Stochastic persistence and stability analysis of a
modified Holling-Tanner 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
The article aims to study the basic dynamical features of a modified Holling-Tanner preypredator model with ratio-dependent functional response. We have proved the global existence
of the solution for the deterministic model. The parametric restriction for persistence of both
species is also obtained along with the proof of local asymptotic stability of the interior equilibrium point(s). Conditions for local bifurcations of interior equilibrium points are provided.
The global dynamic behavior is examined thoroughly with supportive numerical simulation
results. Next we have formulated the stochastic model by perturbing the intrinsic growth rates
of prey and predator populations with white noise terms. The existence-uniqueness of solutions for stochastic model is established. Further we have derived the parametric restrictions
required for the persistence of the stochastic model. Finally we have discussed the stochastic
stability results in terms of first and second order moments. Numerical simulation results are
provided to support the analytical findings.
Key words : Holling-Tanner, Ratio-dependent, Stability, Persistence, Moment.
AMS Subject Classification : 93E15, 37B25, 34K50.
∗
Tel. +91-512-259-6157, Fax. +91-512-259-7500
1
1. Introduction
The classical Leslie-Gower type prey-predator model is based upon the assumptions that prey
population grows logistically in absence of predators and predators carrying capacity is variable, in
particular directly proportional to prey abundance [1–3]. Within a deterministic environment the
Leslie-Gower prey-predator interaction is described by two coupled non-linear ordinary differential
equations
dx
x
= rx 1 −
− cxy,
dt
K
dy
y
= sy 1 − h
,
dt
x
(1)
(2)
subjected to the initial conditions x(0) > 0 and y(0) > 0. Here x ≡ x(t) and y ≡ y(t) stand
for prey and predator population (or density) at time ‘t’. All parameters involved with the model
are positive. ‘r’ and ‘s’ denote intrinsic growth rates for prey and predator respectively. ‘K’ is the
environmental carrying capacity for prey population, ‘c’ is the per capita capturing rate of prey by
a predator per unit time and ‘h’ is proportionality constant indicates the prey amount needed to
y
feed a predator in equilibrium conditions [3, 4]. The term h which is known as the Leslie-Gower
x
term [3,5,6] and a detailed discussion on basic ecological features of the model (1) - (2) and related
dynamics can be found in [7–14]. In this formulation the net growth rate of predator population is
affected by the relative sizes of the two populations at any instant of time. Lowering in prey density
increases the ratio y/x and consequently the net growth rate of predators declines.
The Holling-Tanner model for predator-prey interactions is the modification of the Leslie-Gower
model where the per capita capturing rate is replaced by a saturation function of prey population,
which is known as Holling type-II functional response. The Holling-Tanner model is given by
dx
x
axy
= rx 1 −
−
,
dt
K
b+x
dy
y
= sy 1 − h
,
dt
x
(3)
(4)
subjected to the same initial conditions as described above. Two new positive parameters a and b
denote capturing rate and half-saturation constant respectively. Detailed local and global stability
analysis for equilibrium points along with the existence of Hopf-bifurcating periodic solution and
its stability are discussed by several researches [5, 6, 8–11].
2
Liang and Pan [15] proposed and analyzed the following ratio-dependent Holling-Tanner preypredator model
dx
x
axy
= rx 1 −
−
,
dt
K
my + x
dy
y
= sy 1 − h
,
dt
x
(5)
(6)
subjected to ecologically feasible initial condition x(0), y(0) > 0. This model is obtained by
replacing prey-dependent functional response term in (3) by ratio-dependent functional response
ay/(my + x). Here ‘a’ and ‘m’ are two positive parameters characterizing the Michaelis-MentenHolling type functional response [16]. Prey-predator models with prey-dependent functional response are challenged by several ecologists based upon biological and physiological evidences that
in many situations the consumptions of prey by the predators should not depend upon prey abundance. Rather the predator has to search and compete for food and hence predator abundance also
has the ability to influence the functional response [15,17,18]. To overcome this situation Arditi and
Ginzburg [19] have suggested that the functional response can be approximated by a function of the
prey-to-predator ratio and their claim is already supported by numerous field data and laboratory
experiments [20–22]. Detailed discussion and rigorous dynamical analysis of prey-predator models
with ratio-dependent functional response can be found in [16,18,23–26] and references cited therein.
Recently the Holling-Tanner prey-predator model with modified form of growth law for predator
population has received significant attention [13, 27, 28]. In their study, the authors have replaced
the growth equation (6) by the following equation to describe the growth law for the predators,
h1 y
dy
= y s1 −
,
(7)
dt
k1 + x
where s1 is the intrinsic growth rate of predator, h1 is the intra-specific competition term and k1 is the
measure of environmental carrying capacity for predators in absence of its most favorable food [13].
This modification is based upon the assumption that in case severe scarcity of the favorable food
source for predators, they can switch to other resources and hence their environmental carrying
capacity depends upon the abundance of their favorable food and other resources. In the present
paper we will focus on the following model system,
dx
x
axy
= rx 1 −
−
,
dt
K
my + x
dy
h1 y
= y s1 −
,
dt
k1 + x
3
(8)
(9)
which is a modified Holling-Tanner model with ratio-dependent predation term.
Most of the work on Leslie-Gower, Holling-Tanner and modified Holling-Tanner prey-predator
models are concerned with the dynamic behavior of solution trajectories within a deterministic
environment. Dynamic behavior is limited to either convergence of solutions to an equilibrium
point or a limit set. In the first case we see that once the population reaches the steady state it
remains there at all future time. On the other hand if the concerned model system has unique stable
limit cycle then we observe oscillatory coexistence of both prey and predator with fixed period and
amplitude. In reality, the population should not stay at a constant density at all future time or in
the case of oscillatory coexistence the period and amplitude is not a fixed quantity over a longer
period of time. This realistic feature can be captured if we consider the stochastic formalism instead
of a deterministic approach. As most of the ecosystems are exposed within the open environment,
we cannot ignore the randomly fluctuating environmental forces. To take into account of the
environmental driving forces the deterministic model system can be extended to a stochastic model
system by introducing additive and/or multiplicative noise terms. This approach is well-known
and is adopted in several investigations. But the lack of availability of suitable mathematical tools
to analyze the concerned stochastic model is the main obstacle in this area of research. Most of
the investigations for stochastic models are based upon different linearization techniques which
are valid for the solution trajectories starting from the nearby points of deterministic equilibrium
points. In some recent works, authors have considered the global existence and uniqueness of
solutions for the stochastic model system and also obtained the long time behavior of the solution
trajectories along with the stochastic persistence. Recently, Ji et al. [29] obtained the stationary
distribution of prey population with the help of suitable Lyapunov function for a modified LeslieGower model in presence of multiplicative noise where authors have used some results from their
earlier investigations [30]. However, this approach is not applicable for a general prey-predator
model.
The main purpose of this paper is to study the dynamics of the model system (8)-(9) in the
presence of environmental fluctuation. Firstly we consider some basic dynamical behavior of the
‘modified Holling-Tanner model with ratio-dependent predation term’ in section 2. Next we modify
the deterministic model system to the stochastic differential equation model system by introducing
two multiplicative white noise terms into the growth equations of both species. For the stochastic
4
model system we prove the existence and uniqueness of solutions and stochastic persistence in
section 3. Section 4 includes the moment based stability analysis for the stochastic model system
around co-existing equilibrium point. Basic outcomes of our analytical findings and their ecological
interpretations are provided in the concluding section.
2. Deterministic Model
In this section we present some results for the modified Holling-Tanner model of prey-predator
interaction with ratio-dependent functional response for the consumption of prey by predators
within a deterministic environment. The dynamical model for prey-predator interaction is governed
by the following system of nonlinear coupled ordinary differential equations
dx(t)
ν1 y(t)
= x(t) α1 − β1 x(t) −
,
dt
m1 y(t) + x(t)
dy(t)
ν2 y(t)
= y(t) α2 −
,
dt
m2 + x(t)
(10)
(11)
subjected to the ecologically feasible initial conditions x(0) ≡ x0 > 0 and y(0) ≡ y0 > 0. All
parameters involved with the model are fixed positive constants and their ecological interpretations
has been already discussed in the introduction with a different nomenclature. Biomass of the prey
and the predator species can be measured in terms of ‘kg/ha’ and ‘yr’ is the time unit. Accordingly,
units of the parameters are as follows: α1 , α2 → yr−1 , β1 → kg/ha−1 yr−1 , ν1 → kg yr−1 ind.−1 ,
ν2 → kg/ind., m2 → kg/ha and m1 is a dimensionless quantity. Here ‘ha’ and ‘ind’ stand for
hector and individual respectively [24, 31].
The model (10) - (11) has a singularity at the origin. To avoid this singularity we can redefine
the growth equation for prey as follows,
dx(t)
ν1 y(t)
= x(t) α1 − β1 x(t) −
, (x, y) 6= (0, 0),
dt
m1 y(t) + x(t)
= 0, (x, y) = (0, 0).
(12)
This definition ensures the existence of the trivial equilibrium point (0, 0) (see [23] for details).
We have defined this for mathematical correctness of the prey-predator model, in the rest of the
paper we work on the model (10) - (11). Existence-uniqueness and positivity of solutions for
the above model system subjected to positive initial conditions can be verified easily, we have
omitted those proofs for the sake of brevity, rather we prove here the existence of global solution
5
for the model under consideration. A simple argument shows that the interior of first quadrant,
R2+ = {(x, y) : x, y > 0} is an invariant set.
2.1 Existence of Global Solution
To prove the global existence of solution we use the following lemma (see [32–34] for details).
Lemma 2.1 : If a, b > 0, and
du
dt
≤ u(t)(b − au(t)) with u(0) > 0, then, for all t ≥ 0,
b
,
a − ce−bt
b
b
where c = a − u(0) . In particular, u(t) ≤ max u(0),
for all t ≥ 0.
a
Now using continuous induction we want to prove the global existence and uniqueness of solution
u(t) ≤
for the system (10) - (11). Consider the following interval
I := {t ≥ 0 : ∀ m < t, ∃ a unique solution of (10) - (11) on [0, m)}.
Now we will show that (1) I is non-empty, (2) I is open and finally, (3) I is closed. Indeed, this
implies I = R+ , which ensures the existence of unique global solution for the system (10) - (11).
System (10) - (11) can be put into a matrix differential equation as follows,
dX(t)
= f (t, X(t)),
dt
(13)
where X(t) = (x(t), y(t)) and
f : [0, ∞) × R2 → R2 ,
(t, x, y) → x(t) α1 − β1 x(t) −
ν1 y(t)
m1 y(t) + x(t)
, y(t) α2 −
ν2 y(t)
m2 + x(t)
.
The function f is locally Lipschitz continuous in (x, y). Hence by the Picard-Lindelöf theorem,
there exists a unique local solution of (10) - (11) on an open interval [0, ) for some > 0. Hence I
is non-empty.
Let α0 ∈ I, we need to show that there exists an > 0 such that α0 + ∈ I. Since α0 ∈ I, there
exists a unique local solution of (10) - (11) on [0, m) for all m < α0 . Consider a strictly increasing
sequence {mi }∞
i=1 converging to α0 . Then for each i, there exists a unique local solution (xi , yi ) of
6
(10) - (11) on [0, mi ). By uniqueness, the restriction of (xi+1 , yi+1 ) on [0, mi ) is (xi , yi). We can
extend (xi , yi ) to [0, mi ] by setting (xi (mi ), yi(mi )) = (xi+1 (mi ), yi+1 (mi )). Therefore we have
0
|xi+1 (mi+1 ) − xi (mi )| = |xi+1 (mi+1 ) − xi+1 (mi )| = |xi+1 (ξ)||mi+1 − mi |,
(14)
for some ξ ∈ (mi , mi+1 ). From the first equation of (10) - (11) and using positivity of local solution,
we get
dx(t)
≤ x(t) (α1 − β1 x(t)) .
dt
(15)
(16)
Using Lemma 2.1, we have
α1
0 < xi (t) ≤ max x(0),
β1
:= α11 ,
for all i and all t ∈ [0, mi ]. Proceeding in a similar fashion, we have
0
|yi+1(mi+1 ) − yi (mi )| = |yi+1(mi+1 ) − yi+1 (mi )| = |yi+1 (ξ)||mi+1 − mi |,
(17)
for some ξ ∈ (mi , mi+1 ). From the second equation of (10) - (11) and using (16), we get
ν2
dy(t)
≤ y(t) α2 −
y(t) .
dt
m2 + α11
Again with the help of Lemma 2.1, we get
α2 (m2 + α11 )
0 < yi(t) ≤ max yi (0),
.
ν2
(18)
Using the bounds for xi and yi from (16) and (18), and using the triangle inequality we get from
0
0
(10) - (11) that there exists a uniform bound L such that |xi (t)| ≤ L and |yi (t)| ≤ L for all i and
all t ∈ [0, mi ]. Hence it follows from (14) and (17) that (xi (mi ), yi (mi )) is a Cauchy sequence in R2+
as mi is a Cauchy sequence. Hence (xi (mi ), yi(mi )) converges to a point (l1 , l2 ) as i → ∞. Therefore
there exists a unique local solution of (10) - (11) with prescribed x(α0 ) = l1 and y(α0 ) = l2 . It can
be clearly seen that this solution extends the solution on [0, α0 ) and therefore there exists > 0
such that α0 + ∈ I.
Let {ti }∞
i=1 be a sequence in I converging to a point r. Then according to the definition of I,
there exists a unique local solution of (10) - (11) on [0, m) for all m < ti . Therefore for any arbitrary
m < r, there exists an index i such that m < ti and a unique solution of (10) - (11) exists on [0, m).
7
Hence r ∈ I implying I is closed. This completes the proof of the global existence of the solution
for the model under consideration.
2.2 Boundedness and Permanence
In this subsection, we prove the boundedness of solutions and deduce the criteria for permanence
of solutions for the model (10) - (11) starting from an interior point of R2+ . Now we state a
useful result without proof which is required for the proofs of boundedness and permanence of
solutions [32, 34, 35], which is a modified version of Lemma 2.1.
Lemma 2.2 : For a, b > 0 consider the differential inequality v 0 (t) ≤ (≥) v(t)(a − bv(t)) with
v(0) > 0 then,
a
lim sup v(t) ≤
b
t→+∞
a
lim inf v(t) ≥
.
t→+∞
b
Using positivity of the variables and parameters, we get from (10)
x0 (t) < x(t) (α1 − β1 x(t)) , x(0) > 0,
and hence using above lemma we obtain,
lim sup x(t) ≤
t→+∞
α1
≡ x.
β1
For arbitrary > 0 , there exists a positive quantity T ≡ T () such that
x(t) ≤ x + , ∀ t ≥ T.
Then from (11), we get
y (t) ≤ y(t) α2 −
0
ν2 y(t)
m2 + + x
, y(0) > 0.
Applying Lemma 2.2, we get
lim sup y(t) ≤
t→+∞
α2 (m2 + + x)
.
ν2
Since > 0 is arbitrary, we finally have
lim sup y(t) ≤
t→+∞
α2 (m2 + x)
≡ y.
ν2
8
(19)
Hence the model system (10) - (11) is bounded. Now we recall the definition of permanence [36,37].
Definition : System (10) - (11) is said to be permanent if there exist two positive constants λ
and µ satisfying 0 < λ ≤ µ (irrespective of initial condition) such that all solutions starting from
interior of R2+ satisfy,
λ ≤ min lim inf x(t), lim inf y(t) ≤ max lim sup x(t), lim sup y(t) ≤ µ.
t→+∞
t→+∞
t→+∞
t→+∞
To prove the permanence of solutions we have to find out the lower limits of solutions and to
check their positivity. From (10) we get after some algebraic manipulation,
ν1
0
, x(0) > 0,
x (t) > x(t) α1 − β1 x(t) −
m1
and again using Lemma 2.2, we get
lim inf x(t) ≥
t→+∞
(α1 m1 − ν1 )
≡ x.
β1 m1
Under the parametric restriction α1 m1 > ν1 , we have x > 0. As x(t) is always positive, we can
write from (11),
ν2 y(t)
y (t) > y(t) α2 −
, y(0) > 0,
m2
0
and with help of Lemma 2.2, we get
lim inf y(t) ≥
t→+∞
α2 m2
≡ y > 0.
ν2
Hence we can choose two positive real numbers λ and µ defined by λ = min x, y , µ =
max {x, y}. The results we have established so far can be summarized as follows:
Theorem 2.1 : If α1 m1 > ν1 , then the model (10) - (11) is permanent in the interior of R2+ .
Now we discuss the existence of various equilibria for the model (10) - (11). Apart from the
trivial equilibrium point E0 = (0, 0) (with
reference
to (12)
as the growth equation for prey),
α1
α2 m2
the system has axial equilibria E10 ≡
, 0 and E01 ≡ 0,
. The interior equilibrium
β1
ν2
y
point(s) is(are) the point(s) of intersection of two zero growth isoclines α1 − β1 x − mν1 1y+x
= 0 and
9
y
α2 − mν22+x
= 0 in the interior of the first quadrant. The predator nullcline is a straight line passing
2
) and having the slope
through the point (0, α2νm
2
α2
.
ν2
To understand the shape and position of the
prey nullcline, we put the equation of the prey nullcline as follows
y =
x(α1 − β1 x)
.
m1 β1 x + (ν1 − α1 m1 )
The numerator is positive when 0 < x < αβ11 . Now we consider two cases: if (H1): α1 m1 < ν1
α1
then y > 0 for 0 < x <
and the prey nullcline is a continuous smooth curve joining the points
β1
(0, 0) and ( αβ11 , 0) and having the maximum in the interval 0, αβ11 ; and if (H2): α1 m1 > ν1 then
i
α1
α1 m1 −ν1 α1
1
y > 0 for α1mm11β−ν
<
x
<
and
the
prey
lies
in
the
first
quadrant
for
x
∈
,
and
β1
m1 β1
β1
1
having a vertical asymptote x =
α1 m1 −ν1
.
m1 β1
Sample plots for prey nullclines under the two parametric
y→
restrictions and the predator nullcline are presented in Fig. 1.
x →
Figure 1: Plots of the prey-nullclines (green curve for ν1 > α1 m1 and red curve for ν1 < α1 m1 ) and
the predator nullcline (blue straight line).
Clearly, we find the unique interior equilibrium point under the parametric restriction (H2)
and number of interior equilibrium point varies from zero to two if (H1) is satisfied. In the latter
case, the existence of the interior equilibrium point depends upon further parametric restrictions
as discussed below. If E∗ (x∗ , y∗ ) denotes the co-existing equilibrium point then its components are
given by y∗ =
α2 (m2 +x∗ )
ν2
where x∗ is positive root of the quadratic equation
(β1 m1 α2 + β1 ν2 )x2 + (β1 m1 α2 m2 + α2 ν1 − α1 ν2 − α1 α2 m1 )x + α2 m2 (ν1 − α1 m1 ) = 0.
10
(20)
Clearly, the above quadratic equation has one positive real root when (H2) is satisfied and x∗
is given by
x∗ =
−b +
√
b2 − 4ac
,
2a
where a = β1 (ν2 + m1 α2 ), b = β1 m1 α2 m2 + α2 ν1 − α1 ν2 − m1 α1 α2 , c = α2 m2 (ν1 − α1 m1 ).
Existence of an interior equilibrium point under the parametric restriction (H1) demands the
following two additional parametric restrictions:
(H3): b < 0, i.e. β1 m1 m2 α2 + α2 (ν1 − α1 m1 ) < α1 ν2 .
(H4): b2 ≥ 4ac, i.e. [α2 (ν1 − α1 m1 ) + β1 m1 m2 α2 − α1 ν2 ]2 ≥ 4α2 β1 m2 (ν1 − α1 m1 )(ν2 + α2 m1 ).
Thus under conditions (H1), (H3) and (H4), we have two interior equilibrium points E1∗ =
(x1∗ , y1∗ ) and E2∗ = (x2∗ , y2∗ ) where
x1∗ =
and yj∗ =
α2 (m2 +xj∗ )
,
ν2
−b −
√
√
b2 − 4ac
−b + b2 − 4ac
, x2∗ =
2a
2a
j = 1, 2. When the equality holds in (H4), we find unique interior equilibrium
point E∗SN = (x∗SN , y∗SN ) whose components are x∗SN =
−b
2a
and y∗SN =
α2 (m2 +x∗SN )
,
ν2
a and b
are already mentioned above. Two interior equilibrium points are generated through saddle-node
bifurcation under the restriction (H1) and the parametric equation for the saddle-node bifurcation
curve is
[α2 (ν1 − α1 m1 ) + β1 m1 m2 α2 − α1 ν2 ]2 = 4α2 β1 m2 (ν1 − α1 m1 )(ν2 + α2 m1 ).
(21)
2.3 Local Asymptotic Stability
In this section, we discuss the stability properties of two axial equilibrium points E10 , E01 and
interior equilibrium point(s) based upon the standard linearization technique and then using the
well-known Routh-Hurwitz criteria [38].
Calculating the Jacobian matrix for (10) - (11) at E10 ( αβ11 , 0), we find
J(x, y)|E10 =
−α1 −ν1
0
α2
,
(22)
having one negative eigenvalue λ1 = −α1 and a positive eigenvalue λ2 = α2 > 0. Hence E10 is always
2
a saddle-point. The Jacobian matrix evaluated at the second axial equilibrium point E01 (0, α2νm
)
2
11
is
J(x, y)|E01 =
having eigenvalues λ1 = α1 −
ν1
m1
"
α1 −
α22
ν2
ν1
m1
0
−α2
#
,
(23)
and λ2 = −α2 < 0. Hence E01 is locally asymptotically stable
if (H1) holds. E01 is a saddle point if (H2) is satisfied, and instability of E01 implies the feasible
existence of a unique interior equilibrium point.
Computing the Jacobian matrix for the system (10) - (11) at E∗ we obtain,
#
"
ν1 (x∗ )2
∗ y∗
−β1 x∗ + (mν11yx∗ +x
−
2
2
)
(m
y
+x
)
∗
∗
1 ∗
,
J∗ ≡ J(x, y)|E∗ =
ν2 (y∗ )2
ν 2 y∗
−
2
(m2 +x∗ )
m2 +x∗
(24)
and the characteristic equation for J∗ is
λ2 + λ∆1 + ∆2 = 0,
where
∆1 = −Trace(J∗ ) = β1 x∗ −
ν2 x∗ y∗
∆2 = Det(J∗ ) =
m2 + x∗
ν1 x∗ y∗
ν2 y∗
+
,
2
(m1 y∗ + x∗ )
m2 + x∗
ν1 y∗
β1 −
(m1 y∗ + x∗ )2
+
ν1 ν2 (x∗ )2 (y∗ )2
.
(m2 + x∗ )2 (m1 y∗ + x∗ )2
According to the Routh-Hurwitz criteria, local asymptotic stability of E∗ demands the satisfaction
of the restrictions ∆1 > 0 and ∆2 > 0.
Under the restriction (H2), the interior equilibrium point E∗ = (x∗ , y∗ ) is always locally asymptotically stable as the following sufficient condition for local asymptotic stability
β1 −
ν1 y∗
> 0,
(m1 y∗ + x∗ )2
(25)
is automatically satisfied if (H2) holds. The components of E∗ = (x∗ , y∗ ) satisfy the following
expression
α1 − β1 x∗ =
ν1 y∗
,
m1 y∗ + x∗
(26)
and hence, (25) is equivalent to
β1 ν1 y∗ > (α1 − β1 x∗ )2 .
12
(27)
Solving (26) for y∗ we get,
y∗ =
x∗ (α1 − β1 x∗ )
,
ν1 − m1 (α1 − β1 x∗ )
and then substituting in (27) we find
β12 m1 (x∗ )2 − 2β1 (α1 m1 − ν1 )x∗ + α1 (α1 m1 − ν1 ) > 0.
(28)
If (H2) is satisfied, the discriminant of the quadratic expression is negative,
D = −4β12 [ν1 (α1 m1 − ν1 )] < 0.
Hence the inequality (28) holds good which in turn imply that (25) is satisfied. This ensures the
local asymptotic stability of E∗ under the parametric restriction (H2).
Next we consider the local asymptotic stability condition of two interior equilibrium points under
the parametric restriction (H1). Note that, 0 < x1∗ < x∗SN < x2∗ and after some tedious algebraic calculation one can verify that Det(J(x, y)|E∗SN ) = 0. Considering Det(J(x, y)|E∗ ) as a continuous function of x∗ (using the result y ∗ = α2 (m2 +x∗ )/ν2 ), one can verify that Det(J(x, y)|E1∗ ) < 0
and hence E1∗ is always a saddle point. Further, Det(J(x, y)|E2∗ ) > 0 and local asymptotic stability
of E2∗ depends upon the sign of Tr(J(x, y)|E2∗ ) and small amplitude periodic solution bifurcates
from E2∗ through Hopf-bifurcation and the Hopf-bifurcation condition is given by
β1 x2∗ −
ν1 x2∗ y2∗
ν2 y2∗
+
= 0.
2
(m1 y2∗ + x2∗ )
m2 + x2∗
(29)
The saddle-node bifurcation curve (21) and Hopf-bifurcation curve (29) intersect at the BogdanovTakens (BT) bifurcation point which is a co-dimension two bifurcation point. The condition given
in equation (21) together with the following condition
β1 x∗SN −
ν1 x∗SN y∗SN
ν2 y∗SN
+
(m1 y∗SN + x∗SN )2 m2 + x∗SN
= 0.
(30)
are the required conditions for BT bifurcation. For the sake of brevity we proceed further without giving the proofs for the transversality conditions of co-dimension one and co-dimension two
bifurcations, but these can be proved with the help of standard bifurcation theory [39]. Now we
are in a position to describe the global dynamical behavior of the model under consideration. The
local and non-local bifurcation curves are shown in Fig. 2 where ν1 and m2 are considered as the
13
bifurcation parameters. This bifurcation diagram is a schematic diagram as the distances between
the bifurcation curves are not visible for any chosen set of parameter values. In the bifurcation
diagram, the vertical blue line divides the parametric domain into two parts, in the domain (R1 )
lying on the left of it the condition ν1 < α1 m1 is satisfied and we have unique stable interior
equilibrium point. On the right of the vertical blue line, the number of interior equilibrium points
depends upon the further parametric restrictions. The green curve is the saddle-node bifurcation
curve, in the region bounded from the left by vertical blue line and from the right by the green
curve we always find two interior equilibrium points. In the domain (R7 ), lying on the right of the
saddle-node bifurcation curve, the axial equilibrium point E01 is stable and no interior equilibrium
point exists. The Hopf-bifurcation curve (plotted in red colour) meets the saddle-node bifurcation
curve at the BT-point and the homoclinic bifurcation curve also emerges from the BT-point. On
the Hopf-bifurcation curve, we find the generalized Hopf-bifurcation (GH) point at which the first
Lyapunov number is zero. A non-local bifurcation curve, the saddle-node bifurcation curve of limit
cycle, emerges from the GH-point. The first Lyapunov number is negative for the points lying on
Hopf-bifurcation curve in between BT-point and GH-point. Two non-local bifurcation curves and
the Hopf-bifurcation curve divide the domain lying between the vertical blue curve and the saddlenode bifurcation curve into five different regions, R2 - R6 . One interior equilibrium point, namely
E1∗ is a saddle-point in all these domains. Another interior equilibrium point E2∗ is locally asymptotically stable below the Hopf-bifurcation curve and is unstable within the domain bounded by the
Hopf-bifurcation curve and the saddle-node bifurcation curve. In R3 , the stable interior equilibrium
point is surrounded by an unstable limit cycle, we find two nearby limit cycles surrounding the
unstable interior equilibrium point E2∗ in R4 . We find a stable limit cycle enclosing the unstable
spiral point E2∗ in the domain R5 and E2∗ is an unstable spiral point and there is no limit cycle for
parameter values in the domain R6 .
Finally, we present the phase portraits for parameter values lying in various domains of the
bifurcation diagram except in the domain R4 as two limit cycles are not clearly visible in the phase
portrait. To draw the phase portraits we fix the parameter values α1 = 2, β1 = 0.8, m1 = 0.2,
α2 = .5 and ν2 = .45. The chosen parameter set is hypothetical but similar values are used
in [13, 14, 29, 30] for numerical simulations of a similar type of prey-predator models. Parameter
values for ν1 and m2 for different domains are given in the caption of the Fig. 3. In all these
14
figures, attractors are marked with red colour (stable equilibrium or stable limit cycle) and unstable
equilibrium points are marked with small black circle. Manifolds of equilibrium points are marked
with green colours but the separatrix between two domains of attractions are marked with magenta
colour. For parameter values in R1 , the unique interior equilibrium point is a global attractor.
We find bistability, locally stable axial equilibrium point and one stable interior equilibrium point
E2∗ for parameter values within the domains R2 and R3 . Interestingly, bistability is observed for
parameter values in domain R5 where the stable limit cycle is one attractor. For parameter values in
R6 , both the interior equilibria are unstable and the axial equilibrium point E01 is the only attractor
for parameter values in domain R6 and R7 . Existence of two limit cycles is presented in Fig. 4 and
this figure is obtained with the help of Matcont. This bifurcation diagram shows the saddle-node
bifurcation bifurcation of limit cycle. Continuation of limit cycle in Matcont with m2 as parameter
shows the existence of two limit cycles for .027541283 ≤ m2 ≤ .027541283 when ν1 = 1.49. Two
limit cycles exists between the region bounded by two LPC curves (red coloured dotted lines in
Fig. 4) [40]. Two curves corresponding to two limit cycles are not clearly visible as their range of
existence is too narrow.
R7
R
R
2
BT point
m →
1
2
GH point
R
R
3
6
R
4
R
5
ν →
1
Figure 2: Bifurcation diagram in ν1 m2 -parameter space. The vertical blue line is the plot of
ν1 = α1 m1 , The green curve is the Saddle-node bifurcation curve, The Hopf-bifurcation curve is
plotted in red colour, The cyan blue curve emerging from the BT-point is the homoclinic bifurcation
curve and the violet colured curve is a non-local bifurcation curve emerging from the GH-point.
15
2
3
5
1.8
4.5
2.5
1.6
4
1.4
3.5
2
1.2
2.5
y →
y →
y →
3
1.5
1
0.6
1.5
0.4
1
0.5
0.2
0.5
0
0
1
0.8
2
0.5
1
1.5
x→
2
2.5
0
0
3
1.2
0.5
1
1.5
x→
2
2.5
0
0
3
3
3
2.5
2.5
2
2
0.5
1
1.5
x→
2
2.5
3
0.5
1
1.5
x→
2
2.5
3
1
y →
y →
y →
0.8
1.5
1.5
0.6
1
1
0.5
0.5
0.4
0.2
0
0
0.5
1
x→
1.5
2
2.5
0
0
0.5
1
1.5
x→
2
2.5
3
0
0
Figure 3: Phase portraits for the parameter values chosen in six different domains of the bifurcation
diagram. Upper panel: (ν1 , m2 ) = (.2, .2) ∈ R1 (left); (ν1 , m2 ) = (.8, .68) ∈ R2 (middle); (ν1 , m2 ) =
(1.35, .12) ∈ R3 (right); lower panel: (ν1 , m2 ) = (1.49, .02) ∈ R5 (left); (ν1 , m2 ) = (1.4, .1) ∈ R6
(middle); (ν1 , m2 ) = (1.4, .5) ∈ R7 (right).
3. Stochastic Model
To study the effect of environmental driving forces on the dynamics of the model (10) - (11),
now we formulate the stochastic model by introducing multiplicative noise terms into the growth
equations of prey and predator population. It is worthy to mention here that the formulation of
stochastic model based upon the existing deterministic model is not unique, so relevant approaches
can be found in [41–48] and references cited there in. To formulate the stochastic model, in this
paper we will perturb intrinsic growth rates of both population by white noise terms [44]. This
approach is based upon the assumption that in reality all parameters involved with the deterministic
16
2
x
1.5
1
LPC
0.5
LPC
0
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
m2
Figure 4: Bifurcation of limit cycle for ν1 = 1.49 and a range of values of m2 , other parameter
values are mentioned in the text.
model exhibit random variations to a greater or lesser extent [18]. Introducing white noise terms
into the intrinsic growth rate parameters for two population we get the following extended model
system,
ν1 y(t)
dx(t)
= x(t) α1 − β1 x(t) −
+ σ1 x(t)ξ1 (t),
dt
m1 y(t) + x(t)
dy(t)
ν2 y(t)
= y(t) α2 −
+ σ2 y(t)ξ2(t),
dt
m2 + x(t)
(31)
(32)
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 the Kronecker delta and δ(.) is the ‘Dirac-δ’ function [18, 41, 42, 47]. Two new parameters σ1 and σ2 are the intensities of environmental driving forces. The noise terms introduced in
the stochastic model are multiplicative noises. Presence of these multiplicative noise terms do not
hamper the positivity of solutions as here we prove the global existence of non-negative solution
independent to the intensity of noises. We can write the stochastic model system (31) - (32) into
the standard form of stochastic differential equations as follows,
ν1 y(t)
dx(t) = x(t) α1 − β1 x(t) −
dt + σ1 x(t)dB1 (t)
m1 y(t) + x(t)
ν2 y(t)
dy(t) = y(t) α2 −
dt + σ2 y(t)dB2(t)
m2 + x(t)
17
(33)
(34)
where B1 (t) and B2 (t) are two standard one-dimensional independent Wiener processes defined over
the complete probability space (Ω, F , P ) having filtration {Ft }t≥0 and satisfy the usual conditions
(like both are increasing and right continuous while F0 contains all P -null sets) [49]. The relations
between the white noise terms and Wiener processes are defined by dBr = ξr (t)dt, r = 1, 2 [50].
The solution of (33) - (34) subjected to the positive initial condition is an Ito process [47, 51–54].
Now we can start our analysis of the stochastic model system and we start with the proof for
existence and uniqueness of solutions.
3.1 Existence and uniqueness
Since x(t) and y(t) denote the population densities of the prey and predator at time t, we
are only interested with the positive solutions. Our first task will be to prove the positivity of
solutions for the model (33) - (34) starting with positive initial conditions. Moreover, a stochastic
differential equation will have a unique global (i.e. no explosion in a finite time) solution for any
given initial condition if the coefficients of the equation satisfy the linear growth condition and local
Lipschitz condition [55–57]. Now we can clearly see that the coefficients of system (33) - (34) do not
satisfy both the linear growth condition and local Lipschitz condition. In this section, changing the
variables and using the comparison theorem of stochastic equations [58], we will prove that there is
a unique positive solution with positive initial value of the system (33) - (34).
Before going to prove the main theorem, first of all we prove the local existence of the positive
solution of the system(33) - (34) through the following lemma.
Lemma 3.1 : For positive initial condition x0 > 0 and y0 > 0, there exists unique positive local
solution ((x(t), y(t)) of the model (33) - (34) for t ∈ [0, τe ) a.s. where τe is the explosion time [55,56].
Proof: Introducing new variables u(t) = ln x(t) and v(t) = ln y(t) in (33) - (34) and then applying
the Ito’s formula [48] we get the transformed system as follows
ν1 ev(t)
σ12
u(t)
− β1 e −
dt + σ1 dB1 (t)
du(t) =
α1 −
2
m1 ev(t) + eu(t)
σ22
ν2 ev(t)
dv(t) =
α2 −
−
dt + σ2 dB2 (t)
2
m2 + eu(t)
(35)
(36)
subjected to the initial conditions u(0) = ln x0 and v(0) = ln y0 . Clearly, the functions involved
with the system (35) - (36) satisfy the local Lipschitz criteria and hence there exists a unique
local solution (u(t), v(t)) on t ∈ [0, τe ) where τe is a finite positive number. Now it is clear that
18
x(t) = eu(t) , y(t) = ev(t) is the unique positive local solution of system (33) - (34) subjected to the
positive initial conditions.
Before proceeding further we will show that unique positive solutions are not only local solutions
rather they are global solutions. To prove this we need to show that τe = ∞ a.s.
Theorem 3.1 : The parameters α1 , α2 , ν1 , ν2 , β1 are positive real numbers. Then for any given
initial value (x0 , y0 ) ∈ Int(R2+ ), there is a unique solution X(t) ≡ (x(t), y(t)) of the given stochastic
system on t ≥ 0.
Proof : Since x(t) and y(t) are always positive, we can write following result from (33)
dx(t) ≤ x(t)(α1 − β1 x(t))dt + σ1 x(t)dB1 (t).
(37)
Consider the following stochastic differential equation
dφ1 (t) = φ1 (t)(α1 − β1 φ1 (t))dt + σ1 φ1 (t)dB1 (t)
(38)
with positive initial condition φ1 (0) = x0 . The unique solution of (38) is given by [59, 60]
2
σ1
φ1 (t) =
1
x0
e(α1 − 2 )t+σ1 B1 (t)
.
Z t
σ2
(α1 − 21 )s+σ1 B1 (s)
+ β1
e
ds
(39)
0
Now using the comparison theorem of stochastic differential equations, we get x(t) ≤ φ1 (t), for
all t ∈ [0, τe ), a.s.
On the other hand, from (34) we can write
ν2 y(t)
dy(t) ≤ y(t) α2 −
dt + σ2 y(t)dB2(t).
m2 + φ1 (t)
(40)
Proceeding in a similar fashion as above, we get y(t) ≤ ψ1 (t), t ∈ [0, τe ) a.s. where
2
σ2
ψ1 (t) =
1
y0
+ ν2
Z
0
t
e(α2 − 2 )t+σ2 B2 (t)
.
σ2
1
(α2 − 22 )s+σ2 B2 (s)
e
ds
m2 + φ1 (s)
Using positivity of x(t), we get following inequality from (34)
ν2
dy(t) ≥ y(t) α2 −
y(t) dt + σ2 y(t)dB2(t).
m2
19
(41)
(42)
Clearly
2
σ2
exp(α2 − 2 )t+σ2 B2 (t)
,
Z t
σ2
(α2 − 22 )s+σ2 B2 (s)
ν2
+ m2
e
ds
ψ2 (t) =
1
y0
(43)
0
is the unique solution to the initial value problem
ν2
dψ2 (t) = ψ2 (t) α2 −
ψ2 (t) dt + σ2 ψ2 (t)dB2 (t), ψ2 (0) = y0
m2
(44)
and y(t) ≥ ψ2 (t), for all t ∈ [0, τe ), a.s.
From (33) we can write
dx(t) ≥ x(t)
ν1
α1 −
m1
− β1 x(t) dt + σ1 x(t)dB1 (t),
(45)
and then using similar arguments as above, we obtain
x(t) ≥
1
x0
e
Z
+ β1
0
ν
α1 − m1
t
1
e
−
2
σ1
2
ν
α1 − m1
1
t+σ1 B1 (t)
−
2
σ1
2
:= φ2 (t),
s+σ1 B1 (s)
(46)
ds
for all t ∈ [0, τe ), a.s.
The results obtained so far can be summarized as
φ2 (t) ≤ x(t) ≤ φ1 (t), ψ2 (t) ≤ y(t) ≤ ψ1 (t),
(47)
for all t ∈ [0, τe ) a.s. From the expressions for φ1 (t), φ2 (t), ψ1 (t), ψ2 (t), it is clear that all these
functions are well defined for all t ∈ [0, τe ) a.s and arbitrarily large magnitude of τe , which in turn
imply τe = ∞. Hence the unique positive solutions are global solutions.
3.2 Stochastic Persistence
In this subsection we are intended to prove the stochastic persistence of the model system (33)
- (34) under certain parametric restriction(s) that we have to derive. Stochastic persistence means,
if we start from a positive initial condition, that is, from an interior point of the first quadrant
then solutions trajectories of the stochastic model will always remain within the interior of the first
quadrant and remain bounded at all future time. Here we prove the strong persistence result and
20
for that we recall the definition of strong persistence, detailed discussion is available at [61, 62].
Definition : The population x(t) is said to be strongly persistent in the mean if hx(t)i∗ > 0 where
Rt
Rt
hx(t)i∗ := lim inf t→+∞ 1t 0 x(s) ds and hx(t)i∗ is defined by hx(t)i∗ := lim supt→+∞ 1t 0 x(s) ds.
above definition, hx(t)i stands for the time average of x(t) and is defined by hx(t)i =
Z In
t
1
x(s) ds. Proof of strong persistence result for the stochastic model (33) - (34) is based upon
t 0
the following lemma [62].
Lemma 3.2 : Suppose that x(t) ∈ C[Ω × R+ , R0+ ], where R0+ = {a|a > 0, a ∈ R}.
(i) If there exist 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)
(48)
i=1
for t ≥ T , where βi ’s are constants, 1 ≤ i ≤ n, then hxi∗ ≤ µλ , a.s.
(ii) If there exist 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)
(49)
i=1
for t ≥ T , where βi ’s are constants, 1 ≤ i ≤ n, then hxi∗ ≥ µλ , a.s.
The following theorem is the strong persistence result for the stochastic model system (33) (34).
Theorem 3.2 : If α1 −
ν1
m1
>
σ12
2
and α2 >
σ22
,
2
then x(t) and y(t) are strongly persistent in mean.
Proof : Define V (x(t)) = ln(x(t)) for x(t) ∈ (0, ∞). Then using Ito’s formula, we get from (33),
ν1 y(t)
σ12
−
dt + σ1 dB1 (t))
d (ln(x(t))) = α1 − β1 x(t) −
m1 y(t) + x(t)
2
Integrating both sides from 0 to t and dividing by t, we get
h
i
x(t)
ln x(0)
σ12
σ1 B1 (t)
ν1 y(t)
=
α1 −
+
− β1 hx(t)i −
t
2
t
m1 y(t) + x(t)
2
ν1
σ
σ1 B1 (t)
≥
α1 − 1 −
+
− β1 hx(t)i
2
m1
t
21
Using the previous lemma we get from the above inequality,
σ12
ν1
α1 − 2 − m1
hxi∗ ≥
.
β1
Hence hxi∗ > 0 whenever (α1 −
σ12
)
2
−
ν1
m1
(50)
> 0. Now from equation (34), we get
ν2 y(t)
ν2 x(t)y(t)
dy(t) = y(t) α2 −
+
dt + σ2 y(t)dB2 (t)
m2
m2 (m2 + x(t))
(51)
Let V (y(t)) = ln(y(t)) for y(t) ∈ (0, ∞). Applying Ito’s formula on (51), we obtain
ν2 x(t)y(t)
σ22 ν2 y(t)
d (ln(y(t))) = α2 −
−
+
dt + σ2 dB2 (t)
2
m2
m2 (m2 + x(t))
Again integrating both sides from 0 to t and dividing by t, we obtain
h i
y(t)
ln y(0)
σ22
σ2 B2 (t)
ν2
ν2 x(t)y(t)
= α2 −
+
−
hy(t)i +
t
2
t
m2
m2 (m2 + x(t))
ln
h
y(t)
y(0)
t
i
σ22
σ2 B2 (t)
ν2
≥ α2 −
+
−
hy(t)i
2
t
m2
Applying the previous lemma, we get the following result from the above inequality
σ2
m2 α2 − 22
hyi∗ ≥
,
ν2
and the bound for hyi∗ is positive if α2 >
σ22
.
2
(52)
This completes the proof of the theorem.
Now we present some numerical simulation results to validate the analytical findings based upon
the following stochastic model,
.7y(t)
dx(t) = x(t) 2 − .8x(t) −
dt + σ1 x(t)dB1 (t)
2y(t) + x(t)
.9y(t)
dt + σ2 y(t)dB2(t)
dy(t) = y(t) 1.6 −
.3 + x(t)
(53)
(54)
for different values of forcing intensities σ1 and σ2 . We have performed the numerical simulations
for the stochastic model with the help of Milstein’s method [50], the method has strong order
of convergence equal to 1. For all numerical simulations we have taken the time step equal to
.001. Firstly we take σ1 = .1 = σ2 and both the populations show some fluctuation around
22
their deterministic steady state. The simulation result is presented in Fig-5. It is interesting to
observe that the amplitude of fluctuation for the predator population is greater than that for the
prey population although the intensity of environmental driving forces are the same for both the
populations. We have perturbed the growth rates of the prey and the predator population with
same amplitude of noise terms. The growth rate of predator is much sensitive to determine the
equilibrium level of the population density in the absence of noise terms. This claim can be verified
numerically for chosen set of parameter values. This sensitivity is reflected in the presence of
environmental driving force. When population density of the prey increases then it promotes the
growth of predators apart from their prey-independent growth rate. As a result the growth of
predator population is accelerated with a magnitude higher than the prey growth. This accelerated
growth in the predator population also increases the grazing pressure on prey and we can observe
a quick decay in prey population density which in turn results in the sharp decrease in the density
for predator population. The amplitude of fluctuation increases significantly for the higher values
of σ1 and σ2 . Numerical simulation result for σ1 = 1.2 and σ2 = .8 is presented in Fig-6. It is
clear that neither population fluctuates around their deterministic steady state value. Both the
populations exhibit large amplitude fluctuations but neither of them goes to extinction as σ1 and
σ2 satisfy the parametric restrictions required for the stochastic persistence. For chosen parameter
values in (53) - (54), we have α1 −
ν1
m1
= 1.65 and α2 = 1.6. Hence the parametric restrictions
required for stochastic persistence will be violated if we choose σ1 ≥ 1.82 or σ2 ≥ 1.79. In Fig-7 we
have presented the extinction scenario for prey population as the magnitude of σ1 is taken greater
than the threshold value.
23
6
Prey
Predator
5
population →
4
3
2
1
0
0
10
20
30
40
50
time →
60
70
80
90
100
Figure 5: Numerical simulation result for the solution of (53) - (54) for σ1 = .1, σ2 = .1 and starting
from the initial point (1,1). The solution of the corresponding deterministic model is presented with
broken curves.
12
Prey
Predator
10
population →
8
6
4
2
0
0
10
20
30
40
50
time →
60
70
80
90
100
Figure 6: Numerical simulation for the model (53) - (54) with σ1 = 1.2, σ2 = .8 and starting from
the initial point (1,1) shows high amplitude fluctuation for prey as well as predator population.
Solution of corresponding deterministic model is presented with broken curves.
4. Stochastic Stability Analysis
In section 2, we have discussed about the local asymptotic stability of the coexisting equilibrium
point and in the previous section we obtained the conditions under which the stochastic model
24
12
Prey
Predator
10
population →
8
6
4
2
0
0
10
20
30
time →
40
50
60
Figure 7: Numerical simulation of (53) - (54) for σ1 = 1.9 and σ2 = 1.2 shows that prey population
goes to extinction but predator survives.
system under consideration is strongly persistent in mean. We have observed that the stochastic
persistence depends upon the intensity of environmental fluctuation. Here we are interested to
see whether the introduction of multiplicative noise results in a loss of regularity in the dynamical
behavior of the system. One can easily see that the interior equilibrium point of the deterministic
system (10) - (11) is not an equilibrium point of the corresponding stochastic system (33) - (34)
but the trivial equilibrium point is an equilibrium point of the stochastic system and we are not
interested in stochastic stability of the trivial equilibrium point.
Rather we are interested to see the behavior the solution trajectories for the stochastic model
around the deterministic steady state value. For this purpose, we introduce small perturbation
around the deterministic steady state values and then derive the governing differential equations for
the first and second order moments of the perturbation variables. This approach will result in the
deterministic coupled ordinary differential equations for first and second order moments. Stability
of the equilibrium point for the system of differential equations for first and second order moments
will determine the amplitude of fluctuation of the population around their deterministic steady
state values in presence of environmental noise terms. This technique is used earlier in [63, 64].
To obtain the differential equations for the moments first of all we introduce perturbations around
deterministic steady states with help of the transformation x(t) = x∗ + x1 (t), y(t) = y∗ + y1 (t) where
25
|x1 (t)|, |y1 (t)| 1. Substituting this transformation in (33) - (34) we get the following linearized
version by neglecting the second and higher order terms of small quantities,
dx1 (t) = (A1 x1 + A2 y1 ) dt + σ1 (x1 (t) + x∗ )dB1 (t)
(55)
α22
x1 − α2 y1 dt + σ2 (y1 (t) + y∗ )dB2 (t)
ν2
(56)
dy1 (t) =
where
A1 = −β1 x∗ +
ν1 (x∗ )2
ν1 x∗ y∗
,
A
=
−
.
2
(m1 y∗ + x∗ )2
(m1 y∗ + x∗ )2
From (55) and (56), integrating both sides from 0 to t, taking expectation and then using the mean
zero property of Ito’s integral [48] we can write the system of ordinary differential equations for first
order moments as follows:
dE [x1 (t)]
= A1 E [x1 (t)] + A2 E [y1 (t)]
dt
(57)
dE [y1 (t)]
α2
= 2 E [x1 (t)] − α2 E [y1 (t)] .
dt
ν2
(58)
Next we calculate the differentials of x21 , y12 and x1 y1 . Using Ito’s formula and with help of (55) (56) we get the following stochastic differential equations:
dx21 (t) = 2x1 (A1 x1 + A2 y1 ) + σ12 (x1 + x∗ )2 dt + 2x1 σ1 (x1 + x∗ )dB1 (t)
dy12(t)
dx1 (t)y1 (t) =
2
α2
2
2
= 2y1
x1 − α2 y1 + σ2 (y1 + y∗ ) dt + 2y1 σ2 (y1 + y∗ )dB2 (t)
ν2
(59)
(60)
α22 2
2
x + (A1 − α2 )x1 y1 + A2 y1 dt + σ1 y1 (x1 + x∗ )dB1 (t) + σ2 x1 (y1 + y∗ )dB2 (t).(61)
ν2 1
Integrating the above equations from 0 to t and then taking mathematical expectation of both sides
with help of Fubini’s theorem [42, 65, 66] and finally differentiating with respect to t, we get the
system of differential equations for second order moments as follows:
dE [x21 (t)]
= (σ12 + 2A1 )E x21 (t) + 2A2 E [x1 (t)y1 (t)] + 2x∗ σ12 E [x1 (t)] + σ12 (x∗ )2
dt
2α2
dE [y12 (t)]
= (σ22 − 2α2 )E y12(t) + 2 E [x1 (t)y1 (t)] + 2y∗ σ22 E [y1 (t)] + σ22 (y∗)2
dt
ν2
26
(62)
(63)
α2 dE [x1 (t)y1 (t)]
= 2 E x21 (t) + (A1 − α2 )E [x1 (t)y1 (t)] + A2 E y12 (t) .
dt
ν2
(64)
Here we consider the stochastic stability of the model under consideration in terms of the stability
of first and second order moments. For this purpose, we have to find the steady-states of the first
and second order moments and find the conditions for their stability. The steady states for the first
and second order moments are solutions of the following system of equations
A1 E [x1 (t)] + A2 E [y1 (t)] = 0,
α22
E [x1 (t)] − α2 E [y1 (t)] = 0,
ν2
(σ12 + 2A1 )E x21 (t) + 2A2 E [x1 (t)y1 (t)] + 2x∗ σ12 E [x1 (t)] + σ12 (x∗ )2 = 0,
2 2α22
2
(σ2 − 2α2 )E y1 (t) +
E [x1 (t)y1 (t)] + 2y∗ σ22 E [y1 (t)] + σ22 (y∗ )2 = 0,
ν2
α22 2 E x1 (t) + (A1 − α2 )E [x1 (t)y1 (t)] + A2 E y12 (t) = 0,
ν2
(65)
(66)
(67)
(68)
(69)
which are denoted by E[x1 ]∗ , E[y1 ]∗ , E[x21 ]∗ , E[y12 ]∗ and E[x1 y1 ]∗ . Stability of these steady-states
depends upon the nature of the eigenvalues of the matrix M given by

A1
A2
0
0
0
 α22
−α2
0
0
0
 ν2
 2x σ 2
2
0
σ1 + 2A1
0
2A2
M =  ∗ 1

2α22
2
2
2y∗ σ2
0
σ2 − 2α2
 0
ν2
0
0
α22
ν2
A2
A1 − α2




.


(70)
Applying the Routh-Hurwitz criteria one can find the conditions for the negative real parts of
all eigenvalues of the matrix M but the obtained conditions cannot be put into explicit conditions.
Avoiding the algebraic complication we will validate the result with the help of a numerical example.
For this purpose, first we choose the parameter values α1 = 2, β1 = .8, ν1 = .7, m1 = 2, α2 = 1.6,
ν2 = .9, m2 = .3 and we fix the values of σ1 and σ2 at .1. For this choice x∗ = 2.149, y∗ = 4.354
and we find the steady states for the second order moments as E[x21 ]∗ = .01374, E[y12 ]∗ = .0796
and E[x1 y1 ]∗ = .0113, steady states of first order moments are identically equal to zero. The
stability of the first and second order moments follows from eigenvalues of the matrix M which are
λ1 = −3.2536, λ2,3 = −3.2588 ± .555i, λ4,5 = −1.6319 ± .2775i. Hence the stochastic model is stable
in terms of first and second order moments. In Fig. 8 we have presented the time evolution of the
first and second order moments for chosen parameter values.
27
0.2
E[x1]
E[y ]
1
E[x2]
0.15
1
E[y2]
1
E[x y ]
1 1
moments →
0.1
0.05
0
−0.05
−0.1
0
1
2
3
4
5
time →
6
7
8
9
10
Figure 8: Time evolution of first and second order moments obtained by solving system of differential
equations (60) - (64) numerically for the parametric values as mentioned in text. All moments
converge to their steady state values.
5. Conclusion
In this paper, first we have investigated the global dynamics of a ratio-dependent modified
Holling-Tanner type prey-predator model. Then we have studied the dynamics of the same model
in the presence of environmental driving forces which affect the growth rates of two species. There
are several localized prey and predator populations which follow the Holling-Tanner type interaction,
see [31] and references cited therein for a detailed discussion. Again, insect pest-spider interaction
follows the ratio-dependent Holling-Tanner model [7, 68]. So here we propose this ratio-dependent
Holling-Tanner model with additional food source for the predators, which may be employed for
the control/eradication of the pest. Here we have analyzed the model under certain parametric
restrictions for which the deterministic model system is permanent. Here we have established
the global existence of positive solution for the concerned model. This ensures that the solution
trajectory exists uniquely once it originated from an interior point of the first quadrant. We have
identified the parametric restriction which regulates the dynamics of the system. The dynamics
of the interaction between the prey and their predator is independent of the growth rate of the
predator population and the abundance of additional food available for the predators when the rate
of predation is low (ν1 < α1 m1 ). In this situation, both the species are capable to maintain their
28
equilibrium levels irrespective of the initial population densities and the coupling term (functional
response) is a ratio-dependent function. In contrary, the coexistence (stable or oscillatory) of both
the species or extinction of prey and/or predators are very much sensitive to the initial densities
when the predation rate is high. Interestingly, the predator species can survive in the absence of
its most favorite food as they can switch to the alternative food source and the availability of this
alternative food source is constant. Presence of alternative food for predators and ratio-dependent
functional response leads to a bistable scenario where coexistence of both the species solely depends
upon the initial densities of the prey and the predator populations.
To understand the effect of environmental driving forces we have formulated a stochastic model
system by perturbing the intrinsic growth rates of prey and predator population with the help of
white noise terms. The existence and uniqueness of the global solution for the stochastic model
is established and we obtained the critical magnitudes for the environmental forcing terms for the
stochastic persistence of the prey and predator species. Numerically we have shown that either
population may goes to extinction if the magnitude of environmental fluctuations are very large or
greater than the threshold value. During numerical simulations the positiveness of the parameters
are checked at every time step and for the chosen intensity of environmental fluctuation the values
of αj + σj ξj (t), j = 1, 2 remain positive all the time. In order to maintain this positivity, the choice
of the time stepping is very much important. Finally we have discussed the stochastic stability
in terms of second order moments. Numerical simulations reveal that the solution trajectories for
the stochastic model may move around the deterministic steady state value whenever the intensity
of environmental forces are significantly small but large amplitude fluctuation can be observed for
strong environmental driving forces. Sometimes it results in the extinction of either population.
A similar type of model was investigated by Maiti and Pathak [69] with the help of Langevin
equations. But the intensity of noise terms did not appear in their stochastic stability results.
Variability in the growth rates of two species is responsible for the coexistence of two species or
extinction of one or both the species but the intensity of interaction between them is also a decisive
factor. Under the parametric restriction ν1 < α1 m1 , the persistence or extinction of two species are
solely dependent upon the intensity of environmental driving forces. But when the predation rate
is high, the coexistence of two species also depends upon the initial density of the prey and the
predator population. Sudden decay in prey growth rate may be compensated if the initial density
29
of the predators is low and the amount of alternative food available for the predators are reasonably
high. Validation of the considered model with some realistic dataset will further strengthen our
investigation and also suggest some new scenario which will be worth to investigate. In the near
future we will consider the effect of environmental driving forces on other parameters involved with
the deterministic model system. We hope that the presence of more than two noise terms may lead
to further interesting dynamic behavior compared to the results reported here.
Acknowledgement : First author [P.S.M.] wishes to thank the C. S. I. R., India for their senior
research fellowship. We are very much thankful to the anonymous reviewers for their careful reading
of the manuscript and detailed suggestions to upgrade the quality of the manuscript significantly.
We highly appreciate Dr. A. Morozov (University of Leicester, UK) for fruitful discussion and
suggestions.
References
[1] May R.M. Stability and Complexity in Model Ecosystems. Princeton University Press: New
Jersey; 2001.
[2] Renshaw E. Modelling Biological Populations in Space and Time. Cambridge University Press:
Cambridge; 1991.
[3] Turchin P. Complex Population Dynamics. Princeton University Press: Princeton, 2003.
[4] Malchow H., Petrovskii S.V., Venturino E. Spatiotemporal Pattern in Ecology and Epidemiology. New York, NY: Chapman and Hall; 2008.
[5] Leslie P.H., Gower J. C. The properties of a stochastic model for the predator–prey type of
interaction between two species. Biometrika 1960; 47: 219 - 234.
[6] Pielou E.C. An Introduction to Mathematical Ecology. Wiley - Interscience. New York; 1969.
[7] Aziz-Alaoui M. A., Daher-Okiye M. Boundedness and global stability for a predator-prey model
with modified LeslieGower and Holling-type II schemes, Applied Mathematics Letters 2003;
16: 1069 - 1075.
30
[8] Korobeinikov A. A Lyapunov function for Leslie Gower predator–prey model, Applied Mathematics Letters 2001; 14: 697 – 699.
[9] Hsu S.B., Hwang T.W. Global stability for a class of predator-prey systems, SIAM Journal on
Applied Mathematics. 1995; 55: 763 - 783.
[10] Saez E., Gonzalez-Olivares E. Dynamics of predator-prey model, SIAM Journal on Applied
Mathematics. 1999; 59: 1867 - 1878.
[11] Gasull A., Kooij R.E., Torregrosa J., Limit cycles in the Holling-Tanner model. Publicacions
Matemtiques 1997; 41: 149 - 167.
[12] Leslie P.H. Some further notes on the use of matrices in population mathematics, Biometrika
1948; 35: 213 - 245.
[13] Nindjin A.F., Aziz-Alaoui M.A., Cadivel M. Analysis of a predator-prey model with modified
Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Analysis: Real World
Applications 2006; 7: 1104 - 1118.
[14] Aziz-Alaoui M. A. Study of Leslie-Gower-type tritrophic population. Chaos Solitons Fractals
2002; 14: 1275 - 1293.
[15] Liang Z., Pan H. Qualitative analysis of a ratio-dependent Holling-Tanner model, Journal of
Mathematical Analysis and Applications. 2007; 34: 954 - 964.
[16] Jost C., Arino O., Arditi R. About deterministic extinction in ratio-dependent predator-prey
model. Bulletin of Mathematical Biology. 1999; 61: 19 - 32.
[17] Banerjee M. Spatial pattern formation in ratio-dependent model: higher-order stability analysis. Mathematical Medicine and Biology. 2011; (in press) .
[18] Bandyopadhyay M., Chattopadhyay J. Ratio-dependent predator–prey model: effect of environmental fluctuation and stability. Nonlinearity. 2005; 18: 913 - 936.
[19] Arditi R., Ginzburg L. R. Coupling in predatorprey dynamics: ratio-dependence. Journal of
Theoritical Biology 1989; 139: 311 - 326.
31
[20] Arditi R., Saiah H. Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology. 1992; 73: 1544 - 1551.
[21] Arditi R., Ginzburg L.R., Akcakaya H.R. Variation in plankton densities among lakes: A case
for ratio-dependent models. American Naturalist 1991; 138: 1287 - 1296.
[22] Gutierrez A.P. The physiological basis of ratio-dependent predator-prey theory: A metabolic
pool model of Nicholson’s blowflies as an example. Ecology. 1992; 73: 1552 - 1563.
[23] Xiao D., Ruan S. Global dynamics of a ratio-dependent predator-prey systems. Journal of
Mathematical Biology. 2001; 3: 268 – 290.
[24] Arditi R., Berryman A. The biological control paradox. Trends Ecological Evolution. 1991; 6:
32 - 43.
[25] Freedman H. I., Mathsen A. M. Persistence in predator-prey systems with ratio-dependent
predator influence. Bulletin of Mathematical Biology. 1993; 55: 817 - 827.
[26] Hsu S. B., Hwang T. W., Kuang Y., Global analysis of the Michaelis-Menten type ratiodependent predator-prey system, Journal of Mathematical Biology. 2001; 42: 489 - 506.
[27] Upadhyay R.K., Rai V. Crisis-limited chaotic dynamics in ecological systems. Chaos Solitons
Fractals 2001; 12: 205 - 218.
[28] Upadhyay R.K., Iyengar S.R.K. Effect of seasonality on the dynamics of 2 and 3 species preypredator system. Nonlinear Analysis: Real World Applications 2005; 6: 509 - 530.
[29] Ji C., Jiang D., Shi N. A note on a predator-prey model with modified Leslie-Gower and
Holling-type II schemes with stochastic perturbation. Journal of Mathematical Analysis and
Applications. 2011; 377: 435 – 440.
[30] Ji C., Jiang D., Shi N. Analysis of a predator-prey model with modified Leslie-Gower and
Holling-type II schemes with stochastic perturbation. Journal of Mathematical Analysis and
Applications. 2009; 359: 482 - 498.
[31] Turchin P., Batzli, G. O. Availability of Food and the Population Dynamics of Arvicoline
Rodents. Ecology. 2001; 82: 1521 – 1534.
32
[32] Birkhoff G., Rota G.C. Ordinary Differential Equations, Wiley. Boston; 1982.
[33] Coddington E. A., Levinson N. Theory of Ordinary Differential Equations. McGraw Hill; 1984.
[34] Abbas S., Banerjee M., Hungerbuhler N. Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model. Journal of Mathematical Analysis and Applications.
2010; 367: 249 – 259.
[35] Chen F.D. On a nonlinear non-autonomous predator-prey model with diffusion and distributed
delay. Journal of Computational and Applied Mathematics. 2005; 180: 33 - 49.
[36] Kuang Y. Delay Differential Equations, with Applications in Population Dynamics. Academic
Press: New York; 1993.
[37] Thieme H. R. Mathematics in Population Biology. Princeton University Press: New Jersey;
2003.
[38] Perko, L. Differential Equations and Dynamical Systems. Springer, New York; 2001.
[39] Kuznetsov, Yu.A. Elements of Applied Bifurcation Theory, Appl. Math. Sci., Springer-Verlag,
New York, 2004.
[40] Dhooge, A., Govaerts, W., Kuznetsov, Yu. A., Mestrom, W., Riet, A. M., Sautois, B.
MATCONT and CL MATCONT: Continuation toolboxes in matlab, http://www.ricam.oeaw.
ac.at/people/page/jameslu/Teaching/LectureNotes/II Lecture2/manual.pdf, 2006.
[41] Gardiner C.W. Handbook of Stochastic Methods. Springer-Verlag: New York; 1983.
[42] Gard T.C. Introduction to Stochastic Differential Equations. Marcel Dekker: New York; 1987.
[43] Allen L.J.S. An Introduction to Stochastic Processes with Applications to Biology. Pearson
Education Inc: New Jersey; 2003.
[44] Horsthemke W.; Lefever R. Noise Induced Transitions. Springer-Verlag: Berlin; 1984.
[45] Bandyopadhyay M., Chakrabarti C.G. Deterministic and stochastic analysis of a nonlinear
preypredator system. Journal of Biological Systems 2003; 11: 161 - 172 .
33
[46] Samanta G.P. Influence of environmental noises on the Gomatam model of interacting species.
Ecological Modelling. 1996; 91: 283 - 291.
[47] Carletti M. On the stability properties of a stochastic model for phagebacteria interaction in
open marine environment. Mathematical Biosciences 2002; 175: 117 - 131.
[48] Allen E. Modeling With Ito Stochastic Differential Equations. Dordrecht: The Netherlands;
2007.
[49] Mao X., Marion G., Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Process and Applications. 2002; 97: 95 – 110.
[50] Higham D.J. An algorithmic introduction to numerical simulation of stochastic differential
equations. SIAM Review. 2001; 43: 525 – 546.
[51] Abbas S., Bandyopadhyay M. Stochastically perturbed allelopathic phytplankton model. Electronic Journal of Differential Equations 2010; 98: 1 - 15.
[52] Kolmanovskii V., Shaikhet L. Some peculiarities of the general method of Lyapunov functionals
construction. Applied Mathematics Letters. 2002; 15: 355 – 360.
[53] Kolmanovskii V., Shaikhet L. Construction of Lyapunov functionals for stochastic hereditary
systems: a survey of some recent results. Mathematical and Computer Modelling. 2002; 36:
691 – 716.
[54] Kolmanovskii V. B., Shaikhet L.E. Control of systems with aftereffect, Translations of mathematical monographs. American Mathematical Society: Providence, RI, 1996.
[55] Arnold L. Stochastic Differential Equations: Theory and Applications. Wiley: New York; 1972.
[56] Friedman A. Stochastic Differential Equations and their Applications. Academic Press: New
York; 1976.
[57] Mao X. Stochastic Differential Equations and Applications. Horwood: New York; 1997.
[58] Ikeda N., Watanabe S. Stochastic Differential Equations and Diffusion Processes. NorthHolland: Amsterdam; 1981.
34
[59] Ji C., Jiang D., Shi N. A note on nonautonomous logistic equation with random perturbation.
Journal of Mathematical Analysis and Applications. 2005; 303: 164 – 172.
[60] Ji C., Jiang D., Liu H., Yang Q. Existence, uniqueness and ergodicity of positive solution of
mutualism system with stochastic perturbation. Mathematical Problems in Engineering. 2010;
Article ID 684926.
[61] Chen L., Chen J. Nonlinear Biological Dynamical System. Science Press: Beijing; 1993.
[62] Liu M., Wang K., Wu Q. Survival analysis of stochastic competitive models in a polluted
envioronment and stochastic competitive exclusion principle. Bulletin of Mathematical Biology.
2011; 73: 1969 – 2012.
[63] Mackey M. C., Nechaeva I. G. Solution moment stability in stochastic differential delay equations. Physical Review E. 1995; 52: 3366 - 3376.
[64] Mackey M.C., Lei j. Stochastic differential delay equation, moment stability, and application
to hematopoietic stem cell regulation system, SIAM Journal of Applied Mathematics. 2007;
67: 387 – 407.
[65] Hutson V., Pym J. S. Applications of Functional Analysis and Operator Theory. Academic
Press: London; 1980.
[66] Kolmogorov A.N., Fomin S.V. Introductory Real Analysis. Dover Publications: New York;
1970.
[67] Jiang D., Shi N., Li X. Global stability and stochastic permanence of a non-autonomous logistic
equation with random perturbation. Journal of Mathematical Analysis and Applications. 2008;
340: 588 - 597.
[68] Guo H., Song X. An impulsive predatorprey system with modified LeslieGower and Holling
type II schemes. Chaos, Solitons and Fractals 2008; 36: 13201331.
[69] Maiti A., Pathak S. A modified Holling - Tanner model in stochastic environment. Nonlinear
Analysis Modeling and Control. 2009; 14: 51 – 71.
35