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Global character of a host-parasite model
Chaos, Solitons & Fractals 54 (2013) 1–7
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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Global character of a host-parasite model Qamar Din a,⇑, Tzanko Donchev b a b
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan Department of Mathematics, ‘‘Al. I. Cuza’’ University, Iasßi 700506, Romania
a r t i c l e
i n f o
Article history: Received 28 December 2012 Accepted 15 May 2013 Available online 8 June 2013
a b s t r a c t In this paper, we study the qualitative behavior of a discrete-time host-parasite model. Moreover, the periodicity nature of positive solutions, local asymptotic stability, global behavior of unique positive equilibrium point, and its rate of convergence is discussed. Some numerical examples are given to verify our theoretical results. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Many ecological competition models are governed by differential and difference equations. We refer to [1,2] and the references therein for some interesting results related to the global character and local asymptotic stability. As it is pointed out in [3,4] the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations are of non-overlapping generations. Particularly, the persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions can be discussed more easily in case of difference equations as compared to differential equations. Discrete dynamical systems are commonly applied to the mathematical models such as insect population in which there is a natural division of time in the sense of discrete generations. One of the earliest applications of a discrete-time model to a biological system involved two insects, a parasitoid and its host. The model was developed by Nicholson and Bailey and applied it to the parasitoid (Encarsia formosa), and the host (Trialeurodes vaporarioum) (1935). The term ‘‘parasitoid’’ means a parasite which is freely moving as an adult but lays eggs in the larvae or pupae of their host. Host that are not parasitized give rise to their own progeny. Host that are successfully parasitized die but the eggs laid by the parasitoid may survive to ⇑ Corresponding author. Tel.: +92 3003964842. E-mail addresses: [email protected] gmail.com (T. Donchev).
(Q.
Din),
tzankodd@
0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.05.011
the next generation of parasitoids. The models which have received more attention from experimental and theoretical biologists are the host-parasitoid and host-parasite systems. Parasitoids lay eggs into their hosts and thus the completion of the parasitoid life cycle requires that their hosts be killed. Parasitoids appear like parasites as they grow inside a host, but also appear like predators when they are killers of their hosts. Predators kill their prey for the sake of their food. Parasites live in or on a host and draw food, shelter, or other requirements from that host, often without killing it. Host-parasite models are similar to host-parasitoid models, except that the parasites do not necessarily kill their hosts. A host-parasite model was formulated by Leslie–Grower in 1960 and has a particularly simple form. Let xn denote the host and yn the parasite population at time n. Then the host-parasite model is defined as follow
xnþ1 ¼
axn ; 1 þ byn
ynþ1 ¼
cxn yn ; xn þ dyn
n ¼ 0; 1; 2; . . . ;
ð1Þ
where parameters a; b; c; d 2 Rþ , and the initial conditions x0 ; y0 are positive real numbers. Moreover, the parameters a, b are growth rates of the host and parasite populations in the absence of the other population. If the quantity byn increases, then there will be a reduction in the host population. If the ratio yxnn increases, then there will be a reduction in number of host per parasite resulting a reduction in the parasite population. More precisely, we will investigate local asymptotic stability of unique positive equilibrium point, the periodicity nature of prime period-two solutions, the global asymptotic character of equilibrium
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Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
point, and the rate of convergence of positive solutions of the system (1). For more detail of such biological models, one can see [5–7]. In biological research the data can be obtained only in time-discrete manner. One can see many natural biological events which have discrete nature. For example, in case of some insect population dynamics with non-overlapping generations, the mathematical models can be studied with the help of discrete dynamical systems such as Nicholson–Bailey model for host-parasite relationship. The study of discrete dynamical systems described by difference equations has now been paid great attention since these models are more reasonable than the continuous time models when populations have non-overlapping generations. In case of discrete dynamical systems one has more efficient computational results for numerical simulations, and also has rich dynamics as compared to the continuous ones. In recent years, many papers have been published on the mathematical models of biology that discussed the system of difference equations generated from the associated system of differential equations as well as the associated numerical methods. In mathematical biology, the models such as the hostparasite or host-parasitoid have attracted many researchers during the last few decades. Difference equations or discrete dynamical systems is diverse field which impact almost every branch of pure and applied mathematics. We refer [8–13] for basic theory of difference equations and rational difference equations. It is very interesting to investigate the qualitative behavior of the discrete dynamical systems of non-linear difference equations. Particularly, to discuss the local asymptotic stability of equilibrium points and their global character. For more results for the systems of difference equations, we refer the interested reader to [16–23]. In this paper, our aim is to investigate the qualitative behavior exhibited by interacting host populations of discretely growing organisms in the presence of another species, the parasite, which has some trophic interaction with it, and is capable to change the host’s population dynamics nature. 2. Linearized stability
ynþ1 ¼ gðxn ; yn Þ;
Þ be an equilibrium point of a map F ¼ ðf ; gÞ, x; y Let ð where f and g are continuously differentiable functions at Þ. The linearized system of (2) about the equilibrium ð x; y Þ is point ð x; y
X nþ1 ¼ FðX n Þ ¼ F J X n ; where X n ¼
xn yn
ð2Þ n ¼ 0; 1; . . . ;
and F J is Jacobian matrix of system (2)
Þ. x; y about the equilibrium point ð ; y Þ be the equilibrium point of the system (1), Let ðx then one has
x ¼
ax ; 1 þ by
¼ y
cxy : x þ dy
Þ ¼ ðbðac1Þd is the unique posix; y Then, it follows that ð ; a1 b 1Þ tive equilibrium point of the system (1) for a > 1; c > 1. Þ of the system (1) Moreover, the Jacobian matrix F J ð x; y Þ is given by about the equilibrium point ð x; y
2 Þ ¼ 4 F J ðx; y
b 1þy
ð1þxaybbÞ2
2 cd y dÞ2 ðxþy
x2 c dÞ2 ðxþy
a
3 5:
Þ is given by The characteristic polynomial of F J ð x; y
PðkÞ ¼ k
where f : I J ! I and g : I J ! J are continuously differentiable functions and I; J are some intervals of real numbers. Furthermore, a solution fðxn ; yn Þg1 n¼0 of system (2) is uniquely determined by initial conditions ðx0 ; y0 Þ 2 I J. Þ that satisfies An equilibrium point of (2) is a point ð x; y
x ¼ f ðx; y Þ Þ y ¼ gðx; y
; y Þ is said to be stable if for (i) An equilibrium point ðx every e > 0 there exists d > 0 such that for every iniÞk < d implies tial condition ðx0 ; y0 Þ, if kðx0 ; y0 Þ ð x; y Þk < e for all n > 0, where k:k is usual kðxn ; yn Þ ð x; y Euclidian norm in R2 . Þ is said to be unstable if it (ii) An equilibrium point ð x; y is not stable. Þ is said to be asymptoti(iii) An equilibrium point ð x; y cally stable if there exists g > 0 such that Þk < g and ðxn ; yn Þ ! ð Þ as n ! 1. kðx0 ; y0 Þ ð x; y x; y Þ is called global attractor if (iv) An equilibrium point ð x; y Þ as n ! 1. ðxn ; yn Þ ! ð x; y Þ is called asymptotic glo(v) An equilibrium point ð x; y bal attractor if it is a global attractor and stable.
a
2
To discuss the linearized stability of unique positive equilibrium point of the system (1), first we consider some basic definitions and formulation for such type of stability. Let us consider two-dimensional discrete dynamical system of the form
xnþ1 ¼ f ðxn ; yn Þ
Þ be an equilibrium point of the Definition 1. Let ð x; y system (2).
þ
1 þ yb
þ
!
x2 c ðx þ ydÞ2
x2 ac 2
ð1 þ ybÞðx þ ydÞ
þ
k xy2 abcd
ð1 þ ybÞ2 ðx þ ydÞ2
:
ð3Þ
Theorem 1 [14]. For the system X nþ1 ¼ FðX n Þ; n ¼ 0; 1; . . ., be a fixed point of F. If all of difference equations such that X lie inside the eigenvalues of the Jacobian matrix J F about X is locally asymptotically stable. open unit disk jkj < 1, then X is If one of them has a modulus greater than one, then X unstable. Theorem 2 ([8,13]). Consider the second-degree polynomial equation
k2 þ pk þ q ¼ 0;
ð4Þ
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Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
where p and q are real numbers. Then, the necessary and sufficient condition for both roots of the Eq. (4) to lie inside the open disk jkj < 1 is
After some calculations one has
ðp1 p2 Þ2 ¼ ðp1 þ p2 Þ2 4p1 p2
jpj < 1 þ q < 2:
¼
ð1 þ aÞð1 þ a c þ 3acÞðd þ acdÞ2
a2 b2 ð1 þ cÞc2 ð1 þ cÞ2
<0
and
3. Main results
ðq1 q2 Þ2 ¼ ðq1 þ q2 Þ2 4q1 q2 Theorem 3. The unique positive equilibrium point a1 of the system (1) is locally asymptotiÞ ¼ ðbðac1Þd ð x; y 1Þ ; b
¼
ð1 þ aÞð1 þ cÞð1 þ a c þ 3acÞ b2 c2
cally stable for all a > 1; c > 1.
Which is a contradiction. h
Þ about posiProof. The characteristic of F J ð x; y polynomial a1 is given by tive equilibrium point ðbðac1Þd ; 1Þ b
3.1. Global character
2
PðkÞ ¼ k
1þc
c
ca c þ 1 kþ : ca
ð5Þ
Let
1þc ; p¼
Assume that a > 1, and c > 1. Then, one has
1þq¼2þ
1c
ca
(i) f ðx; yÞ is non-decreasing in x, and non-increasing in y. (ii) gðx; yÞ is non-decreasing in both arguments. (iii) If ðm1 ; M1 ; m2 ; M2 Þ 2 I2 J 2 is a solution of the system
<2
and
jpj ¼
1þc
c
2ac þ að1 cÞ
¼
ac
<
2ac þ ð1 cÞ
ac
¼ 1 þ q:
Hence, from Theorem 2 both roots
k1 ¼ and
k2 ¼
We want to analyze the global stability of the unique positive equilibrium point of the system (1). For this, first we shall prove a general result for such systems. Theorem 5. Let I ¼ ½a; b and J ¼ ½c; d be real intervals, and let f : I J ! I and g : I J ! J be continuous functions. Consider the system (2) with initial conditions ðx0 ; y0 Þ 2 I J. Suppose that following statements are true:
ca c þ 1 q¼ : ca
c
2ac qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ ac þ ða acÞ2 4acð1 c þ acÞ 2ac
m1 ¼ f ðm1 ; M 2 Þ;
M1 ¼ f ðM 1 ; m2 Þ
m2 ¼ gðm1 ; m2 Þ;
M2 ¼ gðM 1 ; M 2 Þ
such that m1 ¼ M 1 , and m2 ¼ M 2 . Then, there exists exactly Þ of the system (2) such that one equilibrium point ð x; y Þ. limn!1 ðxn ; yn Þ ¼ ð x; y
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ ac ða acÞ2 4acð1 c þ acÞ
;
of (5) lie in an open disk jkj < 1, and it followsfrom Theois locally rem 1 that the equilibrium point ðbðac1Þd ; a1 1Þ b asymptotically stable. h Theorem 4. Assume that a > 1 and c > 1, then the system (1) has no prime period-two solutions.
Proof. According to Brouwer fixed point theorem, the function F : I J ! I J defined by Fðx; yÞ ¼ F ðf ðx; yÞ; Þ, which is a fixed point of gðx; yÞÞ has a fixed point ð x; y the system (2). Assume that m01 ¼ a; M 01 ¼ b; m02 ¼ c; M 02 ¼ d such that
miþ1 ¼ f ðmi1 ; M i2 Þ; 1
Miþ1 ¼ f ðM i1 ; mi2 Þ 1
and
miþ1 ¼ gðmi1 ; mi2 Þ; 2
M iþ1 ¼ gðM i1 ; M i2 Þ: 2
Then, Proof. On contrary suppose that the system (1) has a distinctive prime period-two solutions
where p1 – p2 ; q1 – q2 , and pi ; qi are positive real numbers for i 2 f1; 2g. Then, from system (1) one has
p1 ¼
1 þ bq2
p2 ¼
;
ap1
m02 ¼ c 6 gðm01 ; m02 Þ 6 gðM 01 ; M02 Þ 6 d ¼ M 02 : Moreover, one has
m01 6 m11 6 M 11 6 M 01
1 þ bq1
and
and
q1 ¼
m01 ¼ a 6 f ðm01 ; M02 Þ 6 f ðM 01 ; m02 Þ 6 b ¼ M 01 and
. . . ; ðp1 ; q1 Þ; ðp2 ; q2 Þ; ðp1 ; q1 Þ; . . .
ap2
cp2 q2 p2 þ dq2
;
q2 ¼
< 0:
cp1 q1 p1 þ dq1
:
m02 6 m12 6 M 12 6 M 02 : We similarly have
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Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
m11 ¼ f ðm01 ; M02 Þ 6 f ðm11 ; M12 Þ 6 f ðM 11 ; m12 Þ 6 f ðM 01 ; m02 Þ 6
Proof. The proof Theorem 6. h
M 11
and
follows
from
Theorem
3,
and
3.2. Rate of convergence
m12 ¼ gðm01 ; m02 Þ 6 gðm11 ; m12 Þ 6 gðM 11 ; M12 Þ 6 gðM01 ; M 02 Þ
In this section we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (1). The following result gives the rate of convergence of solutions of a system of difference equations
6 M 12 : Now observe that for each i P 0,
a ¼ m01 6 m11 6 6 mi1 6 Mi1 6 M 1i1 6 6 M01 ¼ b
X nþ1 ¼ ðA þ BðnÞÞX n ;
and
ð10Þ mm
c ¼ m02 6 m12 6 6 mi2 6 M i2 6 M 2i1 6 6 M 02 ¼ d: Hence, mi1 6 xn 6 M i1 , and mi2 6 yn 6 M i2 for n P 2i þ 1. Let m1 ¼ limn!1 mi1 ; M1 ¼ limn!1 M i1 ; m2 ¼ limn!1 mi2 , and M2 ¼ limn!1 M i2 . Then, a 6 m1 6 M 1 6 b, and c 6 m2 6 M 2 6 d. By continuity of f and g, one has
m1 ¼ f ðm1 ; M 2 Þ;
M1 ¼ f ðM 1 ; m2 Þ
m2 ¼ gðm1 ; m2 Þ;
M 2 ¼ gðM 1 ; M2 Þ
where X n is an m-dimensional vector, A 2 C is a constant matrix, and B : Zþ ! C mm is a matrix function satisfying
kBðnÞk ! 0
ð11Þ
as n ! 1 ,where k k denotes any matrix norm which is associated with the vector norm
kðx; yÞk ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y 2
Hence, m1 ¼ M 1 ; m2 ¼ M2 . h Theorem 6. Assume that a > 1, and the unique c > 1, then Þ ¼ ðbðac1Þd of the system positive equilibrium point ð ; a1 x; y 1Þ b (1) is a global attractor. cxy
ax
Proof. Let f ðx; yÞ ¼ 1þby, and gðx; yÞ ¼ xþdy. Then, it is easy to see that f ðx; yÞ is non-decreasing in x and non-increasing in y. Moreover, gðx; yÞ is non-decreasing in both x and y. Let ðm1 ; M 1 ; m2 ; M 2 Þ be a solution of the system
m1 ¼ f ðm1 ; M 2 Þ;
M1 ¼ f ðM 1 ; m2 Þ
m2 ¼ gðm1 ; m2 Þ;
M 2 ¼ gðM 1 ; M2 Þ
am1 1 þ bM 2
;
M1 ¼
aM 1 1 þ bm2
ð6Þ
and
m2 ¼
cm1 m2 ; m1 þ dm2
M2 ¼
cM1 M2 : M 1 þ dM 2
ð7Þ
1 þ bM 2 ¼ a ¼ 1 þ bm2 :
ð8Þ
Hence, from (8) we have bðM2 m2 Þ ¼ 0; i:e., m2 ¼ M2 . Now (7) implies that
M 1 þ dM 2 ¼ cM1 :
ð12Þ
exists and is equal to the modulus of one the eigenvalues of matrix A. Proposition 2 [15]. Suppose that condition (11) holds. If X n is a solution of (10), then either X n ¼ 0 for all large n or
kX nþ1 k kX n k
ð13Þ
exists and is equal to the modulus of one the eigenvalues of matrix A. Let fðxn ; yn Þg be any solution of the system (1) such that , where ð Þ ¼ ðbðac1Þd . limn!1 xn ¼ ; a1 x, and limn!1 yn ¼ y x; y 1Þ b To find the error terms, one has from the system (1)
xnþ1 x ¼
From (6), one has
m1 þ dM2 ¼ cm1 ;
q ¼ n!1 lim ðkX n kÞ1=n
q ¼ n!1 lim
Then, one has
m1 ¼
Proposition 1 (Perron’s Theorem [15]). Suppose that condition (11) holds. If X n is a solution of (10), then either X n ¼ 0 for all large n or
axn 1 þ byn
ax 1 þ by
¼
aðxn xÞ 1 þ byn
abxðyn yÞ Þð1 þ byn Þ ð1 þ by
and
cxn yn cxy xn þ dyn x þ dy cdy2n ðxn xÞ cx2 ðyn yÞ ; ¼ þ ðxn þ dyn Þðx þ dyn Þ ðxn þ dyn Þðx þ dyn Þ
¼ ynþ1 y ð9Þ
On subtraction (9) implies that ðc 1ÞðM 1 m1 Þ ¼ 0; i:e., m1 ¼ M1 . Hence, from Theorem 5 the equilibrium point ða1Þd a1 ; b of the system (1) is a global attractor. h bðc1Þ
, then one has x, and e2n ¼ yn y Let e1n ¼ xn
e1nþ1 ¼ an e1n þ bn e2n and
Lemma 1. Assume that a > 1, and c > 1. Then, the unique a1 Þ ¼ ðbðac1Þd is globally positive equilibrium point ð ; x; y 1Þ b asymptotically stable.
e2nþ1 ¼ cn e1n þ dn e2n ; where
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Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
an ¼
cn ¼
a
bn ¼
;
1 þ byn
abx ; Þð1 þ byn Þ ð1 þ by
cdy2n ; ðxn þ dyn Þðx þ dyn Þ
dn ¼
cx2 : ðxn þ dyn Þðx þ dyn Þ
Moreover,
lim an ¼
a
;
n!1
b 1þy
lim cn ¼
2 cd y ; dÞ2 ðx þ y
n!1
lim bn ¼
n!1
lim dn ¼
n!1
xab bÞ2 ð1 þ y
;
x2 c : dÞ2 ðx þ y
Now the limiting system of error terms can be written as
"
e1nþ1 e2nþ1
2
#
¼4
b 1þy
ð1þxaybbÞ2
2 cd y dÞ2 ðxþy
x2 c dÞ2 ðxþy
a
3" 5
e1n e2n
:
Which is similar to linearized system of (1) about the equiÞ. librium point ð x; y Using proposition (1), one has following result. Theorem 7. Assume that fðxn ; yn Þg be a positive solution of , x, and limn!1 yn ¼ y the system (1) such that limn!1 xn ¼ 1 e ða1Þd a1 n Þ ¼ bðc1Þ ; b . Then, the error vector en ¼ x; y where ð e2n of every solution of (1) satisfies both of the following asymptotic relations 1 n
Þj; lim ðken kÞ ¼ jk1;2 F J ðx; y
n!1
shown in Fig. 2. Example 3. Let a ¼ 10; b ¼ 3:1; c ¼ 12; d ¼ 3:5. Then, the system (1) can be written as
xnþ1 ¼
10xn ; 1 þ 3:1yn
ynþ1 ¼
12xn yn ; xn þ 3:5yn
shown in Fig. 3. Example 4. Let a ¼ 50; b ¼ 15; c ¼ 40; d ¼ 20. Then, the system (1) can be written as
In order to verify our theoretical results and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to the system of nonlinear difference Eq. (1). All plots in this section are drawn with mathematica. Example 1. Let a ¼ 2; b ¼ 0:1; c ¼ 2:5; d ¼ 0:8. Then, the system (1) can be written as
ynþ1 ¼
2:5xn yn ; xn þ 0:8yn
xnþ1 ¼
50xn ; 1 þ 15yn
ynþ1 ¼
40xn yn ; xn þ 20yn
the system (17) is shown in Fig. 4. An attractor of the system (17) is shown in Fig. 5.
ð14Þ
in Fig. 1. pffiffiffi pffiffiffi Example 2. Let a ¼ 2; b ¼ 1:1; c ¼ 3; d ¼ 1:2. Then, the system (1) can be written as
pffiffiffi 2xn ; 1 þ 1:1yn
ynþ1 ¼
pffiffiffi 3xn yn ; xn þ 1:2yn
ð17Þ
with initial conditions x0 ¼ 1:5; y0 ¼ 2:4. In this case the unique equilibrium point ða1Þd a1 ¼ ð1:67524; 3:26672Þ. Moreover, the plot of bðc1Þ ; b
with initial conditions x0 ¼ 2; y0 ¼ 1:2. In this case the unique equilibrium point ða1Þd a1 ¼ ð5:33333; 10Þ. Moreover, the plot is shown bðc1Þ ; b
xnþ1 ¼
ð16Þ
4. Examples
2xn ; 1 þ 0:1yn
with initial conditions x0 ¼ 0:2; y0 ¼ 0:1. In this case the unique equilibrium point ða1Þd a1 ¼ ð0:617265; 0:376558Þ. Moreover, the plot is bðc1Þ ; b
with initial conditions x0 ¼ 3; y0 ¼ 2:5. In this case the unique equilibrium point ða1Þd a1 ¼ ð0:923754; 2:90323Þ. Moreover, the plot is bðc1Þ ; b
kenþ1 k Þj; lim ¼ jk1;2 F J ðx; y n!1 ken k
Þ are the characteristic roots of Jacobian mawhere k1;2 F J ð x; y Þ. trix F J ð x; y
xnþ1 ¼
Fig. 1. Plot of system (14).
#
ð15Þ Fig. 2. Plot of system (15).
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Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
Fig. 3. Plot of system (16).
Fig. 6. Plot of solutions of the system (18).
Fig. 4. Plot of system (17).
Fig. 7. An attractor of the system (18).
5. Conclusion
Fig. 5. An attractor of the system (17).
Example 5. Let a ¼ 500; b ¼ 150; c ¼ 510; d ¼ 170. Then, the system (1) can be written as
xnþ1 ¼
500xn ; 1 þ 150yn
ynþ1 ¼
510xn yn ; xn þ 170yn
ð18Þ
with initial conditions x0 ¼ 3; y0 ¼ 2:5. In this case the unique equilibrium point ða1Þd a1 ¼ ð1:11109; 3:32654Þ. Moreover, the plot of bðc1Þ ; b
the system (18) is shown in Fig. 6. An attractor of the system (18) is shown in Fig. 7.
This work is related to the qualitative behavior of a discrete-time host-parasite model. We proved that the system (1) has a unique positive equilibrium point, which is locally asymptotically stable. The method of linearization is used to prove the local asymptotic stability of unique equilibrium point. Linear stability analysis shows that the steady states of the system (1) will be stable for all a > 1; c > 1. Moreover, the system (1) has no prime period-two solutions. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining the possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. In case of nonlinear dynamical systems, it is very crucial to discuss global behavior of the system. In the paper, we prove the global asymptotic stability of the unique equilibrium point the system (1). Moreover, we investigated the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (1). Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions. The main result of this paper is to prove the global asymptotic stability of the unique positive equilibrium point of the system (1). More precisely, we have proved that under the conditions that a > 1 and c > 1, then the
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Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
unique equilibrium solution of original system (1) is globally asymptotically stable. From our investigations it is obvious that, if the host population is a pest, then according to the Leslie–Gower model, a fast-growing parasite population with a growth rate larger than that of the host that significantly reduces the host population, b > a and d > c, would help in reducing the pest population. Furthermore, if the quantity byn increases, then there will be a reduction in the host population. If the ratio yxnn increases, then there will be a reduction in number of host per parasite resulting a reduction in the parasite population. Furthermore, the results indicate that the observed population dynamics of host species is more critically dependent on the infectivity of the parasite and the growth rate of the host plays a less significant role. We have also shown, by using linear stability analysis, how the dynamics of this host and parasite populations attains local asymptotic stability under certain conditions of parameters. Provided the host population’s intrinsic growth rate is positive, i:e., c > 1, and d > 0, then the parasites are capable of regulating the growth of the host population only, if a > 1, and c > 1. Acknowledgements The authors thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Higher Education Commission of Pakistan. The second author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE2011–3-0154. References [1] Ahmad S. On the nonautonomous Lotka–Volterra competition equation. Proc Am Math Soc 1993;117:199–204. [2] Tang X, Zou X. On positive periodic solutions of Lotka–Volterra competition systems with deviating arguments. Proc Am Math Soc 2006;134:2967–74.
[3] Zhou Z, Zou X. Stable periodic solutions in a discrete periodic logistic equation. Appl Math Lett 2003;16(2):165–71. [4] Liu X. A note on the existence of periodic solution in discrete predator–prey models. Appl Math Model 2010;34:2477–83. [5] Allen Linda JS. An Introduction to Mathematical Biology. Prentice Hall; 2007. [6] Brauer F, Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology. Springer; 2000. [7] Edelstein-Keshet L. Mathematical Models in Biology. McGraw-Hill; 1988. [8] Kulenovic´ MRS, Ladas G. Dynamics of Second Order Rational Difference Equations. Chapman and Hall/CRC; 2002. [9] Camouzis E, Ladas G. Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures. Boca Raton: Chapman and Hall/CRC Press; 2007. [10] Elaydi S. An Introduction to Difference Equations. 3rd ed. New York: Springer-Verlag; 2005. [11] Agarwal RP, Wong P. Advanced Topics in Difference Equations. Kluwer Academic Publishers; 1997. [12] Agarwal RP. Difference Equations and Inequalities. New York: Marcel Dekker; 1992. second ed., (2000). [13] Grove EA, Ladas G. Periodicities in Nonlinear Difference Equations. Boca Raton: Chapman and Hall/CRC Press; 2004. [14] Sedaghat H. Nonlinear Difference Equations: Theory with Applications to Social Science Models. Dordrecht: Kluwer Academic Publishers; 2003. [15] Pituk M. More on Poincare’s and Perron’s theorems for difference equations. J Differ Equ Appl 2002;8:201–16. [16] El-Metwally E, Grove EA, Ladas G, Levins R, Radin M. On the difference equation xnþ1 ¼ a þ bxn1 exn . Nonlinear Anal 2001;47:4623–34. [17] Kalabusˆic´ S, Kulenovic´ MRS, Pilav E. Dynamics of a two-dimensional system of rational difference equations of Leslie–Gower type. Add Differ Equ 2011. http://dx.doi.org/10.1186/1687-1847-2011-29. [18] Kurbanli AS, Çinar C, Yalçinkaya I. On the behavior of positive solutions of the system of rational difference equations yn1 n1 xnþ1 ¼ y xxn1 þ1 ; ynþ1 ¼ xn y þ1. Math Comput Model 2011;53:1261–7. n
n1
[19] Din Q. Dynamics of a discrete Lotka-Volterra model. Add Differ Equ 2013;1:95. [20] Q. Din, On a system of rational difference equation, Demon Math, 2013. [21] Din Q, Qureshi MN, Khan AQ. Dynamics of a fourth-order system of rational difference equations. Add Differ Equ 2012;1:1–15. [22] Zhang Q, Yang L, Liu J. Dynamics of a system of rational third-order difference equation. Add Differ Equ 2012;1:1–6. [23] Shojaei M, Saadati R, Adibi H. Stability and periodic character of a rational third order difference equation. Chaos Solitons Fract 2009;39:1203–9.
Contents lists available at SciVerse ScienceDirect
Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Global character of a host-parasite model Qamar Din a,⇑, Tzanko Donchev b a b
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan Department of Mathematics, ‘‘Al. I. Cuza’’ University, Iasßi 700506, Romania
a r t i c l e
i n f o
Article history: Received 28 December 2012 Accepted 15 May 2013 Available online 8 June 2013
a b s t r a c t In this paper, we study the qualitative behavior of a discrete-time host-parasite model. Moreover, the periodicity nature of positive solutions, local asymptotic stability, global behavior of unique positive equilibrium point, and its rate of convergence is discussed. Some numerical examples are given to verify our theoretical results. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Many ecological competition models are governed by differential and difference equations. We refer to [1,2] and the references therein for some interesting results related to the global character and local asymptotic stability. As it is pointed out in [3,4] the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations are of non-overlapping generations. Particularly, the persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions can be discussed more easily in case of difference equations as compared to differential equations. Discrete dynamical systems are commonly applied to the mathematical models such as insect population in which there is a natural division of time in the sense of discrete generations. One of the earliest applications of a discrete-time model to a biological system involved two insects, a parasitoid and its host. The model was developed by Nicholson and Bailey and applied it to the parasitoid (Encarsia formosa), and the host (Trialeurodes vaporarioum) (1935). The term ‘‘parasitoid’’ means a parasite which is freely moving as an adult but lays eggs in the larvae or pupae of their host. Host that are not parasitized give rise to their own progeny. Host that are successfully parasitized die but the eggs laid by the parasitoid may survive to ⇑ Corresponding author. Tel.: +92 3003964842. E-mail addresses: [email protected] gmail.com (T. Donchev).
(Q.
Din),
tzankodd@
0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.05.011
the next generation of parasitoids. The models which have received more attention from experimental and theoretical biologists are the host-parasitoid and host-parasite systems. Parasitoids lay eggs into their hosts and thus the completion of the parasitoid life cycle requires that their hosts be killed. Parasitoids appear like parasites as they grow inside a host, but also appear like predators when they are killers of their hosts. Predators kill their prey for the sake of their food. Parasites live in or on a host and draw food, shelter, or other requirements from that host, often without killing it. Host-parasite models are similar to host-parasitoid models, except that the parasites do not necessarily kill their hosts. A host-parasite model was formulated by Leslie–Grower in 1960 and has a particularly simple form. Let xn denote the host and yn the parasite population at time n. Then the host-parasite model is defined as follow
xnþ1 ¼
axn ; 1 þ byn
ynþ1 ¼
cxn yn ; xn þ dyn
n ¼ 0; 1; 2; . . . ;
ð1Þ
where parameters a; b; c; d 2 Rþ , and the initial conditions x0 ; y0 are positive real numbers. Moreover, the parameters a, b are growth rates of the host and parasite populations in the absence of the other population. If the quantity byn increases, then there will be a reduction in the host population. If the ratio yxnn increases, then there will be a reduction in number of host per parasite resulting a reduction in the parasite population. More precisely, we will investigate local asymptotic stability of unique positive equilibrium point, the periodicity nature of prime period-two solutions, the global asymptotic character of equilibrium
2
Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
point, and the rate of convergence of positive solutions of the system (1). For more detail of such biological models, one can see [5–7]. In biological research the data can be obtained only in time-discrete manner. One can see many natural biological events which have discrete nature. For example, in case of some insect population dynamics with non-overlapping generations, the mathematical models can be studied with the help of discrete dynamical systems such as Nicholson–Bailey model for host-parasite relationship. The study of discrete dynamical systems described by difference equations has now been paid great attention since these models are more reasonable than the continuous time models when populations have non-overlapping generations. In case of discrete dynamical systems one has more efficient computational results for numerical simulations, and also has rich dynamics as compared to the continuous ones. In recent years, many papers have been published on the mathematical models of biology that discussed the system of difference equations generated from the associated system of differential equations as well as the associated numerical methods. In mathematical biology, the models such as the hostparasite or host-parasitoid have attracted many researchers during the last few decades. Difference equations or discrete dynamical systems is diverse field which impact almost every branch of pure and applied mathematics. We refer [8–13] for basic theory of difference equations and rational difference equations. It is very interesting to investigate the qualitative behavior of the discrete dynamical systems of non-linear difference equations. Particularly, to discuss the local asymptotic stability of equilibrium points and their global character. For more results for the systems of difference equations, we refer the interested reader to [16–23]. In this paper, our aim is to investigate the qualitative behavior exhibited by interacting host populations of discretely growing organisms in the presence of another species, the parasite, which has some trophic interaction with it, and is capable to change the host’s population dynamics nature. 2. Linearized stability
ynþ1 ¼ gðxn ; yn Þ;
Þ be an equilibrium point of a map F ¼ ðf ; gÞ, x; y Let ð where f and g are continuously differentiable functions at Þ. The linearized system of (2) about the equilibrium ð x; y Þ is point ð x; y
X nþ1 ¼ FðX n Þ ¼ F J X n ; where X n ¼
xn yn
ð2Þ n ¼ 0; 1; . . . ;
and F J is Jacobian matrix of system (2)
Þ. x; y about the equilibrium point ð ; y Þ be the equilibrium point of the system (1), Let ðx then one has
x ¼
ax ; 1 þ by
¼ y
cxy : x þ dy
Þ ¼ ðbðac1Þd is the unique posix; y Then, it follows that ð ; a1 b 1Þ tive equilibrium point of the system (1) for a > 1; c > 1. Þ of the system (1) Moreover, the Jacobian matrix F J ð x; y Þ is given by about the equilibrium point ð x; y
2 Þ ¼ 4 F J ðx; y
b 1þy
ð1þxaybbÞ2
2 cd y dÞ2 ðxþy
x2 c dÞ2 ðxþy
a
3 5:
Þ is given by The characteristic polynomial of F J ð x; y
PðkÞ ¼ k
where f : I J ! I and g : I J ! J are continuously differentiable functions and I; J are some intervals of real numbers. Furthermore, a solution fðxn ; yn Þg1 n¼0 of system (2) is uniquely determined by initial conditions ðx0 ; y0 Þ 2 I J. Þ that satisfies An equilibrium point of (2) is a point ð x; y
x ¼ f ðx; y Þ Þ y ¼ gðx; y
; y Þ is said to be stable if for (i) An equilibrium point ðx every e > 0 there exists d > 0 such that for every iniÞk < d implies tial condition ðx0 ; y0 Þ, if kðx0 ; y0 Þ ð x; y Þk < e for all n > 0, where k:k is usual kðxn ; yn Þ ð x; y Euclidian norm in R2 . Þ is said to be unstable if it (ii) An equilibrium point ð x; y is not stable. Þ is said to be asymptoti(iii) An equilibrium point ð x; y cally stable if there exists g > 0 such that Þk < g and ðxn ; yn Þ ! ð Þ as n ! 1. kðx0 ; y0 Þ ð x; y x; y Þ is called global attractor if (iv) An equilibrium point ð x; y Þ as n ! 1. ðxn ; yn Þ ! ð x; y Þ is called asymptotic glo(v) An equilibrium point ð x; y bal attractor if it is a global attractor and stable.
a
2
To discuss the linearized stability of unique positive equilibrium point of the system (1), first we consider some basic definitions and formulation for such type of stability. Let us consider two-dimensional discrete dynamical system of the form
xnþ1 ¼ f ðxn ; yn Þ
Þ be an equilibrium point of the Definition 1. Let ð x; y system (2).
þ
1 þ yb
þ
!
x2 c ðx þ ydÞ2
x2 ac 2
ð1 þ ybÞðx þ ydÞ
þ
k xy2 abcd
ð1 þ ybÞ2 ðx þ ydÞ2
:
ð3Þ
Theorem 1 [14]. For the system X nþ1 ¼ FðX n Þ; n ¼ 0; 1; . . ., be a fixed point of F. If all of difference equations such that X lie inside the eigenvalues of the Jacobian matrix J F about X is locally asymptotically stable. open unit disk jkj < 1, then X is If one of them has a modulus greater than one, then X unstable. Theorem 2 ([8,13]). Consider the second-degree polynomial equation
k2 þ pk þ q ¼ 0;
ð4Þ
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Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
where p and q are real numbers. Then, the necessary and sufficient condition for both roots of the Eq. (4) to lie inside the open disk jkj < 1 is
After some calculations one has
ðp1 p2 Þ2 ¼ ðp1 þ p2 Þ2 4p1 p2
jpj < 1 þ q < 2:
¼
ð1 þ aÞð1 þ a c þ 3acÞðd þ acdÞ2
a2 b2 ð1 þ cÞc2 ð1 þ cÞ2
<0
and
3. Main results
ðq1 q2 Þ2 ¼ ðq1 þ q2 Þ2 4q1 q2 Theorem 3. The unique positive equilibrium point a1 of the system (1) is locally asymptotiÞ ¼ ðbðac1Þd ð x; y 1Þ ; b
¼
ð1 þ aÞð1 þ cÞð1 þ a c þ 3acÞ b2 c2
cally stable for all a > 1; c > 1.
Which is a contradiction. h
Þ about posiProof. The characteristic of F J ð x; y polynomial a1 is given by tive equilibrium point ðbðac1Þd ; 1Þ b
3.1. Global character
2
PðkÞ ¼ k
1þc
c
ca c þ 1 kþ : ca
ð5Þ
Let
1þc ; p¼
Assume that a > 1, and c > 1. Then, one has
1þq¼2þ
1c
ca
(i) f ðx; yÞ is non-decreasing in x, and non-increasing in y. (ii) gðx; yÞ is non-decreasing in both arguments. (iii) If ðm1 ; M1 ; m2 ; M2 Þ 2 I2 J 2 is a solution of the system
<2
and
jpj ¼
1þc
c
2ac þ að1 cÞ
¼
ac
<
2ac þ ð1 cÞ
ac
¼ 1 þ q:
Hence, from Theorem 2 both roots
k1 ¼ and
k2 ¼
We want to analyze the global stability of the unique positive equilibrium point of the system (1). For this, first we shall prove a general result for such systems. Theorem 5. Let I ¼ ½a; b and J ¼ ½c; d be real intervals, and let f : I J ! I and g : I J ! J be continuous functions. Consider the system (2) with initial conditions ðx0 ; y0 Þ 2 I J. Suppose that following statements are true:
ca c þ 1 q¼ : ca
c
2ac qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ ac þ ða acÞ2 4acð1 c þ acÞ 2ac
m1 ¼ f ðm1 ; M 2 Þ;
M1 ¼ f ðM 1 ; m2 Þ
m2 ¼ gðm1 ; m2 Þ;
M2 ¼ gðM 1 ; M 2 Þ
such that m1 ¼ M 1 , and m2 ¼ M 2 . Then, there exists exactly Þ of the system (2) such that one equilibrium point ð x; y Þ. limn!1 ðxn ; yn Þ ¼ ð x; y
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ ac ða acÞ2 4acð1 c þ acÞ
;
of (5) lie in an open disk jkj < 1, and it followsfrom Theois locally rem 1 that the equilibrium point ðbðac1Þd ; a1 1Þ b asymptotically stable. h Theorem 4. Assume that a > 1 and c > 1, then the system (1) has no prime period-two solutions.
Proof. According to Brouwer fixed point theorem, the function F : I J ! I J defined by Fðx; yÞ ¼ F ðf ðx; yÞ; Þ, which is a fixed point of gðx; yÞÞ has a fixed point ð x; y the system (2). Assume that m01 ¼ a; M 01 ¼ b; m02 ¼ c; M 02 ¼ d such that
miþ1 ¼ f ðmi1 ; M i2 Þ; 1
Miþ1 ¼ f ðM i1 ; mi2 Þ 1
and
miþ1 ¼ gðmi1 ; mi2 Þ; 2
M iþ1 ¼ gðM i1 ; M i2 Þ: 2
Then, Proof. On contrary suppose that the system (1) has a distinctive prime period-two solutions
where p1 – p2 ; q1 – q2 , and pi ; qi are positive real numbers for i 2 f1; 2g. Then, from system (1) one has
p1 ¼
1 þ bq2
p2 ¼
;
ap1
m02 ¼ c 6 gðm01 ; m02 Þ 6 gðM 01 ; M02 Þ 6 d ¼ M 02 : Moreover, one has
m01 6 m11 6 M 11 6 M 01
1 þ bq1
and
and
q1 ¼
m01 ¼ a 6 f ðm01 ; M02 Þ 6 f ðM 01 ; m02 Þ 6 b ¼ M 01 and
. . . ; ðp1 ; q1 Þ; ðp2 ; q2 Þ; ðp1 ; q1 Þ; . . .
ap2
cp2 q2 p2 þ dq2
;
q2 ¼
< 0:
cp1 q1 p1 þ dq1
:
m02 6 m12 6 M 12 6 M 02 : We similarly have
4
Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
m11 ¼ f ðm01 ; M02 Þ 6 f ðm11 ; M12 Þ 6 f ðM 11 ; m12 Þ 6 f ðM 01 ; m02 Þ 6
Proof. The proof Theorem 6. h
M 11
and
follows
from
Theorem
3,
and
3.2. Rate of convergence
m12 ¼ gðm01 ; m02 Þ 6 gðm11 ; m12 Þ 6 gðM 11 ; M12 Þ 6 gðM01 ; M 02 Þ
In this section we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (1). The following result gives the rate of convergence of solutions of a system of difference equations
6 M 12 : Now observe that for each i P 0,
a ¼ m01 6 m11 6 6 mi1 6 Mi1 6 M 1i1 6 6 M01 ¼ b
X nþ1 ¼ ðA þ BðnÞÞX n ;
and
ð10Þ mm
c ¼ m02 6 m12 6 6 mi2 6 M i2 6 M 2i1 6 6 M 02 ¼ d: Hence, mi1 6 xn 6 M i1 , and mi2 6 yn 6 M i2 for n P 2i þ 1. Let m1 ¼ limn!1 mi1 ; M1 ¼ limn!1 M i1 ; m2 ¼ limn!1 mi2 , and M2 ¼ limn!1 M i2 . Then, a 6 m1 6 M 1 6 b, and c 6 m2 6 M 2 6 d. By continuity of f and g, one has
m1 ¼ f ðm1 ; M 2 Þ;
M1 ¼ f ðM 1 ; m2 Þ
m2 ¼ gðm1 ; m2 Þ;
M 2 ¼ gðM 1 ; M2 Þ
where X n is an m-dimensional vector, A 2 C is a constant matrix, and B : Zþ ! C mm is a matrix function satisfying
kBðnÞk ! 0
ð11Þ
as n ! 1 ,where k k denotes any matrix norm which is associated with the vector norm
kðx; yÞk ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y 2
Hence, m1 ¼ M 1 ; m2 ¼ M2 . h Theorem 6. Assume that a > 1, and the unique c > 1, then Þ ¼ ðbðac1Þd of the system positive equilibrium point ð ; a1 x; y 1Þ b (1) is a global attractor. cxy
ax
Proof. Let f ðx; yÞ ¼ 1þby, and gðx; yÞ ¼ xþdy. Then, it is easy to see that f ðx; yÞ is non-decreasing in x and non-increasing in y. Moreover, gðx; yÞ is non-decreasing in both x and y. Let ðm1 ; M 1 ; m2 ; M 2 Þ be a solution of the system
m1 ¼ f ðm1 ; M 2 Þ;
M1 ¼ f ðM 1 ; m2 Þ
m2 ¼ gðm1 ; m2 Þ;
M 2 ¼ gðM 1 ; M2 Þ
am1 1 þ bM 2
;
M1 ¼
aM 1 1 þ bm2
ð6Þ
and
m2 ¼
cm1 m2 ; m1 þ dm2
M2 ¼
cM1 M2 : M 1 þ dM 2
ð7Þ
1 þ bM 2 ¼ a ¼ 1 þ bm2 :
ð8Þ
Hence, from (8) we have bðM2 m2 Þ ¼ 0; i:e., m2 ¼ M2 . Now (7) implies that
M 1 þ dM 2 ¼ cM1 :
ð12Þ
exists and is equal to the modulus of one the eigenvalues of matrix A. Proposition 2 [15]. Suppose that condition (11) holds. If X n is a solution of (10), then either X n ¼ 0 for all large n or
kX nþ1 k kX n k
ð13Þ
exists and is equal to the modulus of one the eigenvalues of matrix A. Let fðxn ; yn Þg be any solution of the system (1) such that , where ð Þ ¼ ðbðac1Þd . limn!1 xn ¼ ; a1 x, and limn!1 yn ¼ y x; y 1Þ b To find the error terms, one has from the system (1)
xnþ1 x ¼
From (6), one has
m1 þ dM2 ¼ cm1 ;
q ¼ n!1 lim ðkX n kÞ1=n
q ¼ n!1 lim
Then, one has
m1 ¼
Proposition 1 (Perron’s Theorem [15]). Suppose that condition (11) holds. If X n is a solution of (10), then either X n ¼ 0 for all large n or
axn 1 þ byn
ax 1 þ by
¼
aðxn xÞ 1 þ byn
abxðyn yÞ Þð1 þ byn Þ ð1 þ by
and
cxn yn cxy xn þ dyn x þ dy cdy2n ðxn xÞ cx2 ðyn yÞ ; ¼ þ ðxn þ dyn Þðx þ dyn Þ ðxn þ dyn Þðx þ dyn Þ
¼ ynþ1 y ð9Þ
On subtraction (9) implies that ðc 1ÞðM 1 m1 Þ ¼ 0; i:e., m1 ¼ M1 . Hence, from Theorem 5 the equilibrium point ða1Þd a1 ; b of the system (1) is a global attractor. h bðc1Þ
, then one has x, and e2n ¼ yn y Let e1n ¼ xn
e1nþ1 ¼ an e1n þ bn e2n and
Lemma 1. Assume that a > 1, and c > 1. Then, the unique a1 Þ ¼ ðbðac1Þd is globally positive equilibrium point ð ; x; y 1Þ b asymptotically stable.
e2nþ1 ¼ cn e1n þ dn e2n ; where
5
Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
an ¼
cn ¼
a
bn ¼
;
1 þ byn
abx ; Þð1 þ byn Þ ð1 þ by
cdy2n ; ðxn þ dyn Þðx þ dyn Þ
dn ¼
cx2 : ðxn þ dyn Þðx þ dyn Þ
Moreover,
lim an ¼
a
;
n!1
b 1þy
lim cn ¼
2 cd y ; dÞ2 ðx þ y
n!1
lim bn ¼
n!1
lim dn ¼
n!1
xab bÞ2 ð1 þ y
;
x2 c : dÞ2 ðx þ y
Now the limiting system of error terms can be written as
"
e1nþ1 e2nþ1
2
#
¼4
b 1þy
ð1þxaybbÞ2
2 cd y dÞ2 ðxþy
x2 c dÞ2 ðxþy
a
3" 5
e1n e2n
:
Which is similar to linearized system of (1) about the equiÞ. librium point ð x; y Using proposition (1), one has following result. Theorem 7. Assume that fðxn ; yn Þg be a positive solution of , x, and limn!1 yn ¼ y the system (1) such that limn!1 xn ¼ 1 e ða1Þd a1 n Þ ¼ bðc1Þ ; b . Then, the error vector en ¼ x; y where ð e2n of every solution of (1) satisfies both of the following asymptotic relations 1 n
Þj; lim ðken kÞ ¼ jk1;2 F J ðx; y
n!1
shown in Fig. 2. Example 3. Let a ¼ 10; b ¼ 3:1; c ¼ 12; d ¼ 3:5. Then, the system (1) can be written as
xnþ1 ¼
10xn ; 1 þ 3:1yn
ynþ1 ¼
12xn yn ; xn þ 3:5yn
shown in Fig. 3. Example 4. Let a ¼ 50; b ¼ 15; c ¼ 40; d ¼ 20. Then, the system (1) can be written as
In order to verify our theoretical results and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to the system of nonlinear difference Eq. (1). All plots in this section are drawn with mathematica. Example 1. Let a ¼ 2; b ¼ 0:1; c ¼ 2:5; d ¼ 0:8. Then, the system (1) can be written as
ynþ1 ¼
2:5xn yn ; xn þ 0:8yn
xnþ1 ¼
50xn ; 1 þ 15yn
ynþ1 ¼
40xn yn ; xn þ 20yn
the system (17) is shown in Fig. 4. An attractor of the system (17) is shown in Fig. 5.
ð14Þ
in Fig. 1. pffiffiffi pffiffiffi Example 2. Let a ¼ 2; b ¼ 1:1; c ¼ 3; d ¼ 1:2. Then, the system (1) can be written as
pffiffiffi 2xn ; 1 þ 1:1yn
ynþ1 ¼
pffiffiffi 3xn yn ; xn þ 1:2yn
ð17Þ
with initial conditions x0 ¼ 1:5; y0 ¼ 2:4. In this case the unique equilibrium point ða1Þd a1 ¼ ð1:67524; 3:26672Þ. Moreover, the plot of bðc1Þ ; b
with initial conditions x0 ¼ 2; y0 ¼ 1:2. In this case the unique equilibrium point ða1Þd a1 ¼ ð5:33333; 10Þ. Moreover, the plot is shown bðc1Þ ; b
xnþ1 ¼
ð16Þ
4. Examples
2xn ; 1 þ 0:1yn
with initial conditions x0 ¼ 0:2; y0 ¼ 0:1. In this case the unique equilibrium point ða1Þd a1 ¼ ð0:617265; 0:376558Þ. Moreover, the plot is bðc1Þ ; b
with initial conditions x0 ¼ 3; y0 ¼ 2:5. In this case the unique equilibrium point ða1Þd a1 ¼ ð0:923754; 2:90323Þ. Moreover, the plot is bðc1Þ ; b
kenþ1 k Þj; lim ¼ jk1;2 F J ðx; y n!1 ken k
Þ are the characteristic roots of Jacobian mawhere k1;2 F J ð x; y Þ. trix F J ð x; y
xnþ1 ¼
Fig. 1. Plot of system (14).
#
ð15Þ Fig. 2. Plot of system (15).
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Q. Din, T. Donchev / Chaos, Solitons & Fractals 54 (2013) 1–7
Fig. 3. Plot of system (16).
Fig. 6. Plot of solutions of the system (18).
Fig. 4. Plot of system (17).
Fig. 7. An attractor of the system (18).
5. Conclusion
Fig. 5. An attractor of the system (17).
Example 5. Let a ¼ 500; b ¼ 150; c ¼ 510; d ¼ 170. Then, the system (1) can be written as
xnþ1 ¼
500xn ; 1 þ 150yn
ynþ1 ¼
510xn yn ; xn þ 170yn
ð18Þ
with initial conditions x0 ¼ 3; y0 ¼ 2:5. In this case the unique equilibrium point ða1Þd a1 ¼ ð1:11109; 3:32654Þ. Moreover, the plot of bðc1Þ ; b
the system (18) is shown in Fig. 6. An attractor of the system (18) is shown in Fig. 7.
This work is related to the qualitative behavior of a discrete-time host-parasite model. We proved that the system (1) has a unique positive equilibrium point, which is locally asymptotically stable. The method of linearization is used to prove the local asymptotic stability of unique equilibrium point. Linear stability analysis shows that the steady states of the system (1) will be stable for all a > 1; c > 1. Moreover, the system (1) has no prime period-two solutions. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining the possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. In case of nonlinear dynamical systems, it is very crucial to discuss global behavior of the system. In the paper, we prove the global asymptotic stability of the unique equilibrium point the system (1). Moreover, we investigated the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (1). Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions. The main result of this paper is to prove the global asymptotic stability of the unique positive equilibrium point of the system (1). More precisely, we have proved that under the conditions that a > 1 and c > 1, then the
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unique equilibrium solution of original system (1) is globally asymptotically stable. From our investigations it is obvious that, if the host population is a pest, then according to the Leslie–Gower model, a fast-growing parasite population with a growth rate larger than that of the host that significantly reduces the host population, b > a and d > c, would help in reducing the pest population. Furthermore, if the quantity byn increases, then there will be a reduction in the host population. If the ratio yxnn increases, then there will be a reduction in number of host per parasite resulting a reduction in the parasite population. Furthermore, the results indicate that the observed population dynamics of host species is more critically dependent on the infectivity of the parasite and the growth rate of the host plays a less significant role. We have also shown, by using linear stability analysis, how the dynamics of this host and parasite populations attains local asymptotic stability under certain conditions of parameters. Provided the host population’s intrinsic growth rate is positive, i:e., c > 1, and d > 0, then the parasites are capable of regulating the growth of the host population only, if a > 1, and c > 1. Acknowledgements The authors thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Higher Education Commission of Pakistan. The second author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE2011–3-0154. References [1] Ahmad S. On the nonautonomous Lotka–Volterra competition equation. Proc Am Math Soc 1993;117:199–204. [2] Tang X, Zou X. On positive periodic solutions of Lotka–Volterra competition systems with deviating arguments. Proc Am Math Soc 2006;134:2967–74.
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