DOI : 10.17577/IJERTCONV5IS05039
- Open Access
- Total Downloads : 20
- Authors : Harkaran Singh, H.S. Bhatti
- Paper ID : IJERTCONV5IS05039
- Volume & Issue : ESDST – 2017 (Volume 5 – Issue 05)
- Published (First Online): 24-04-2018
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Dynamical Behaviors of Discrete-time Prey-Predator System
Harkaran Singh
Department of Applied Sciences,
Khalsa College of Engineering and Technology, Amritsar-143001, Punjab, India
H.S. Bhatti
Department of Applied Sciences,
-
Engineering College, Fatehgarh Sahib, Punjab, India
AbstractIn the present study, the dynamical behaviors of discrete-time prey-predator system. Global stability of the model at the fixed points has been discussed. The specific conditions for existence of flip bifurcation and Hopf
Applying forward Eulers scheme to the system of equations (1.1), we obtain the system as
+ [( ) ],
bifurcation have been derived by using center manifold {
[ ( ) ](1.2)
theorem and bifurcation theory. To analyse our results, numerical simulations have been carried out.
+
+ .
KeywordsPrey-predator system, Center manifold theorem, Flip bifurcation, Hopf bifurcation, Chaos.
-
INTRODUCTION
The Prey-Predator model is a topic of great interest for
-
STABILITY OF THE FIXED POINTS
The fixed points of the system (1.2) are
(0,0), ( , 0) , (, ),
where = + , = .
many mathematicians and biologists which starts with the
+
+
pioneer work of Lotka [1] and Volterra [2]. The dynamic relationship between predators and prey living in the same environment will continue to be one of the important themes in mathematical ecology [3-4]. Many authors [5-11]
The jacobian matrix of (1.2) at the fixed point (, ) is given by
1 + ( 2 )
have suggested that the discrete time models are more
appropriate than the continuous ones and provide efficient results when the populations have non-overlapping
= [
1 + ( 2 + )].
generations. However, there are few articles [12-18] discussing the dynamical behaviors of discrete-time predatorprey models by involving bifurcations and chaos phenomena.
In the present study, we investigated the dynamical behaviors of discrete-time prey-predator model by involving bifurcations and chaos phenomena. This paper is organized as follows: In Section 2, we obtained the fixed points of the discrete-time model and discussed the stability criterion of the discrete-time model at fixed points. In Section 3, the specific conditions of existence of flip bifurcation and Hopf bifurcation have been derived. In Section 4, to analyse our results, numerical simulations have been carried out and further discussion on the period doubling bifurcation and chaotic behavior has been carried out.
The discrete-time prey-predator model [19] is
= ( ) ,
The characteristic equation of the jacobian matrix can be
written as
2 + (, ) + (, ) = 0, (2.1) where
(, ) = = 2 + + (2 ) + ( + 2),
(, ) = = [1
+ ( 2 )][1 + ( 2
+ )] + 2
Lemma 2.1: Let () = 2 + + . Suppose that
(1) > 0, 1 and 2 are roots of () = 0. Then
(i) |1| < 1 and |2| < 1 if and only if (1) > 0 and < 1;
{
= ( ) + ,
(1.1)
(ii) |1| < 1 and |2| > 1 (or |1| > 1 and
where and represents the densities of prey and predator respectively; , , , denotes the intrinsic growth rate of prey, capture rate, death rate of predator and the conversion rate respectively; and , denotes the intra-specific competition coefficients of prey and predator respectively.
|2| < 1) if and only if (1) < 0;
(iii) |1| > 1 and |2| > 1 if and only if (1) > 0 and > 1;
(iv) 1 = 1 and |2| 1 if and only if (1) = 0 and 0, 2;
Proposition 2.4. When > , there exists different
topological types of (, ), where = + , =
+
(v)
and
are complex and | | = | | = 1 if
for all possible parameters.
1 2 1 2 +
and only if 2 4 < 0 and = 1.
Let 1 and 2 be the roots of eq. (2.1), which are known as eigen values of the fixed point (, ). Then (, ) is called a sink or locally asymptotically stable if |1| < 1and
|2| < 1. (, ) is called a source or locally unstable if
|1| > 1 and |2| > 1. (, ) is non-hyperbolic if either
|1| = 1 or |2| = 1. (, ) is called a saddle if |1| > 1
and |2| < 1 (or |1| < 1 and |2| > 1). (see [11])
-
(, ) is sink if either condition (i.1) or (i.2) holds:
(i.1) 0 < < , 0.
(+)()
(i.2) 0 < < , < 0,
(+)()
Proposition 2.2. The fixed point (0,0) is source if > 2,
saddle if 0 < < 2, and non-hyperbolic if = 2.
It has been observed that when = 2, one of the eigen
values of the critical point (0,0) is -1 and magnitude of other is not equal to 1. Thus the flip bifurcation occur when
parameter changes in small neighborhood of = 2.
Proposition 2.3. There exists different topological types of
( , 0) for possible parameters.
(i) ( , 0) is sink if > and 0 < <
where = + + and = 2 4( + )( )( + ).
-
(, ) is source if either condition (ii.1) or (ii.2) holds:
(ii.1) > + , 0.
(+)()
(ii.2) > + , < 0.
(+)()
-
(, ) is non-hyperbolic if either condition (iii.1) or (iii.2) holds:
(iii.1) = ± , 0.
(+)()
{2 , 2 }.
(iii.2) = ± , < 0.
(ii) ( , 0) source if > and >
(+)()
{2 , 2 }.
-
( , 0) is non-hyperbolic if = 2 =
-
(, ) is saddle for all values of the parameters, except for that which lies in (i) to (iii).
2
and > .
From lemma (2.1), it has been observed that one of the eigen values of the fixed point (, ) is -1 and
(iv) ( , 0) is saddle for all values of the
parameters, except for that which lies in (i) to (iii).
The term (iii) of proposition 2.3 implies that the parameters lie in the set
magnitude of other is not equal to 1, if the term (iii.1) of proposition 2.4 holds. The term (iii.1) of proposition 2.4 may be written as follows:
1
= {(, , , , , ): = , 0,
(+)()
> 0, , , , , , > 0}, 2 = {(, , , , , ): =
= {(, , , , ), = 2 , 2
,
+ , 0, > 0, , , , , , > 0},
, , , , > 0}.
(+)()
where = + + and = 2
If the term (iii) of proposition 2.3 holds, than one of the eigen values of the fixed point ( , 0) is -1 and magnitude
of other is not equal to 1. The point ( , 0) undergoes flip
bifurcation when the parameter changes in small
neighbourhood of .
4( + )( )( + ).
From lemma (2.1), it has been observed that the eigen values of the fixed point (, ) as a pair of conjugate complex numbers with modulus 1, if the term (iii.2) of
proposition 2. holds. The term (iii.2) of proposition 2.4 may be described as follows:
21 = 11, 22 = 1 + 1[1 21 +
1], 23 = 11, 24 = 11,
= {(, , , , , ): =
, < 0,
21 = 1, 22 = 1 21 + 1 ,
> 0, , , , , , > 0}.
(+)()
23
= 1
, 24
= 1.
-
-
BIFURCATION BEHAVIOR
In this section, we study the flip bifurcation and Hopf bifurcation at the fixed point (, ).
-
FLIP BIFURCATION
Consider the system (1.2) with arbitrary parameter
(1, 1, 1, 1, 1, 1, 1) 1, which is described as
Consider the following translation:
(
() = ),
where = (
12 12
1 11 2 11).
Taking 1 on both sides of eq. (3.3), we get
follows:
1 0
(, , )
{ + 1[(1 1) 1],
+ 1[(1 1) + 1].
(3.1)
() ( 0 2) () + ((, , )), (3.4) where
Eq. (3.1) has fixed point B(, ), whose eigen values are
(, , ) = [ 13(211)1223] + [14(211)]2
1 = 1, 2 = 3
1(111+111+111111) with
(11+11)
12(2+1)
12242 + [11(211)1221] +
12(2+1)
| | 1 by proposition (2.4), where = 11+11 , =
12(2+1)
12(2+1)
2 11+11
[12(211)1222] + [13(211)1223] +1111 and
11+11
12(2+1) 12(2+1)
[14(211)]2 12242,= 11 .
12(2+1)
12(2+1)
2
1 (11+11)(1111)
(, , ) = [ 13(1+11)+1223] + [ 14(1+11)] +
12(2+1) 12(2+1)
where 1 = 111 + 111 + 111 111 and
12242 + [11(1+11)+1221] + [12(1+11)+1222] +
= 2 4(
+ )(
)(
+ ).
12(2+1)
12(2+1)
12(2+1)
1 1 1 1
1 1 1 1
1 1 1 1 1 1
[13(1+11)+1223] + [14(1+11)]2 + 12242,Consider the perturbation of (3.1) as below:
12(2+1)
12(2+1)
12(2+1)
+ (
+ )[( ) ],
= 12( + ), and = (1 + 11) + (2 11).
{ ( 1
1 1 1
)[ ( )
] (3.2)
+
1 +
1 1
+ 1 ,
where || 1 is a limited perturbation parameter. Let = and = .
After transformation of the fixed point (, ) of map (3.2) to the point (0, 0), we obtained
11 + 12 + 13 + 142 + 11 + 12 +
13 + 142 and 21 + 22 + 23 +
242 + 21 + 22 + 23 + 242, (3.3) where
11 = 1 + 1[1 21 1], 12 = 11 ,
Applying center manifold theorem to eq. (3.4) at the origin in limited neighborhood of = 0. The center manifold
(0,0) can be approximately presented as:
(0,0) = {(, ): = 0 + 12 + 2 + 32 +
((|| + ||)3)},
where ((|| + ||)3) is a function with at least third order in variables(, ).
By simple calculations for center manifold, we have
0 = 0,
= , = ,
= [13(1+11)+1223](1+11)1214(1+11)24(1+11)2,
13 1 1
14 1 1
1 (2+1)(2+3)
= [11(1+11)+1221]12+[12(1+11)+1222](1+11) ,
11 = 1 21 1 , 12 = 1 , 13 = 2
12(2+1)2
1 , 14 = 1 ,
3 = 0.
Now, consider the map restricted to the center manifold
(0,0) as below:
: + 12 + 2 + 32 + 42 + 53 +
((|| + ||)4), (3.5)
-
HOPF BIFURCATION
Consider the system (1.2) with arbitrary parameter (2, 2, 2, 2, 2, 2, 2) , which is described as follows:
where
1 = 1
{(1 + 11)[13(2 11) 1223] +
{ + 2[(2 2) 2],
+ 2[(2 2) + 2]
(3.6)
(2+1)
1214(2 11) 24(2 11)2},
(3.6) has fixed point B(, ), where = 22+22, =
22+22
2222 and = (222+222+222222).
2
= 1 (2+1)
[11(2
11
) 12
21
] 1
12(2+1)
(1 +
22+22 2
(22+22)(2222)
11)[12(2 11) 1222] ,
Consider the perturbation of (3.6) as follows:
3
= 1 (2+1)
{[13(2 11) 1223](2 211
{ + (2 + )[(2 2) 2],
+ (2 + )[(2 2) + 2],
(3.7)
1)2 + 21214(2 11)2 + 24(1 + 11)(2
11)2 + [11(2 11) 1221]1 [13(2 11)
1223](1 + 11) + 1214(2 11) 24(1 + 11)2} +
where || 1 is small perturbation parameter.
1
12(2+1)
[12(2
11
) 12
22
](2
11
)1,
The characterization equation of map (3.7) at B(, ) is given by 2 + () + () = 0,
4
= 1 (2+1)
[11(2 11) 1221]2 + 1( +1)
12 2
(2
where
11)[12(2 11) 1222]2,
() = 2 + (2+)(222+222+222222) ,
= 1 {[ ( )
](
2
(22+22)
5 (2+1)
13 2 11
12 23 2 11
( +)( + + )
1)1
+ 212
14
(2
11
)1
+ 24
(1 + 11
)(2
() = 1 2 2 2 2 2 2 2 2 2 2 2 2 2 +
(22+22)
11)1}.
According to Flip bifurcation, the discriminatory quantities
1 and 2 are given by:
(2+)2(22+22)(2222) (22+22)
Since the parameter (2, 2, 2, 2, 2, 2, 2) , the eigen values of B(, ) are a pair of conjugate complex
1
= ( 2
+ 1 2)| 2
2
2
,
(0,0)
numbers and with modulus 1, where
, = ()4()2().
= (1 3 + (1 2) )| . 2
2 6 3
2 2
(0,0)
Now we have
1
After simple calculations, we obtain 1 = 2 and 2 = 5 + 2.
|| = (())
12,
Analyzing above and the flip bifurcation [20], we write a theorem similar to [11] as below:
= |||
=0
= (222+222+222222) > 0.
2(22+22)
Theorem 3.1. If 2 0, and the parameter alters in the limiting region of the point (0, 0), then the system (3.2) passes through flip bifurcation at the point B(x, y). Also, the period-2 points that bifurcate from fixed point B(x, y) are stable (resp., unstable) if 2 > 0 (resp., 2 < 0).
When varies in small neighborhood of = 0, then
, = .
Hopf bifurcation requires that when = 0, , 1 ( = 1, 2, 3, 4) which is equivalent to (0) 2,0,1,2.
Since the parameter (2, 2, 2, 2, 2, 2, 2) , therefore (0) 2, 2. We only require that (0) 0, 1, which leads to
(222 + 222 + 222 222)2 = (22 +
22)(22 + 22)(22 22), = 2,3. (3.8) Let = and = .
After transformation the fixed point (, ) of map (3.7) to the point (0, 0). We have
+ (2 + )[2 22 2 22
2( + )] and + (2 + )[2 22 +
2 22 + 2( + ) (3.9)
After that we discuss the normal form of (3.9) when = 0. Consider the following translation:
() = (), where = (0 1).
Taking 1on both sides of (3.9), we get
(, )
Theorem 3.2. If the condition (3.8) holds, and the parameter alters in the limited region of the point (0, 0), then the system (3.7) passes through Hopf bifurcation at the point (, ). Moreover, if < (resp., > ), then an attacting (resp., repelling) invariant closed curve bifurcates from the fixed point (, ) for > (resp.,
< ).
-
-
NUMERICAL SIMULATIONS
To verify the theoretical analysis, we draw the bifurcation diagrams, largest Lypunov exponents and phase portraits for the system (1.2). This shows the complete dynamical behavior and the global stability of prey-predator system at the fixed points. We discuss bifurcation in following cases:
Case 1: In this case, we draw the bifurcation diagram of the model (1.2) taking = . , = , = , = . , =
, = . , the initial value of (, ) = (. , . ) and
covering [0.5, 0.9]. From Fig. 4.1 (a), we see that from the fixed point (. , . ), flip bifurcation appears
at = . having = 7.1235 and 2 = 18.8 and
( ) ( ) ( ) + (
(, )
where
), (3.10)
(, , , , , ) = (3.7, 1, 2, 0.9, 1, 0.9) 1. It shows that the Theorem 3.1 is correct.
(, ) = 2 [ 2 + (
+ ) 2] ,
The phase portraits in the Fig. 4.2 shows that there
2
2 2 2
are chaotic sets at = 0.79, 0.8. Moreover, the Largest Lypunov exponents corresponding to = 0.79, 0.8 are
(, ) = 2[22 + 2],
= and = + .
According to Hopf bifurcation, the discriminatory quantity
is given by
positive that confirm the chaotic sets.
= [()
]
+
() , (3.11) where
= [ + + ( )],
= [ + + ( + )],
= [ + + ( + )],
= [ + + + + ( +
)],
On solving (3.11), we obtain the value of .
Analyzing above and Hopf bifurcation [20], we write a theorem similar to [11] as below:
Fig.4.1.(a)
Fig. 4.1. (b)
Fig. 4.1 (a) Bifurcation diagram of system (1.2) with covering [0.5, 0.9],
a = 3.7, b = 1, c = 2, d = 0.9, l = 1, m = 0.9, the initial value of (x, y) = (0.86, 0.63), where horizontal axis, vertical axis presents , x respectively. (b) Largest Lyapunov exponents related to 4.1 (a).
= 0.790
= 0.8
Fig. 4.2. Phase portraits for several values of corresponding to Fig. 4.1
-
where horizontal axis, vertical axis presents x, y respectively.
Case 2: In this case, we draw the bifurcation diagram of the model (1.2) taking = 1, = 0.9, = 0.4, = 0.7, = 0.5, = 1, the initial value of (, ) = (0.7, 0.3) and covering [1, 1.4]. We see from Fig. 4.3 (a) that from the fixed point (1.4743, 0.8205), Hopf bifurcation emerges at
= 1.2336 with = 0.28192, = 0.95943, =
0.3669 and (, , , , , ) = (1, 0.9, 0.4, 0.7, 0.5, 1)
. It shows that the Theorem 3.2 is accurate.
From the Fig 4.3(a), it has been observed that when < 1.2336, the fixed point (1.4743, 0.8205) of the system (1.2) is stable. At = 1.2336 the fixed point loses its stability and as exceeds from 1.2336, an invariant circle generates.
The largest Lyapunov exponents in Fig. 4.3(b) shows that for the parameter (1, 1.2336), the Lyapunov exponents are negative. For (1.2336, 1.34200), some Lyapunov exponents are positive and some negative, it means that there exist stable period windows in chaotic region.
The phase portraits in Fig. 4.4, shows that a smooth invariant circle bifurcates from the fixed point (1.4743, 0.8205). There appears an invariant circle for some values of and its radius becomes larger with the growth of e.g. when varies from 1.2336 to 1.23787. There appears a closed curve for some values of and its shape changes with the growth of e.g. when varies from 1.25123 to 1.342.
Fig. 4.3(a)
Fig. 4.3(b)
Fig. 4.3(a) Bifurcation diagram of system (1.2) with covering [1, 1.4], a = 1, b = 0.9, c = 0.4, d = 0.7, l = 0.5, m = 1 the initial value of (x, y) = (0.7, 0.3), where horizontal axis, vertical axis presents , x respectively. (b) Largest Lyapunov exponents related to (a).
= 1.2336
= 1.2378
= 1.25123
= 1.342
Fig. 4.4. Phase portraits for several values of related to Fig. 4.3 (a) where horizontal axis, vertical axis presents x, y respectively.
-
-
CONCLUSIONS
-
+
In this paper, we investigated the dynamical behaviors of discrete-time prey-predator model in the closed first quadrant 2 . Global stability of the model at the fixed points has been discussed. The map undergoes flip bifurcation and Hopf bifurcation at the fixed point under specific conditions, when varies in small neighbourhood of 1or 2 and . Numerical simulations display cascade of period doubling bifurcation, chaotic sets in case of flip bifurcation, and smooth invariant circle and closed curves in case of Hopf bifurcation. The complexity of dynamical behaviors is confirmed by computation of Lyapunov exponents.
Further, Liu et al. [19] for corresponding continuous predator-prey model concluded that the system is globally asymptotically stable under certain conditions, whereas we observed that the discrete system has a rich and complex dynamical behavior than the continuous system.
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