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Functional-Integral Based Perturbation Theory for the Malthus-Verhulst Process

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Functional-Integral Based Perturbation Theory for the Malthus-Verhulst Process 1238 Brazilian Journal of Physics, vol. 36, no. 4A, December, 2006 Functional-Integral Based Perturbation Theory for the Malthus-Verhulst Process Nicholas R. Moloney1,2 and Ronald Dickman2 1 Institute of Theoretical Physics, Eotv¨ os¨ University, Pazm´ any´ Peter´ set´ any´ 1/A, 1117 Budapest, Hungary 2Departamento de F´ısica, ICEx, Universidade Federal de Minas Gerais, 30123-970 Belo Horizonte - Minas Gerais, Brasil Received on 21 August, 2006 We apply a functional-integral formalism for Markovian birth and death processes to determine asymptotic corrections to mean-field theory in the Malthus-Verhulst process (MVP). Expanding about the stationary mean- field solution, we identify an expansion parameter that is small in the limit of large mean population, and derive a diagrammatic expansion in powers of this parameter. The series is evaluated to fifth order using computational enumeration of diagrams. Although the MVP has no stationary state, we obtain good agreement with the associated quasi-stationary values for the moments of the population size, provided the mean population size is not small. We compare our results with those of van Kampen’s Ω-expansion, and apply our method to the MVP with input, for which a stationary state does exist. We also devise a modified Fokker-Planck approach for this case. Keywords: Birth-and-death process; Master equation; Path integral; Omega-expansion; Doi formalism I. INTRODUCTION A lattice of coupled MVPs exhibits (in the infinite-size limit) a phase transition belonging to the directed percolation uni- The need to analyze Markov processes described by a mas- versality class. ter equation arises frequently in physics and related fields In the perturbation approach developed here [6, 9], mo- [1, 2]. Since such equations do not in general admit an exact ments of the population size n are expressed as functional in- solution, approximation methods are of interest. A widely ap- tegrals over a pair of functions, ψ(t) and ψ˜ (t), involving an plied approximation scheme is van Kampen’s ‘Ω-expansion’, effective action. The latter, obtained from an exact mapping which furnishes corrections to the (deterministic) mean-field of the original Markov process, generally includes a part that or macroscopic description in the limit of large effective sys- is bilinear in the functions ψ(t) and ψ˜ (t), whose moments can tem size [1]. Another approximation method, based on a path- be determined exactly, and ‘interaction’ terms of higher or- integral representation for birth-and-death type processes, was der, that are treated in an approximate manner. In the present proposed by Doi [3] and then by Grassberger and Scheunert approach, the interaction terms are analyzed in a perturbative [4]. Later Peliti [6] and Goldenfeld [5] revived it. Renewed fashion, leading to a diagrammatic series. With increasing interest in this type of representation has been stimulated by order, the number of diagrams grows explosively, so that it Cardy and coworkers [7, 8]. This approach was recently re- becomes convenient to devise a computational algorithm for viewed and extended [9], and applied to derive a series expan- their enumeration and evaluation. Elaboration of such an algo- sion for the activity in a stochastic sandpile [10], and to study rithm does not, however, require any very sophisticated tech- metastability in the contact process [11]. niques, and could in fact be applied to a variety of problems. It should be noted that while effectively exact results can This is the approach that was applied to the stochastic sand- be obtained via numerical analysis of the master equation, the pile in Ref. [10]. In the latter case, the evaluation of diagrams calculations become extremely cumbersome for large popula- involves calculating multidimensional wave-vector integrals. tions or multivariate processes. A further limitation of numer- The present example is free of this complication, allowing us ical analyses is that they do not furnish algebraic expressions to derive a slightly longer series than for the sandpile. that may be required in theoretical developments. For these In this work we focus on stationary moments of the MVP. reasons it is highly desirable to study approximation methods The series expressions are apparently divergent, but neverthe- for stochastic processes. less provide nearly perfect predictions away from the small- In the present work we apply the path-integral based pertur- population regime, as compared with direct numerical evalu- bation approach to a simpler problem, namely, the Malthus- ation of quasi-stationary properties. One might suppose that Verhulst process (MVP), a birth-and-death process in which the divergent nature of the perturbation series is due to the unlimited population growth is prevented by a saturation ef- MVP not possessing a true stationary state. (The process must fect. (The death rate per individual grows linearly with pop- eventually become trapped in the absorbing state, although the ulation size.) This is an important, though highly simplified lifetime grows exponentially with the mean population [12].) model in population dynamics. Although the master equa- Applying our method to the MVP with a steady input, which tion for this process is readily solved numerically, the model does possess a stationary state, we find however that the per- serves as a useful testing ground for approximation methods. turbation series continues to be divergent, although again pro- Nicholas R. Moloney and Ronald Dickman 1239 viding excellent predictions over most of parameter space. (see Eq. (54) of [9]). In the following section we define the Malthus-Verhlst Our goal is to expand each factorial moment about its mean- process, explain the perturbation method and report the se- field value. To this end, consider the shift of variable, ries coefficients for the first four moments, up to fifth order in the expansion parameter. In Sec. III we briefly compare ψ(t)=n + φ(t) (6) these results to those of the Ω-expansion. Then in Sec. IV we present numerical comparisons of our method (and of the where n is a real constant. On performing this shift and letting =(λ − )/ν ∝ ψφ0 Ω-expansion) against quasi-stationary properties. We apply n 1 , which eliminates the term ˆ , the argu- F [ψ,ψ] our method to the MVP with input in Sec. V, and also discuss ment of the exponential (i.e., of the factors ˆ z=1 and [− ] an approximation based on the Fokker-Planck equation. We exp SI ) in Eq. (3) becomes, summarize our findings in Sec. VI. t 2 ζ + dt {−ψˆ [∂t +w]φ + nψˆ 0 II. PERTURBATION THEORY FOR THE +(2−λ)ψˆ 2φ − νψφˆ 2 − νψˆ 2φ2}, (7) MALTHUS-VERHULST PROCESS where we have introduced w ≡ λ−1 (equal to −w as defined ζ Consider the Malthus-Verhulst process (MVP) n(t),in in [9]). We recognize the exponential of plus the first term in F [ψ,φ] which each individual has a rate λ to reproduce, and a rate the integrand as ˆ z=1; the remaining terms then repre- − of µ + ν(n − 1) to die, if the total population is n. By an ap- sent SI, the new effective interaction, which will be treated − propriate choice of time scale we can eliminate one of these perturbatively. The four terms of SI are represented graphi- parameters; we choose to set µ = 1. Then in what follows cally, following the conventions of Ref. [9], in Fig. 1. we use a dimensionless time variable t = µt and dimension- less rates λ = λ/µ and ν = ν/µ. From here on we drop the primes. The mean-field or rate equation description of the process is x˙ =(λ − 1)x − νx2 (1) FIG. 1: Vertices in the perturbation series for the MVP. where x ≡n(t), with the nontrivial stationary solution x = (λ − 1)/ν for λ > 1. We are interested in deriving systematic corrections to this result, and in calculating higher moments We refer to these vertices as source, bifurcation, conjunction of the process. and 4-vertex, respectively. Our starting point is the expression for the r-th factorial mo- ment of a general process taking non-negative integer values, A. Perturbation expansion r( ) = ( − )···( − + ) = −p (r)(ζ= ) , n t f n n 1 n r 1 e Ut p (2) The analysis of the MVP now follows the lines of [9]. Let where, for simplicity, we have assumed an initial Poisson distribution with parameter p (see Eq. (108) of Ref. [9]), [A] ≡ Dφ Dψˆ A(φ,ψˆ )F [φ,ψˆ ], (8) so that the probability generating function at time zero is Φ ( )= p(z−1) (r) 0 z e . Here, the kernel Ut is given by the func- denote the free expectation of any function A of φ and ψˆ . tional integral, From the discussion of Sec. 4 Ref. [9] we have, [φ( )r]=( − )r −rwt, ( ) ∂rU (z,ζ) t p n e (9) U r (ζ) ≡ t t ∂ r z z=1 (here we used φ(0)=p − n, and have already canceled the ζ 0 factor e in Ut with the corresponding factor in the inital gen- r erating function, Φ0(ζ)), = Dψ Dψψˆ (t) F [ψ,ψˆ ]z=1 exp[−SI] , (3) [ψˆ (t)] = 0, (10) (equivalent to Eq. (106) of [9]), with t and the propagator, F [ψ,ψˆ ]z=1 = exp − dt ψˆ [∂t +(1−λ)]ψ +ζ (4) 0 −w(t1−t2) [φ(t1)ψˆ (t2)] = Θ(t1 −t2)e . (11) containing the bilinear part of the action. In the case of the ( ) MVP, the “interaction” part is, Consider now the series for the mean population size n t . The zeroth-order term is simply t t 2 2 SI = dt [−λψˆ ψ+νψˆ (1+ψˆ )ψ ] ≡ dt LI(t ), (5) ( ) = +[φ( )] = +( − ) −wt 0 0 n t 0 n t n p n e (12) 1240 Brazilian Journal of Physics, vol.
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