Global stability of a population model

Global stability of a population model

Chaos, Solitons & Fractals 59 (2014) 119–128 Contents lists available at ScienceDirect Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibr...

2MB Sizes 4 Downloads 90 Views

Chaos, Solitons & Fractals 59 (2014) 119–128

Contents lists available at ScienceDirect

Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Global stability of a population model q Q. Din Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan

a r t i c l e

i n f o

Article history: Received 12 July 2013 Accepted 21 December 2013 Available online 20 January 2014

a b s t r a c t In this paper, we study the qualitative behavior of a discrete-time population model. More precisely, we investigate boundedness character, existence and uniqueness of positive equilibrium point, local asymptotic stability and global asymptotic stability of unique positive equilibrium point, and the rate of convergence of positive solutions of a population model. In particular, our results solve an open problem proposed by Kulenvic´ and Ladas in their monograph (Kulenvic´ and Ladas, 2002) [8]. Some numerical examples are given to verify our theoretical results. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Many population models are governed by differential and difference equations. We refer to [12–14] and the references therein for some interesting results related to the global character and local asymptotic stability. As it is pointed out in [6,7] the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations are of non-overlapping generations. The study of discrete-time models 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. Discrete-time models give rise to more efficient computational models for numerical simulations and also show rich dynamics compared to the continuous ones. In recent years, many papers have been published on the mathematical models of population dynamics that discussed the system of difference equations generated from the associated system of differential equations as well as the associated numerical methods. Mathematical models of population dynamics have created a major area of research interest during the last few decades. Exponential difference equations can be used to study the models in population dynamics. For more detail of some interesting

q

This work was supported by HEC of Pakistan. E-mail address: [email protected]

0960-0779/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.chaos.2013.12.008

biological models, one can see [1–3]. Nonlinear difference equations are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering and economics. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points. For more results for the qualitative behavior of difference equations, we refer the reader to [4,5,15–20]. Metwally et al. [12] investigated boundedness character, asymptotic behavior, periodicity nature of the positive solutions and stability of equilibrium point of following population model:

xnþ1 ¼ a þ bxn1 exn : Papaschinopoulos et al. [13] studied the boundedness, the persistence and the asymptotic behavior of positive solutions of following two directional interactive and invasive species model:

xnþ1 ¼ a þ bxn1 eyn ;

ynþ1 ¼ c þ dyn1 exn :

Recently, Papaschinopoulos et al. [14] study the asymptotic behavior of the positive solutions of the systems of the two difference equations:

xnþ1 ¼ a þ byn1 eyn ;

ynþ1 ¼ c þ dxn1 exn ;

120

Q. Din / Chaos, Solitons & Fractals 59 (2014) 119–128

and

xnþ1 ¼ a þ byn1 exn ;

y1 ¼ ax0 ð1  ey0 Þ 6

ynþ1 ¼ c þ dxn1 eyn :

We consider here an open problem proposed in [8] related to the global character of solutions of the population model

xnþ1 ¼ axn eyn þ b;

ynþ1 ¼ axn ð1  eyn Þ;

ð1Þ

where 0 < a < 1 and 0 < b < 1. In 2002, Kulenvic´ and Ladas [8] proposed the following open problem: Open Problem 6.10.16 (A Population Model). Assume a 2 ð0; 1Þ and b 2 ð1; 1Þ. Investigate the global character of all positive solutions of the system

xnþ1 ¼ axn e

yn

þ b;

ynþ1 ¼ axn ð1  e

yn

ab : 1a

Hence, x1 2 I and y1 2 J. Suppose that result is true for n ¼ k > 1; i:e., xk 2 I and yk 2 J. Now we will use mathematical induction to prove that the result is true at n ¼ k þ 1. From the system (1), we obtain

xkþ1 ¼ axk eyk þ b 6

b ; 1a

and

ykþ1 ¼ axk ð1  eyk Þ 6

ab : 1a

Hence, the proof is completed. h

Þ;

which may be viewed as a population model. In this paper, our aim is to investigate boundedness character, existence and uniqueness of positive equilibrium point, local asymptotic stability, global asymptotic stability of unique positive equilibrium point, and the rate of convergence of positive solutions of the system (1). For more detail of such population models, one can see [1–3]. 2. Boundedness Theorem 1. Assume that 0 < a < 1, then every positive solution fðxn ; yn Þg1 n¼0 of the system (1) is bounded.

3. Linearized stability Let us consider two-dimensional discrete dynamical system of the form

xnþ1

¼ f ðxn ; yn Þ

ynþ1

¼ gðxn ; yn Þ;

n ¼ 0; 1; . . . ;

ð2Þ

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 Þ ðx0 ; y0 Þ 2 I  J. An equilibrium point of (2) is a point ð x; y that satisfies

x ¼ f ðx; y Þ  ¼ gðx; y Þ y

Proof. Let fðxn ; yn Þg1 n¼0 be any positive solution of the system (1). From xnþ1 ¼ axn eyn þ b, one has

xn P b;

n ¼ 1; 2; . . . ;

xnþ1 6 axn þ b;

n ¼ 0; 1; 2; . . . :

Þ be an equilibrium point of the Definition 1. Let ð x; y system (2).

Consider the difference equation

znþ1 ¼ azn þ b;

n ¼ 0; 1; 2; . . . ; n

a Þ with an initial condition z0 . Then, zn ¼ an z0 þ bð1 . 1a Assume that 0 < a < 1, then it follows that zn 6 z0 þ 1b a for all n ¼ 1; 2; . . .. Hence, fzn g is a bounded sequence. Taking z0 ¼ x0 , then by comparison one has xn 6 1b a for all n ¼ 1; 2; . . .. Moreover, from ynþ1 ¼ axn ð1  eyn Þ, we obtain

ynþ1 6 axn 6

ab : 1a

ab Hence, xn 6 1b a and yn 6 1 a for all n ¼ 1; 2; . . .. h

Theorem 2. Let fðxn ; yn Þg1 be a positive solution of the   n¼0  ab  system (1). Then, b; 1b a  0; 1 a is invariant set for system (1). Proof. Let fðxn ; yn Þg1 of thesystem n¼0 be a positive solution  (1) with initial conditions x0 2 I ¼ b; 1b a and  ab  y0 2 J ¼ 0; 1 a . Then, from the system (1)

x1 ¼ ax0 ey0 and

b þb6 ; 1a

; 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 be an equilibrium point of a map x; y Definition 2. Let ð Fðx; yÞ ¼ ðf ðx; yÞ; gðx; yÞÞ, where f and g are continuously Þ. The linearized system of differentiable functions at ð x; y Þ is given by (2) about the equilibrium point ð x; y

X nþ1 ¼ FðX n Þ ¼ F J X n ; where X n ¼



xn yn



and F J is Jacobian matrix of system (2)

Þ. about the equilibrium point ð x; y The Jacobian matrix of linearized system of (1) about Þ is given by the fixed point ð x; y

121

Q. Din / Chaos, Solitons & Fractals 59 (2014) 119–128

Þ ¼ F J ðx; y



 axey :   y að1  e Þ axe

aey

Proof

 y

Proposition 1 [11]. Assume that X nþ1 ¼ FðX n Þ; n ¼ 0;  be a fixed 1; . . ., is a system of difference equations and X  point of F. If all eigenvalues of the Jacobian matrix J F about X  is locally lie inside the open unit disk jkj < 1, then X asymptotically stable. If one of them has a modulus greater  is unstable. than one, then X

Lemma 1 ([8,9]). Consider the second-degree polynomial equation

k2 þ pk þ q ¼ 0;

(i) The Jacobian matrix of linearized system of (1) about  the fixed point 1b a ; 0 is given by

FJ

#  " ab a  1 b a : ;0 ¼ ab 1a 0



1a

Obviously, the roots of characteristic polynomial of  F J 1b a ; 0 are given by

k1 ¼ a < 1;

k2 ¼

ab < 1; 1a

iff

ab < 1  a:

(ii) The proof of (ii) is obvious. h

ð3Þ

where p and q are real numbers. Then, the necessary and sufficient condition for both roots of the Eq. (3) to lie inside the open disk jkj < 1 is

Arguing as in [3], we take the following theorems for local asymptotic stability of positive equilibrium point of the system (1).

jpj < 1 þ q < 2:

Theorem 5. The unique positive equilibrium point h i h i ab Þ 2 b; 1b a  0; 1 ð x; y a of system (1) is locally asymptoti-

Theorem 3. Assume that 0 < a < 1 and ab > 1  a. Then, the system (1) has a unique positive equilibrium point    ab  Þ 2 b; 1b a  0; 1 ð x; y a .

r cally stable if bð1 þ aÞ þ r < 1r , where r ¼ bx.

Proof. Consider the following system of equations

x ¼ axey þ b;

y ¼ axð1  ey Þ:

Clearly, ðx; yÞ ¼



b ;0 1a



ð4Þ

is a solution of the system (4).

Moreover, from (4), one has

y x¼ ; að1  ey Þ



lim f ðxÞ ¼ 1;

x!bþ

ax

where

f ðxÞ ¼ ln





ax xb

and

lim FðxÞ ¼ 1;

x!1b a

  Hence, FðxÞ has at least one positive solution in b; 1b a . Moreover,

F0ðxÞ ¼

aðef ðxÞ  1Þ2

 ¼ axð1  ey Þ: y

As pointed out in [3], it is convenient to discuss stability behavior in terms of the quantity r where r ¼ bx. The equilibrium value r ¼ bx is of interest in modeling as being the x with b. This gives ratio of the steady-state 

1r

a

;

b  ¼ ða þ r  1Þ: y r

In terms of the quantity r the positive equilibrium point ; y Þ of (1) is given by ðx

1  a  ab < 0: að1  aÞ

  2 ef ðxÞ ef ðxÞ  f ðxÞ  1 f 0ðxÞ  a ef ðxÞ  1

ð5Þ

Þ be unique positive equilibrium point of Assume that ð x; y the system (1), then form the system (1) we have

ey ¼

x!bþ

and

lim F ðxÞ ¼

 PðkÞ ¼ k2  axey þ aey k þ a2 xey :

x ¼ axey þ b;

 : y ¼ ln xb

f ðxÞ Set FðxÞ ¼ a 1e  x, ð f ðxÞ Þ  b  x 2 b; 1a . Then,

Proof. The characteristic equation of Jacobian matrix Þ about ð Þ is given by F J ð x; y x; y

;

b where f 0ðxÞ ¼ xðbxÞ < 0 for b < x < 1b a. It follows that F0ðxÞ < 0 for b 6 x 6 1b a. Hence, FðxÞ has a unique positive   solution in b; 1b a . The proof is therefore completed. h

Þ ¼ ðx; y

In order to verify the consistency of the ratio r ¼ bx, it suffices to show that

      b b b ab  0; : ; ða þ r  1Þ 2 b; r r 1a 1a From 0 < ey < 1, it follows that

0< Theorem 4. Assume that 0 < a < 1, then following statements are true:  (i) The equilibrium point 1b a ; 0 of the system (1) is locally asymptotically stable if and only if ab < 1  a.  (ii) The equilibrium point 1b a ; 0 of the system (1) is unstable if ab > 1  a.

  b b ; ða þ r  1Þ : r r

1r

a

< 1:

ð6Þ

From (6), one can easily obtain that 1  a < r < 1; i:e., b < br < 1b a. Furthermore, using the inequality 1  a < r < 1, we obtain that

b ab 0 < ða þ r  1Þ < : r 1a

122

Q. Din / Chaos, Solitons & Fractals 59 (2014) 119–128

Let

RðkÞ ¼ k2 ;



SðkÞ ¼ axey þ ae

 y

k  a2 xey :

r Assume that bð1 þ aÞ þ r < 1r , and jkj ¼ 1. Then, one has

jSðkÞj 6 axey þ aey þ a2 xey ¼ aey ðx þ 1 þ axÞ   1r ðab þ b þ r Þ < 1: ¼ r Then, by Rouche’s Theorem RðkÞ and RðkÞ  SðkÞ have same number of zeroes in an open unit disk jkj < 1. Hence, both roots

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k1 ¼ ey a þ ax þ ðax þ aÞ2  4a2 xey ; 2 and

k2 ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 y e a þ ax  ðax þ aÞ2  4a2 xey ; 2

of (5) lie in an open disk jkj < 1, and it follows from Þ is locally Proposition 1 that the equilibrium point ð x; y asymptotically stable. h The following theorem shows necessary and sufficient condition for local asymptotic stability of unique positive equilibrium point of the system (1). Theorem 6. The unique positive equilibrium point h i h i ab Þ 2 b; 1b a  0; 1 ð x; y a of system (1) is locally asymptotically stable if and only if

ð1  rÞðr þ bÞ < r þ abð1  rÞ < 2r; where r ¼

ð7Þ

b .  x

xey þ aey and q ¼ a2  xey , then (5) can be Proof. Let p ¼ a written as 2

PðkÞ ¼ k  pk þ q:

Þ is called global attractor if (i) An equilibrium point ð x; y Þ as n ! 1. ðxn ; yn Þ ! ð x; y Þ is called asymptotic glo(ii) An equilibrium point ð x; y bal attractor if it is a global attractor and stable. We want to analyze the global stability of the unique positive equilibrium point of the system (1). For this, first we prove a general result for such systems. Arguing as in [9], we take the following theorem: Theorem 7. 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: (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 ; M 2 Þ 2 I2  J 2 is a solution of the system

m1 ¼ f ðm1 ; M2 Þ;

M 1 ¼ f ðM1 ; m2 Þ

m2 ¼ gðm1 ; m2 Þ;

M 2 ¼ gðM1 ; M 2 Þ

such that m1 ¼ M 1 , and m2 ¼ M2 . Then, there exists Þ of the system (2) em such exactly one equilibrium point ð x; y Þ. that limn!1 ðxn ; yn Þ ¼ ð x; y 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; M02 ¼ d such that

m1iþ1 ¼ f ðmi1 ; M i2 Þ;

Miþ1 ¼ f ðMi1 ; mi2 Þ; 1

and

m2iþ1 ¼ gðmi1 ; mi2 Þ;

M 2iþ1 ¼ gðMi1 ; M i2 Þ:

Then,

Then, jpj ¼ a xey þ aey and 1 þ a2  xey ¼ 1 þ q. It follows that

m01 ¼ a 6 f ðm01 ; M 02 Þ 6 f ðM 01 ; m02 Þ 6 b ¼ M 01 ;

jpj < 1 þ q < 2;

and

if and only if

m02 ¼ c 6 gðm01 ; m02 Þ 6 gðM 01 ; M02 Þ 6 d ¼ M 02 :

axey þ aey < 1 þ a2 xey < 2;

Moreover, one has

if and only if

m01 6 m11 6 M 11 6 M 01 ;

  rþb ab <1þ ð1  rÞ ð1  rÞ < 2; r r

and

if and only if ð1  rÞðr þ bÞ < r þ abð1  rÞ < 2r. Hence, from Lemma 1 the unique positive equilibrium point    ab  Þ 2 b; 1b a  0; 1 ð x; y a of the system (1) is locally asymptotically stable if and only if ð1  rÞðr þ bÞ < r þabð1  rÞ < 2r. h

m02 6 m12 6 M 12 6 M 02 : We similarly have

m11 ¼ f ðm01 ;M 02 Þ 6 f ðm11 ;M 12 Þ 6 f ðM11 ;m12 Þ 6 f ðM01 ;m02 Þ 6 M11 ; and

m12 ¼ gðm01 ; m02 Þ 6 gðm11 ; m12 Þ 6 gðM 11 ; M 12 Þ 6 gðM 01 ;M 02 Þ 6 M12 : 4. Global stability Þ be an equilibrium point of the Definition 3. Let ð x; y system (2).

Now observe that for each i  0,

a ¼ m01 6 m11 6    6 mi1 6 Mi1 6 M 1i1 6    6 M01 ¼ b; and

123

Q. Din / Chaos, Solitons & Fractals 59 (2014) 119–128

c ¼ m02 6 m12 6    6 mi2 6 M i2 6 M 2i1 6    6 M02 ¼ d: mi1

M i1 ,

mi2

and

M i2

Hence, 6 xn 6 and 6 yn 6 for n P 2i þ 1. Let m1 ¼ limn!1 mi1 ; M 1 ¼ limn!1 M i1 ; m2 ¼ limn!1 mi2 , and M 2 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;

M 1 ¼ f ðM 1 ; m2 Þ

m2 ¼ gðm1 ; m2 Þ;

M 2 ¼ gðM 1 ; M2 Þ

m2 ¼ am1 ð1  em2 Þ;

 M2 ¼ aM 1 1  eM2 :

ð9Þ

From the system (8), one has

em2 ¼

M1  b ; aM1

eM2 ¼

m1  b : am1

ð10Þ

From (9), one has

am1  m2 aM1  M2 ; eM2 ¼ : am1 aM1

Hence, m1 ¼ M1 ; m2 ¼ M 2 . h

em2 ¼

Theorem 8. Assume that 0 < a < 1, then the unique positive    ab  Þ 2 b; 1b a  0; 1 x; y equilibrium point ð is a global a attractor.

Furthermore, assuming as in the proof of Theorem 1.16 of [9], it suffices to suppose that

Proof. Let f ðx; yÞ ¼ axey þ b, and gðx; yÞ ¼ axð1  ey Þ. Then, it is easy to see that f ðx; yÞ is non-decreasing in x and non-increasing in y. Moreover, gðx; yÞ is nondecreasing 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;

M 1 ¼ f ðM 1 ; m2 Þ

m2 ¼ gðm1 ; m2 Þ;

M 2 ¼ gðM 1 ; M2 Þ

Then, one has

m1 ¼ am1 eM2 þ b;

M1 ¼ aM 1 em2 þ b;

ð8Þ

m1 6 M 1 ;

ð11Þ

m2 6 M 2 :

Moreover,

b 6 m1 6

b ; 1a

b 6 M1 6

b : 1a

ð12Þ

On subtracting (9) and using (10), we obtain

M 2  m2 ¼ aðM 1  m1 Þ þ am1 em2  aM1 eM2     M1  b m1  b  aM 1 ¼ aðM 1  m1 Þ þ am1 aM1 am1   b ðM1 þ m1 Þ ðM1  m1 Þ: ¼ a1þ M 1 m1

Fig. 1. Plots for the system (21).

124

Q. Din / Chaos, Solitons & Fractals 59 (2014) 119–128

Using (12), it follows that

 1 M2  m2 6 a  1 þ ðM 1 þ m1 Þ ðM 1  m1 Þ b ! 2 ð1  aÞ  2 ðm1  M 1 Þ: 6 1a

From (13) and (16), it follows that m2 ¼ M 2 . Similarly, one can show that m1 ¼ M1 . Hence, from Theorem 7 the unique positive equilibrium point of the system (1) is a global attractor. h



ð13Þ

Using the fact that eM2 6 em2 , from (10) and (11) we obtain

eM2 ¼

aM1  M2 M1  b 6 em2 ¼ ; aM1 aM1

Proof. The proof follows from Theorem 6 and Theorem 8. h

which implies that

aM1  M2 6 M1  b;

ð14Þ

and

eM2

Lemma 2. Assume that 0 < a < 1 and ð1  rÞðr þ bÞ < r þabð1  rÞ < 2r. Then, the unique positive equilibrium point ab Þ 2 b; 1b a  0; 1 ð x; y a of the system (1) is globally asymptotically stable.

5. Rate of convergence

m1  b am1  m2 ¼ 6 em2 ¼ ; am1 am1

which implies that

m1  b 6 am1  m2 :

ð15Þ

Adding (14) and (15) gives

1 m1  M 1 6 ðM2  m2 Þ: 1a

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

X nþ1 ¼ ðA þ BðnÞÞX n ; ð16Þ

Fig. 2. Plots for the system (22).

ð17Þ

125

Q. Din / Chaos, Solitons & Fractals 59 (2014) 119–128

where X n is an m-dimensional vector, A 2 C mm is a constant matrix, and B : Zþ ! C mm is a matrix function satisfying

kBðnÞk ! 0

ð18Þ

as n ! 1, where k  k denotes any matrix norm which is associated with the vector norm

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  x, and limn!1 yn ¼ y x 2 b; 1b a and limn!1 xn ¼   ab   2 0; 1a . To find the error terms, one has from the y system (1)

xnþ1  x ¼ axn eyn  axey

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kðx; yÞk ¼ x2 þ y2 :

axðeyn ey Þ Þ; ðy n  y ¼ aeyn ðxn  xÞ þ  yn  y

and Proposition 2 (Perron’s theorem [10]). Suppose that condition (18) holds. If X n is a solution of (17), then either X n ¼ 0 for all large n or

q ¼ n!1 lim ðkX n kÞ1=n

ð19Þ

exists and is equal to the modulus of one the eigenvalues of matrix A. Proposition 3 [10]. Suppose that condition (18) holds. If X n is a solution of (17), then either X n ¼ 0 for all large n or

q ¼ n!1 lim

kX nþ1 k kX n k

ð20Þ

 ¼ axn ð1  eyn Þ  axð1  ey Þ ynþ1  y axðey eyn Þ Þ: ¼ að1  eyn Þðxn  xÞ þ ðyn  y  yn  y , then one has Let e1n ¼ xn   x, and e2n ¼ yn  y

e1nþ1 ¼ an e1n þ bn e2n ; and

e2nþ1 ¼ cn e1n þ dn e2n ; an ¼ aeyn ; bn ¼ y   axðe eyn Þ . dn ¼  yn y where

Moreover,

Fig. 3. Plots for the system (23).

axðeyn ey Þ  yn y

; cn ¼ að1  eyn Þ,

and

126

Q. Din / Chaos, Solitons & Fractals 59 (2014) 119–128

lim an ¼ aey ;

n!1

lim bn ¼ axey ;

Open Problem: Assume that a  1 and b > 0. Find necessary and sufficient conditions for which the unique positive equilibrium point of the system (1) is globally asymptotically stable.

n!1

and

 lim cn ¼ a 1  ey ;

n!1

lim dn ¼ axey :

n!1

Now the limiting system of error terms can be written as

"

e1nþ1 e2nþ1

#

 ¼

"  y

aey

axe að1  e Þ axey  y

e1n e2n

# ;

which is similar to linearized system of (1) about the equiÞ. librium point ð x; y Using proposition (2), one has following result. Theorem 9. 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  h i h i b a b  2 0; 1a . Then, the error vector where  x 2 b; 1a and y  1 e en ¼ n2 of every solution of (1) satisfies both of the en following asymptotic relations 1

Þj; lim ðken kÞn ¼ jk1;2 F J ðx; y

n!1

lim

n!1

kenþ1 k Þj; ¼ jk1;2 F J ðx; y ken k

Þ are the characteristic roots of Jacobian x; y where k1;2 F J ð Þ. matrix F J ð x; y For the readers interested in the area we leave the following open problem.

6. Examples In order to verify our theoretical results and to support our mathematical discussions, we consider some interesting numerical examples in this section. These examples represent different types of qualitative behavior of the system (1). First three examples show the existence and uniqueness of positive equilibrium point of the system (1). Furthermore, the unique positive equilibrium point of the system (1) is locally asymptotically stable if and only if ð1  rÞðr þ bÞ < r þ abð1  rÞ < 2r with 0 < a < 1. The last two examples show that the system (1) has unique positive equilibrium point which is globally asymptotically stable even for a  1. Hence, last two examples support our open problem presented in this paper. Example 1. Let a ¼ 0:93, and b ¼ 0:16. Then, the system (1) can be written as

xnþ1 ¼ 0:93xn eyn þ 0:16;

ynþ1 ¼ 0:93xn ð1  eyn Þ;

with initial conditions x0 ¼ 1:1;

Fig. 4. Plots for the system (24).

y0 ¼ 0:08.

ð21Þ

Q. Din / Chaos, Solitons & Fractals 59 (2014) 119–128

In this case the unique positive equilibrium point Þ ¼ ð1:11975; 0:0816178Þ. Furthermore, ð1  rÞðr þ bÞ x; y ð ¼ 0:25961; r þ abð1  rÞ ¼ 0:270428 and 2r ¼ 0:285779. Hence, necessary and sufficient condition (7) for local asymptotic stability is satisfied, i:e., ð1  rÞðr þ bÞ ¼ 0:25961 < r þ abð1  rÞ ¼ 0:270428 < 2r ¼ 0:285779. Moreover, in Fig. 1 the plot of xn is shown in Fig. 1a, plot of yn is shown in Fig. 1b and an attractor of the system (21) is shown in Fig. 1c. Example 2. Let a ¼ 0:985, and b ¼ 0:031. Then, the system (1) can be written as

xnþ1 ¼ 0:985xn eyn þ 0:031; ynþ1 ¼ 0:985xn ð1  eyn Þ;

ð22Þ

with initial conditions x0 ¼ 1:001; y0 ¼ 0:001. In this case the unique positive equilibrium point Þ ¼ ð1:02319; 0:0156521Þ. Furthermore, ð1  rÞðr þ bÞ x; y ð ¼ 0:0594403; r þ abð1  rÞ ¼ 0:0599073 and 2r ¼ 0:060 5948. Hence, necessary and sufficient condition (7) for local asymptotic stability is satisfied, i:e., ð1  rÞ ðr þ bÞ ¼ 0:0594403 < r þ abð1  rÞ ¼ 0:0599073 < 2r ¼ 0:0605948. Moreover, in Fig. 2 the plot of xn is shown in Fig. 2a, plot of

127

yn is shown in Fig. 2b and an attractor of the system (22) is shown in Fig. 2c.

Example 3. Let a ¼ 0:999999, and b ¼ 0:006. Then, the system (1) can be written as

xnþ1 ¼ 0:999999xn eyn þ 0:006; ynþ1 ¼ 0:999999xn ð1  eyn Þ;

ð23Þ

with initial conditions x0 ¼ 1; y0 ¼ 0:004. In this case the unique positive equilibrium point Þ ¼ ð1:003; 0:005999Þ. Furthermore, ð1  rÞðr þ bÞ ¼ x; y ð 0:0119104; r þ abð1  rÞ ¼ 0:0119462 and 2r ¼ 0:011 9641. Hence, necessary and sufficient condition (7) for local asymptotic stability is satisfied, i:e., ð1  rÞðr þ bÞ ¼ 0:0119104 < r þabð1  rÞ ¼ 0:0119462 < 2r ¼ 0:0119641. Moreover, in Fig. 3 the plot of xn is shown in Fig. 3a, plot of yn is shown in Fig. 3b and an attractor of the system (23) is shown in Fig. 3c.

Example 4. The computer simulations indicate that for some values of parameter a  1 the system (1) has unique positive equilibrium point. As an example, we take

Fig. 5. Plots for the system (25).

128

Q. Din / Chaos, Solitons & Fractals 59 (2014) 119–128

a ¼ 1:0001 and b ¼ 0:0005. In this case the system (1) can be written as xnþ1 ¼ 1:0001xn eyn þ 0:0005;

ynþ1 ¼ 1:0001xn ð1  eyn Þ; ð24Þ

with initial conditions x0 ¼ 1:0002; y0 ¼ 0:0006. From computer simulations the unique positive equilibrium point of the system (23) is approximated by ð1:0002 000200009757; 0:0006000200020054085Þ (after 500000 iterations). Moreover, in Fig. 4 the plot of xn is shown in Fig. 4a, plot of yn is shown in Fig. 4b and an attractor of the system (24) is shown in Fig. 4c. Example 5. Taking a ¼ 1:05 and b ¼ 0:09. In this case the system (1) can be written as

xnþ1 ¼ 1:05xn eyn þ 0:09;

ynþ1 ¼ 1:05xn ð1  eyn Þ;

ð25Þ

with initial conditions x0 ¼ 1:05; y0 ¼ 0:5. From computer simulations the unique positive equilibrium point of the system (25) is approximated by ð1:02113; 0:141056Þ. Moreover, in Fig. 5 the plot of xn is shown in Fig. 5a, plot of yn is shown in Fig. 5b and an attractor of the system (25) is shown in Fig. 5c.

7. Conclusions This work is related to the qualitative behavior of a discrete-time population model. More precisely, we solve an open problem proposed in [8] related to the global behavior of the positive solutions of the population model (1). We proved that the system (1) has a unique positive equi   ab  Þ 2 b; 1b a  0; 1 librium point ð x; y a , 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 positive steady states of the system (1) are locally asymptotically stable if and only if ð1  rÞðr þ bÞ < r þ abð1  rÞ < 2r.  Moreover, the equilibrium point 1b a ; 0 of the system (1) is locally asymptotically stable if and only if ab < 1  a. 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. Particularly, the condition for global stability in population biology is a very interesting mathematical problem. Usually the biologists believe that a unique, positive, locally asymptotically stable equilibrium point in an ecological system is very important in biological point of view. Therefore, it is very important to find necessary and sufficient conditions which may guarantee the global stability of the unique positive equilibrium point of the given model. In the paper, we prove the necessary and sufficient condition for the global asymptotic stability of the    ab  Þ 2 b; 1b a  0; 1 unique positive equilibrium point ð x; y a of 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 necessary and sufficient condition for global asymptotic Þ stability of the unique positive equilibrium point ð x; y    ab  of the system (1). The experimental ver2 b; 1b a  0; 1 a ifications from computer simulations show that the system (1) has stable steady states for some parametric values a  1 (see Fig. 4 and Fig. 5) but proofs are yet to be completed. Hence, to investigate the global character of all positive solutions of the system (1) for a  1; b > 0 is an other open problem. Acknowledgments The author thanks 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. References [1] Allen Linda JS. An introduction to mathematical biology. Prentice Hall; 2007. [2] Brauer F, Castillo-Chavez C. Mathematical models in population biology and epidemiology. Springer; 2000. [3] Edelstein-Keshet L. Mathematical models in biology. McGraw-Hill; 1988. [4] Ahmad S. On the nonautonomous Lotka-Volterra competition equation. Proc Amer Math Soc 1993;117:199–204. [5] Tang X, Zou X. On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments. Proc Amer Math Soc 2006;134:2967–74. [6] Zhou Z, Zou X. Stable periodic solutions in a discrete periodic logistic equation. Appl Math Lett 2003;16(2):165–71. [7] Liu X. A note on the existence of periodic solution in discrete predator-prey models. Appl Math Model 2010;34:2477–83. [8] Kulenovic´ MRS, Ladas G. Dynamics of second order rational difference equations. Chapman and Hall/CRC; 2002. [9] Grove EA, Ladas G. Periodicities in nonlinear difference equations. Boca Raton: Chapman and Hall/CRC Press; 2004. [10] Pituk M. More on Poincare’s and Perron’s theorems for difference equations. J Differ Equ Appl 2002;8:201–16. [11] Sedaghat H. Nonlinear difference equations: theory with applications to social science models. Dordrecht: Kluwer Academic Publishers; 2003. [12] 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. [13] Papaschinopoulos G, Radin MA, Schinas CJ. On the system of two difference equations of exponential form: xnþ1 ¼ a þ bxn1 eyn ; ynþ1 ¼ c þ dyn1 exn . Math Comput Model 2011;54:2969–77. [14] Papaschinopoulos G, Schinas CJ. On the dynamics of two exponential type systems of difference equations. Comput Math Appl 2012;64(7):2326–34. [15] Din Q. Dynamics of a discrete Lotka-Volterra model. Adv Differ Equ 2013;1:95. [16] Din Q, Donchev T. Global character of a host-parasite model. Chaos Solitons Fractals 2013;54:1–7. [17] Din Q, Qureshi MN, Khan AQ. Dynamics of a fourth-order system of rational difference equations. Adv Differ Equ 2012;1:1–15. [18] Din Q. Global behavior of a rational difference equation. Acta Univ Apulensis 2012;30:35–49. [19] Din Q, Khan AQ, Qureshi MN. Qualitative behavior of a hostpathogen model. Adv Differ Equ 2013;1:263. [20] Khan AQ, Qureshi MN, Din Q. Global dynamics of some systems of higher-order rational difference equations. Adv Differ Equ 2013;2013:354.