always oliear. However, for ease of compariso with other papers, we provide i Table the cumulat closure fuctios correspodig to the differet momet closure fuctios 7 The th order cumulat, κ is give as follows i terms of the ucetered momets κ 1 = µ 1, κ = µ µ 1 κ 3 = µ 3 3µ 1 µ + µ 1 3, κ 4 = µ 4 4µ 1 µ 3 3µ + 1µ µ 1 6µ 1 4,... 13

14 Table 3 Momet Closure Fuctios (MCF) for secod order trucatio ( = ) ad correspodig Cumulat Closure Fuctios (CCF) for µ 3 ad κ 3, respectively, correspodig to the differet Momet Closure Techiques (MCT) discussed i this paper. SDM refers to separable derivative-matchig. MCT MCF CCF SDM ϕ3 s(µ) = µ3 µ 1 3 φ3 s(κ) = 3κ + κ3 κ 1 κ1 3 Normal ϕ g 3 (µ) = 3µ µ 1 µ 1 3 φ g 3 (κ) = 0 Logormal ϕ l 3 (µ) = µ3 µ 3 1 Nasell-Poisso New-Poisso φ l 3 (µ) = 3κ κ 1 + κ3 κ 3 1 ϕ p1 3 (µ) = µ 1 + 3µ 1 µ µ 3 1 φ p1 3 (κ) = κ 1 ϕ p 3 (µ) = µ µ 1 + 3µ 1µ µ 3 1 φ p 3 (κ) = κ Biomial ϕ b 3 (µ) = (µ µ 1 ) µ 1 (µ µ 1 ) + 3µ 1µ µ 3 1 φ b 3 (µ) = κ κ 1 κ discussed so far for =. We use superscripts s, l, g, p1, p ad b to deote separable derivative-matchig, logormal, ormal, Nasell-poisso, ew-poisso ad biomial momet closure fuctios, respectively. 6 Compariso of Momet closures I this sectio, we itroduce two criteria to compare the differet momet closure techiques. The first criterio is the error where e +1 (t) := µ +1 (t) ϕ +1 (µ(t)) = i=0 (t t 0 ) i ε+1 i i! (x 0), (31) ε+1 i (x 0) := di µ +1 (t) t=t0 dt i di ϕ +1 (µ(t)) t=t0 dt i. (3) We call ε i +1 (x 0) the derivative matchig error. Ideally, oe would like to have ε i +1 (x 0) = 0, but as already poited out i Sectio 4, this is geerally ot possible. With determiistic iitial coditios µ (t 0 ) = [x 0,x 0,...]T as i Theorem, the derivative matchig error is typically a polyomial i x 0. The lesser the order of this polyomial, the better is ϕ +1 (µ) i approximatig µ +1. The secod criterio is the steady-state solutio of the trucated momet dyamics (17). I particular, we are iterested i determiig if there exists a uique otrivial positive real steady-state which is physically meaigfull. This is importat, 14

15 because it is well-kow that ormal momet closures ca have spurious, imagiary ad sometimes eve stable egative steady-states, which lead to biologically meaigless solutios (Keelig, 000). 6.1 Derivative Matchig Error Momet Closures for = We recall from Table 3 that ϕ3 l (µ) = ϕs 3 (µ), ad therefore we do ot eed to discuss logormal momet closure separately. By substitutig ϕ3 s(µ), ϕg p1 3 (µ), ϕ3 (µ), ϕ p 3 (µ) ad ϕb 3 (µ) from Table 3 i (31) (3), oe obtais the correspodig derivative matchig errors, which will be deoted usig the appropriate superscripts. Usig Table 3 ad symbolic maipulatio i Mathematica, we ca show that s ε 0 3 (x 0) = g ε 0 3 (x 0) = p ε 0 3 (x 0) = b ε 0 3 (x 0) = 0 p1 ε3 0 (x 0) = x 0. ε3 i (x 0) P x0 (i + 1), = {s,g, p1, p,b}, i {1,,...} (33a) (33b) (33c) where P x0 ( j) deotes the set of polyomials i x 0 of order j. Sice p1 ε 0 3 (x 0) = x 0, the Nasell-poisso momet closure fuctio will have a large iitial error, especially for large iitial coditios, whe compared to all other momet closure fuctios. For all, i {1,,...}, all of these momet closure fuctios match derivatives, with the derivative matchig error beig of the same order i x 0. The simulatio results discussed below show that with the exceptio of Nasell-poisso momet closure fuctio, which cosistetly provides the worst estimates, all other momet closure fuctios perform fairly well. Example: We cosider the stochastic logistic model with a 1 =.30, a =.0, b 1 =.015, b =.001, (34) which is used by Matis et al. (1998) to model the populatio dyamics of the Africa Hoey Bee. Usig (17) with the matrices A ad B computed i (11), we have the followig trucated momet dyamics ν = ν ϕ 3 (ν). ν ν 0.03 The time evolutio of the momets correspodig to differet momet closure techiques is obtaied by substitutig the appropriate momet closure fuctio from Table 3 i place of ϕ 3 (ν). Figure 1 plots the differet errors (31) durig the time iterval [0,.5] for x 0 = 5. The error is approximated by the first ie terms of 15

16 ' ( % $!, - *%!$ +!%!(!'!# *$!&!!"# $ $"# % %"# ) Fig. 1. Evolutios of the error e +1 i (31) for the differet momet closure techiques i Table 3, with parameters as i (34) ad x 0 = 5. the series i (31). The letters s, g, p1, p ad b are used to deote the errors correspodig to separable derivative-matchig, ormal, Nasell-poisso, ew-poisso ad biomial momet closure fuctios, respectively. I order to evaluate the performace of these momet closures techiques past the iitial period, we also compute the exact evolutio of the momets. This is oly possible because the populatio limit U = 5 is small ad oe ca obtai the exact solutio by umerically solvig the Kolmogorov equatio (6). Figure cotais plots of the mea ad variace errors, respectively, for the differet momet closure fuctios with x 0 = 5 ad x 0 = 0. The letters s, g, p1, p ad b are used to deote the errors correspodig to separable derivative-matchig, ormal, Nasell-poisso, ew-poisso ad biomial momet closure fuctios, respectively. For x 0 = 0 the biomial momet closure fuctio provides the best estimate both iitially ad at steady-state, whereas for x 0 = 5 the ew-poisso momet closure fuctio does best iitially, but the biomial momet closure fuctio cotiues to provide the most accurate steady-state estimate. As oe would expect from (33), the Nasell-poisso momet closure fuctio performs the worst Momet Closures for 3 All distributios D discussed i Sectio 5 are characterized by at most parameters (m ), hece the correspodig distributio-based momet closure fuctios ϕ+1 D are ot uique for 3. I this sectio we illustrate for logormal ad ormal momet closures, how derivative matchig ca be used as a tool for gaugig their performaces ad choosig the oes that yield the least derivative matchig errors. We also propose a ew momet closure fuctio for = 3, which as we will see guaratees better approximatio at least locally i time, as compared to other oes. 16

17 !&!'!&!#!&!$!!!&!$ + -$,!!!&!#!!&!$.$ + -,!!&!'!!&!) -#.!!&!%!!&!(!!&!)!!&#!!&!".#!!&#$!!&!(!!&#'! " #! #" $! $" %! %" * (a) mea error for x 0 = 5!!&!'! " #! #" $! $" %! %" * (b) mea error for x 0 = 0 $&" &'" $ & )&!'% #&" # ($ + (#!'$!&" * )!'#!'" )",!! + *!!&"! " #! #" $! $" %! %" ' (c) variace error for x 0 = 5!!'"! " # $ % &! &" ( (d) variace error for x 0 = 0 Fig.. Evolutios of the mea error µ 1 ν 1 ad of the variace error (µ µ 1 ) (ν ν 1 ) for the differet momet closures i Table 3 for =, with parameters as i (34) ad x 0 = 5 (left) x 0 = 0 (right). Logormal momet closure: Oe ca see from Example 3 of Sectio 5, that a family of logormal momet closure fuctios for µ +1 is give by ϕ l +1 (µ) = µt 1 1 µ t... µ t, C m 1 t m = C +1 1, m=1 C m t m = C +1. (35) m=1 The derivative matchig errors for these momet closure fuctios are give by l ε 0 +1 (x 0) = 0, l ε i +1 (x 0) P x+0 ( + i + 1 w), i {1,,...} (36) where the costat w {1,,...,} is such that m=1 C m k t m = C +1 k, k {1,...,w}, Ck m t m C +1 k, k {w + 1,...,} m=1 with higher values of w correspod to better approximatios. From (35) ad (36), we coclude that the logormal momet closure guaratees that w. But from (B.1) we see that this ca be improved ad we actually set w = for the separable 17

18 derivatives-matchig momet closure, for which s ε 0 +1 (x 0) = 0, s ε i +1 (x 0) P x+0 (i + 1), i {1,,...}. (37) Thus, the separable derivative-matchig momet closure fuctio leads to the best estimate for µ +1, i the sese that it has the least derivative matchig error amog all the momet closure fuctios cosistet with the logormal distributio. Normal momet closure: We first restrict our attetio to the case = 3. We recall from Example that a family of ormal momet closure fuctios for µ 4 were give by 4µ 1 µ 3 + 3µ 1µ µ 1 + 6µ h(µ 1, µ µ 1,µ 3 3µ µ 1 + µ 3 1 ). Its straight forward to see that the fuctio h(κ 1,κ,κ 3 ) correspods to the cumulat closer fuctio for κ 4. However, oe eeds to be careful i pickig the fuctios h. To demostrate this we cosider the followig ormal momet closure fuctios ϕ g 4 (µ) = 4µ 1µ 3 + 3µ 1µ µ 1 + 6µ 1 4 (38) ϕ g1 4 (µ) = 4µ 1µ 3 + 3µ 1µ µ 1 + 6µ µ 3 3µ 1 µ + µ 1 3 (39) ϕ g 4 (µ) = 4µ 1µ 3 + 3µ 1µ µ 1 + 6µ (µ 3 3µ 1 µ + µ 1 3 )µ 1 (40) which correspod to cumulat closure fuctios h(κ 1,κ,κ 3 ) = 0, κ 3 ad κ 3 κ 1 respectively. Usig symbolic maipulatio i Mathematica we have the followig derivative matchig errors for the ormal momet closure fuctios (38) (40): ε i 4 (x 0) P x0 (i + 1), = {g,g1} g ε i 4 (x 0) P x0 (i + ) for all i {1,,...} ad zero for i = 0. Thus, the ormal momet closure fuctios (38) (39) have derivative matchig errors of the same form as (37), ad provide better approximates for µ 4 as compared to (40), for which the derivative matchig errors are a order higher. I geeral for 3, a family of ormal momet closure fuctios for µ +1 is obtaied usig the cumulat closure fuctio κ +1 = h(κ 1,κ,...,κ ) where the fuctio h is zero if ay of κ 3,...,κ is zero. Usig symbolic maipulatio i Mathematica we foud that choosig cumulat closure fuctios as h(κ 1,κ,...,κ ) = f d κ d d=3 18

19 for some costats f d, typically lead to derivative matchig error of the form (37) ad provides good estimates for µ +1. Zero first-order error momet closure: We ow propose a slight modificatio of the ormal momet closure fuctio (38) which is give by ϕ z 4 (µ) = 4µ 1µ 3 + 3µ 1µ µ 1 + 6µ µ µ 1 (41) ad correspods to the cumulat closure fuctio κ for κ 4. It yields the followig derivative matchig errors z ε i 4 (x 0) = 0, i {0,1} z ε i 4 (x 0) P x0 (i + 1), i. Hece, ulike the separable derivative-matchig ad ormal momet closure fuctios, ϕ z 4 (µ) yields zero 0th ad 1 st order derivative matchig errors, ad hece, provides the best estimates for µ 4 as compared to them, at least ear t = 0. I order to gauge the performaces of the above momet closure fuctios we cosider the stochastic logistic model with parameters as i (34), = 3 ad x 0 = 0. We recall from Table 1 that for = 3, we have the followig separable derivativematchig momet closure fuctio ϕ s 4 (µ) = µ4 1 µ4 3 µ 6 Usig (11), we have the followig trucated momet dyamics ν ν = ν ν 1 ν ν 3. (4) ϕ 3(ν) Substitutig the momet closer fuctios (38) (4) i place of ϕ 3 (ν), we obtai the correspodig approximate time evolutio of momets. Figure 3 cotais plots of the mea, variace, ad third cumulat errors. The letters g, g1, g, z ad s represet these errors for the momet closure fuctios (38) (4), respectively. As expected the ormal momet closure fuctios (38) (39) perform much better tha the ormal momet closure fuctio (40). The separable derivative-matchig momet closure fuctio (4) also provide good estimates. Oe ca also see that the zero first-order error momet closure fuctio (41), which guaratees the best approximatio ear t = 0 actually provides i this case the most accurate estimate for µ 4 for all time. As ca be see from Figure ad 3, the mea ad variace errors for x 0 = 0 with = 3 are a order of magitude smaller as compared to the oes with =. 19

20 !'" ()#!!%!!!'"!#!#'"!$!$'"!%!%'",$!&! " #! #" $! $" %! %" &! * (a) mea error for x 0 = 0 -,# +,!'!)!'!(!'!"!'!&!'!%!'!$!'!#!,$ +!!'!#! " #! #" $! $" %! %" &! * (b) variace error for x 0 = 0,,# -!'$!, +!!'$!!'& -,#!!')!!'(,$!#!#'$! " #! #" $! $" %! %" &! * (c) third cumulat error for x 0 = 0 Fig. 3. Evolutios of the mea error µ 1 ν 1, variace error (µ µ 1 ) (ν ν1 ), ad third cumulat error (µ 3 3µ µ 1 + µ 1 3) (ν 3 3ν ν 1 + ν1 3 ) for the differet momet closure fuctios (38) (4) for = 3, with parameters as i (34) ad x 0 = Steady-State Solutios of the Trucated Momet Dyamics We ow look at the steady-state solutios of the trucated momet dyamics (17) for the differet momet closure techiques Separable Derivative-Matchig Momet closure Cosider the trucated momet dyamics of order N with momet closure fuctios as give i Table 1. From (17), at steady-state we have 0 = Aν s ( ) + Bϕ s +1 (νs ( )) (43) where ν s ( ) deotes the steady-state solutio. Usig (16) at steady-state we have ν s ( ) = c 1ν s 1 ( ). ν s ( ) = c 1 ν s 1 ( ) (44a) (44b) 0

21 ϕ +1 (ν s ( )) = c ν s 1 ( ) (44c) for some positive real umbers c 1,...,c. Usig the above equalities ad Table 1 we coclude that ϕ3 s (νs ( )) = c 3 1, ϕs 4 (νs ( )) = c4 c 6 ν1 s ( ), 1 ϕ s 5 (νs ( )) = c10 1 c5 3 c 10,... (45) for {,3,4,...} respectively. Substitutig (45) i (44) yields the followig uique o-trivial solutio for ν s 1 ( ) c = ν1 s ( ) = = 3 c 3 1 c 6 1 c 3 c 4 c 10 1 c5 3 c 10 c 4 = 4 ad the correspodig ν s ( ),...,νs ( ) ca be calculated from (44). Hece, the separable derivative-matchig momet closure always yields a uique o-trivial positive real steady-state N. I terms of the parameters a 1, b 1, a ad b, the costats c 1, c ad c 3 are give as follows c 1 = K, c = K + σ, c 3 = K 3 + 3Kσ + σσ K = a 1 a b 1 + b, σ = a 1b + b 1 a (b 1 + b ), σ = b b 1 b + b 1, ad hece, ν1 s ( ) = K 1+ σ ) K (1+ 3σ K + σσ K 3 K = (1+ σ K ) 4 = Other Momet closure techiques I this sectio we will see that the momet closure fuctios which lack the separable structure of (0), ca lead to scearios of biologically meaigless steadystate solutios for the trucated momet dyamics (17). Substitutig the momet closure fuctios from Table i (43), we get the followig o-trivial steady-state solutios for = : [ ν g 3 1 ( ) = K 4 ± 1 (1 8σ ) 1 4 K ν p1 1 ( ) = K [ 3 4 ± 1 4 ] ( 1 8(σ 1) ) 1 K ] 1

22 [ ν p 3K 1 1 ( ) = ± 1 ((K + 1) 8σ ) 1 ] 4 4 K ν1 b ( ) = K σ K 1. From the above steady-states we coclude the followig for = : (1) The biomial momet closure fuctio leads to a uique o-trivial attractig real steady-state, which ca be egative for a rage of parameters. () Normal, Nasell-poiso ad ew-poiso momet closure fuctios, yield two o-trivial steady-states, with the oe with the sig beig a spurious steadystate. Followig the defiitio by Nasell (003b), a steady state is spurious, if lim M ν 1 ( ) K, where σ ad K are both O(M). For =, all these spurious steady-states happe to be ustable, ad hece, the trucated model will ot coverge to them. Whe the parameters are chose, such that the term uder the square root sig is egative, the both the o-trivial steady-states would be imagiary, ad hece, biologically meaigless. I geeral for 3, the ormal momet closure fuctios itroduced i Sectio 6.1. yield K 3 o-trivial steady-states with K 1 of them beig spurious. For example, for = 3, the ormal ad the zero first-order error momet closure fuctios (38) ad (41) yield 3 o-trivial steady-state solutios give as the roots of the third order polyomials ad (3c 1 + 4c )ν 1 ( ) 1c 1 ν 1 ( ) + 6ν 1 ( ) 3 = c 3, (3c 1 + 4c 1)ν 1 ( ) 1c 1 ν 1 ( ) + 6ν 1 ( ) 3 = c 3 c 1, respectively, with of them beig spurious. Also for a rage of parameter values, biologically meaigless scearios of a combiatio of imagiary ad egative steady states ca happe. O the other had, the separable derivative-matchig momet closure fuctio with a uique o-trivial positive real steady-state N has a clear advatage. 7 Coclusio ad Future Work A procedure for costructig momet closures for the stochastic logistic model was preseted. This was doe by first assumig a separable form for the momet closure fuctio ϕ +1 (ν), ad the, matchig its time derivatives with µ +1, at some iitial time t 0. We showed that for iitial coditios x(t 0 ) = x 0 with probability oe, there exists a uique separable derivative-matchig momet closure

23 fuctio for which the i th derivative matchig error is a polyomial i x 0 of order i + 1 for all i {1,,...} ad zero for i = 0. Explicit formulas to costruct these momet closure fuctios were provided. Comparisos with alterative momet closure techiques available i literature were carried out based o trasiet performace ad steady-state solutios of the trucated momet dyamics, which led to the followig coclusios: (1) Derivative matchig ca be used as a effective tool for gaugig the performace of momet closure fuctios. We showed that for =, with the exceptio of the Nasell-poisso, all other momet closure fuctios i Table perform derivative matchig ad give fairly good estimates of µ 3. For 3, we showed that amog the momet closure fuctios cosistet with the logormal distributio, the separable derivatives-matchig momet closure fuctio provides the best estimate for µ +1. We also proposed families of ormal momet closure fuctios that perform derivative matchig ad result i good estimates. For = 3, a ew zero first-order error momet closure fuctio was also proposed which guarateed better approximatios, at least locally i time, as compared to the other momet closure fuctios discussed i this paper. () The separable derivative-matchig momet closure, always yields a uique o-trivial positive real steady-state N, ad hece, i some sese superior to the other momet closures, which ca have spurious, imagiary ad eve stable egative steady-states. Possible directios for future research iclude the developmet of a systematic procedure to costruct momet closure fuctios of the form (41) which ca yield zero derivative matchig errors up to ay order ad the extesio of the results i this paper to multi-specie birth-death Markov processes. Primary results regardig the latter ca be foud i Hespaha ad Sigh (005). A Appedix Substitutig (4), (7) ad (8) i (10) ad doig a biomial expasio we have dµ m dt = E[(x + 1) m x m )η(x) + (x 1) m x m )χ(x)] = E = E [ m j=1c mj [(a 1 x b 1 x ) + ( 1) j (a x + b x )]x m j ] [ ] m C m j f ( j, p)x m j+p j=1 (A.1) where C m j ad f ( j, p) are defied i (1) ad (13). Usig (A.1), (1) ad (13) we coclude that the evolutio of µ m, m N, ca be writte as 3

24 µ m = = = m C m j f ( j, p)µ m j+p j=1 m+p 1 C m m+p r f (m + p r, p)µ r, r=p m+1 C m m+p r f (m + p r, p)µ r. r=1 r = m j + p (A.) B Appedix Proof of Theorem : STEP 1: We first show that with γ m chose as the solutio to the liear system of equatios C +1 k = γ m C m k, k = {1,...,}. (B.1) m=1 equalities (3) hold. Usig (15), (16) ad (17) oe ca see that the followig equalities imply (3): µ +1 (t 0 ) = ϕ+1 s (ν(t 0)) (B.) dµ +1 (t) dt t=t0 = dϕs +1 (ν(t)) dt t=t0 + s ε+1(x 1 0 ), (B.3) where s ε 1 +1 (x 0) is a polyomial i x 0 of order. Let γ m be chose as the solutio of (B.1). We show ext that equalities (B.) ad (B.3) hold. With iitial coditios startig o the set of determiistic distributios, we have µ m (t 0 ) = x m 0. From µ(t 0) = ν(t 0 ), (0) ad (B.1) we have µ +1 (t 0 ) = x +1 0 = x m=1 γ mc m 1 0 = ϕ s +1 (µ(t 0)) = ϕ s +1 (ν(t 0)) which proves (B.). From (A.) we have µ +1 (t) = µ +1 (t 0 ) = + r=1 + r=3 C p r f ( p r, p)µ r C p r f ( p r, p)xr 0 + s 1 ε1 +1(x 0 ), (B.4) where s 1 ε1 +1 (x 0) is a polyomial i x 0 of order. Usig (A.) ad (17) we have 4

25 ν m = m+1 C m m+p r f (m + p r, p)ν r, ν +1 = ϕ+1 s (ν). r=1 (B.5) Thus, from (0), (B.1), (B.5) ad ν(t 0 ) = [x 0,...,x 0 ]T dϕ+1 s (ν(t)) = dt = dϕ+1 s (ν(t)) dt t=t0 = = m=1 γ m ν γ 1 1 νγ m 1 m m=1 m=1 m=1 m+1 γ m ν γ 1 r=1 ν γ ν m 1 νγ m 1 m ν γ C m m+p r f (m + p r, p)ν r, m+1 γ m C m m+p r f (m + p r, p)x ( m=1 γ mm)+r m 0 r=1 m+1 γ m C m m+p r f (m + p r, p)x0 +1+r m. r=1 With a chage of variable q = r m, the last equality becomes dϕ+1 s (ν(t)) dt t=t0 = = + q=3 m=1 m=1 + q=+ m γ m C m +1+p q f ( p q, p)xq 0 γ m C m +1+p q f ( p q, p)xq 0 + s ε1 +1(x 0 ), (B.6) where s ε1 +1 (x 0) is a polyomial i x 0 of order. Usig (B.1), equality (B.6) reduces to dϕ+1 s (ν(t)) t=t0 = dt + q=3 C p q f ( p q, p)xq 0 + s ε1 +1(x 0 ). (B.7) Comparig (B.7) with (B.4) oe ca see that (B.3) holds. STEP : We ow show that solutio to (B.1) is uique ad give by (). Towards that ed, we observe that for all z R, oe ca write usig biomial expasio, [1 (1 + z)] +1 = +1 v=0 = 1 + C +1 v ( 1) v (1 + z) v +1 v=1 C +1 v ( 1) v v C v wz w. w=0 (B.8) Equatig coefficiets for z s, s {1,...,} o both sides of (B.8) we have 0 = +1 v=1 C +1 v ( 1) v C v s 5

26 ( 1) C +1 s = C +1 v ( 1) v C v s. (B.9) v=1 Comparig (B.9) with (B.1) oe ca see that a solutio to (B.1) will be (). Also the system of liear equatios (B.1) ca be put ito the form C = Πγ where γ = [γ 1,...,γ ] T, C = [C1 +1,...,C +1 ] T ad C 1 1 C 1... C 1 0 C Π =... C Cm m As the upper triagular matrix Π is o-sigular, the above solutio to γ m is uique. Refereces Bailey, N. T. J., The Elemets of Stochastic Processes. Wiley, New York. Barlett, M. S., Gower, J. S., Leslie, P. H., A compariso of theoretical ad empirical results for some stochastic models. Biometrika 47, Hespaha, J. P., Mar Stochastic hybrid systems: Applicatios to commuicatio etworks. I: Alur, R., Pappas, G. J. (Eds.), Hybrid Systems: Computatio ad Cotrol. No. 993 i Lect. Notes i Comput. Sciece. Spriger-Verlag, Berli, pp Hespaha, J. P., Sigh, A., 005. Stochastic models for chemically reactig systems usig polyomial stochastic hybrid systems. It. J. of Robust ad Noliear Cotrol 15, Keelig, M. J., 000. Multiplicative momets ad measures of persistece i ecology. J. of Theoretical Biology 05, Matis, J. H., Kiffe, T. R., O apprroximatig the momets of the equilibrium distributio of a stochastic logisitc model. Biometrics 5, Matis, J. H., Kiffe, T. R., 00. O iteractig bee/mite populatios: a stochastic model with aalysis usig cumulat trucatio. Evirometal ad Ecological Statistics 9, Matis, J. H., Kiffe, T. R., Parthasarathy, P. R., O the cumulat of populatio size for the stochastic power law logisitc model. Theoretical Populatio Biology 53, Nasell, I., 001. Extictio ad quasi-statioarity i the verhulst logistic model. J. of Theoretical Biology 11, Nasell, I., 003a. A extesio of the momet closure method. Theoretical Populatio Biology 64,

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