Time dependence of moments of an exactly solvable Verhulst model under random perturbations

Time dep endence of momen ts of an ex actly solv able V erh ulst mo del under random p ert urbations ∗ V.M. Loginov No v em b er 8, 2018 Cen ter of In terdisciplinary Researc hes of Krasnoy ars k State Pedagogical Univ ersit y ul. Leb edev oi, 89. Krasno y arsk 660049 Russia e-mail: loginov@imf i.kspu.ru Abstract Explicit expressions for one p oin t momen ts corresp onding to stoc has- tic V erhulst m o del driv en by Marko vi an coloured dic hotomo us noise are presented. It is sh o wn that the momen ts are the giv en functions of a decreasing exp onent. The asymptotic b eha vior (for large time) of the momen ts is d escrib ed b y a single decreasing exp onent . Keywor ds : S to chastic V erhulst mo del, one p oint m oments, explicit expressions. 1 In tro du ction There are a lot of pap ers dev oted to description of the temp ora ry ev o lu- tion of momen ts with exactly solv able nonlinear sto c ha stic equations. In [1] w e gav e some general pro cedure to explicitly solv e the master equations of h yp erb olic type corresp onding to nonlinear sto c hastic equations driv en by ∗ This pap er w a s written with partia l financial supp or t from the RFBR g rant 0 6 -01- 00814 . 1 dic hotomous noise. The metho d is based on a generalization of Laplace fac- torization method [2, 3]. As an example w e hav e considered complete exact nonstationary solution of the master equations for proba bility distribution corresp onding to sto c hastic V erh ulst mo del. In this pa p er w e calculate one p oin ts moments of arbitrar y degree and discuss its time ev olution. Let us consider nonlinear stochastic dynamical system ˙ x = p ( x ) + α ( t ) q ( x ) , (1) where x ( t ) is the dynamical v ariable, p ( x ), q ( x ) are giv en functions of x , α ( t ) is the random function with know n statistical c haracteristics. The mo del (1) arises in differen t applications (see for example [4, 5] a nd bibliograph y therein). An import an t application of this mo del consists in study of noise- induced tra nsitions in phys ics, c hemistry and biology . The functions p ( x ), q ( x ) are often take n p o lynomial. F or example, if we set p ( x ) = p 1 x + p 2 x 2 , q ( x ) = q 2 x 2 , p 1 > 0, p 2 < 0, | p 2 | > q 2 > 0, then the eq uation (1) desc rib es the p opulation dynamics when r esources (n utrition) fluctuate ( V erh ulst model). In the f ollo wing w e will assume α ( t ) to b e binary (dic hotomic) noise α ( t ) = ± 1 with switc hing frequency 2 ν > 0. As o ne can sho w (see [6]), the a v erag es W ( x, t ) = h f W ( x, t ) i and W 1 ( x, t ) = h α ( t ) f W ( x, t ) i f o r the pro babilit y densit y f W ( x, t ) in the space of p ossible tra jectories x ( t ) of the dynamical system (1) satisfy a system (also called “master equations”):    W t + ( p ( x ) W ) x + ( q ( x ) W 1 ) x = 0 , ( W 1 ) t + 2 ν W 1 + ( p ( x ) W 1 ) x + ( q ( x ) W ) x = 0 . (2) W e supp ose that the initial condition W ( x, 0) = W 0 ( x ) for the proba bility distribution is no nr a ndom. This implies that the initial condition for W 1 ( x, t ) at t = 0 is zero: W 1 ( x, 0) = h α (0) f W ( x , 0) i = h α (0) i W 0 ( x ) = 0. The pro b- abilit y distribution W ( x, t ) should b e nonnegativ e and normalized for all t : W ( x, t ) ≥ 0 , R ∞ −∞ W ( x, t ) dx ≡ 1. In [1] w e ha v e obtained t he f o llo wing explicit fo rm of the complete solution of the system (2) for probabilit y distribution W ( x, t ): W ( x, τ ) = 1 2 e − τ n δ  x − e − τ x ∗ 1+( p 2 + q 2 )( e τ − 1) x ∗  + δ  x − e − τ x ∗ 1 − ( p 2 − q 2 )( e τ − 1) x ∗ o + 1 2 q 2 x 2 n H  x e τ (1+( p 2 − q 2 ) − x ( p 2 − q 2 )) − x ∗  − H  x e τ (1+( p 2 + q 2 ) − x ( p 2 + q 2 )) − x ∗ o , (3) where τ = ν t is the dimensionless time and x ∗ is an initial v alue for (1), H ( z ) = R z −∞ δ ( θ ) dθ is the Hea viside function. The solution (3) corr esp o nds to Cauc h y pro blem W 0 ( x ) = δ ( x − x ∗ ). Here w e set that ν = 1 . The function W ( x, τ ) is in fact a conditio na l proba bilit y 2 distribution, that is W ( x, τ ) ∆ x ≡ W ( x, τ | x ∗ , τ = 0)∆ x is the probability that at t he time τ the dynamical v ariable x b elongs to interv a l ( x, x + ∆ x ) under condition that at some previous initial time τ = 0 the v ariable x is equal to x ∗ . F rom the equation (1) follo ws t ha t the dynamical v ariable has three sta- tionary p oints: x 1 = 1 | p 2 | + q 2 , x 2 = 1 | p 2 | − q 2 , x 3 = 0 . It is conv enien t to use the definition x 1 and x 2 for transformatio n of the expression (3) to the form: W ( x, τ ) = 1 2 e − τ ( δ x − e τ x ∗ x 2 x 2 + ( e τ − 1 ) x ∗ ! + δ x − e τ x ∗ x 1 x 1 + ( e τ − 1 ) x ∗ !) + x 1 x 2 ( x 2 − x 1 ) x 2 ( H xx 1 e − τ x 1 + x ( e − τ − 1 ) − x ∗ ! − H xx 2 e − τ x 2 + x ( e − τ − 1 ) − x ∗ !) . (4) 2 Calculation of o ne p oin t momen t s The one p oint conditional momen ts of n -th order one defines as κ n ( τ ) = h x n ( τ ) | x (0) = x ∗ , τ = 0 i = Z ( D ) x n W ( x, τ ) dx, (5) where ( D ) is the suppo r t of the probability distribution. F urther w e consider the case ( D ) = ( x 1 , x 2 ). After simple calculations one o btains κ n ( τ ) = 1 2 e − τ ( e τ x ∗ x 2 x 2 + ( e τ − 1 ) x ∗ ! n + e τ x ∗ x 1 x 1 + ( e τ − 1) x ∗ ! n ) + x 1 x 2 ( x 2 − x 1 )( n − 1) n ( x 2 β ( τ )) n − 1 − ( x 1 γ ( τ )) n − 1 o , (6) where β ( τ ) = x ∗ x ∗ + ( x 2 − x ∗ ) e − τ , γ ( τ ) = x ∗ x ∗ − ( x ∗ − x 1 ) e − τ . 3 Let τ → 0 , then β ( τ ) → x ∗ x 2 and γ ( τ ) → x ∗ x 1 . In this limit from (6) one has κ n ( τ ) → x n ∗ . Let us consider another asymptotic τ → ∞ . F rom (6) o ne obtains for n = 1 the following statio na r y v alue o f t he moment κ 1 ( τ ) = x 1 x 2 x 2 − x 1 (ln x 2 − ln x 1 ) , and for n 6 = 1 stationar y v alues of momen ts are equal to κ n ( τ ) = x 1 x 2 n − 1  x n − 2 2 + x n − 3 2 x 1 + ... + x 2 x n − 3 1 + x n − 2 1  . Generally the mo ments κ n ( τ ) are g iv en functions depending on the de- creasing ex p onen t e − τ and can b e represen ted b y a series o ve r the pow ers of e − τ . In [8] a similar represen tation w as found for the sto c hastic V erh ulst mo del with fluctuating co efficien t at the first degree of the dynamical v ariable x . In t he limit for large τ ≫ 1 t he time behavior of κ n ( τ ) can b e described b y a single exp o nent. Imp ortant r o le is pla ye d b y the first t w o initial momen ts ( n = 1 , 2). Let us consider the momen t of first or der. In this case κ 1 ( τ ) = x ∗ 2 1 1 + ( e τ − 1) x ∗ x 2 + 1 1 + ( e τ − 1 ) x ∗ x 1 ! + x 1 x 2 x 2 − x 1 ln x 2 β ( τ ) x 1 γ ( τ ) . (7) In the asymptotics τ → ∞ from (7) in first order ov er the infinitesimal exp( − τ ), one has κ 1 ( τ ) ≈ x 1 x 2 x 2 − x 1 ln x 2 x 1 +  x 1 + x 2 2 − x 1 x 2 x ∗  e − τ . (8) It is inte resting that there exists the initial v alue x ∗ = 2 x 1 x 2 x 1 + x 2 ≡ 1 | p 2 | . In this case the co efficien t at exp( − τ ) is equal to zero. Therefore we should tak e in to account the next o rder, i.e. exp( − 2 τ ). Ph ysically it means that in p oin t x ∗ = 1 | p 2 | the correlatio n of v ariable x ( τ ) with the give n initial v alue of v ariable x ( x (0) = x ∗ ) decreas es more rapidly at τ → ∞ . Whe n x ∗ 6 = 1 | p 2 | the correlations tends to stationary lev el a s exp( − τ ). Here w e g iv e also a n explicit expression f or the case n = 2: κ 2 ( τ ) = 1 2 e − τ   e τ x ∗ x 2 x 2 + ( e τ − 1 ) x ∗ ! 2 + e τ x ∗ x 1 x 1 + ( e τ − 1) x ∗ ! 2   + x 2 ∗ x 1 x 2 (1 − e − τ ) ( x ∗ + ( x 2 − x ∗ ) e − τ )( x ∗ − ( x ∗ − x 1 ) e − τ ) . (9) 4 3 Conclud ing remarks W e ha v e considered the time ev olution of one p o in t momen ts of dynamical v ariable corresp onding to the sto c hastic V erh ulst mo del. The explic it for m of the momen ts sho ws that the momen ts are the functions of exp( − τ ). In [8] it w as sho wn for V erhulst mo del when parameter fluctuates at dynamical v ari- able x (not x 2 ), that the exact solution for one p oin t momen ts is presen ted b y a series o ver pow ers o f exp ( − τ ). F rom formulae o btained here one can write the momen ts κ n ( τ ) in the same form. It should b e noted that in this comm unication we ha ve obtained an explicit fo r m o f the solution. Under the condition τ → ∞ the momen ts decre ase in time as single exp onential function. It is sho wn t hat the time dependence of the momen t κ 1 ( τ ) whic h ph ysically describ es the correlatio n b et wee n the v alue of the dynamical v ari- able x at the time τ with its g iven ( no nrandom v alue x ∗ ) at the initial time τ = 0 c hanges. This time b ehav ior dep ends on the choice of the initial v alue x ∗ . The critical v alue is x ∗ = 1 / | p 2 | . F or this v alue of x ∗ in the limit τ → ∞ the correlations decrease as exp( − 2 τ ), no t as exp( − τ ). It should be noticed that the one-p oint momen ts for some special type o f the dynamical system (1) with p olynomial functions p ( x ) and q ( x ) Gaussian white noise fluctuation co efficien t a t the first p o w er of x we re considere d in [7, 9, 10], where it was shown tha t the asymptotic behavior of t he momen ts is described b y a pow er function. References [1] G anzha E.I., Logino v V.M., Tsare v S.P. Exact solutions of hy p er- b olic systems of kinetic equations. Application to V erh ulst mo del with random p erturbation. accepted fo r publication in Mathematics of Com- putation. E-print http://www.arxi v.org/ , 2006, math.AP/06 1 2793. [2] S .P. Tsarev. Generalized Laplace T ransformations and In tegration of Hyp erb olic Systems of Linear P artial D iff erential Equations Pr o c. IS- SA C’2005 (July 24–27, 2 005, Beijing, China) A CM Press, 20 05, p. 325– 331; also e- prin t cs.SC/0501030 at http://ww w.archiv.org/ . [3] S .P. Tsare v. 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