Stochastic persistence and stationary distribution
in a Holling-Tanner type prey-predator model
Partha Sarathi Mandal and Malay Banerjee∗
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Kanpur - 208016, INDIA
e-mail : [email protected]
Abstract
In this paper, we study a stochastic predator-prey model with Beddington-DeAngelis type
functional response and logistic growth for predators. The deterministic model is already wellstudied and we recall some important results here. We construct the stochastic model from
the deterministic model by introducing multiplicative noise terms into the growth equations
of prey and predator populations. For the stochastic model, we show that the system admits
unique positive global solution starting from the positive initial value. Then we prove that the
system is strongly persistent in mean when the intensity of environmental forcing is less than
some threshold magnitudes. Finally, we show that the system has a stationary distribution
under certain parametric restrictions. Numerical simulations are carried out to substantiate
the analytical results.
Key words : Beddington-DeAngelis functional response, stability, Itô’s formula, global solution, persistence in mean, stationary distribution.
∗
Tel. +91-512-259-6157, Fax. +91-512-259-7500
1
1. Introduction
Traditional prey-dependent predator-prey models have been challenged by several ecologists
based on the fact that functional and numerical response over ecological time scale depend on the
densities of both prey an predator populations. Most general model describing the dynamics of preypredator population over continuous time and within deterministic environment can be represented
by
dx
= xf (x) − g(x, y)y,
dt
dy
= h(x, y)y,
dt
(1)
(2)
where x(t) and y(t) are the densities of prey and predator populations at time t. f (x) is the per
capita net prey growth in absence of predator, g(x, y) which plays the most important role in preypredator model (Solomon 1949) represents the functional response of predators and h(x, y) is the
numerical response of predators (measures the growth rate of predators). Function f can be taken
either of the form f (x) = r (exponential growth) or f (x) = r(1 −
x
)
K
(logistic growth). Generally
we assume that the functional response of predator is a function of prey density only i.e g ≡ g(x),
without any dependence on predator density [1, 2]. This formalism is based upon the assumption
that prey and predator individuals encounter each other randomly in space and time [3]. Hence
the prey dependent model will be applicable on spatially homogeneous systems in which the time
scale of prey removal by predators is of the same order of magnitude as that of the reproduction of
predator populations [4].
In the area of mathematical ecology, an adequate amount of thorough investigation is carried
out by several researchers to study the local and global dynamical behavior of several predator-prey
models within deterministic environment, see [5–17] and references cited therein. In [5], authors
have studied the following prey-predator model:
dx
x
= rx 1 −
− yp(x),
dt
K
dy
hy
= y s 1−
,
dt
x
(3)
(4)
subjected to positive initial conditions x(0) > 0, y(0) > 0 and r, s, K, h are all positive parameters.
The prey population grows logistically with carrying capacity K and intrinsic growth rate r in
absence of predator and the predator population grows logistically with intrinsic growth rate s and
2
carrying capacity proportional to the density of prey. The consumption of prey by the predators
is governed by the prey dependent functional response p(x). Parameter h is the number of prey
required to feed one predator at equilibrium conditions [10, 18]. The term h xy , involved with the
predator growth equation, is known as Leslie-Gower term [5, 19–25].
Classical Holling-Tanner model for predator-prey interaction is obtained from (3) - (4) for specific
choice of p(x) and is governed by the following system of non-linear coupled ordinary differential
equations:
dx
x
axy
= rx 1 −
−
,
dt
K b + x
dy
hy
= y s 1−
,
dt
x
(5)
(6)
subjected to the same initial conditions as given above. The functional response is known as
Holling type-II functional response and parameters a > 0 and b > 0 represent capturing rate and
half-saturation constant respectively. For this model, exhaustive analysis and detailed discussion
regarding the local and global stability of equilibrium points along with the existence of Hopfbifurcating periodic solution and it’s stability have been reported by several researchers [5,9,19,21–
23, 25].
In [26], authors have proposed and studied the following ratio-dependent Holling-Tanner type
prey-predator model
x
axy
dx
= rx(1 − ) −
,
dt
K
my + x
(7)
dy
hy
=y s 1−
,
dt
x
(8)
subjected to the biologically feasible initial condition x(0) > 0, y(0) > 0. This model is obtained
by replacing prey-dependent functional response term in (5) by ratio-dependent functional response
ay
.
my+x
Here two positive parameters a and m characterize the Michaelis-Menten Holling type func-
tional response [27]. In many situations, predator has to search and compete for food and hence
predator abundance also has the ability to influence the functional response [26, 28]. Arditi and
Ginzburg [4] have suggested that, in many cases, one can simplify this predator dependence as a
ratio dependent model instead of modeling explicitly all conceivable interference mechanisms and
their claim is already supported by numerous field data and laboratory experiments [29–31].
3
Recently, several researchers [6, 32, 33] have drawn their attention on a modified Holling-Tanner
prey-predator model. They have modified
the growth
equation for predator population by considhy
ering the per-capita growth rate as s 1 −
. This means that k1 /h is the environmental
x + k1
carrying capacity of predators due to severe scarcity of the favorite food [34]. This modification is
justified as it prevents the extinction of predator population in absence of prey and it is a realistic feature from ecological point of view as they have alternative food source. Incorporating this
modification in (7) - (8) we get the following modified system
dx
= rx 1 −
dt
dy
= y s1 −
dt
x
axy
−
,
K
my + x
h1 y
,
k1 + x
(9)
(10)
where s1 is the intrinsic growth rate of predators and h1 is the intra-specific competition term
and k1 is the environmental carrying capacity for predators in absence of their most favorite food
source [6].
A major mathematical difficulty with ratio-dependent functional response is that the concerned
model fails to satisfy the continuity condition at origin. This difficulty can be avoided if we consider
the Beddington-DeAngelis-type functional response instead of ratio-dependent form. But the basic
dynamical behaviors of the concerned prey-predator interaction remains unaltered. It is well known
that Beddington-DeAngelis-type functional response is most suitable than other available response
functions and is capable to take care of a number of ecological mechanisms [35, 36]. It also produce
very rich and biologically reasonable dynamics [37]. Hence we are interested to study the dynamics
of prey-predator interaction with Beddington-DeAngelis functional response. In this paper, we
will consider the following model to describe the interaction between prey and predators within
deterministic environment,
dX(τ )
m1 Y (τ )
= X(τ ) a1 − b1 X(τ ) −
,
dτ
α1 X(τ ) + β1 Y (τ ) + γ1
dY (τ )
m2 Y (τ )
= Y (τ ) a2 −
,
dτ
k1 + X(τ )
(11)
(12)
subjected to the feasible initial conditions X(0) ≡ X0 > 0 and Y (0) ≡ Y0 > 0. All the parameters
a1 , a2 , b1 , m1 , m2 , α1 , β1 , γ1 and k1 are assumed to be positive and constant with respect to time.
Here a1 , a2 stand for the growth rate of prey and predator species respectively, b1 is the rate of
4
intra-specific competition for prey species, m1 is the maximum value at which per capita reduction
rate of prey can attain, γ1 (resp., k1 ) measures the extent to which environment provides protection
to prey (resp., to predator), m2 has the similar meaning as m1 . The functional response mentioned
in (11)-(12) was introduced by Beddington [35] and DeAngelis et al. [38] in 1975.
Significant amount of works have been done with the deterministic setup which do not incorporate the effect of either environmental fluctuations or demographic stochasticity into the modeling
approach which are important components for ecosystems exposed within open environment. In
deterministic autonomous models, we always assume that parameters involved in the system are
absolute constant irrespective of the environmental periodicity and fluctuations. There are some
literature [39–42] where authors have considered the non-autonomous ordinary differential equation
models to study the models with seasonally varying parameters. In [39], authors have identified
six basic seasonality mechanisms related with classical two dimensional prey-predator model and
demonstrated the existence of multiple attractors and catastrophic transitions with the help of continuation technique. Two dimensional prey-predator models with seasonally varying parameters are
capable to demonstrate torus destruction and period doubling route to chaos [43, 44]. But in real
situation, parameters involved with the model always fluctuate around some average value due to
continuous fluctuation in the environment. May [2] pointed out that all the parameters involved in
the population model exhibit random fluctuation as the factors controlling them are not constant.
Hence equilibrium distributions obtained from the deterministic analysis are not realistic rather
they fluctuate randomly around some average value. Sometimes we can observe large amplitude
fluctuation in population density which leads to extinction of certain species. Further the oscillatory
co-existence of two or more species with varying periodicity can not be captured within the deterministic setup apart from quasi-periodic or chaotic oscillation for a range of parameter values. In
order to study the dynamics of interacting population under realistic situation we need to analyze
the associated stochastic model. However, there is no unified approach to formulate a stochastic
model for the interacting populations under consideration. The available stochastic formalisms can
be divided into two broad classes, one of them is known as discrete or continuous time Markov
Chain modeling (see [45–50]) and other is the noise added systems [28, 51–65]. However, both of
this two classes can be sub-divided further based upon the concerned formulation principle.
Stochastic differential equation models obtained from existing deterministic models by intro5
ducing additive or multiplicative noise terms have been received moderate attention from several
researchers and it is established that the resulting dynamics solely depend upon the intensity of
environmental fluctuations. A larger part of noise added models have no steady-states and hence
the stochastic stability analysis can not be carried out. However, there are some exceptional cases
where the noise added into the system in such a way that the coexisting equilibrium point of deterministic model is also an equilibrium point for the noise added model [28, 55, 66]. The stochastic
stability of such models are carried out with the help of suitable Lyapunov functions. But this
type of modeling approach fails to capture the most interesting scenario known as noise induced
transitions. There are a handful number of literature for population models where the change in
bifurcation scenario or noise induced bifurcation is addressed [61, 62]. The method suggested by
Nisbet and Gurney [67] is suitable for analyzing the dynamics in case of small perturbation from
the deterministic steady-state. With the said formalism, it is difficult to predict anything in case
of far from equilibrium situations. During the last one decade, some significant investigations are
carried out for classical Lotka-Volterra type population models within fluctuating environment to
study the existence and uniqueness of solutions, stochastic persistence and stationary distribution
to understand the asymptotic behavior of solution trajectories [64, 65]. Apart from these, some
Gause-type and Leslie-Gower type models with multiplicative noise terms are analyzed in recent
years [33, 68–71]. Till to date there are limited number of tools to analyze the stochastic models
having colored noises [61, 62, 72, 73].
The present paper is organized as follows: in section 2, we recall the basic stability results for
the dimensionless version of system (11)-(12). In section 3, we formulate the stochastic model by
incorporating white noise terms into the growth rates of prey and predator populations, keeping all
other parameters fixed and then establish the existence of unique positive global solution for the
stochastic model. In section 4, we derive the parametric restrictions required for strong stochastic
persistence in mean for both prey and predator species. Existence of stationary distribution for
two population which exist under certain parametric restrictions is provided in section 5. Results
obtained for the stochastic model are validated and explained with numerical experiments are
presented in section 6. Ecological interpretations of obtained results are provided in the concluding
section.
6
2. Deterministic Model
In this section, we discuss some basic dynamical properties of the dimensionless model corresponding to the original model (11)-(12) in brief as those results are already established in [74].
Using the transformation of variables t = a1 τ, x(t) =
b1
X(τ ),
a1
y(t) =
m2 b1
Y
a1 a2
(τ ) in (11)-(12) we get
the following dimensionless version,
dx(t)
ay(t)
= x(t) 1 − x(t) −
,
(13)
dt
αx(t) + βy(t) + γ
y(t)
dy(t)
= by(t) 1 −
,
(14)
dt
k + x(t)
a2 m1
a2
a2
b1
where a = a1 m2 , b = a1 , α = α1 , β = β1 m2 , γ = γ1 a1 , k = k1 ab11 are the dimensionless parameters and they are all positive. Above model system is subjected to positive initial
conditions x(0) ≡ x0 > 0, y(0) ≡ y0 > 0. For convenience we use t as dimensionless time instead
of τ . For the sake of convenience, here we introduce the following notations:
R2+ = (x, y) ∈ R2 |x ≥ 0, y ≥ 0 , R+ = {x ∈ R|x ≥ 0} ,
and the interior of the first quadrant is denoted by Int(R2+ ). It has been known that the solutions
of system (13)-(14) are non-negative and bounded for all t ∈ R+ (see [74] for details). Here we
recall the following lemma related with dissipativeness of above deterministic model, detailed proof
are available in [74].
Theorem 2.1 Let B be the set defined by
B = (x, y) ∈ R2+ : 0 ≤ x ≤ 1, 0 ≤ x + y ≤ L1 ,
1 5b + (1 + b)2 (1 + k) , then
4b
(1) B is positively invariant;
with L1 =
(2) all solutions of (13)-(14) initiating in R2+ are ultimately bounded and eventually enter the
attracting set B.
It is easy to verify that the model system (13)-(14) has three equilibrium points belonging to
the boundary of R2+ irrespective of any parametric restrictions. E0 ≡ (0, 0) is trivial equilibrium
point, E10 = (0, k) is the first axial equilibrium point and the second axial equilibrium point is given
7
by E20 = (1, 0). Interior equilibrium point refers to the point(s) of intersection of two zero growth
isoclines
ay
αx+βy+γ
= 1 − x and y = x + k in Int(R2+ ). If we denote the unique interior equilibrium
point by E∗ ≡ (x∗ , y ∗ ), then
y ∗ = x∗ + k,
and x∗ is the positive real root of the quadratic equation
(α + β)ξ 2 + (βk + γ + a − α − β)ξ + (ak − βk − γ) = 0.
Depending upon the parametric restrictions the above quadratic equation may have one or two
positive roots and the system under consideration will have two coexisting steady-states in case of
two positive roots of the quadratic equation. Here we assume the parametric restriction ak < kβ+γ
ensuring the existence of unique interior equilibrium point. The entire analysis of this paper is based
upon this parametric restriction.
2.1 Local Stability Analysis for E∗
Detailed discussion about the local asymptotic stability analysis of axial equilibriums and interior
equilibrium point corresponding to the system (13)-(14) are analyzed in details by Khellaf and
Hamri [74]. Now for convenience and smooth readability of this paper, we briefly discuss the basic
features of local asymptotic stability of unique interior equilibrium point E∗ .
The Jacobian matrix associated with (13)-(14) and evaluated at E∗ is given by
#
"
∗ y∗
−ax∗ (αx∗ +γ)
−x∗ + (αx∗aαx
+βy ∗ +γ)2
(αx∗ +βy ∗ +γ)2
J(E∗ ) =
.
b(y ∗ )2
−by ∗
(k+x∗ )2
k+x∗
Characteristic equation for J(E∗ ) is
λ2 + λA1 + A2 = 0,
where
aαx∗ y ∗
by ∗
+
,
(αx∗ + βy ∗ + γ)2 k + x∗
bx∗ y ∗
aαy ∗
abx∗ (αx∗ + γ)(y ∗ )2
A2 = Det(J(E∗ )) =
1
−
+
.
k + x∗
(αx∗ + βy ∗ + γ)2
(k + x∗ )2 (αx∗ + βy ∗ + γ)2
A1 = −Tr(J(E∗ )) = x∗ −
8
According to the Routh-Hurwitz criteria, local asymptotic stability of E∗ demands the restriction
A1 > 0 and A2 > 0. Both of which are satisfied under the sufficient condition,
1−
aαy ∗
> 0.
(αx∗ + βy ∗ + γ)2
(15)
Now we derive the explicit parametric restriction satisfying the inequality (15). Components of
unique interior equilibrium E∗ satisfy
1 − x∗ =
ay ∗
> 0.
αx∗ + βy ∗ + γ
(16)
and hence (15) is now equivalent to the condition
ay ∗ > α(1 − x∗ )2 .
(17)
Solving (16) for y ∗ we get,
y∗ =
(αx∗ + γ)(1 − x∗ )
.
a − β(1 − x∗ )
(18)
Substituting this expression for y ∗ in (17), we get the required condition in terms of x∗ as follows,
a(αx∗ + γ)(1 − x∗ )
> α(1 − x∗ )2 ,
∗
a − β(1 − x )
(19)
αβ(x∗ )2 + 2α(a − β)x∗ + αβ + aγ − aα > 0.
(20)
or equivalently,
The discriminant of the quadratic expression in terms of x∗ is given by
D = −4aα [β(α + γ) − aα] .
Therefore D is negative if
β(α + γ) − aα > 0.
So the inequality (20) holds good if the conditions αβ + a(γ − α) > 0 and β(α + γ) − aα > 0
are satisfied simultaneously. So the parametric restriction (15) is satisfied whenever the above
two conditions hold simultaneously. Therefore these two conditions together with the existence
condition give the sufficient condition for local asymptotic stability of unique interior equilibrium
point E∗ . Now we validate the analytical findings with the help of numerical example. For this
9
purpose we choose the parameters a = 1, α = 4, β = 6, γ = 2, b = 2 and k = 1. One can easily
verify that conditions kβ +γ −ak = 7 > 0, αβ +a(γ −α) = 22 > 0 and β(α+γ)−aα = 32 > 0 are
satisfied and hence E∗ (0.888, 1.888) is locally asymptotically stable. Fig-1 depicts the time evolution
of solution trajectory starting from the initial point (.4, .2) and hence E∗ is locally asymptotically
stable.
2
1.8
x(t)
1.6
y(t)
population→
1.4
1.2
1
0.8
0.6
0.4
0.2
0
10
20
30
40
50
60
70
80
90
100
time→
Figure 1: Time evolution of x(t) and y(t) obtained from numerical simulation, local asymptotic
stability of E∗ (0.888, 1.888).
3. Stochastic Model
The main focus of this paper is to understand the dynamics of modified Holling-Tanner type
prey-predator model in presence of environmental driving forces. To incorporate the effect of environmental fluctuations, here we formulate the stochastic model by introducing the multiplicative
noise terms into the growth equations of both the predator and prey populations. In several existing
literature [75], authors have demonstrated that one or more system parameter(s) can be perturbed
stochastically with white noise term to derive environmentally perturbed system. For example, a
parameter r involved with the concerned deterministic model can be perturbed stochastically, with
r → ζ(t) where ζ(t) is white noise and > 0 stands for the noise intensity [28, 75]. By perturbing
the intrinsic growth rates of prey and predator populations we will get our desired stochastic model
from the deterministic system (13)-(14). It is worthy to mention here that the concerned approach
to formulate the stochastic model based upon existing deterministic model is not unique and other
relevant approaches can be found in [45, 46, 51, 55, 72, 76–78]. Introducing randomness into the
10
deterministic system (13)-(14) by perturbing dimensionless growth rate of prey by 1 ξ1 (t) and that
for predators by 2 ξ2 (t) we get the following system,
dx(t)
ay(t)
= x(t) 1 + 1 ξ1 (t) − x(t) −
,
dt
αx(t) + βy(t) + γ
y(t)
dy(t)
= by(t) 1 + 2 ξ2 (t) −
,
dt
k + x(t)
(21)
(22)
subjected to the positive initial conditions x(0), y(0) > 0. ξ1 (t) and ξ2 (t) are two mutually independent white noise terms, characterized by hξ1(t)i = 0 = hξ2 (t)i and hξi(t)ξj (t1 )i = δij δ(t − t1 )
where δij is Kronecker delta and δ(.) is the ‘Dirac-δ’ function [28, 55, 76–78]. Here parameters 1
and 2 denote the intensities of environmental forcing. Now we can write the system of equations
(21)-(22) into the form of stochastic differential equations as follows:
ay(t)
dx(t) = x(t) 1 − x(t) −
dt + σ1 x(t)dB1 (t),
αx(t) + βy(t) + γ
y(t)
dt + σ2 y(t)dB2 (t),
dy(t) = by(t) 1 −
k + x(t)
(23)
(24)
where σ1 = 1 , σ2 = b2 and B1 (t), B2 (t) are two standard one-dimensional independent Wiener
processes defined over the complete probability space (Ω, F , P ) with a filtration {Ft }t≥0 satisfying
the usual conditions (i.e. it is increasing and right continuous while F0 contains all P -null sets [75]).
The relations between the white noise terms and Wiener processes are defined by dBr = ξr (t)dt,
r = 1, 2 [79]. The stochastic model takes care of the fact that dimensionless intrinsic growth rates
of prey and predators are randomly fluctuating due to variability of environmental conditions. This
formulation is also capable to capture the density dependent random immigration and emigration
of populations. Similar type of formulation for dimensionless Lotka-Volterra model are discussed
elaborately in [80, 81]. Any solution of (23)-(24) subjected to the positive initial condition is an
Itô process [55–58, 66]. Without any loss of generality we can assume, σ1 > 0 and σ2 > 0. To
investigate the dynamical behavior, first we have to establish the global existence of solutions.
Moreover, as the model under consideration is a population dynamics model, we are concerned
with non-negative solutions only. Hence in the next section, we first show that the solution of the
system (23) - (24) is positive and global.
3.1 Existence and uniqueness of the positive global solution
11
As we know, in order for a stochastic differential equation to have a unique global (i.e. no explosion in a finite time) solution for any given initial value, the functions involved with stochastic system
are generally required to satisfy the linear growth condition and local Lipschitz condition [82–84].
However, the functions of system (23) - (24) do not satisfy the linear growth condition. So the
solution of system (23) - (24) may explode at a finite time. In this section, using the change of
variables [33], first we show that there exists a unique positive local solution of the system (23)
- (24) and then using the Lyapunov analysis method [85], we prove that this solution is global.
Explanation for ‘explosion time’ used in following lemma can be found in [82, 83].
Lemma 3.1.1 For (x0 , y0) ∈ Int(R2+ ), there is a unique positive local solution (x(t), y(t)) of the
system (23) - (24) for t ∈ [0.τe ) a.s. where τe is the explosion time.
Proof: Using the transformation of variables u(t) = log x(t), v(t) = log y(t) and applying Itô’s
formula we get from (23) - (24),
aeu(t)
σ12
u(t)
− e − u(t)
dt + σ1 dB1 (t),
du(t) =
1−
2
αe + βev(t) + γ
σ22
bev(t)
dv(t) =
b−
−
dt + σ2 dB2 (t),
2
k + eu(t)
(25)
(26)
subjected to the initial condition u(0) = log x0 , v(0) = log y0 . The functions involved with drift
part of above stochastic differential system satisfy the linear growth condition and they are locally
Lipchitz. Hence there exist unique local solution (u(t), v(t)), for t ∈ [0, τe ) where τe is any finite
positive real number. Clearly, x(t) = eu(t) , y(t) = ev(t) is the unique positive local solution of
stochastic differential system (23) - (24) starting from an interior point of the first quadrant.
Now we are in a position to show that this unique solution is not only a local solution rather it
is global solution. To prove this we need to show that τe = ∞ almost surely (a.s.). To prove this
we utilize the following lemma (see [85] for proof).
Lemma 3.1.2 For all z > 0 the following inequality holds
z ≤ 2(z + 1 − log z) − (4 − 2 log 2).
12
Theorem 3.1.1 For any initial condition (x0 , y0 ) ∈ Int(R2+ ), there exist unique solution (x(t), y(t))
for the stochastic system (23)-(24) for all t ≥ 0 and the solution will remain in Int(R2+ ) with probability one, namely (x(t), y(t)) ∈ Int(R2+ ) for all t ≥ 0 a.s.
Proof : We choose a sufficiently large non-negative number r0 such that both of x0 and y0 lie
within the interval [ r10 , r0 ]. For each integer r ≥ r0 , we can define the stopping time
1
1
τr = inf{t ∈ [0, τe ) : x ∈
/ ( , r) or y ∈
/ ( , r)},
r
r
where inf ∅ = ∞ (as usual ∅ denotes the empty set). Clearly, τr is increasing as r → ∞. Set
τ∞ = lim τr whence τ∞ ≤ τe a.s. To prove that τe = ∞, it is sufficient to prove that τ∞ = ∞ a.s. If
r→∞
possible, let us assume the statement be false. Then there exists two constants T > 0 and ∈ (0, 1)
such that
P {τ∞ ≤ T } > .
(27)
Hence we can find an integer r1 ≥ r0 such that
P {τr ≤ T } ≥ ,
(28)
for all r ≥ r1 . Define a C 2 -function V : Int(R2+ ) → Int(R+ ) by
V (x, y) = (x + 1 − log x) + (y + 1 − log y).
As z + 1 − log z ≥ 0, for all z > 0, the function V (.) is positive definite for all (x, y) ∈ Int(R2+ ).
Calculating the differential of V along the solution trajectories of the system (23)-(24) using Itô’s
formula, we get
(x − 1) 1 − x −
ay
dV (x, y) =
αx + βy + γ
+ σ1 (x − 1)dB1 + σ2 (y − 1)dB2 .
+ b(y − 1) 1 −
y
x+k
σ12 σ22
+
+
dt
2
2
Positivity of x(t) and y(t) implies,
ay
by
σ12 + σ22
dV (x, y) ≤ 2x +
+ by +
+
dt + σ1 (x − 1)dB1 + σ2 (y − 1)dB2 ,
αx + βy + γ
x+k
2
a σ12 σ22
1
≤
+
+
+ 2x + b 1 +
y dt + σ1 (x − 1)dB1 + σ2 (y − 1)dB2 .
(29)
β
2
2
k
13
Defining three positive constants
a σ12 σ22
1
c1 = +
+ , c2 = b 1 +
, c3 = max{4, 2c2},
β
2
2
k
and using lemma (3.1.1), we can write
2x + c2 y ≤ 4(x + 1 − log x) + 2c2 (y + 1 − log y) ≤ c3 V (x, y).
(30)
Using (29) and (30) we can write,
dV (x, y) ≤ (c1 + c3 V (x, y)) dt + σ1 (x − 1)dB1 + σ2 (y − 1)dB2 .
Finally assume, c4 = max{c1 , c3 }, and hence
dV (x, y) ≤ c4 (1 + V (x, y)) dt + σ1 (x − 1)dB1 + σ2 (y − 1)dB2 .
Therefore for t1 ≤ T ,
Z τr ∧t1
Z
dV (x, y) ≤ c4
0
τr ∧t1
(1 + V (x, y)) dt + σ1
0
Z
τr ∧t1
(x − 1) dB1 + σ2
0
Z
0
τr ∧t1
(y − 1) dB2 ,
where τr ∧ t1 = min{τr , t1 }. Using the property of Itô’s integral [46], we get from the above inequality
Z τr ∧t1
V (x(τr ∧ t1 ), y(τr ∧ t1 )) ≤ V (x0 , y0) + c4
(1 + V (x, y)) dt.
0
Taking expectation of both sides of the above inequality and applying Fubini’s theorem [77,86,87]
we get,
EV (x(τr ∧ t1 ), y(τr ∧ t1 )) ≤ V (x0 , y0) + c4 E
Z
τr ∧t1
(1 + V (x, y)) dt,
Z τr ∧t1
≤ V (x0 , y0) + c4 t1 + c4 E
V (x, y) dt,
0
Z t1
≤ V (x0 , y0) + c4 T + c4 E
V (x(τr ∧ t), y(τr ∧ t)) dt,
0
Z t1
= V (x0 , y0) + c4 T + c4
EV (x(τr ∧ t), y(τr ∧ t)) dt.
0
0
Using Gronwall inequality [84], we get from the above inequality
EV (x(τr ∧ T ), y(τr ∧ T )) ≤ c5 ,
14
(31)
where
c5 = (V (x0 , y0) + c4 T ) ec4 T .
Set
Ωr = {τr ≤ T }
for r ≥ r1 . So by (28), we have P (Ωr ) ≥ . Note that for every ω ∈ Ωr , there is at least one of
x(τr , ω), y(τr , ω) which is equal either r or 1r and hence V (x(τr ), y(τr )) is no less than the smallest
of r + 1 − log r and 1r + 1 − log 1r = 1r + 1 + log r. Consequently,
1
V (x(τr ), y(τr )) ≥ (r + 1 − log r) ∧
+ 1 + log r .
r
Therefore from (27) and (31), it follows that
c5 ≥ E [1Ωr (ω)V (x(τr , ω), y(τr , ω))] ,
1
≥ (r + 1 − log r) ∧
+ 1 + log r ,
r
where 1Ωr is the indicator function of Ωr .
Letting r → ∞ we get ∞ > c5 = ∞ which leads us to a contradiction. So we must have τ∞ = ∞
a.s. This completes the proof.
4. Stochastic persistence
In this section we are intended to prove the stochastic persistence of the model system (23)-(24)
under certain parametric restriction(s). Stochastic persistence means, if we start from a positive
initial condition, that is, from an interior point of the first quadrant then solution trajectories of the
stochastic model will always remain within the interior of the first quadrant and remain bounded at
all future time. There are several concepts of stochastic persistence [88], here we use the notion of
stochastic persistence in mean. Before proving the main result of this section we define stochastic
persistence in mean.
Definition : The population x(t) is said to be strongly persistent in the mean if hx(t)i∗ > 0, where
Z
Z
1 t
1 t
hx(t)i :=
x(s) ds, hx(t)i∗ := lim inf
x(s) ds.
t→+∞ t 0
t 0
15
The proof of strong persistence result for the stochastic model (23)-(24) is based upon the following lemma (see Lemma 4 in [88]).
Lemma 4.1 Suppose x(t) ∈ C[Ω × R+ , R0+ ], where R0+ = {a|a > 0, a ∈ R}.
(i) If there are positive constants µ, T and λ ≥ 0 such that
ln x(t) ≤ λt − µ
Z
t
x(s) ds +
0
n
X
βi Bi (t)
i=1
for t ≥ T , where βi ’s are constants, 1 ≤ i ≤ n, then hx(t)i∗ ≤ µλ , a.s.
(ii) If there are positive constants µ, T and λ ≥ 0 such that
ln x(t) ≥ λt − µ
Z
t
x(s) ds +
0
n
X
βi Bi (t)
i=1
for t ≥ T , where βi ’s are constants, 1 ≤ i ≤ n, then hx(t)i∗ ≥ µλ , a.s.
Remark : hx(t)i∗ is defined by hx(t)i∗ := lim supt→+∞
1
t
Rt
0
x(s) ds.
Following theorem is the strong stochastic persistence result for the stochastic system (23)-(24).
Theorem 4.1 Assuming β > a, if 1 −
a
β
>
σ12
2
and b >
σ22
,
2
then solutions of stochastic model
system (23)-(24) starting from any interior point of first quadrant are strongly persistent in mean.
Proof : Define V (x(t)) = ln(x(t)) for x(t) ∈ (0, ∞). Then using Itô’s formula, we get from (23),
σ12
ay(t)
d (ln(x(t))) = 1 − x(t) −
−
dt + σ1 dB1 (t)).
βy(t) + αx(t) + γ
2
Integrating both sides from 0 to t and diving by t, we get
h i
x(t)
ln x(0)
σ12
σ1 B1 (t)
ay(t)
=
1−
+
− hx(t)i −
,
t
2
t
βy(t) + αx(t) + γ
a σ12
σ1 B1 (t)
≥
1− −
+
− hx(t)i.
β
2
t
16
Using Lemma 4.1 we get from the above inequality,
a σ12
hx(t)i∗ ≥ 1 − −
.
β
2
σ2
Hence hx(t)i∗ > 0 whenever 1 − βa − 21 > 0. Now from equation (24), we get
by(t)
bx(t)y(t)
dy(t) = y(t) b −
+
dt + σ2 ydB2 (t).
k
k(k + x(t))
(32)
(33)
Let V (y(t)) = ln(y(t)) for y(t) ∈ (0, ∞). Applying Itô’s formula, we obtain
bx(t)y(t)
σ22 by(t)
d (ln(y(t))) = b −
−
+
dt + σ2 dB2 (t).
2
k
k(k + x(t))
Again integrating both sides from 0 to t and diving by t, we get
h i
y(t)
ln y(0)
σ22
σ2 B2 (t) b
bx(t)y(t)
=
b−
− hy(t)i +
+
,
t
2
t
k
k(k + x(t))
σ22
σ2 B2 (t) b
≥
b−
+
− hy(t)i.
2
t
k
Applying Lemma 4.1, we get the following result from above inequality
σ2
k b − 22
hy(t)i∗ ≥
,
b
and the bound for hy(t)i∗ is positive if b >
σ22
.
2
This completes the proof of the theorem.
The strong persistence result of the prey-predator model under consideration within fluctuating
environment solely depends upon the intensity of environmental
The threshold magr fluctuation.
√
nitudes of environmental driving forces are given by σ1∗ ≡ 2 1 − βa and σ2∗ ≡ 2b. Situations
may arise that either prey population or predator population or both of them become extinct if
forcing intensities are significantly high. We will explain this result in details with help of numerical
example. This persistence result tells us that under certain condition population neither explode
nor going to extinction whenever forcing intensities of environmental driving forces are below the
threshold value. But nature of distribution of population at future time still remains in vein. To
get a satisfactory answer in this direction we will establish that there exists stationary distribution
of prey and predator populations under certain parametric restriction.
17
5. Existence of stationary distribution
In this section, we prove the existence of stationary distribution of prey and predator populations.
For this purpose we find the stationary distribution for solutions of system (23)-(24), which in turn
imply the stability in stochastic sense. Before proving the main theorem related with the stationary
distribution we state a useful lemma from [89] which will be useful to prove the theorem.
Let X(t) be a homogeneous Markov process defined in the l dimensional Euclidean space, denoted by, El and is described by the following system of stochastic differential equations:
dX(t) = b(X)dt +
k
X
fr (X)dBr (t).
(34)
r=1
The diffusion matrix is defined by [89],
A3 (x) = (aij (x)) , aij (x) =
k1
X
fri (x)frj (x).
r=1
We assume there exists a bounded domain U ⊂ El with regular boundary Γ, having the following
properties:
P1: In the domain U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix
A3 (x) is bounded away from zero.
P2: If x ∈ El \ U, the mean time τ at which a path emerging from x reaches the set U is finite,
and sup Ex τ < ∞ for every compact subset S ⊂ El .
x∈S
Lemma 5.1 If above assumptions hold, then the Markov process X(t) has a stationary distribution
µ(.). Let g(.) be a function integrable with respect to the measure µ. Then
Px
for all x ∈ El .
1
lim
T →∞ T
Z
T
g (X(t)) dt =
0
Z
g(x)µ(dx)
El
=1
Remark 5.1 The proof of the above lemma is given in [89]. For the existence of a stationary
distribution with suitable density function, we refer to Theorem 4.1 at p. 119 and Lemma 9.4 at p.
138 in [89].
18
To validate (P1), it is sufficient to prove that F is uniformly elliptical in U, where F u =
P
b(x)ux + (tr(A3 (x)uxx )) /2, i.e. there is a positive number M such that li,j=1 aij (x)ξi ξj ≥ M|ξ|2
for any x ∈ U and ξ ∈ Rl (see chapter 3, p. 103 of [77] and Rayleigh’s principle in [90], chapter 6, p.
349). To verify (P2), it is enough to show that there exist some neighborhood U and a non-negative
C 2 -function V such that for any x ∈ El \ U, LV (x) is negative definite function (for more details
see p. 1163 in [91]).
Remark 5.2 We can write the system (23)-(24) in the form of (34) as follows,


ay(t)
x(t)
1
−
x(t)
−
x(t)
σ1 x(t)
0
αx(t)+βy(t)+γ


d
=
dt +
dB1 (t) +
dB2 (t).
y(t)
0
σ2 y(t)
by(t) 1 − y(t)
k+x(t)
Here the diffusion matrix is
A3 = diag σ12 x2 , σ22 y 2 .
∗
A− aαy
γ
n
a(αx∗ +γ+1)
2
> 0, δ < min M1 −
(x∗ +
Theorem 5.1 Assume k ≤ γ, α ≥ 1, k(a−β) ≤ γ,
)
2 ∗
M2
, a 1 − (αx 2+γ+1) (y ∗ )2 where (x∗ , y ∗ ) is the equilibrium of system (13)-(14),
a(αx∗ +γ+1)
2(M1 −
)
2
M22
aαy ∗
a ∗ 2
+
kM
,
M
=
k
A
−
A = αx∗ + βy ∗ + γ, δ = M2 x∗ +
, M2 = A2 x∗ σ12 + 2b
y σ2
∗ +γ+1)
2
1
γ
4(M1 − a(αx 2
)
∗
∗
and is a positive numbers satisfying M1 − a(αx 2+γ+1) > 0 and 1 − (αx +γ+1)
> 0 simultaneously.
2
Then there exists stationary distribution µ(.) for the stochastic system (23)-(24).
Proof : Since k(a − β) ≤ γ, then there exists a positive equilibrium point of system (13)-(14).
Hence we have
x∗ +
y∗
ay ∗
=
1,
= 1.
αx∗ + βy ∗ + γ
x∗ + k
Define a positive definite function V : E2 → R+ , where E2 = Int(R2+ ) as follows
x a y
∗
∗
∗
∗
∗
∗
V (x, y) = (αx + βy + γ) x − x − x ln ∗
+
y − y − y ln
≡ V1 + V2 .
x
b
y∗
19
(35)
Applying Itô’s formula and using (35), we can calculate,
x∗
A x∗
dV1 = A 1 −
dx +
(dx)2
x
2 x2
A ∗ 2
ay
∗
= A(x − x ) 1 − x −
+ x σ1 dt + σ1 A(x − x∗ )dB1
αx + βy + γ
2
∗
ay
ay
A ∗ 2
∗
∗
= A(x − x ) x +
−x−
+ x σ1 dt + σ1 A(x − x∗ )dB1
αx∗ + βy ∗ + γ
αx + βy + γ
2
∗
∗
aαy
Ax∗ σ12
∗ 2
∗
∗ a(αx + γ)
= − A−
(x − x ) − (x − x )(y − y )
++
dt
αx + βy + γ
αx + βy + γ
2
+σ1 A(x − x∗ )dB1 ,
and
dV2 =
=
=
=
y∗
a y∗
a
1−
dy +
(dy)2
b
y
2b y 2
y
a ∗ 2
a
∗
+ y σ2 dt + σ2 (y − y ∗ )dB2
a(y − y ) 1 −
x+k
2b
b
y∗
y
a
a
a(y − y ∗ )
−
+ y ∗ σ22 dt + σ2 (y − y ∗)dB2
∗
x +k x+k
2b
b
∗ 2
∗
∗
a(y − y )
a(x − x )(y − y )
a
a
−
+
+ y ∗ σ22 dt + σ2 (y − y ∗ )dB2 .
x+k
x+k
2b
b
Therefore
dV
= dV1 + dV2
a
= LV dt + σ1 A(x − x∗ )dB1 + σ2 (y − y ∗)dB2 ,
b
where
LV
a(αx∗ + γ)
aαy ∗
= −A(x − x ) − (x − x )(y − y )
+
(x − x∗ )2
αx + βy + γ αx + βy + γ
∗ 2
∗
∗
a(y − y )
a(x − x )(y − y )
−
+
+ M2 .
x+k
x+k
a(αx∗ + γ)
aαy ∗
≤ −A(x − x∗ )2 + |x − x∗ ||y − y ∗|
+
(x − x∗ )2
αx + βy + γ αx + βy + γ
a(y − y ∗ )2 a|x − x∗ ||y − y ∗ |
−
+
+ M2 .
x+k
x+k
∗ 2
∗
∗
20
Using k ≤ γ, α ≥ 1 and the positivity of the solution, we get
LV
aαy ∗
a(αx∗ + γ + 1)|x − x∗ ||y − y ∗ | a(y − y ∗ )2
(x − x∗ )2 +
−
+ M2
γ
x+k
x+k
∗
−(A − aαy
)(x + k)(x − x∗ )2 + a(αx∗ + γ + 1)|x − x∗ ||y − y ∗ | − a(y − y ∗ )2
γ
=
+ M2
x+k
∗
)(x − x∗ )2 + a(αx∗ + γ + 1)|x − x∗ ||y − y ∗| − a(y − y ∗)2
−k(A − aαy
γ
≤
+ M2 .
x+k
≤ −A(x − x∗ )2 +
Note that
≤ −M1 (x − x∗ )2 + a(αx∗ + γ + 1)|x − x∗ ||y − y ∗| − a(y − y ∗)2 + M2 (k + x).
(k + x)LV
By Young’s inequality, we have
(x − x∗ )2 (y − y ∗ )2
a(αx + γ + 1)|x − x ||y − y | ≤ a(αx + γ + 1)
+
.
2
2
∗
∗
∗
∗
Therefore
(k + x)LV
a(αx∗ + γ + 1)
(αx∗ + γ + 1)
∗ 2
≤ − M1 −
(x − x ) − a 1 −
(y − y ∗)2 + M2 (k + x)
2
2


2
∗
a(αx + γ + 1) 
M2

= − M1 −
x − x∗ + a(αx∗ +γ+1)
2
2 M1 −
2
∗
(αx + γ + 1)
−a 1 −
(y − y ∗)2 + δ.
2
Now if δ satisfies the following condition,



2


∗
∗
a(αx + γ + 1)  ∗
M2
 , a 1 − (αx + γ + 1) (y ∗ )2 ,
δ < min
M1 −
x + ∗


2
2
2 M1 − a(αx 2+γ+1)
then the ellipsoid


2
∗
a(αx + γ + 1) 
M2
(αx∗ + γ + 1)
∗



M1 −
x− x + +a 1 −
(y−y ∗)2 = δ,
a(αx∗ +γ+1)
2
2
2 M1 −
2
lies entirely in Int(R2+ ). We can take U to be a neighborhood of the ellipsoid with Ū ⊆ E2 = Int(R2+ ),
where Ū is the compact closure of U. So for x ∈ U \ E2 , LV < 0, which implies condition (P2) in
lemma 5.1 is satisfied.
21
Besides, there is
M = min σ12 x2 , σ22 y 2, (x, y) ∈ Ū > 0
such that
2
X
i,j=1
aij ξi ξj = σ12 x2 ξ12 + σ22 y 2ξ22 ≥ M|ξ|2 ,
for all (x, y) ∈ Ū , ξ ∈ R2 , which shows that condition (P1) of lemma 5.1 is also satisfied.
Therefore we can conclude that the stochastic system (23)-(24) has a stationary distribution
µ(.).
6. Numerical simulation results
In this section we provide numerical simulation results to substantiate the analytical findings
for the stochastic model system reported in the previous sections. It is worthy to mention here that
the approximated sample paths of the stochastic model (23)-(24) obtained from direct simulation
must be close to those of the actual Itô process and these idea is linked with the concept of a
strong solution for a system of stochastic differential equations [28]. To find the approximate strong
solution of the system (23)-(24) in Itô sense with positive initial condition we use the Milsteins
method having strong order of convergence γ = 1 [79]. This Milstein’s scheme is obtained from
Euler-Maruyama scheme by incorporating a correction term for stochastic increment.
For numerical simulations of the stochastic model (23)-(24) we choose the parameters a = 1,
α = 4, β = 6, γ = 2, b = 2 and k = 1. We use different values of σ1 and σ2 in order to understand
their role on the dynamics. For all numerical simulations reported here are carried out with the
choice of time stepping ∆t = 0.01, we have checked our simulations with smaller time steps also.
The numerical scheme obtained through Milsteins method applied on the stochastic model under
consideration is given by,
1 − xj −
xj+1 = xj + xj
yj+1
√
1 2
ayj
2
∆t + σ1 1j ∆t + σ1 ∆t(1j − 1) ,
αxj + βyj + γ
2
√
byj
1 2
2
= yj + yj
b−
∆t + σ2 2j ∆t + σ2 ∆t(2j − 1) ,
k + xj
2
where 1j and 2j are two independent Gaussian random variables N(0, 1) for j = 1, 2, ..., n [92].
We start our numerical simulation with environmental forcing intensities σ1 = 0.01, σ2 = 0.02
and starting from the initial point (0.4, 0.2). Result of one simulation run is reported in Fig - 2.
22
After some initial transients the population densities fluctuate around the deterministic steady state
values x∗ = .888 and y ∗ = 1.888 respectively. Parameter values chosen above and the choice of σ1
and σ2 are consistent with the conditions required for stochastic persistence and the existence of
stationary distribution (see Th. 4.1 and Th. 5.1). We have repeated the simulation 10,000 times
keeping all parameters fixed and never observed any extinction scenario up to t = 100. One can
verify that the extinction scenario will not appear at any future time for chosen parameter values
and noise intensities. The stationary distribution of prey and predator population is also provided
at the lower panel in Fig - 2. From stationary distribution of two populations it is clear that they
are distributed normally around the mean values .888 and 1.888 respectively.
1
2
1.8
0.9
1.6
1.4
y(t)→
x(t)→
0.8
0.7
0.6
1.2
1
0.8
0.6
0.5
0.4
0.4
0
20
40
60
80
0.2
0
100
20
40
t→
5
14
4.5
100
1.9
1.95
2
x 10
4
relative frequency density→
12
relative frequency density→
80
5
x 10
10
8
6
4
2
0
0.86
60
t→
3.5
3
2.5
2
1.5
1
0.5
0.87
0.88
0.89
0.9
0.91
0
1.75
0.92
x(t)→
1.8
1.85
y(t)→
Figure 2: Upper panel: Results of one simulation run of the stochastic system (23)-(24) with
parameter values as mentioned in the text and σ1 = 0.01, σ2 = 0.02 (green curves). Solution
trajectories are plotted along with the solutions of corresponding deterministic model (13) - (14)
to see the variation of population around the deterministic steady state (red broken curve). Lower
panel: Stationary distribution of prey and predator population obtained at t = 100 from 10,000
simulation run.
Next we increase strengths of environmental forcing to σ1 = 0.1 and σ2 = 0.2 and again we
observe that the population distribution fluctuate around the deterministic steady-state value but
23
amplitude of fluctuation is more compared to earlier case. This fluctuation is also reflected at the
stationary distribution as prey population is distributed within (.6, 1.2) and predator population
remains within the range (1.3, 2.6). In earlier case they was within (.86, .92) and (1.82, 1.97)
respectively. In this case also we have not observed any extinction scenario. The result of one
simulation run and stationary distribution obtained from 10,000 simulation run at t = 100 is
presented in Fig - 3.
1.3
3
1.2
2.5
1.1
2
y(t)→
x(t)→
1
0.9
0.8
0.7
1.5
1
0.6
0.5
0.5
0.4
0
20
40
60
80
0
0
100
20
t→
4.5
100
x 10
4
relative frequency density→
relative frequency density→
80
4
x 10
12
10
8
6
4
2
0
60
t→
4
14
40
3.5
3
2.5
2
1.5
1
0.5
0.7
0.8
0.9
1
1.1
1.2
0
1
1.3
x(t)→
1.5
2
2.5
3
y(t)→
Figure 3: Upper panel: Time evolution of prey and predator population obtained from one simulation run of the system (23)-(24) σ1 = 0.1, σ2 = 0.2 with other parameter values as mentioned
in the text. Solutions of stochastic model system (green curve) is plotted against the solution of
deterministic model (red broken curve). Lower panel: Stationary population distributions of prey
and predator populations at t = 100.
If we choose σ1 = 1.3 and σ2 = 0.2, then the first condition of Theorem (4.1) will be violated. As
a result, prey population goes to extinction and extinction time for this simulation is 80 and it will
vary with another simulation run. However, it is ensured that prey population will goes to extinction
at a future time. In Fig-4, we have presented the extinction scenario for prey population. Again
for σ1 = 0.1 and σ2 = 2.1, second condition of Theorem (4.1) is violated and as a result predator
population goes to extinction as depicted in Fig-5. It is interesting to see that the components
24
which are not going to extinction are not fluctuating around the deterministic steady-state rather
they fluctuate around some fuzzy value. This fuzziness can be observed from more simulation runs.
These results lead us to conclude that the amplitude of oscillation and average value of populations
around which they fluctuate at all future time depends upon the strength of environmental driving
force when their magnitudes are significantly high.
4
3
3.5
2.5
3
2
y(t)→
x(t)→
2.5
2
1.5
1.5
1
1
0.5
0.5
0
0
20
40
60
80
0
0
100
20
40
t→
60
80
100
t→
Figure 4: Numerical simulation for the system (23)-(24) with σ1 = 1.3, σ2 = 0.2 shows that prey
population goes to extinction but predator population survives.
1.3
12
1.2
10
1.1
8
y(t)→
x(t)→
1
0.9
0.8
0.7
6
4
0.6
2
0.5
0.4
0
20
40
60
80
0
0
100
t→
20
40
60
80
100
t→
Figure 5: Numerical simulation for the system (23)-(24) with σ1 = 0.1, σ2 = 2.1 shows that predator
population goes to extinction but prey population survives.
Finally we choose σ1 = 1.3 and σ2 = 2.1 such that both the conditions required for persistence
are violated. In this situation prey as well as predator population go to extinction after initial
large amplitude oscillation. One such simulation result is presented in Fig - 6. In this figure
prey population goes to extinction before predators but reverse situation will arise in case of more
simulation runs.
25
3
4.5
4
2.5
3.5
3
y(t)→
x(t)→
2
1.5
1
2.5
2
1.5
1
0.5
0.5
0
0
20
40
60
80
0
0
100
t→
20
40
60
80
100
t→
Figure 6: Numerical simulation for the system (23)-(24) with σ1 = 1.3, σ2 = 2.1 shows that both
prey and predator populations go to extinction.
7. Conclusion
In this paper, we have considered the basic features of a modified Holling-Tanner type preypredator model with Beddington-DeAngelis type functional response in presence of multiplicative
noise terms to understand the dynamics in presence of environmental driving forces. We have established the existence of non-negative global solution of the stochastic model and obtained conditions
required for the strong persistence in mean for both the species. Our analytical findings reveal that
the forcing intensity of fluctuating environment plays a crucial role behind the survival of prey and
predator species. Obtained analytical results are verified with supportive numerical simulations.
Although we are considering a prey-predator model, the survival of predator species in absence of
prey population is justified as we have assumed that the predators have alternative food source
and their growth follows the logistic growth law. The existence of stationary distribution for both
populations are established under certain parametric restrictions. These parametric restrictions
reflect the idea that large amplitude environmental noise can destabilize the system and in that
situation one can not find any stationary distribution. The existence of stationary distribution
imply stochastic stability to some extent. It is worthy to mention here that the mathematical investigations carried out in this paper is based upon the complete noise added model without using
any approximation technique.
For deterministic models one can find out time independent equilibrium levels and study the
local and global stability properties. For stochastic systems obtained from deterministic setup
by adding environmental stochasticity, most of the time it is impossible to obtain steady states
26
for the governing system of stochastic differential equations. In this situation, one can look at
the probabilistic smoke cloud, obtained from the stationary distribution of populations and verify
whether they are concentrated within a small domain or not. In case of noise added models,
there is a continuous spectrum of disturbances generated by the added environmental noise terms,
and the system is always in tension between deterministic stabilization and random environmental
fluctuations. If we consider the dynamics of stochastic model satisfying the parametric restriction(s)
required for stability of deterministic equilibrium point, then random environmental driving forces
are responsible for spreading the population distribution around the deterministic steady-state.
The spread of stationary population distribution solely depends upon the strength of environmental
forcing. Stochastic models having compact cloud of population distribution are called stochastically
stable systems [2,28]. For a clear understanding of this idea we can look at the probability cloud of
stationary distribution for parameter values as mentioned in the last section and for two different
set of forcing intensities, σ1 = .01, σ2 = .02 and σ1 = .1, σ2 = .2. The distribution of population
obtained through numerical simulation at t = 100 for two different set of forcing intensities is
reported in Fig - 7. It is clear that the probability cloud is distributed around the deterministic
steady-state (0.888, 1.888) but their dispersion is controlled by σ1 and σ2 . The distributions of
population reported in Fig - 7 remain unaltered at all future time.
1.94
2.4
1.92
2.2
y(t=100) →
y(t=100) →
1.96
1.9
1.88
2
1.8
1.86
1.6
1.84
1.4
1.82
0.86
0.88
0.9
x(t=100) →
1.2
0.92
0.8
1
x(t=100) →
1.2
Figure 7: Stationary distribution of prey and predator population around deterministic steady-state
at time t = 100 for a = 1, α = 4, β = 6, γ = 2, b = 2, k = 1 and σ1 = .01, σ2 = .02 (left panel);
σ1 = .1, σ2 = .2 (right panel).
27
Finally we like to mention that the results we have reported here are not exhaustive rather starting point of many further investigations. For example, the proof for pathwise persistence, estimation
of persistence time and estimation of extinction time remain open problems. Further, the deterministic model system is capable to exhibit bistable scenario as well as oscillatory co-existence of prey
and predator populations arising through Hopf-bifurcation within certain parametric domain. Here
we have considered the noise perturbation on intrinsic growth rates only but one can think about
the consequences where other parameters, like environmental carrying capacity, rate of capture of
prey by predators, half-saturation constants are also stochastically varying. We must conface here
that the discussed technique is not applicable to the cases when noise term appear in denominator.
So far as our knowledge goes this is a limitation of the mathematical technique discussed here. Of
course some mean-field approximation techniques can be employed in such situations (for details
see [61, 62, 93, 94]). The noise induced transition scenario for noise driven prey-predator model
obtained by perturbing all system parameters with bounded noises [95, 96] is also an interesting
problem and will be addressed in near future.
Acknowledgement: The work of first author is supported by the Council of Scientific and Industrial Research, India. We are also thankful to the anonymous reviewer for his/her constructive
suggestions which helped us to write a better description of our present work.
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