bifurcations of a class of singular biological economic models

Available online at www.sciencedirect.com

Chaos, Solitons and Fractals 40 (2009) 1309–1318

www.elsevier.com/locate/chaos

Bifurcations of a class of singular biological economicmodels q

Xue Zhang *, Qing-ling Zhang, Yue Zhang

Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110004, PR China

Key Laboratory of Integrated Automation of Process Industry(Northeastern Univ.), Ministry of Education, Shenyang,

Liaoning 110004, PR China

Accepted 3 September 2007

Communicated by Prof. Ji-Huan He

Abstract

This paper studies systematically a prey–predator singular biological economic model with time delay. It shows thatthis model exhibits two bifurcation phenomena when the economic profit is zero. One is transcritical bifurcation whichchanges the stability of the system, and the other is singular induced bifurcation which indicates that zero economicprofit brings impulse, i.e., rapid expansion of the population in biological explanation. On the other hand, if the eco-nomic profit is positive, at a critical value of bifurcation parameter, the system undergoes a Hopf bifurcation, i.e., theincrease of delay destabilizes the system and bifurcates into small amplitude periodic solution. Finally, by using Matlabsoftware, numerical simulations illustrate the effectiveness of the results obtained here. In addition, we study numeri-cally that the system undergoes a saddle-node bifurcation when the bifurcation parameter goes through critical value ofpositive economic profit.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

At present, mankind is facing the problems about shortage of resource and worsening environment. So there hasbeen rapidly growing interests in the analysis and modeling of biological system. From the view of human needs,the exploitation of biological resources and harvest of population are commonly practiced in the fields of fishery, wild-life and forestry management. Kar and Pahari [1], Xiao et al. [2] and Kumar et al. [3] have studied the dynamics inpredator–prey models with harvesting and obtained complex dynamic behavior, such as stability of equilibrium points,Bogdanov–Takens bifurcation, Hopf bifurcation, limit cycle, heteroclinic bifurcation and so on. Refs. [4–7] have dis-cussed the predator–prey model with time delay and analyzed the effects of time delay on model dynamics, such asthe time delay may change the stability of equilibrium points and even cause a switching of stabilities. In addition, there

0960-0779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2007.09.010

q This work was supported by National Science Foundation of China (60574011).* Corresponding author. Tel.: +86 24 83689929.

E-mail addresses: [email protected] (X. Zhang), [email protected] (Q.-l. Zhang), [email protected](Y. Zhang).

mailto:[email protected]



1310 X. Zhang et al. / Chaos, Solitons and Fractals 40 (2009) 1309–1318

is also a considerable literature on discrete dynamical systems, e.g., see [8–11] and references therein, modeling somespecies whose generations are non-overlapping by applying Euler scheme to differential equations. Most of these dis-cussions on biological models are based on normal systems governed by differential equations or difference equations.

In daily life, economic profit is a very important factor for governments, merchants and even every citizen, so it isnecessary to research biological economic systems, which are always described by differential–algebraic equations.However, to our best knowledge, the reports of such systems are few in biology. Considering the economic theoryof fishery resource [12] proposed by Gordon in 1954, this paper studies a class of predator–prey singular biological eco-nomic model with Holling type II functional form as follows:

d~xd~t¼ ~x r1 � a11~x�

~yaþ ~x

� �;

d~yd~t¼ ~y �r2 þ b~x

aþ ~x� a22~y� �

� eE~y;

0 ¼ eEð~p~y � ~cÞ � m;

8>>>><>>>>: ð1Þ

where ~x and ~y represent the prey density and predator density at time ~t, respectively; the Holling type II function re-sponse given by b~x

aþ~x represents the attack rate per predator; r1 > 0 and �r2 < 0 are the intrinsic growth rate and intrinsicmortality rate of respective species; aii > 0, i = 1, 2 represent the strength of respective intraspecific competition; eE rep-resents capture capability, eE~y indicates that the harvest of predator is proportional to its density at time ~t; ~p > 0, ~c > 0and m > 0 are the harvest reward, the cost and the economic profit, respectively.

In this paper, we assume reasonable biological conditions as follows:

b > r2; r1 > aa11: ð2Þ
In order to model the predator’s gestation period, we will replace ~x in predator’s equation by
~zð~tÞ ¼Z ~t

�1~xðsÞ 1eT exp � 1eT ð~t � sÞ

� �ds;

where the entire past history of ~x is taken into account and the importance of individual events is measured by1eT exp 1eT ð~t � sÞ

� �; where eT is the measure of the influence of the past, s < ~t is some particular time in the past and ~t

is current time. Therefore, we will study the following singular system:

d~xd~t¼ ~x r1 � a11~x�

~yaþ ~x

� �;

d~yd~t¼ ~y �r2 þ b~z

aþ ~z� a22~y� �

� eE~y;

d~zd~t¼ 1eT ð~x� ~zÞ;

0 ¼ eEð~p~y � ~cÞ � m:

8>>>>>>>><>>>>>>>>:ð3Þ

We non-dimensionalize the system (3) with the following scaling:

x ¼ ~xa; y ¼ ~y

ar1

; z ¼ ~za; E ¼

eEr1

; t ¼ r1~t;

and then obtain the form

dxdt¼ x 1� a1x� y

1þ x

� �;

dydt ¼ y �r þ b1z

1þ z� a2y � E� �

;

dzdt ¼

1T ðx� zÞ;

0 ¼ Eðpy � cÞ � m;

8>>>>>>><>>>>>>>:ð4Þ

where the non-dimensional parameters are defined as

a1 ¼a11ar1

; a2 ¼ aa22; b1 ¼br1

; c ¼ ~cr1; r ¼ r2

r1

; p ¼ ~par21; T ¼ r1

eT :
Now, the reasonable conditions (2) become
b1 > r; a1 < 1: ð5Þ

X. Zhang et al. / Chaos, Solitons and Fractals 40 (2009) 1309–1318 1311

For simplicity, let

f ðX ;E; lÞ ¼f1ðX ;E; lÞf2ðX ;E; lÞf3ðX ;E; lÞ

0B@1CA ¼

x 1� a1x� y1þ x

� �y �r þ b1z

1þ z� a2y � E� �

1T ðx� zÞ

0BBBB@1CCCCA;

gðX ;E; lÞ ¼ Eðpy � cÞ � m;

where X = (x,y,z)T, l is a bifurcation parameter, which will be defined in what follows.In this paper, we mainly discuss the effects of time delay and economic profit on the model dynamics in the region

R3þ ¼ fðx; y; z;EÞjx P 0; y P 0; z P 0;E P 0g. In Section 2, under the condition of zero economic profit, we study the

existence and stability of equilibrium points and give sufficient conditions of existence for transcritical bifurcation andsingular induced bifurcation. Considering positive economic profit, Section 3 discusses Hopf Bifurcation and provesthat the increase of time delay destabilizes the system and bifurcates into small amplitude periodic solution. In Section4, numerical simulations verify the effectiveness of mathematical conclusions and existence of saddle-node bifurcation.Finally, we give remarks to conclude this paper in Section 5.

2. The model with zero economic profit

When the economic profit is zero, the system (4) can be written by


1þ x

� �;

dydt ¼ y �r þ b1z

1þ z� a2y � E� �

;

dzdt ¼

1T ðx� zÞ;

0 ¼ Eðpy � cÞ:

8>>>>>>>><>>>>>>>>:ð6Þ

By the analysis of roots for the system (6), we obtain the following theorem:

Theorem 1

(1) The system (6) has two equilibrium points P0(0,0,0,0) and P 11a1; 0; 1

a1; 0

� �for any positive parameters.

(2) If b1 > r(1 + a1), there exists another equilibrium point P 2ð�x; �y;�z; 0Þ, where �x is the intersection of the following null-

clines:

y ¼ ð1þ xÞð1� a1xÞ;

y ¼ 1

a2

�r þ b1x1þ x

� �;

�y ¼ ð1þ �xÞð1� a1�xÞ and �z ¼ �x.(3)

(i) If D1 > 0 andb1xð1Þ

3

1þxð1Þ3

� r � a2cp > 0, there exist two positive equilibrium points P ð1Þ3 ðx

ð1Þ3 ; y�; zð1Þ3 ;Eð1Þ3 Þ and

P ð2Þ3 ðxð2Þ3 ; y�; zð2Þ3 ;Eð2Þ3 Þ;

(ii) If D1 > 0;b1xð1Þ

3

1þxð1Þ3

� r � a2cp < 0 and

b1xð2Þ3

1þxð2Þ3

� r � a2cp > 0, there exists an unique positive equilibrium point

P 3ðxð2Þ3 ; y�; zð2Þ3 ;Eð2Þ3 Þ. In both the cases (i) and (ii), D1 = p2(1 � a1)2 � 4pa1(c � p), and xðiÞ3 ; i ¼ 1; 2 are roots

of the following equation:

pa1x2 þ pða1 � 1Þxþ c� p ¼ 0;

and satisfy xð1Þ3 < xð2Þ3 ; y� ¼ cp ; zðiÞ3 ¼ xðiÞ3 ; EðiÞ3 ¼

b1xðiÞ3

1þxðiÞ3

� r � a2cp ;

(iii) If D1 = 0 and b1ð1�a1Þ1þa1

� r � a2cp > 0, the unique positive equilibrium point is P 3

1�a1

2a1; c

p ;1�a1

2a1; b1ð1�a1Þ

1þa1� r � a2c

p

� �.


From the system (6), we get

A ¼ DX f � DEf ðDEgÞ�1DX g ¼

1� 2a1x� yð1þxÞ2 � x

1þx 0

0 �r þ b1z1þz� 2a2y � E þ pEy

py�cb1yð1þzÞ2

1T 0 � 1

T

0BB@1CCA:

The characteristic polynomial of the matrix A is

jkI � Aj ¼ k3 þ p1ðX ;EÞk2 þ p2ðX ;EÞkþ p3ðX ;EÞ ¼ 0;

where

p1ðX ;EÞ ¼ �1þ 2a1xþ y

ð1þ xÞ2þ 1

Tþ r � b1z

1þ zþ 2a2y þ E � pEy

py � c;

p2ðX ;EÞ ¼ r � b1z1þ z

þ 2a2y þ E � pEypy � c

� ��1þ 2a1xþ y

ð1þ xÞ2þ 1

T

!

þ 1

T�1þ 2a1xþ y

ð1þ xÞ2

!;

p3ðX ;EÞ ¼1

Tr � b1z

1þ zþ 2a2y þ E � pEy

py � c

� ��1þ 2a1xþ y

ð1þ xÞ2

!

þ b1xy

T ð1þ xÞð1þ zÞ2:

So we obtain easily that the equilibrium point P0 is a saddle for any positive parameters and if b1 < r(1 + a1), theequilibrium point P1 is a stable focus or node; otherwise, P1 is a saddle. For equilibrium points P2 and P3, if they satisfyp1 > 0 and p1p2 > p3 > 0, they are stable focus or node.

From the analysis above, we can see that the stability of equilibrium point P1 changes from stable to unstable whenb1 increases through r(1 + a1). Therefore, if b1 is regarded as bifurcation parameter, i.e., l = b1, the following Theorem2 is obtained:

Theorem 2. If a2ð1þ a1Þ2 þ ra21–0, the system (6) undergoes transcritical bifurcation at the equilibrium point P1 when

bifurcation parameter l is increased through r(1 + a1).

Proof. When the bifurcation parameter l = b1 = r(1 + a1),

ðDX fRÞP 1¼ AP 1

¼�1 � 1

1þa10

0 0 01T 0 � 1

T

0B@1CA

has a simple zero eigenvalue with right eigenvector x2 = (�1,1 + a1,�1)T and left eigenvector x1 = (0,1,0); and

ðx1ðDX DlfRÞx2ÞP 1¼ 0 1 0ð Þ

0 0 0

0 z1þz

yð1þzÞ2

0 0 0

0B@1CA �1

1þ a1

�1

0B@1CA ¼ 1;

ðx1D2X fRðx2;x2ÞÞP 1

¼ x1

X3

i¼1

ðeixT2 DX ðDX fiÞTx2Þ

!P 1

¼ �2a2ð1þ a1Þ2 � 2ra21;

where ei is unit vector.According to the literature [13], the system (6) undergoes transcritical bifurcation at the equilibrium point P1. This

completes the proof. h

When the economic profit m = 0, detDEg = 0 at the equilibrium point P3. In succession, we discuss the bifurcationbehavior regarding m as the bifurcation parameter, i.e., l = m.

Theorem 3. Assume �1þ 2cpð1þa1Þ < 0 and �a1 þ y�

ð1þx�Þ2 –0. When the bifurcation parameter l increases through 0, the

system (6) undergoes singular induced bifurcation at the equilibrium point P3 and the stability of the equilibrium point P3

changes, i.e., from stable to unstable.


Proof. Let D = DEg = py � c. It has a simple zero eigenvalue and

DX f DEf

DX g DEg

�� P 3

¼

1� 2a1x� yð1þ xÞ2

� x1þ x 0 0

0 �r þ b1z1þ z� 2a2y � E b1y

ð1þ zÞ2�y

1T 0 � 1

T 0

0 pE 0 py � c

��

��P 3

¼ � pT

E�x�y� �a1 þy�

ð1þ x�Þ2

!;

DX f DEf Dlf

DX g DEg Dlg

DX D DED DlD

��P 3

¼

1� 2a1x� yð1þ xÞ2

� x1þ x 0 0 0

0 �r þ b1z1þ z� 2a2y � E b1y

ð1þ zÞ2�y 0

1T 0 � 1

T 0 0

0 pE 0 py � c �1

0 p 0 0 0

��

��P 3

¼ � pT

x�y� �a1 þy�

ð1þ x�Þ2

!;

traceðDEf adjðDEgÞDX gÞP 3¼ trace

0

�y

0

0B@1CA 0 pE 0ð Þ

0B@1CA

P 3

¼ �pE�y�:

On the other hand, we get

c1 ¼ �traceðDEf adjðDEgÞDX gÞP 3¼ pE�y� > 0;

c2 ¼ DlD� DX D DEDð ÞDX f DEf

DX g DEg

!�1Dlf

Dlg

!0@ 1AP 3

¼ 1

E�> 0:

According to the Theorem 3 in [14], the system (6) undergoes singular induced bifurcation at the equilibrium point P3

when the bifurcation parameter l = m = 0. When l increases through 0, one eigenvalue of the system (6) moves fromC� to C+ along the real axis by diverging through1. It brings impulse, i.e., rapid expansion of the population in bio-logical explanation. By a simple calculation, the others remain bounded and stay at C� away from the origin. There-fore, the stability of the system (6) changes from stable to unstable at the equilibrium point P3 when the economic profitincreases through 0. This completes the proof. h

3. The model with positive economic profit

In this section, we investigate how time delay influences the stability of the system (4). For simplicity, we discuss thesystem (4) without intraspecific competition of predator, i.e., a2 = 0.

3.1. The system (4) without time delay

We consider the system (4) without time delay, which can be written as follows:


1þ x

� �;

dydt ¼ y �r þ b1x

1þ x� E� �

;

0 ¼ Eðpy � cÞ � m:

8>>>><>>>>: ð7Þ


If the equilibrium point of the system (7) is eP 4ðx; y;EÞ, coordinate of x is the root of the following equation:

f ðxÞ ¼ Ax3 þ Bx2 þ Cxþ D ¼ 0; ð8Þ

where A = pa1(b1 � r), B = � p(b1 � r � a1b1 + 2a1r), C = m � pb1 + 2pr + cb1 � cr � a1rp, D = r(p � c) + m.By the analysis of the roots for the Eq. (8), we can obtain that if there exists m = m* > 0 satisfying B2 > 3AC and

f �BþffiffiffiffiffiffiffiffiffiffiffiffiB2�3ACp

3A

� �¼ 0, the number of equilibrium point changes from two to zero when m increases through m*, i.e., there

are two positive equilibrium points when m < m*, no positive equilibrium points when m > m*, and only one positiveequilibrium point when m = m*. It is difficulty to compute analytically the value m*, hence we will study saddle-nodebifurcation numerically in the next section.

For the equilibrium point eP 4, we can get

A ¼�a1xþ xy

ð1þ xÞ2� x

1þ x

b1yð1þ xÞ2

pEypy � c

0BB@1CCA:

Its characteristic polynomial is

jkI � Aj ¼ k2 � ðr1 þ r4Þkþ r1r4 � r2r3 ¼ 0;

where r1 ¼ �a1xþ xyð1þxÞ2 ; r2 ¼ � x

1þx ; r3 ¼ b1yð1þxÞ2 ; r4 ¼ pEy

py�c.Assume

r1 þ r4 < 0 and r1r4 � r2r3 > 0; ð9Þ

then eP 4 is an asymptotically stable equilibrium point of the system (7).

3.2. The analysis of the system (4)

Assume that the positive equilibrium point of the system (4) is P4(x,y,z,E), where z = x and others coordinates arethe same as those of eP 4, then the characteristic polynomial, at the equilibrium point P4, is

jkI � Aj ¼ k3 þ m1ðx; y; z;EÞk2 þ m2ðx; y; z;EÞkþ m3ðx; y; z;EÞ ¼ 0;

where

m1ðx; y; z;EÞ ¼ a1x� xy

ð1þ xÞ2þ 1

T� pEy

py � c; ð10Þ

m2ðx; y; z;EÞ ¼ �pEy

T ðpy � cÞ þ x a1 �y

ð1þ xÞ2

!� pEy

py � cþ 1

T

� �; ð11Þ

m3ðx; y; z;EÞ ¼ �pExy

T ðpy � cÞ a1 �y

ð1þ xÞ2

!þ b1xy

T ð1þ xÞ3: ð12Þ

If the condition (9), r1r4 < 0 and T < r1þr4

r1r4hold, we obtain mi > 0, i = 1,2,3.

Furthermore,

m1m2 � m3 ¼1

T 2ðn1T 2 þ n2T þ n3Þ;

where n1 = �r1r4(r1 + r4), n2 = (r1 + r4)2 + r2r3, n3 = �(r1 + r4).Let

T � ¼ �n2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

2 � 4n1n3

p2n1

;

then according to the Routh–Hurwitz criterion, the equilibrium point P4 is stable when T < T*; when T > T*, the equi-librium point P4 is unstable.

Theorem 4. As time delay T is increased through T*, the system (4) undergoes a Hopf bifurcation at equilibrium point P4 if

the conditions (9), r1r4 < 0 and T < r1þr4

r1r4hold, i.e., for T > T* and j T � T*j � 1, an attracting invariant closed curve

bifurcates from the equilibrium point P4.


Proof. For T = T*, we have the following eigenvalues

k1 ¼ffiffiffiffiffiffim2

pi; �k1 ¼ �

ffiffiffiffiffiffim2

pi; k2 ¼ �m1:

Assume

k1 ¼ a1ðT Þ þ ib1ðT Þ; �k1 ¼ a1ðT Þ � ib1ðT Þ; k2 ¼ a2ðT Þ;

then the transversality condition is

da1ðT ÞdT

� �T �¼ m03 � m02m1 � m2m01

2ðm21 þ m2Þ

� �T �:

From the Eqs. (10)–(12), we get

da1ðT ÞdT

� �T �¼ ðr1 þ r4Þðr1r4T �2 � 1Þ

2T �½ð1� ðr1 þ r4ÞT �Þ2 þ ðr1r4 � r2r3ÞT ��> 0:

Thus the system (4) undergoes a Hopf bifurcation at equilibrium point P4. Moreover, for T > T* and jT � T*j � 1, anattracting invariant closed curve bifurcates from the equilibrium point P4. This completes the proof. h

4. Numerical simulation

To illustrate the results obtained, let us consider the following particular cases.

Case I. For the parameter values a1 = 0.8, a2 = 1, b1 = 6, p = 7, c = 1, r = 2, T = 0.2, m = 0, the system (4) has apositive equilibrium point P �1ð1:1676; 0:1429; 1:1676; 1:0891Þ. When economic profit m = �0.001, the eigenvalues are�0.9, �5, �2721.8, and then become �0.899, �5, 3631.3 when the parameter value m = 0.001. It is obvious that twoeigenvalues remain almost constant and another moves from C� to C+ along the real axis by diverging through 1.

Case II. Take a1 = 0.8, a2 = 0, b1 = 6, p = 7, c = 1, r = 2, m = 0.5 for example, then the bifurcation value of the timedelay is T* = 0.1494.

0.537 0.5372 0.5374 0.5376 0.5378 0.5380.875

0.8755

0.876

0.8765

0.877

0.8775

x(t)

y(t)

Fig. 1. When T = 0.1 < T*, the equilibrium point P �2 is stable.


Fig. 1 illustrates that equilibrium point P �2ð0:5374; 0:8764; 0:5374; 0:0974Þ of the system (4) is stable when the timedelay T < T*. When T passes through the critical value T*, the equilibrium point P �2 loses its stability and a Hopf bifur-cation occurs. The bifurcating period solution from P �2 at T* is stable, which is depicted in Fig. 2.

0.535 0.5355 0.536 0.5365 0.537 0.5375 0.538 0.5385 0.539 0.5395 0.540.87

0.872

0.874

0.876

0.878

0.88

0.882

x(t)

y(t)

0 10 20 30 40 50 60 70 80 90 1000.4

0.5

0.6

0.7

0.8

0.9

1

1.1

t

x(t)y(t)

a

b

Fig. 2. When T = 0.15 > T*, bifurcation period solutions from the equilibrium point P �2 occur, and (a) is phase trajectories, (b) is thetime response.

0.49 0.5 0.51 0.52 0.53 0.54

0.84

0.86

0.88

0.9

0.92

0.94

0.96

x(t)

y(t)

Fig. 3. Phase trajectories for m = 0.2, remaining other parameters invariable.


When m = 0.2 and other parameters remain invariable, both the prey–predator populations converge to their equi-librium values (see Fig. 3). This indicates that the economic profit has an effect of stabilizing the equilibrium point of thepredator–prey system.

Case III. We discuss another bifurcation phenomenon by numerical simulation, saddle-node bifurcation. Whenr = 2.723, T = 0.2, m = m* = 0.5 and the others are the same as those of Case II, there is an unique positive equilibriumpoint P �3ð0:9995; 0:4008; 0:9995; 0:2772Þ. By numerical simulations, we can obtain that there are two positiveequilibrium points P ð1Þ3 ð0:9247; 0:5009; 0:9247; 0:1596Þ, and P ð2Þ3 ð1:0744; 0:2914; 1:0744; 0:3847Þ when m = 0.4 and nopositive equilibrium points when m > m*. Furthermore,

ðDX f � DEf ðDEgÞ�1DX gÞP�3¼

�0:6993 �0:5001 0

0 0:4302 0:6015

5 0 �5

0B@1CA;

which has a geometrically simple zero eigenvalue with the right eigenvector v = (0.5028,�0.7032,0.5028)T and the lefteigenvector u = (1,1.1625,0.1399). And

ðuDlfRÞP�3¼ ðuðDlf � DEf ðDEgÞ�1DlgÞÞP �

3¼ �0:258;

ðuD2X fRðv; vÞÞP �

3¼ u

X3

i¼1

ðeivTDX ðDX fiÞTvÞ !

P�3

¼ 1:4606;

where ei, i = 1,2,3 is unit vector.

According to the literature [14], the system (6) undergoes saddle-node bifurcation at the equilibrium point P �3.

5. Conclusions

Assume that the predator growth rate depends on past quantities of prey in an exponentially decreasing way. Thispaper discusses a class of predator–prey singular biological economic model with Holling type II functional form, whichis a differential–algebraic system. The algebraic equation is inspired by economic theory of fishery property resourceproposed by Gordon. It shows that this model exhibits two bifurcation phenomena when the economic profit is zero.One is transcritical bifurcation which changes the stability of the system, and the other is singular induced bifurcation


which indicates that zero economic profit brings impulse, i.e., rapid expansion of the population in biological explana-tion. On the other hand, under the condition of positive economic profit, the combination of time delay and economicprofit may change the stability of the system or cause the population and capture capability to oscillate periodically. Atthe same time, the system undergoes a saddle-node bifurcation if the value of economic profit goes through a criticalvalue of positive economic profit.

Acknowledgments

The authors gratefully thank the contribution of National Research Organization and the anonymous authorswhose work largely constitutes this sample file.

References

[1] Kar TK, Pahari UK. Modeling and analysis of a prey–predator system with stage-structure and harvesting. Nonlinear Anal2007;8(2):601–9.

[2] Xiao DM, Li WX, Han MA. Dynamics in ratio-dependent predator–prey model with predator harvesting. Math Anal Appl2006;324(1):14–29.

[3] Kumar S, Srivastava SK, Chingakham P. Hopf bifurcation and stability analysis in a harvested one-predator–two-prey model.Appl Math Comput 2002;129(1):107–18.

[4] Huang CX, He YG, Huang LH, Yuan ZH. Hopf bifurcation analysis of two neurons with three delays. Nonlinear Anal2007;8(3):903–21.

[5] Sun CJ, Lin YP, Han MA. Stability and hopf bifurcation for an epidemic disease model with delay. Chaos Solitons & Fractals2006;30(1):204–16.

[6] Kar TK, Pahari UK. Non-selective harvesting in prey–predator models with delay. Commun Nonlinear Sci Numer Simul2006;11(4):499–509.

[7] Zhang CR, Zu YG, Zheng BD. Stability and bifurcation of a discrete red blood cell survival model. Chaos Solitons & Fractals2006;28(2):386–94.

[8] Summers D, Cranford JG, Healey BP. Chaos in periodically forced discrete-time ecosystem models. Chaos Solitons & Fractals2000;11(14):2331–42.

[9] Jing ZJ, Yang JP. Bifurcation and chaos in discrete-time predator–prey system. Chaos Solitons & Fractals 2006;27(1):259–77.[10] Liao XY, Ouyang ZG, Zhou SF. Permanence and stability of equilibrium for a two-prey one-predator discrete model. Appl Math

Comput 2007;186(1):93–100.[11] Zhang Y, Zhang QL, Zhao LC, Yang CY. Dynamical behaviors and chaos control in a discrete functional response model. Chaos

Solitons & Fractals 2007;34(4):1318–27.[12] Gordon HS. Economic theory of a common property resource: the fishery. J Political Econ 1954;62(2):124–42.[13] Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer-

Verlag; 1983. p. 117–56.[14] Venkatasubramanian V, Schattler H, Zaborszky J. Local bifurcation and feasibility regions in differential–algebraic systems. IEEE

Trans Autom Control 1995;40(12):1992–2013.

bifurcations of a class of singular biological economic models

Documents