International Journal of Management and Applied Science, ISSN: 2394-7926
Volume-2, Issue-8, Aug.-2016
EFFECT OF CORRELATED NOISES ON THE STATIONARY
DISTRIBUTION OF POPULATION SIZE: A MODEL BASED STUDY
1
BAPI SAHA, 2AKSHOY PATRA
1,2
Mathematics Department, Govt. College of Engg. & Textile Technology, Berhampore, W. B. India
E-mail: [email protected], [email protected]
Abstract— In this paper, we introduce two correlated noises in single species dynamic models. Here we consider both
−logistic and the Allee effect model for the growth process of the species. The stationary distribution is obtained through
the help of Fokker-Planck equation. The effect of the correlation coefficient, the strength of the correlated noises, the Allee
threshold and the density regulation parameter on the stationary density are analyzed, which can be used as a surrogate for
understanding the extinction status of the species.
Keywords— Theta-Logistic model, Allee Effect, Population Dynamics, Correlated Noise, Stationary Distribution, Steady
State.
stochasticity together needs to be analyzed. Allee
effect is a growth process in which at low population
the per-capita growth rate is an increasing function of
the population size.
In this paper, we shall investigate this dynamics and
study the stationary distribution of the populations.
To start this idea we first investigate such dynamics
on the theta-logistic model, which has not been
investigated before and subsequently get to the core
of the ideas related with the Allee effect.
I. INTRODUCTION
The emphasis on the small population is important, in
a sense that, the overall growth mechanism is mainly
governed by the birth and death process. Hence, the
variation in the birth and death rates gives rise to the
demographic stochasticity. Apart from variability in
demographic rates, there is another source of
disturbance affecting the natural populations, is the
environmental stochasticity. The crucial point is that,
the environmental stochasticity affect the populations
however large or small they are in numbers. The
changes to environmental conditions, e.g. drought,
rain, etc. have a significant impact on the population
sizes. In particular, when the abundance is too small
in numbers, a very small perturbations may lead to
the population to extinction [15]. The demographic
stochasticity plays an important role for the dynamics
of small populations and imposes a higher risk of
extinction for the species. In recent times, the studies
of nonlinear systems with correlated noise terms have
attracted attention in physics, chemistry and biology
[11]. In most of the works the noises are treated as to
be uncorrelated, since it was assumed that they have
different origins. However, the noises in some
stochastic process may have a common origin, and
thus may be correlated [12]. It is quite often that in
these systems the noises affect the dynamics through
a system variable, i.e., the noise is both multiplicative
and additive.
The effect of correlated noises can have a potential
impact on the population growth mechanism, and
even it can hinder the evolution of nonlinear
biological systems [18]. Correlated noise processes
have also found applications in studying steady-state
properties of a single-mode laser, in analyzing
bistable kinetics [7], in giant suppression of activation
rate [19], in producing directed motion in spatially
symmetric periodic potentials [16]; predator-prey
dynamics in ecological interactions etc. Thus in this
aspect the investigation into the interplay of the Allee
effect and both demographic and environmental
II. THE DETERMINISTIC MODEL
The logistic differential equation is given by,
=
−
...........(1)
where x is the size of the population or biomass
concentration of an organism under study. This
deterministic growth rate
declines linearly
from a maximum (intrinsic) growth rate , to zero
when = /b, called the carrying capacity. The
generalized -logistic growth models are defined as:
=
−
……..(2)
where is the population size, is the growth rate,
is the death rate. eqn. (2) can be reformulated as,
=
1−
………(3)
This form is more convenient to work with for
ecological applications as each of the parameters
(untransformed) has ecologically meaningful
characteristics. In eqn. (3) the deterministic growth
rate declines from a maximum growth rate
= at
low abundance, to zero when = , where =
,
quantifies the carrying capacity. However, throughout
this article we shall use eqn. 2.
Effect of Correlated Noises on the Stationary Distribution of Population Size: A Model Based Study
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International Journal of Management and Applied Science, ISSN: 2394-7926
Though most of the species follow -logistic growth
process there is so many empirical evidences on the
Allee effect reported in many natural populations,
including plants [10], insects [14], marine
invertebrates and birds and mammals [6]. Due to the
increasing number of rare and endangered species,
the dynamics of small populations, including the
Allee effect has received significant attention among
ecologists and conservation biologists.
IV. STEADY STATE ANALYSIS
Since the population size cannot be negative, the
Fokker-Planck equation for the evolution of steady
state probability distribution function (SPDF)
corresponding to the stochastic differential equations
(6)-(7) given in the previous section, under the
constraint ≥ 0 is
( , )
The general Allee model is given by,
= ( − ) 1 − ……….(4)
where is the intrinsic growth rate, is the Allee
threshold and is the carrying capacity.
This gives
( )=
The general form of the stochastic differential
equation incorporating the two noise terms is given
by [13],
( )
= ( ) + ( ) ( ) + ( ) ( )....(5)
−
−
+
Here = , = , = (
( ) as
exp ∫
)
( ′)
( ′)
dx′ .........(11)
The explicit expression for SPDF for -logistic
model can not be found and so we solve it
numerically.
In case of the Allee model (4) the expressions are
( )= − − + 2
− 2 √
( )= − 2 √
+ .
For this case the expressions for
The stochastic differential equations incorporating the
two noise terms in the Allee effect model (4) is given
by,
=
( ) ( )…….(9)
where N is the normalizing constant. The expressions
for ( ) and ( ) can be obtained following the
standard method. In case of -logistic model the
expressions for ( ) and ( ) are given by [13],
( ) = −
+ − √ … … (12)
( )= − 2 √ + ........(13)
In this equation ( ) represents the deterministic
trend and ( ) and ( ) are Gaussian white noises.
There are some external effects such as temperature,
change in climate, which might have some effects on
the rate of growth generating a multiplicative noise.
Apart from these, there are some other noises, which
inhibit the growth process of the population. Taking
all these facts under consideration we obtain
=
−
+ ( ) − ( )............(6)
−logistic model. Here = , =
( ) ( )+
=−
where ( , ) is the probability density. The
stationary probability distribution is obtained by
solving
( , )
= 0…..(10)
III. STOCHASTIC MODEL FORMULATION
for
Volume-2, Issue-8, Aug.-2016
( ) is given by,
( ) − ( )....(7)
and = In the above Equations (6) - (7), ( ) and ( ) are
Gaussian white noises with zero mean and
〈 ( )〉 = 〈 ( )〉 = 0
〈 ( ) ( ′)〉 = 2 ( − ′)
〈 ( ) ( ′)〉 = 2 ( − ′)
〈 ( ) ( ′)〉 = 2 √
( − ′)........(8)
where = (1 + ), = , = .
V. RESULT AND DISCUSSION
The prime objective of this paper is to analyze the
growth dynamics of species under the influence of
two correlated noises. The correlated noises may have
a severe impact upon the growth process of the
species. It is observed that the chance of extinction of
population increases as the correlation coefficient
between the two noises increases when the population
undergoes the logistic growth process [2]. The effect
of correlation between the two noises in case when
the population growth process follows -logistic and
the Allee effect model is still not considered as far as
our knowledge goes. In this section, we demonstrate
where and are the strengths of the multiplicative
noise and additive noise respectively. λ is the degree
of correlation between the two types of noises. We
assume the above stochastic differential equations are
Stratonovich stochastic differential equation.
In most of the previous cases ([12] [13]) the noises
are assumed to be delta correlated i.e the correlation
time was zero.
Here also for simplicity we consider the correlation
time to be 0.
Effect of Correlated Noises on the Stationary Distribution of Population Size: A Model Based Study
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International Journal of Management and Applied Science, ISSN: 2394-7926
the effect of different parameters of two different
models viz. -logistic model (2), the Allee effect
models (4).
Volume-2, Issue-8, Aug.-2016
Again, we can see from Fig. 3(a) that for higher
values of
the probability of staying near 0
population size is high. So the strength of
multiplicative noise has the detrimental effect upon
the population. On the other hand, in case of the
strength of the additive noise ( ), its increase leads to
increase in the probability of staying near high
population (Fig. 3(b)).
-logistic model is among the most widely used
growth curve models in population dynamics. The
presence of correlated noises in the -logistic growth
process is an important area of research. When
demographic variation is zero the stationary
distribution of population size of the -logistic model
is a generalized gamma distribution for ≠ 0 [8]. It
is evident from Fig. 1 that the probability to stay
close to the low population size increases as
decreases and the probability of staying at high
population size increases as
increases. The
numerical simulation shows that the above fact is true
irrespective of the sign of as well as whether is
greater than 1 or not. So it can be concluded that the
population is less vulnerable to extinction when is
high in presence of two correlated noises. This result
is quite similar as obtained by [22].
Fig.3a. Plot of Pst(x) (probability density function)
vs.population size ( ) for different values of for the model
(2). The parameter values are r = 1, K = 10, α= 3, λ = 0.5, θ= 0.8
Fig.3b. Plot of Pst(x) (probability density function)
vs.population size ( ) for different values of for the model
(2). The parameter values are r = 1, K = 10, = 0.8, λ = 0.5, θ=
0.8
Fig.1. Plot of
( ) (probability density function) vs.
population size ( ) for different values of for -logistic
modelSThe parameter values are = , = , = . ,
= , = .
In case of the correlation coefficient ( ) between the
two noises, a glimpse on Fig. 2 reveals that there exist
two population sizes ∗ and ∗ (0 < ∗ < ∗ ) such that
the probability of the population size to stay below ∗
increases as decreases. The probability of staying
between ∗ and ∗ decreases as
decreases. The
effect is again reversed when population size
exceeds ∗ .
The effect of correlated noises when the growth
process of the species follows the Allee effect is still
unexplored. In this paper we pay attention to this area
of research. Here we first consider the Allee effect
model (4).
We can see that for the Allee effect model (4) as the
Allee threshold increases the probability of staying
close to low population size increases (see Fig. 4) and
thus chance of extinction becomes high which is a
very general fact.
Fig.2. Plot of
( ) (probability density function) vs.
population size ( ) for different values of λ when θ is greater
than in case of θ-logistic model (2). The parameter values are
= , =
, = . , = , = .
Fig.4. Plot of
( )(probability density function) vs.
population size ( ) for different values of the Allee threshold
( ) for the model (4). The parameter values are = . , =
, = . , α = 0.8, λ= 0.5.
Effect of Correlated Noises on the Stationary Distribution of Population Size: A Model Based Study
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International Journal of Management and Applied Science, ISSN: 2394-7926
Volume-2, Issue-8, Aug.-2016
The effect of the strength of multiplicative noise has
negative impact on the survivability of the species.
We can see from Fig. 5(a) that as D increases the
population size has lesser probability of staying near
high population size. On the contrary the increase in
the strength of additive noise inhibits the chance of
extinction which is clear from Fig. 5(b).
Fig.6b. Plot of
( ) (probability density function) vs.
population size ( ) for different negative values of λ for the
Allee effect model (4). The parameter values are = . , =
, = . , α= 0.8, = .
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Fig.5a Plot of
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