Functional-integral based perturbation theory for the Malthus-Verhulst process
Brazilian Journal of Physics, vol. 36, no. 4A, December, 2006
1238
Functional-Integral Based Perturbation Theory for the Malthus-Verhulst Process
Nicholas R. Moloney1,2 and Ronald Dickman2
1 Institute of Theoretical Physics,
Eötvös University, Pázmány Péter sétány 1/A,
1117 Budapest, Hungary
2 Departamento de Fı́sica, 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 meanfield 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
The need to analyze Markov processes described by a master equation arises frequently in physics and related fields
[1, 2]. Since such equations do not in general admit an exact
solution, approximation methods are of interest. A widely applied approximation scheme is van Kampen’s ‘Ω-expansion’,
which furnishes corrections to the (deterministic) mean-field
or macroscopic description in the limit of large effective system size [1]. Another approximation method, based on a pathintegral representation for birth-and-death type processes, was
proposed by Doi [3] and then by Grassberger and Scheunert
[4]. Later Peliti [6] and Goldenfeld [5] revived it. Renewed
interest in this type of representation has been stimulated by
Cardy and coworkers [7, 8]. This approach was recently reviewed and extended [9], and applied to derive a series expansion for the activity in a stochastic sandpile [10], and to study
metastability in the contact process [11].
It should be noted that while effectively exact results can
be obtained via numerical analysis of the master equation, the
calculations become extremely cumbersome for large populations or multivariate processes. A further limitation of numerical analyses is that they do not furnish algebraic expressions
that may be required in theoretical developments. For these
reasons it is highly desirable to study approximation methods
for stochastic processes.
In the present work we apply the path-integral based perturbation approach to a simpler problem, namely, the MalthusVerhulst process (MVP), a birth-and-death process in which
unlimited population growth is prevented by a saturation effect. (The death rate per individual grows linearly with population size.) This is an important, though highly simplified
model in population dynamics. Although the master equation for this process is readily solved numerically, the model
serves as a useful testing ground for approximation methods.
A lattice of coupled MVPs exhibits (in the infinite-size limit)
a phase transition belonging to the directed percolation universality class.
In the perturbation approach developed here [6, 9], moments of the population size n are expressed as functional integrals over a pair of functions, ψ(t) and ψ̃(t), involving an
effective action. The latter, obtained from an exact mapping
of the original Markov process, generally includes a part that
is bilinear in the functions ψ(t) and ψ̃(t), whose moments can
be determined exactly, and ‘interaction’ terms of higher order, that are treated in an approximate manner. In the present
approach, the interaction terms are analyzed in a perturbative
fashion, leading to a diagrammatic series. With increasing
order, the number of diagrams grows explosively, so that it
becomes convenient to devise a computational algorithm for
their enumeration and evaluation. Elaboration of such an algorithm does not, however, require any very sophisticated techniques, and could in fact be applied to a variety of problems.
This is the approach that was applied to the stochastic sandpile in Ref. [10]. In the latter case, the evaluation of diagrams
involves calculating multidimensional wave-vector integrals.
The present example is free of this complication, allowing us
to derive a slightly longer series than for the sandpile.
In this work we focus on stationary moments of the MVP.
The series expressions are apparently divergent, but nevertheless provide nearly perfect predictions away from the smallpopulation regime, as compared with direct numerical evaluation of quasi-stationary properties. One might suppose that
the divergent nature of the perturbation series is due to the
MVP not possessing a true stationary state. (The process must
eventually become trapped in the absorbing state, although the
lifetime grows exponentially with the mean population [12].)
Applying our method to the MVP with a steady input, which
does possess a stationary state, we find however that the perturbation series continues to be divergent, although again pro-
Nicholas R. Moloney and Ronald Dickman
1239
viding excellent predictions over most of parameter space.
In the following section we define the Malthus-Verhlst
process, explain the perturbation method and report the series coefficients for the first four moments, up to fifth order
in the expansion parameter. In Sec. III we briefly compare
these results to those of the Ω-expansion. Then in Sec. IV
we present numerical comparisons of our method (and of the
Ω-expansion) against quasi-stationary properties. We apply
our method to the MVP with input in Sec. V, and also discuss
an approximation based on the Fokker-Planck equation. We
summarize our findings in Sec. VI.
II.
Consider the Malthus-Verhulst process (MVP) n(t), in
which each individual has a rate λ to reproduce, and a rate
of µ + ν(n − 1) to die, if the total population is n. By an appropriate choice of time scale we can eliminate one of these
parameters; we choose to set µ = 1. Then in what follows
we use a dimensionless time variable t = µt and dimensionless 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
−p
n (t) f = n(n − 1) · · · (n − r + 1) = e
(r)
Ut (ζ = p) ,
=
Dψ D ψ̂ ψ(t)r F [ψ, ψ̂]z=1 exp[−SI ] ,
(3)
0
dt {−ψ̂[∂t +w]φ + nψ̂2
(7)
where we have introduced w ≡ λ−1 (equal to −w as defined
in [9]). We recognize the exponential of ζ plus the first term in
the integrand as F [ψ̂, φ]z=1 ; the remaining terms then represent −SI , the new effectiv (...truncated)
