# stationary distributions of a noisy logistic process[精选推荐pdf]

arXivcond-mat/0502389v1 [cond-mat.stat-mech] 16 Feb 2005Stationary distributions of a noisy logistic processP. F. G´ ora†M. Smoluchowski Institute of Physics and Complex Systems Research Center Jagellonian University, Reymonta 4, 30-059 Krak´ ow, PolandReceived Stationary solutions to a Fokker-Planck equation corresponding to a noisy logistic equation with correlated Gaussian white noises are con- structed. Stationary distributions exist even if the corresponding deter- ministic system displays an unlimited growth.Positive correlations be- tween the noises can lead to a minimum of the variance of the process and to the stochastic resonance if the system is additionally driven by a periodic signal.PACS numbers 05.40.Ca, 02.50.Ey, 87.23.Cc1. IntroductionThe logistic equation˙ x ax1 − x, a 0, x 0,1is one of the best-known and most popular models in population dynamics. Perturbing this equation by a multiplicative noise is an obvious generaliza- tion of the deterministic theory, aiming at describing populations that livein an ever-changing environment. The logistic equation with a fluctuating growth rate˙ x a pξtx1 − x,2has been first discussed by Leung in Ref. [1] and later by many other authors. Recently in Ref. [2] we have discussed a further generalization of 2 in whichboth the growth rate and the limiting population level fluctuate, and thesefluctuations are correlated in time˙ x a pξmtx − b q ξatx2.3†e-mailgoraif.uj.edu.pl12fp2printed on February 2, 2008Here ξm,aare two Gaussian white noises GWNs that satisfy hξiti 0, hξit1ξit2i δt1−t2, i m,a, hξmt1ξat2i cδt1−t2, p, q are theamplitudes of the two noises and the correlation coefficient c ∈ [−1,1]. For the sake of terminology, we will sometimes call the noise ξmt “multiplica- tive” and the noise ξat “additive”; see Eq. 5 below for a rationale behind these names. Please note, though, that on the level of Eq. 3 both these noises are coupled multiplicatively to the process xt. Note also that ifb 0, the corresponding deterministic equation converges to a stable fixed point; accordingly, we will call a system with a positive b “convergent”. If b 6 0, the corresponding deterministic system displays unlimited growth. We will call a system with a negative b “exploding”. Recently Mao et al. have shown in Ref. [3] that in the absence of ξm, the system 2 remains positive and bounded even in the “exploding” case.In Ref. [4] we havediscussed certain difficulties that may arise in numerical simulations of such a system. This paper generalizes the result of Mao et al. to the case of both noises present. In Ref. [2] we have shown how the dynamics 3 is related to the problem of a linear stochastic resonance. We have mapped the nonlinear equation 3 into a linear Langevin equation with two correlated noises and used the solutions of the latter to heuristically explain the behaviour of the noisylogistic equation. Specifically, the substitutiony 1 x4converts Eq. 3 into a linear equation˙ y −a pξmty b q ξat5which can be solved exactly for realizations of the process yt. The process 5 has a convergent mean ifa −1 2p2 06aand a convergent variance if a stronger conditiona − p2 06bholds. Using the properties of the process yt, useful prediction can be made about the noisy logistic process. Since in the presence of correlations,for certain values of parameters the variance of yt first shrinks and then grows as a function of the “multiplicative” noise strength, q, one expects a similar behaviour for the logistic process xt as well. These predictions have been corroborated numerically in Ref. [2]. In particular, if c ±1,fp2printed on February 2, 20083bp ∓ aq 0 and the condition 6b is satisfied, the variance of the process yt vanishes. The relation between the processes yt and xt 1/yt intuitively means that if almost all realizations of the er asymptotically reach the same constant value, so do almost all realizations of the latter. However, as a al relation between moments of these processes is not trivial, the predictions based on the properties of yt have only a heuristic value. In the following we will construct mathematically exact stationary solu- tions to the Fokker-Planck equation corresponding to Eq. 3 and re-examine the above results from the point of view of these stationary solutions. Fur- thermore, we will show numerically that if the parameters undergo periodic for example, circaannual or seasonal oscillations, a positive correlation between the noises leads to a stochastic resonance. In the Appendix we extend to the case of two correlated noises the proof originally proposed by Mao et al. in Ref. [3] that solutions to Eq. 3, when started from a positive initial condition, never become negative almost surely.2. The Fokker-Planck equationThe problem of constructing a Fokker-Planck equation correspondingto a process driven by two correlated Gaussian white noises has been first discussed in Ref. [5], where the two noises have been decomposed into two independent processes. The same result has been later re-derived in [6], where the authors have attempted to avoid an explicit decomposition of the noises but eventually resorted to a disguised of the decomposition. The general Langevin equation˙ x hx g1xξmt g2xξat,7where xt is a one-dimensional process and ξm,aare as in Eq. 3, leads to the following Fokker-Planck equation in the Ito interpretation∂Px,t ∂t −∂ ∂xhxPx,t 1 2∂2 ∂x2BxPx,t,8awhere Bx [g1x]2 cg1xg2x [g2x]2.8bIn the case of Eq. 3 the corresponding Fokker-Planck equation there- fore reads∂Px,t ∂t −∂ ∂x[a − bxxPx,t] 1 2∂2 ∂x2x2p2− 2cpqx q2x2Px,t.94fp2printed on February 2, 2008It is apparent that the absolute signs of the two noise amplitudes do not in-fluence the solutions to the above equations, only their relative sign, sgnpq, does. In the following we will assume that sgnpq 1. This comes at no loss to the generality as Eq. 9 is invariant under a simultaneous change ofsigns of pq and the correlation coefficient, c. Stationary solutions to Eq. 9 are the normalizable solutions to [7, 8]x2p2−2cpqxq2x2dPstxdx2x2q2x2 b−3cpqx − a−p2Pstx 0.10A slight modification of the argument presented originally in Ref. [3] shows that all solutions to Eq. 3 that start from a positive initial condition remain positive almost surely; see the Appendix for a proof. Physically speaking, this results from a presence of an absorbing barrier at x 0 in Eq. 3 should the population suddenly drop to zero, it would stay there forever. The al result of Ref. [3], extended here to the case of two noises present, ensures that the population never actually becomes nonpositive although in certain cases see below it may dynamically cluster in a close proximity of x 0. Therefore, we can divide both sides of 10 by x and obtainxp2−2cpqxq2x2dPstxdx22q2x2 b − 3cpqx − a − p2Pstx 0,11 provided Pstx is normalizable over the x 0 semiaxix. One may be tempted to try to immediately solve Eq. 11 by standards, but a word of caution is needed here should the coefficient at dPst/dt vanish, special care must be taken.2.1. The case of only one noise presentBefore proceeding to the general case, we will discuss the special cases where only one of the amplitudes p, q does not vanish. Purely “multiplicative” noise. If q 0, the Langevin equation 3 re- duces to˙ x a pξmtx − bx212and we find for PstxPstx Nx2a−p2/p2exp −2bxp2 ,13fp2printed on February 2, 20085where N is a normalization constant.This function is normalizable for x 0 if b 0, or if the system is convergent, anda −1 2p2 0,14awhich determines the behaviour of the distribution around x 0. Moreover, ifa − p2 0,14bPstx goes to zero as x → 0and has a maximum at x a − p2/b. Note that the conditions 14a, 14b coincide with the conditions 6a, 6b determining the properties of the linear process 5. For p2 a 1 2p2,the distribution is mildly divergent at x 0 and decreases monotically with an increasing x. Purely “additive” noise. If p 0, the Langevin equation takes the ˙ x a − bxx qx2ξat15and we obtain for the stationary distributionPstx N x4exp −a − 2bxq2x2 16which is normalizable whenever a 0. Note that no bounds on b are im- posed The stationary distribution exists for both convergent and exploding systems. This special case falls into a broad category discussed recently by Mao et al. in Ref. [3], without further generalizations provided by the present paper. The distribution 16 has a maximum at the positive root of2q2x2 bx − a 0.172.2. The general caseIf both noises are present and are not maximally correlated, |c| 6 1, the solution to 11 readsPstx Nx2a−p2/p2p2−2cpqxq2x2ap2/p2exp−2bp−acqarctan qx−cp√1−c2p√1 − c2p2q.18 Since the exponential term is limited, the convergence normalization prop- erties of 18 are determined by those of the fractional term. The denomi- nator is always strictly positive. For x → ∞, Pstx ∼ x−4for all possible6fp2printed on February 2, 200801234567890.00.51.01.52.0Pstxxq 0.101234567890.00.51.01.52.0Pstxxq 0.501234567890.00.51.01.52.0Pstxxq 0.601234567890.00.51.01.52.0Pstxxq 5.0Fig.1. Stationary distributions 18 in a strongly correlated case, c 0.99. Clock- wise, from top-left q 0.1, q 0.5, q 0.6, and q 5.0. Other parameters, common for all panels, are p 0.5, a b 1.values of parameters. Pstx is therefore normalizable if it does not diverge too rapidly at x → 0, or again if the condition 14a holds. Either in this case, no bounds on b are imposed. If the condition 14b holds as well, the distribution 18 approaches zero as x → 0, but this condition is no longer associated with the presence of a maximum. The maximum of Pstx coincides with the positive root of2q2x2 b − 3cpqx − a − p2 0,19cf. Eq. 11, provided such a root exists. It certainly does for a − p2 0, but it can appear also for p2 a 1 2p2, where the distribution is mildlydivergent. Example stationary distributions in a strongly correlated case are presented on Fig. 1. For large values of the additive noise strength, q, the distributions are highly skewed and squeezed against the x 0 axis. For comparison, on Fig. 2 we show example stationary distributions for the un- correlated and a strongly anticorrelated cases. The distributions presented are skewed and much wider than the distribution from Fig. 1 with the samefp2printed on February 2, 200870.00.51.01.52.00.00.51.01.52.0Pstxxc00.00.51.01.52.00.00.51.01.52.0Pstxxc-0.99Fig.2.Stationary distributions for the uncorrelated c 0, left panel and a strongly anticorrelated c −0.99, right panel cases. Other parameters are a b 1, p q 0.5.value of q 0.5. These distributions also get squeezed as the additive noise strength becomes large. Note that the distribution corresponding to the anticorrelated noises is already more squeezed than the distribution for the uncorrelated case. If the distribution 18 is normalizable, it has a convergent mean and a variance. Its higher moments are divergent.2.3. The maximally correlated caseIf the two noises are maximally anticorrelated, c ±1, Eq. 11 takes the xp ∓ qx2dPst dx 2p ∓ qx2 q2x2 b ∓ pqx − aPst 0,20leading to the following candidate solutionPstx Nx2a−p2/p2p ∓ qx2ap2/p2exp ∓2bp ∓ aq pqp ∓ qx .21With our sign convention adopted, sgnpq 1, this solution is nor- malizable if c −1 and the condition 14a holds. The distribution 21 with the “” sign can be obtained from Eq. 18 by taking the limit c → −1. This distribution decreases as x−4with x → ∞. The maximum of 21, if it exists, coincides with the positive root of Eq. 19 with c −1. The case of c 1 is more challenging. First, if the resonant conditionbp − aq 0228fp2printed on February 2, 2008holds, Eq. 20 is solved byPstx δ x −p q 23regardless of the value of p. This result is stronger than that reported in Ref. [2] where we could predict a δ-shaped distribution only in the a−p2 0 case, or when the variance of the corresponding linear system was conver- gent, as otherwise any predictions based on the linear system failed. Notethat with the condition 22 statisfied, Eq. 3 reduces to a rescaled of Eq. 2. If c 1 and the condition 22 does not hold, Eq. 20 does not have a normalizable solution. This observation is slightly surprising, but ally speaking, it results from the fact that the double limitlim x→p/qc→1exp−2bp − acqarctanqx−cp√1−c2p√1 − c2p2q24does not exits Its value depends on which route the singularity is ap- proached. The nonexistence of stationary solutions in the fully correlated, non-resonant case is, therefore, related to the essential singularity of thecomplex exponential at infinity. The fact that with c 1, bp−aq 6 0, thedrift and diffusive terms in Eq. 20 both vanish, but at different points, is the physical reason for this apparent oddity If the population gets locatedaround the point of the vanishing diffusion, it is washed away by the drift,and if it gets located around the point of the vanishing drift, it diffusively leaks from there. Nevertheless, if a−1 2p2 0, in numerical simulations the cases of c 1 and c 1 − ε with 0 - 2q0.000.050.100.150.200.25012345678910- 2qFig.3. The variancex2−hxi2determined the distribution 18 as a function of the additive noise strength, q. Main panel p 0.5, the curves, from bottom to top, correspond to c 0.99, c 0.90, c 0.75, c 0.50, c 0.25, c 0, and c −0.25, respectively. Inset p 1.1, the curves correspond, from bottom to top, to c 0.99, c 0.98, c 0.97, c 0.96, c 0.95, and c 0.94, respectively. Other parameters, common for all curves presented, are a b 1.correlated and resonant case, the stationary distribution becomes δ-shaped and its variance indeed vanishes, much as predicted by the linear system. Since we now know the mathematically exact stationary distributions, we can test the behaviour of the variance in the general case directly.Recall that the distribution 18 has the two first moments convergent whenever it is normalizable. Unfortunately, analytical expressions for these moments cannot be obtained, mainly due to the presence of the complicated exponential term. Therefore, we have calculated the moments by numeri- cally integrating over the distribution 18. Results are presented on Fig. 3. If the distribution approaches zero as x → 0, or when the condition 14bis satisfied, and if the two noises are positively correlated, 0 a 1 2p2,a distinct minimum in the variance of x also appears but it is present only forfairly large and po