Fuzzy Delay Differential Equation in Predator-Prey Interaction: Analysis on Stability of Steady State

Abstract

In this paper, a fuzzy delay predator-prey (FDPP) system is proposed by adopting fuzzy parameter in a delay predator-prey (DPP) system. The steady state and linear stability of FDPP system are determined and analyzed. Here, we show that the trivial steady state is unstable for all value of delays. Mean while the semi trivial steady state is locally asymptotically stable for all values of delays under certain conditions. We prove that the steady state are fuzzy numbers. Several examples are considered to show the results.

Full text

Fuzzy Delay Differential Equation in Predator-Prey
Interaction: Analysis on Stability of Steady State
Normah Maan, Khabat Barzinji, and Nor’aini Aris.
Abstract—In this paper, a fuzzy delay predator-prey (FDPP)
system is proposed by adopting fuzzy parameter in a delay
predator-prey (DPP) system. The steady state and linear stabil-
ity of FDPP system are determined and analyzed. Here, we show
that the trivial steady state is unstable for all value of delays.
Mean while the semi trivial steady state is locally asymptotically
stable for all values of delays under certain conditions. We prove
that the steady state are fuzzy numbers. Several examples are
considered to show the results.
Index Terms—Delay predator-prey (DPP), Steady states,
Stability, Fuzzy delay predator-prey(FDPP).
I. INTRODUCTION
I
N real world, the study of population dynamics including
(stable, unstable, and oscillatory behavior) has become
very important since Volterra and Lotka proposed the sem-
inal models of predator-prey models in 1920. Predator-prey
models represent the basis of many models used today
in the analysis of population dynamics and is one of the
most popular in mathematical ecology. And the dynamics
properties of the predator-prey models which have significant
biological background has been paid a great attention. Some
studies in the area of predator-prey interaction, that treat
population can be extended by including time delay. The time
delay is included into population dynamics when the rate of
changes of population is not only a function of the present
population but also depends on the pervious population.
In 2012, Changjin and Peiluan [1] explained the stability,
the local Hopf bifurcation for the delay predator-prey model
with two delays. In 2008, Toaha et.al [2] showed a determin-
istic and continuous model for predator-prey with time delay
and constant rates of harvesting and studied the combined
effects of harvesting and time delay on the dynamics of
predator-prey model.
Although, the concepts of the steady states refer to the
absence of changes in a system, In some cases, studying the
stability of the steady state solutions become an important
subject since, by examining what happens in a steady state,
we can better understand the behavior of a system.
Forde et.al [3] had studied the stability analysis of
the steady states of delay predator-prey interaction. They
also considered the possibility of existence of the periodic
solutions.
In our real life, we have learned to accept that we are
actually dealing with uncertainty. Scientists also accepted the
fact that uncertainty is very important study in most applica-
tions. Modeling the real life problems in such cases, usually
Manuscript received March 8, 2013; revised April 12, 2013. This work
was supported in part by Fundamental Research Grant Scheme FRGS 4F127
and Research University Grants RUG 07J77
N. Maan is with the Department of Mathematical Sciences, Uni-
versiti Teknologi Malaysia, Johor Bahru, Malaysia. e-mail: (normah-
maan@utm.my).
K. Barzinji and N. Aris are with Universiti Teknologi Malysia.
involves vagueness or uncertainty in some of the parameters.
The concept of fuzzy set and system was introduced by
Zadeh [6] and its development has been growing rapidly
to various situation of theory and application including the
theory of differential equations with uncertainty. The later
is known as fuzzy differential equation. It has been used to
model a dynamical systems under possibility uncertainty [5].
In this paper the fuzzy approach is used to model an uncer-
tainty in dynamical system which then can be represented as
fuzzy delay differential equations. Specifically, the discussion
on the theory and analysis of delay predator-prey differential
equations with uncertainty parameters is considered.
The organization of this paper is as follows. In Sec-
tion II, the basic definitions regarding the fuzzy number,
steady states and characteristic equation are briefly presented.
In Section III delay predator-prey system is introduced,
followed by the formulation of fuzzy delay predator-prey
(FDPP) system. The analysis, of the steady state and linear
stability are also given. And Section IV presents some
numerical examples, finally the conclusion of the finding is
given in Section V .
II. PRELIMINARIES
Definition 1 [6] A fuzzy number is a function such as u :
R → [0, 1] satisfying the following properties:
1) u is normal, i.e ∃ x
0
∈ R with u(x
0
) = 1.
2) u is a convex fuzzy set i.e u(λx + (1 − λ)y) ≥
min{u(x), u(y)}∀x, y ∈ R, λ ∈ [0, 1].
3) u is upper semi-continuous on R.
4) {x ∈ R : u(x) > 0} is compact where
¯
A denotes the
closure of A.
Definition 2 [5]
An α− cut, u
α
, is a crisp set which contains all the
elements of the universal set X that have a membership
function at least to the degree of α and can be expressed
as u
α
= {x ∈ X : µ
u
(x) ≥ α}
Definition 3 [4]
A fuzzy number u is completely determined by any pair
u = (u, u) of functions u(α), u(α) : [0, 1] → R satisfying
the three conditions:
1) u(α), u(α) is a bounded, monotonic, (nondecreas-
ing,nonincreasing) left- continuous function for all
α ∈ (0, 1] and right-continuous for α = 0.
2) For all α ∈ (0, 1] we have: u(α) ≤ u(α) .
For every u = (u, u), v = (v, v) and k > 0,
(u + v)(α) = u(α) + v(α)
(u + v)(α) = u(α) + v(α)
(ku)(α) = ku(α), (ku)(α)) = ku(α)

Fuzzy sets is a mapping from a universal set into [0, 1].
Conversely, every function µ : X −→ [0, 1] can be
represented as a fuzzy set ( [6]). We can define a set
F
1
= {x ∈ ℜx, is about a
2
} with triangular membership
function as below
Definition 4 [6]
µ
F
1
(x) =







x−a
1
a
2
−a
1
, x ∈ [a
1
, a
2
)
1 x = a
2
−x+a
3
a
3
−a
2
x ∈ (a
2
, a
3
]
0 otherwise
So the Fuzzy set F can be written as any ordinary function
F = {(x, µ
F
(x)) : x ∈ X}.
Consider the linear fuzzy delay system as follows:
˙
x
α
(t) =A
α
x
α
(t) + B
α
x
α
(t − τ )
˙
x
α
(t) =A
α
x
α
(t) + B
α
x
α
(t − τ ) 0 ≤ α ≤ 1
x
α
(t) =x
α0
t ∈ [t
0
− τ, t
0
]
x
α
(t) =x
α0
(1)
Suppose (a
ij
)
α
= [(a
ij
)
−
α
, (a
ij
)
+
α
], A
α
= [A
−
α
, A
+
α
] where
A
−
α
= [(a
ij
)
−
α
]
n×n
A
+
α
= [(a
ij
)
+
α
]
n×n
and (b
ij
)
α
=
[(b
ij
)
−
α
, (b
ij
)
+
α
], B
α
= [B
−
α
, B
+
α
] where B
−
α
= [(b
ij
)
−
α
]
n×n
,
B
+
α
= [(b
ij
)
+
α
]
n×n
. Then we introduce the following
definition :
Definition 5 Let A(µ, α) = [a
ij
(µ, α)]
n×n
= (1−µ)A
α
−
+
µA
α
+
, B(µ, α) = [b
ij
(µ, α)]
n×n
= (1 − µ)B
α
−
+ µB
α
+
,
for µ ∈ [0, 1]. The solution of (1) is (x
α
(t), x
α
(t)), if
(x
α
(t), x
α
(t)) is also a solution of the problem below:
˙
x
α
(t) =
1
∪
µ=0
C(µ, α)x
α
(t) +
1
∪
µ=0
D(µ, α)x
α
(t − τ ),
˙
x
α
(t) =
1
∪
µ=0
C(µ, α)x
α
(t) +
1
∪
µ=0
D(µ, α)x
α
(t − τ )
x
α
(t) = x
α0
t ∈ [t
0
− τ, t
0
], 0 ≤ α ≤ 1
x
α
(t) = x
α0
(2)
The elements of the matrices C and D are determined from
of A(µ, α) and B(µ, α) as follws:
c
ij
=
{
ea
ij
(µ, α), a
ij
≥ 0
ga
ij
(µ, α), a
ij
< 0
and
d
ij
=
{
eb
ij
(µ, α), b
ij
≥ 0
gb
ij
(µ, α), b
ij
< 0
where e is the identity operation and g corresponds to
negative value in ℜ and ∀z, w ∈ ℜ,
e : (z, w ) → (z, w),
g : (z, w) → (w, z).
III. DELAY PREDATOR-PREY SYSTEM
Consider the delay predator-prey system (DPP) of equa-
tions as follows:
dx(t)
dt
= x(1 − x) − yp(x)
dy(t)
dt
= be
−d
j
τ
y(t − τ)p(x(t − τ)) − dy
(3)
where x is prey population, y is a predator population, d is a
death rate of predator, p(x) is a predator functional response
to prey and τ is time necessary to change prey biomass into
predator biomass.
A. Fuzzy Delay Predator-Prey System
We propose a new model of system (3) by, first let p(x) =
cx which is the standard mass action or linear response. Then
we fuzzify the linear part of the system (3) by symmetric
triangular fuzzy number and let x(t), y(t) are non negative
fuzzy functions.
Let
e
1 = (1 − (1 − α)σ
1
, 1 + (1 − α)σ
1
)
e
d = (d − (1 − α)σ
2
, d + (1 − α)σ
2
) where 0 ≤ α ≤ 1.
By using Definition 5 system (3) can be written as follows:




˙x
α
(t)
˙
x
α
(t)
˙y
α
(t)
˙
y
α
(t)




=




a
1
0 0 0
0 a
1
0 0
0 0 0 −a
2
0 0 −a
2
0








x
α
x
α
y
α
y
α




+




−x
2
α
(t) − cx
α
(t)y
α
(t)
−x
2
α
(t) − cx
α
(t)y
α
(t)
cbe
−d
j
τ
x
α
(t − τ )y
α
(t − τ )
cbe
−d
j
τ
x
α
(t − τ )y
α
(t − τ )




, (4)
where
a
1
= (1 − µ)(1 − (1 − α)σ
1
) + µ(1 + (1 − α)σ
1
),
a
2
= (1 − µ)(d − (1 − α)σ
2
) + µ(d + (1 − α)σ
2
),
0 ≤ µ ≤ 1.
Then (4) is known as fuzzy delay predator-prey (FDPP)
system.
B. Steady States
To find the steady states of the system (4), we assume
that the constant (x
∗
, x
∗
, y
∗
, y
∗
)
α
, is a solution and we will
determine the values of these constant. The equations for
determining steady states are
x
α
(a
1
− x
α
− cy
α
) = 0
x
α
(a
1
− x
α
− cy
α
) = 0
− a
2
y
α
+ cbe
−d
j
τ
x
α
y
α
= 0
− a
2
y
α
+ cbe
−d
j
τ
x
α
y
α
= 0.
(5)
If x
α
= 0 and x
α
= 0, then the first and the second
equations of (5) are satisfied, from third and the fourth
equations we obtain (0, 0, 0, 0)
α
as trivial steady state.
If we consider y
α
= y
α
= 0, then the third and fourth
equations of (5) are satisfied, and the first and second
equations gives x
α
= a
1
and x
α
= a
1
, where
a
1
= (1 − (1 − α)σ
1
) and a
1
= (1 + (1 − α)σ
1
).
If y
α
and y
α
are not equal zero then the steady state
equations are:
a
1
− x
α
− cy
α
= 0
a
1
− x
α
− cy
α
= 0.
(6)
So, if the equation (6) are satisfied, then the system (4)
has a nontrivial steady state(x
∗
, x
∗
, y
∗
, y
∗
)
α
. Thus, the
system (4) has three steady state solutions such that ;

(0, 0, 0, 0)
α
, (a
1
, a
1
, 0, 0)
α
and the nontrivial steady state
(x
∗
, x
∗
, y
∗
, y
∗
)
α
.
Theorem 1 Consider the DPP system ( 3), if the coefficients
of linear part of x and y are symmetric triangular fuzzy
numbers then the trivial steady state (0, 0, 0, 0)
α
is a fuzzy
number and the semi trivial steady state (a
1
, a
1
, 0, 0)
α
∀α ∈
[0, 1], is also fuzzy number.
Proof: The proof of Theorem 1 is trivial.
Now, we test the stability of the steady states.
C. Linear Stability
The linearization of the fuzzy system (4) about the trivial
steady state (0, 0, 0, 0)
α
is




˙x
α
(t)
˙
x
α
(t)
˙y
α
(t)
˙
y
α
(t)




=




a
1
0 0 0
0 a
1
0 0
0 0 0 −a
2
0 0 −a
2
0








x
α
x
α
y
α
y
α




+




0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0








x
αt
x
αt
y
αt
y
αt




. (7)
Where x
tα
= x
α
(t − τ) and similarly for y from linearized
model we obtain the characteristic equation
(a
1
− λ)
2
(λ
2
− a
2
2
) = 0. (8)
Clearly the linear system has eigenvalues a
1
and ±a
2
which
are two positive and one negative fuzzy numbers. Hence, the
trivial steady state is unstable for all values of τ .
We can conclude the following proposition:
Proposition 1 A trivial steady state (0, 0, 0, 0)
α
with char-
acteristic equation (8) is unstable for all values of delay.
Similarly, for the semi trivial steady state (a
1
, a
1
, 0, 0) where




˙x
α
(t)
˙
x
α
(t)
˙y
α
(t)
˙
y
α
(t)




=




a
1
− 2a
1
0 −ca
1
0
0 a
1
− 2a
1
0 −ca
1
0 0 0 −a
2
0 0 −a
2
0








x
α
x
α
y
α
y
α




+




0 0 0 0
0 0 0 0
0 0 cbe
−d
j
τ
a
1
0
0 0 0 cbe
−d
j
τ
a
1








x
αt
x
αt
y
αt
y
αt




. (9)
The characteristic equation for (9) is
λ
4
+ Aλ
3
+ Bλ
2
+ Cλ + D+
e
−(d
j
+λ)τ
(Eλ
3
+ F λ
2
+ Gλ)+
e
−2(d
j
+λ)τ
(Hλ
2
+ Iλ + J) = 0,
(10)
where
A = − 2a
1
+ 4, B = a
2
1
− 4a
1
+ 4(1 − (1 − α)
2
σ
2
1
),
C =a
2
2
, D = 2a
2
2
(1 + (1 − α)σ
1
) − a
2
2
a
1
,
E = − 2cb, F = 4a
1
cb − 8cb,
G = − 2a
2
1
cb + 8a
1
cb − 8cb(1 − (1 − α)
2
σ
2
1
),
H =c
2
b
2
(1 − (1 − α)
2
σ
2
1
),
I = − 2a
1
c
2
b
2
(1 − (1 − α)
2
σ
2
1
)+
4c
2
b
2
(1 − (1 − α)
2
σ
2
1
),
J =a
2
1
c
2
b
2
(1 − (1 − α)
2
σ
2
1
)−
4a
1
c
2
b
2
(1 − (1 − α)
2
σ
2
1
)+
4c
2
b
2
(1 − (1 − α)
2
σ
2
1
)
2
.
(11)
The steady state is stable in the absence of delay if the roots
of λ
4
+(A +E)λ
3
+(B+F +H)λ
2
+(C+G+I)λ+(D+J) =
0 have negative real parts. This occurs if and only if
(A + E) > 0, (C + G + I) > 0, (D + J) > 0
and (A + E)(B + F + H)(C + G + I) >
(C + G + I)
2
+ (A + E)
2
(D + J).
(12)
Hence, in the absence of time delay, the steady state
(a
1
, a
1
, 0, 0) is stable if and only if (12) are satisfied.
Now for increasing τ, τ ̸= 0, we first assume that the root
of the characteristic equation (10) is λ = iµ and µ > 0.
Substitute λ = iµ in (10), we obtain,
µ
4
− Aiµ
3
− Bµ
2
+ Ciµ + D + e
−d
j
τ
(
cos(µτ) −
isin(µτ)
)(
− iEµ
3
− F µ
2
+ iGµ
)
+ e
−2d
j
τ
(
cos(2µτ) −
isin(2µτ)
)(
− Hµ
2
+ iIµ + J
)
= 0.
Separating the real and imaginary parts, we get
µ
4
− Bµ
2
+ D = e
−d
j
τ
(
cos(µτ)(F µ
2
)
+ sin(µτ )(Eµ
3
− Gµ)
)
− e
−2d
j
τ
(
cos(2µτ)(−Hµ
2
+ J)+
sin(2µτ)(Iµ)
)
,
− Aµ
3
+ Cµ = e
−d
j
τ
(
cos(µτ)(Eµ
3
− Gµ)+
sin(µτ)(−F µ
2
)
)
+ e
−2d
j
τ
(
cos(2µτ)(−Iµ)+
sin(2µτ)(−Hµ
2
+ J)
)
.
Squaring and adding both sides gives the polynomial of
degree eight as follows:
(µ
4
− Bµ
2
+ D)
2
+ (−Aµ
3
+ Cµ)
2
=
(
e
−d
j
τ
(cos(µτ)(F µ
2
) + sin(µτ )(Eµ
3
+ Gµ))
− e
−2d
j
τ
(cos(2µτ)(−Hµ
2
+ J) + sin(2µτ)(Iµ))
)
2
+
(
e
−d
j
τ
(cos(µτ)(Eµ
3
− Gµ) + sin(µτ )(−F µ
2
))
− e
−2d
j
τ
(cos(2µτ)(Iµ) + sin(2µτ )(−Hµ
2
+ J))
)
2
.
(13)
As τ → ∞, the right hand side of (13)→ 0 and let γ = µ
2
the equation (13) can be written in terms of γ as follows:
S(γ) = γ
4
+ (A
2
− 2B)γ
3
+ (B
2
+ 2D − 2AC)γ
2
+ (C
2
− 2BD)γ + D
2
= 0.
(14)

This can be simplified by substituting the known values of
A, B, C and D. For the γ
3
coefficient, we have
A
2
− 2B = 4a
2
1
− 16a
1
+ 16 − 2(a
2
1
− 4a
1
+4(1 − (1 − α)
2
σ
2
1
))
= 2(a
1
− 2)
2
+ 8(1 − α)
2
σ
2
1
which is always positive.
Further, for the γ
2
and γ coefficients, we have
B
2
+ 2D − 2AC =
(
a
2
1
+ 4(1 − (1 − α)
2
σ
2
1
)
)
2
+
(
(a
1
− 2) − 2(1 − α)σ
1
)(
2a
2
2
− 8a
1
((a
1
− 2)
+ 2(1 − α)σ
1
)
)
,
(15)
C
2
− 2BD = a
4
2
+ 2a
2
2
(
(a
1
− 2) − 2(1 − α)σ
1
)
2
(
(a
1
− 2) + 2(1 − α)σ
1
)
(16)
respectively. (15) and (16) are positive coefficient if the right
hand side of (15) and (16) are greater than zero for certain
value of α. Finally, the constant term D
2
is always positive.
Therefore all the coefficients of the polynomial (14) are
positive and it has no positive real roots. In other words
iµ is not a root of the characteristic equation (10) for
increasing delay. Hence, the system (4) cannot lead to a
bifurcation. It means that the semi trivial steady state is
locally asymptotically stable for all values of delay [7]. We
conclude the following proposition:
Proposition 2 A semi trivial steady state (a
1
, a
1
, 0, 0) with
characteristic equation (10) is locally asymptotically stable
for all values of delay if and only if
• (A + E) > 0 , (C + G + I) > 0 , (D + J) > 0 and
(A + E)(B + F + H)(C + G + I) > (C + G + I)
2
+
(A + E)
2
(D + J). A, B, C and D are given by (11).
•
(
(a
2
1
+4(1−(1−α)
2
σ
2
1
))
2
+((a
1
−2)−2(1−α)σ
1
)(2a
2
2
−
8a
1
((a
1
− 2) + 2(1 − α)σ
1
))
)
> 0 for certain value of
α.
•
(
a
4
2
+ 2a
2
2
((a
1
− 2) − 2(1 − α)σ
1
)
2
((a
1
− 2) + 2(1 −
α)σ
1
)
)
> 0 for certain value of α.
IV. NUMERICAL EXAMPLES
To show the behavior and properties of our analysis of
the steady states, two examples will be given in this section.
Example 1
Consider the model (4) with parameters b = 0.2, c = 0.5,
d = 1.2, d
j
= 1, σ
1
= 1.4, σ
2
= 0.1, σ
4
= 0.2, σ
5
= 0.5,
µ = 1 with three initial conditions (x
α
, x
α
) = (m
1
− (1 −
α)σ
4
, m
1
+(1−α)σ
5
) and (y
α
, y
α
) = (m
2
−(1−α)σ
4
, m
2
+
(1 − α)σ
5
) where m
1
= 1, 3, 2 and m
2
= 2, 2, 2. For α = 1
and τ = 0, the semi trivial steady state of the model is
(1, 1, 0, 0) and it is stable. Hence, we conclude that for τ ≥ 0,
it is locally asymptotically stable. The results is shown in
Figures 1 and 2.
Example 2
Consider the same conditions of Example 1 but for α =
0.8, the semi trivial steady state (0.63, 1.63, 0, 0) is locally
asymptotically stable for all values of τ. It means that the
conditions of Proposition 2 are satisfied and it is shown in
Figure 3.
0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
solution x
solution y
Fig. 1. The Steady State Converges to (1, 0) for Different Initial Conditions
and α = 1, τ = 0
0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
solution x
solution y
Fig. 2. The Steady State for α = 1 and τ = 2
0 1 2 3 4 5 6 7 8 9
−20
−15
−10
−5
0
5
10
15
20
solution x
solution y
Fig. 3. The Steady State for α = 0.8 and τ = 2
V. CONCLUSION
In this paper, we proposed a system of fuzzy delay
predator-prey equations by using symmetric triangular fuzzy
number. The crisp delay predator-prey of (2 × 2) system is
extended to a FDPP of (4 × 4) system by using parametric
form of α− cut. The FDPP system has trivial, semi trivial
and nontrivial steady states. In this case the characteristic
equation is of degree 4. The fuzzy system proposed leads
to the difficulty of locating the roots of the characteristic
equation since the system becomes larger compare with the
crisp system. Generally, the situation is more complex to
arrive at general conditions on the coefficients of character-
istic equation such that it describes a locally asymptotically
stable for semi trivial steady state for all values of delay, and
the trivial steady state is always unstable. We conclude the
results as in Propositions 1 and 2. We provide two examples
to demonstrate the results.

ACKNOWLEDGMENT
This research is supported by Research Management Cen-
tre - UTM and Malaysian Organization High Education
(MOHE) Grants through votes 4F127 and 07J77. The authors
are thankful to the financial support.
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