qualitative analysis of an ivlev-type bio-economic system

JAMCJ Appl Math ComputDOI 10.1007/s12190-014-0754-9

O R I G I NA L R E S E A R C H

Qualitative analysis of an Ivlev-type bio-economicsystem

Wei Liu · Yuxian Chen · Qiugen Liao

Received: 26 August 2013© Korean Society for Computational and Applied Mathematics 2014

Abstract The paper considers a bio-economic system with Ivlev-type functional re-sponse. Formally, our bio-economic system takes the form of differential-algebraicequations. We investigate Hopf bifurcation and center stability of the proposed bio-economic system. Some easily verifiable criteria are obtained on these issues. Finally,a numerical example is supplied to illustrate the effectiveness of our criteria.

Keywords Bio-economic system · Hopf bifurcation · Center stability ·Differential-algebraic system · Harvesting

Mathematics Subject Classification 92D25

1 Introduction

From the view of human needs, the biological resources in the predator-prey systemsare commonly harvested. However, many earlier studies have shown that harvest-ing has a strong impact on population dynamics, ranging from rapid depletion tocomplete preservation of biological populations. Concerning the conservation for thelong-term benefits of humanity, there is a wide-range of interest in the analysis andmodelling of predator-prey systems with harvesting. Differential equation models forinteractions between species are one of the classical applications of mathematics tobiology. Many works have been done for the predator-prey systems with harvesting,e.g., see Refs. [1, 3, 5, 8, 11–16, 21–25]. Especially, a class of predator-prey bio-economic systems are proposed in Refs. [13, 14, 21, 22, 24, 25]. The bio-economicsystems are described by differential algebraic equations, since the harvesting efforton the predator-prey systems are considered from an economic perspective. By using

W. Liu (B) · Y. Chen · Q. LiaoSchool of Mathematics and Computer Science, Xinyu University, Xinyu 338004, Chinae-mail: [email protected]

mailto:[email protected]

W. Liu et al.

the modelling method of Zhang et al. [13, 14, 21, 22, 24, 25], we also obtain a bio-economic system. In the research of the dynamic behaviors of bio-economic systems,Zhang et al. [13, 14, 21, 22, 24, 25] have discussed the issues of singularity inducedbifurcation, transcritical bifurcation, saddle-node bifurcation, Neimark-Sacker bifur-cation, state feedback control, etc. Different from the issues studied in Refs. [13, 14,21, 22, 24, 25], we investigate the Hopf bifurcation and center stability of the bio-economic system (1.7). Besides, there are some relevant articles describing bifurca-tion algorithmic techniques [2, 6, 7, 17, 18] and the references therein. A model formethanol electro-oxidation [17] is investigated using stoichiometric network analysisas well as concepts from algebraic geometry revealing the occurrence of a Hopf and asaddle-node bifurcation. Ref. [2] shows that computing threshold conditions for epi-demic models can be done fully algorithmically using quantifier elimination for realclosed fields and related simplification methods for quantifier-free formulas. Sturm etal. [18] have discussed the use of algebraic and logical methods to reduce questionson the existence of Hopf bifurcations in parameterized polynomial vector fields toquantifier elimination problems over the reals combined with the use of the quanti-fier elimination over the reals and simplification techniques available in REDLOG. InRef. [7], Errami et al. present a fully algorithmic technique to compute Hopf bifurca-tion fixed point for reaction systems with linear conservation laws using reaction co-ordinates instead of concentration coordinates. In Ref. [6], Edneral describes a usefulalgorithm for normalization of nonlinear autonomous nonlinear ordinary differentialequations. The research methods used in this paper are different from Refs. [2, 6, 7,17, 18]. Our bifurcation and stability analysis for the bio-economic system is basedmainly on the local parameterization method in Ref. [4] and the formal series methodin Ref. [20]. Our methods enrich the tool box of algorithmic techniques for bifurca-tion and stability analysis. In what follows, we introduce the bio-economic systemthat will be investigated in this paper.

A generalized predator-prey model can be modelled by the following ordinarydifferential equations, indicated by Yodzis [19] and Chen [3]:{

x = xf (x) − yF(x, y),

y = yG(x, y),(1.1)

where x = x(t) and y = y(t) represent the prey density and predator density at time t ,respectively; The function f (x) is the intrinsic growth rate or per capita growth rate;The function F = F(x, y) describes the predator functional response, that is, the de-pendence of its per capita consumption rate upon the sizes of the prey and predatorsclasses, respectively; The function G = G(x,y) describes the predator numerical re-sponse, that is, the per capita growth rate of the predator population, again as a func-tion which depends on the sizes of both population classes.

In this paper, we use the traditional logistic form for the growth function f (x):

f (x) = a

(1 − b

ax

)(a, b > 0), (1.2)

where a is the growth rate of the prey in the absence of predators, b representsthe self-regulation constant of the prey, a/b is the carrying capacity of the prey.

Qualitative analysis of an Ivlev-type bio-economic system

The predator functional response F is chosen as the Ivlev-type functional response[1, 12], which takes the form

F = k(1 − e−cx

)(k, c > 0), (1.3)

where k represents the maximum rate of predation and c denotes the decrease in mo-tivation to hunt. We can see that the above Ivlev-type functional response is mono-tonically increasing as well as uniformly bounded.

In view of the predator numerical response in the well-known Lotka-Volterrapredator-prey model [3, 16], we choose

G = −d + x (d > 0), (1.4)

where d stands for the intrinsic death rate of predator.Substituting Eqs. (1.2)–(1.4) into Eq. (1.1), we get the following predator-prey

system with Ivlev-type functional response:

{x = x(a − bx − ky

x(1 − e−cx)),

y = y(−d + x).(1.5)

Gordon [9] has studied the effect of the harvesting effort on the ecosystem from aneconomic perspective. In Ref. [9], an equation is proposed to investigate the economicinterest of the yield of the harvesting effort, which can be expressed as

Net Economic Revenue (NER) = Total Revenue (TR) − Total Cost (TC).

Let E(t) and z(t) represent the harvesting effort and the density of harvested pop-ulation respectively, then TR = pz(t)E(t) and TC = sz(t)E(t), where p denotesharvesting reward per unit harvesting effort for unit weight, s represents harvestingcost per unit harvesting effort. Associated with the predator-prey system (1.1), an al-gebraic equation which considers the economic profit v of the harvesting effort onprey can be written as

E(t)(px(t) − s

) = v. (1.6)

On the basis of Eqs. (1.5) and (1.6), we obtain the following bio-economic system,which consists of two differential equations and an algebraic equation:

⎧⎪⎨⎪⎩

x = x(a − bx − kyx

(1 − e−cx) − E),

y = y(−d + x),

0 = E(px − s) − v.

(1.7)

In this paper, we mainly discuss the effects of varying the economic profit on thedynamics of the system (1.7) in the region R3+ = {(x, y,E)|x > 0, y > 0,E > 0}.

W. Liu et al.

For the sake of simplicity, let us denote

f (v,X) =(

f1(v,X)

f2(v,X)

)=

(x(a − bx − ky

x(1 − e−cx) − E)

y(−d + x)

),

g(v,X) = E(px − s) − v, X = (x, y,E)T ,

where the economic profit v is chosen as a bifurcation parameter, which will bedefined in the following. When there is no danger of confusion, t is occasionallydropped from the related variables, i.e., x(t) and y(t) are simply expressed by x

and y, respectively.In addition, the structure of this paper is outlined as follows. In Sect. 2, we inves-

tigate the Hopf bifurcation of the system (1.7). In Sect. 3, the center stability analysisfor the system (1.7) is presented. In Sect. 4, we provide an example to demonstrateour analytical results with some numerical simulations. Finally, this paper ends by abrief discussion.

Notations. The notations are fairly standard. Rn denotes the n-dimensional Eu-clidean space. Let A be a matrix. Then AT , A−1 denotes, respectively, the transposeof, and the inverse of A. In denotes an identity matrix of dimension n × n. B repre-sents the closure set of the set B .

2 Hopf bifurcation

In this section, we will present some analytical criteria for the Hopf bifurcation ofthe bio-economic system (1.7). In order to obtain the criteria, we need the followingpreparations.

A point X0 = (x0, y0,E0)T is an equilibrium point of the system (1.7) if and only

if X0 is a solution of the equations⎧⎪⎨⎪⎩

a − bx0 − ky0x0

(1 − e−cx0) − E0 = 0,

−d + x0 = 0,

E0(px0 − s) − v = 0.

By computing, we can easily obtain that the system (1.7) has an equilibrium point

X0 = (x0, y0,E0)T =

(x0,

x0(a − bx0 − E0)

k(1 − e−cx0),

v

px0 − s

)T

,

where x0 = d .In this paper, we only concentrate on the positive equilibrium point of the bio-

economic system (1.7), since the biological interpretation of the positive equilibriumpoint implies that the prey, the predator and the harvesting effort all exist, which arerelevant to our study. Thus, throughout the paper we assume that

v > 0, px0 − s > 0, a > bx0 + E0. (2.1)


In the following, D denotes the differential operator, and DXg denotes the matrixof partial derivatives of the components of g with respect to X.

Since DXg(v,X0)Q = (0,0,px0 − s), we make the transformation X = QX forthe system (1.7), where

Q =⎛⎝ 1 0 0

0 1 0− pE0

px0−s0 1

⎞⎠ , X = (x, y, E)T .

The motivation for the above transformation can be found in Appendix B. Conse-quently, the system (1.7) is converted into

⎧⎪⎨⎪⎩


(1 − e−cx) + pE0px0−s

x − E),

y = y(−d + x),

0 = (− pE0px0−s

x + E)(px − s) − v.

(2.2)

From Sect. 1, we can obtain that

f (v, X) =(

f1(v, X)

f2(v, X)

)=

(x(a − bx − ky

x(1 − e−cx) + pE0

px0−sx − E)

y(−d + x)

),

g(v, X) =(

− pE0

px0 − sx + E

)(px − s) − v.

Obviously, the system (2.2) has a positive equilibrium point X0 = (x0, y0,E0 +px0E0px0−s

)T , where X0 = (x0, y0, E0)T . Besides, we can calculate that DXg(v, X0) =

(0,0,px0 − s).We employ the following local parameterization in Ref. [4] for the system (2.2):

X = ψ(v,Y ) = X0(v) + U0Y + V0h(v,Y ), g(v,ψ(v,Y )

) = 0,

where

U0 =⎛⎝1 0

0 10 0

⎞⎠ , V0 =

⎛⎝0

01

⎞⎠ , Y = (y1, y2)

T ∈ R2,

h is a continuous mapping from R × R2 into R which is smooth with respect to Y .And then, we can deduce that the parametric system of the system (2.2) takes theform of {

y1 = f1(v,ψ(v,Y )),

y2 = f2(v,ψ(v,Y )).(2.3)

For more details about the above local parameterization, refer to Appendix A. Sub-sequently, the Jacobian matrix J (v) of the parametric system (2.3) at Y = 0 can be

W. Liu et al.

given by(Dy1f1(v,ψ(v,Y )) Dy2f1(v,ψ(v,Y ))

Dy1f2(v,ψ(v,Y )) Dy2f2(v,ψ(v,Y ))

)∣∣∣∣Y=0

=(

DXf1(v, X0(v))

DXf2(v, X0(v))

)(DXg(v, X0(v))

UT0

)−1 (0I2

)

=(

DXf1(v, X0(v))

DXf2(v, X0(v))

)(I20

)=

(Dxf1(v, X0(v)) Dyf1(v, X0(v))

Dxf2(v, X0(v)) Dyf2(v, X0(v))

)

=(

ky0x0

+ px0E0px0−s

− (bx0 + ky0e−cx0

x0+ kcy0e

−cx0) −k(1 − e−cx0)

y0 0

).

Hence, the characteristic equation of the Jacobian matrix J (v) is

λ2 + a1(v)λ + a2(v) = 0, (2.4)

where a1(v) = bx0 + ky0e−cx0

x0+ kcy0e

−cx0 − (ky0x0

+ px0E0px0−s

), a2(v) = ky0(1 − e−cx0).

Remark 2.1 The positive equilibrium point X0 of the system (2.2) corresponds to theequilibrium point Y = 0 of the parametric system (2.3). Therefore, the matrix J (v)

can be considered as the Jacobian matrix of the system (2.2) at X0, which can be alsodetermined by the method proposed in Ref. [25].

Suppose that

(1 + cx0)e−cx0 + px0(1 − e−cx0)

px0 − s�= 1 and N1N2 > 0 (2.5)

hold, where

N1 = 2bx0 + cx0(a − bx0)e−cx0

1 − e−cx0− a,

N2 = px0

(px0 − s)2+ (1 + cx0)e

−cx0

(px0 − s)(1 − e−cx0)− 1

(px0 − s)(1 − e−cx0).

Then by a1(v) = 0, we can obtain that the positive bifurcation value is v0 = N1N2

. In

fact, if we let a21(v) < 4a2(v), then Eq. (2.4) has a pair of conjugate complex roots:

λ1,2 = −1

2a1(v) ± i

√a2(v) − a2

1(v)

4:= α(v) ± iω(v).

In view of Eqs. (2.1) and (2.5), we have

α(v0) = 0, α′(v0) = N2

2�= 0, ω(v0) =

√ky0

(1 − e−cx0

) �= 0.


Therefore, a phenomenon of Hopf bifurcation occurs at the bifurcation value v0.For convenience, we use ω0 to denote ω(v0) in the following. To calculate the

Hopf bifurcation, according to Refs. [4, 10], when v = v0, X = X0, we shall lead thenormal form of the system (2.2) as follows:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

y1 = ω0y2 + 12a1

11y21 + a1

12y1y2 + 12a1

22y22 + 1

6a1111y

31 + 1

2a1112y

21y2

+ 12a1

122y1y22 + 1

6a1222y

32 + o(|Y |4),

y2 = −ω0y1 + 12a2

11y21 + a2

12y1y2 + 12a2

22y22 + 1

6a2111y

31 + 1

2a2112y

21y2

+ 12a2

122y1y22 + 1

6a2222y

32 + o(|Y |4).

(2.6)

By computing, we can derive the following normal form of the system (2.2) withv = v0 and X = X0, which takes the form of

⎧⎪⎪⎨⎪⎪⎩

y1 = ω0y2 + (kc2ω0y0e

−cx0

2 − bω0 − psω0E0(px0−s)2 )y2

1 + kcy0e−cx0y1y2

+ (p2sω2

0E0

(px0−s)3 − kc3ω20y0e

−cx0

6 )y31 − kc2ω0y0e

−cx0

2 y21y2 + o(|Y |4),

y2 = −ω0y1 + ω0y1y2.

(2.7)

The calculating process of the above normal form (2.7) is given in Appendix C.Summarizing the above results, we obtain the following theorem on the Hopf bi-

furcation of the bio-economic system (1.7).

Theorem 2.1 For the bio-economic system (1.7), there exist a positive constant ε andtwo small enough neighborhoods of the positive equilibrium point X0(v): O and P ,where 0 < ε � 1, O ⊂ P , such that

(i) If σ0 > 0, that is,

(2b + 2psE0

(px0 − s)2

)e−cx0 + 6p2sE0(1 − e−cx0)

c(px0 − s)3> kc2y0e

−cx0,

then,(1) when v0 < v < v0 + ε, X0(v) repels all the points in P , and X0(v) is unsta-

ble;(2) when v0 − ε < v < v0, there exists at least one periodic solution in O , that

repels all the points in O\{X0(v)}, and there also exists one (may be the sameone) that repels all the points in P \O , and X0(v) is locally asymptoticallystable.

(ii) If σ0 < 0, that is,

(2b + 2psE0

(px0 − s)2

)e−cx0 + 6p2sE0(1 − e−cx0)

c(px0 − s)3< kc2y0e

−cx0,

then,

W. Liu et al.

(1) when v0 −ε < v < v0, X0(v) absorbs all the points in P , and X0(v) is locallyasymptotically stable;

(2) when v0 < v < v0 + ε, there exists at least one periodic solution in O , thatabsorbs all the points in O\{X0(v)}, and there also exists one (may be thesame one) that absorbs all the points in P \O , and X0(v) is unstable.

Proof Comparing the coefficients of Eq. (2.6) with those of Eq. (2.7), we have

a111 = kc2ω0y0e

−cx0 − 2bω0 − 2psω0E0

(px0 − s)2,

a112 = kcy0e

−cx0 , a212 = ω0, a1

22 = a211 = a2

22 = 0,

a1111 = 6p2sω2

0E0

(px0 − s)3− kc3ω2

0y0e−cx0 , a1

122 = a2112 = a2

222 = 0.

According to the Hopf bifurcation theorem in Refs. [4, 10], now we need to judge thesign of the following value σ0:

16σ0 := {a1

11

(a2

11 − a112

) + a222

(a2

12 − a122

) + (a2

11a212 − a1

12a122

)}/ω0

+ (a1

111 + a1122 + a2

112 + a2222

)= 2bcky0e

−cx0 + 2pcsky0E0e−cx0

(px0 − s)2+ 6p2sω2E0

(px0 − s)3− k2c3y2

0e−cx0 .

In the following, there are two cases ought to be studied. That is, σ0 > 0 and σ0 < 0.Since the following process is similar to the proof of the Hopf bifurcation theorem inRefs. [4, 10], the process is omitted here. �

Remark 2.2 The local stability of X0 is equivalent to the local stability of X0.

Remark 2.3 When σ0 = 0, a degenerate Hopf bifurcation takes place.

3 Center stability

In this section, we will use the formal series method in Ref. [20] to investigate thecenter stability of the bio-economic system (1.7). And the following theorem givessome sufficient conditions on the issue.

Theorem 3.1

(i) If

2b + 2psE0

(px0 − s)2> kc2y0e

−cx0 ,

then the center X0(v0) of the bio-economic system (1.7) is an unstable focus;


(ii) If

2b + 2psE0

(px0 − s)2< kc2y0e

−cx0 ,

then the center X0(v0) of the bio-economic system (1.7) is a stable focus.

Proof For simplicity, let τ = ω0t . Using y to denote the derivative of y at τ , then thesystem (2.7) is changed into⎧⎪⎪⎨

⎪⎪⎩y1 = y2 + (

kc2y0e−cx0

2 − b − psE0(px0−s)2 )y2

1 + kcy0e−cx0

ω0y1y2

+ (p2sω0E0(px0−s)3 − kc3ω0y0e

−cx0

6 )y31 − kc2y0e

−cx0

2 y21y2 + o(|Y |4),

y2 = −y1 + y1y2.

(3.1)

Let y1 = z2, y2 = z1, then the system (3.1) becomes⎧⎪⎪⎨⎪⎪⎩

z1 = −z2 + z1z2,

z2 = z1 + (kc2y0e

−cx0

2 − b − psE0(px0−s)2 )z2

2 + kcy0e−cx0

ω0z1z2

+ (p2sω0E0(px0−s)3 − kc3ω0y0e

−cx0

6 )z32 − kc2y0e

−cx0

2 z1z22 + o(|Z|4),

(3.2)

where Z = (z1, z2)T .

For the system (3.2), we consider the following Lyapunov function:

V (z1, z2) = z21 + z2

2 + V3(z1, z2) + V4(z1, z2),

where V3(z1, z2) and V4(z1, z2) represent the third degree homogeneous polynomialand the fourth degree homogeneous polynomial in z1 and z2, respectively. Calculat-ing the derivative of V (z1, z2) along the solution of the system (3.2), we have

dV (z1, z2)

dτ

∣∣∣∣(3.2)

= (2z1 + Dz1V3(z1, z2) + Dz1V4(z1, z2)

)(−z2 + z1z2)

+ (2z2 + Dz2V3(z1, z2) + Dz2V4(z1, z2)

)×

(z1 +

(kc2y0e

−cx0

2− b − psE0

(px0 − s)2

)z2

2 + kcy0e−cx0

ω0z1z2

+(

p2sω0E0

(px0 − s)3− kc3ω0y0e

−cx0

6

)z3

2 − kc2y0e−cx0

2z1z

22 + · · ·

). (3.3)

In Eq. (3.3), let the third degree homogeneous polynomial in z1 and z2 be zero, wecan obtain

z1Dz2V3(z1, z2) − z2Dz1V3(z1, z2)

= −2kcy0e−cx0

ω0z1z

22 − 2z2

1z2 +(

2b + 2psE0

(px0 − s)2− kc2y0e

−cx0

)z3

2.

W. Liu et al.

Let z1 = r cos θ , z2 = r sin θ , then the above equation is transformed into

dV3(cos θ, sin θ)

dθ= −2kcy0e

−cx0

ω0sin2 θ cos θ − 2 sin θ cos2 θ

+(

2b + 2psE0

(px0 − s)2− kc2y0e

−cx0

)sin3 θ

:= −H3(cos θ, sin θ)

= a0

2+

∞∑η=1

(aη cosηθ + bη sinηθ),

where a02 +∑∞

η=1(aη cosηθ + bη sinηθ) is the Fourier series of H3(cos θ, sin θ). It isclear that there exists such a V3(cos θ, sin θ) if and only if a0 = 0, that is,

∫ 2π

0H3(cos θ, sin θ)dθ = 0.

Since

∫ 2π

0

{2kcy0e

−cx0

ω0sin2 θ cos θ + 2 sin θ cos2 θ

−(

2b + 2psE0

(px0 − s)2− kc2y0e

−cx0

)sin3 θ

}dθ = 0,

we can calculate that

V3(z1, z2) =(

2

3+ 2kc2y0e

−cx0

3− 4b

3− 4psE0

3(px0 − s)2

)z3

1

+(

kc2y0e−cx0 − 2b − 2psE0

(px0 − s)2

)z1z

22 − 2kcy0e

−cx0

3ω0z3

2.

Subsequently, we have

Dz1V3(z1, z2) =(

2 + 2kc2y0e−cx0 − 4b − 4psE0

(px0 − s)2

)z2

1

+(

kc2y0e−cx0 − 2b − 2psE0

(px0 − s)2

)z2

2,

(3.4)

Dz2V3(z1, z2) =(

2kc2y0e−cx0 − 4b − 4psE0

(px0 − s)2

)z1z2

− 2kcy0e−cx0

ω0z2

2.


In Eq. (3.3), let the fourth degree homogeneous polynomial in z1 and z2 be zero, wecan obtain

z1Dz2V4(z1, z2) − z2Dz1V4(z1, z2)

= kc2y0e−cx0z1z

32 +

(kc3ω0y0e

−cx0

3− 2p2sω0E0

(px0 − s)3

)z4

2 − z1z2Dz1V3

+{(

b + psE0

(px0 − s)2− kc2y0e

−cx0

2

)z2

2 − kcy0e−cx0

ω0z1z2

}Dz2V3. (3.5)

From Eqs. (3.4) and (3.5), we can calculate that

dV4(cos θ, sin θ)

dθ

= kcy0e−cx0

ω0

(4b + 4psE0

(px0 − s)2− 2kc2y0e

−cx0

)sin2 θ cos2 θ

+ kcy0e−cx0

ω0

(kc2y0e

−cx0 − 2b − 2psE0

(px0 − s)2

)sin4 θ

+(

2b + 2psE0

(px0 − s)2+ 2k2c2y2

0e2(−cx0)

ω20

)sin3 θ cos θ

+(

b + psE0

(px0 − s)2− kc2y0e

−cx0

2

)

×(

2kc2y0e−cx0 − 4b − 4psE0

(px0 − s)2

)sin3 θ cos θ

+(

4b + 4psE0

(px0 − s)2− 2 − 2kc2y0e

−cx0

)sin θ cos3 θ

:= −H4(cos θ, sin θ).

Similar to the discussion about the existence of V3(cos θ, sin θ), we can obtain thatthere exists such a V4(cos θ, sin θ) if and only if

∫ 2π

0H4(cos θ, sin θ)dθ = 0.

However,∫ 2π

0H4(cos θ, sin θ)dθ = πkcy0e

−cx0

4ω0

(2b + 2psE0

(px0 − s)2− kc2y0e

−cx0

)�= 0.

Therefore we need to modify V4(cos θ, sin θ) such that V4(cos θ, sin θ) satisfies thefollowing equation:

dV4(cos θ, sin θ)

dθ= −H4(cos θ, sin θ) + C4

:= −H ′4(cos θ, sin θ),

W. Liu et al.

where C4 = 12π

∫ 2π

0 H4(cos θ, sin θ)dθ .We can verify that ∫ 2π

0H ′

4(cos θ, sin θ)dθ = 0.

And then, the modified V4(cos θ, sin θ) does exist.By the above construction of V (z1, z2), we have

dV (z1, z2)

dτ

∣∣∣∣(3.2)

= C4r4 + o

(r4).

In what follows, there are two cases should be discussed. That is, C4 > 0 and C4 < 0.If C4 > 0, then the origin of the system (3.2) is an unstable focus, i.e., the centerX0(v0) of the system (1.7) is an unstable focus. If C4 < 0, then the origin of thesystem (3.2) is a stable focus, i.e., the center X0(v0) of the system (1.7) is a stablefocus. The proof of Theorem 3.1 is now completed. �

4 A numerical example

In this section, we will give a numerical example to verify the compatibility of thetheorems presented in this paper.

According to (2.1), (2.5), the conditions (i) in Theorems 2.1 and 3.1, we choosethe coefficients of the system (1.7) as follows:

a = 4 − 1.5e−1, b = 1.25 − 0.5e−1,

k = 1, c = 0.5, d = 2, p = 1, s = 1.

And then the system (1.7) becomes⎧⎪⎨⎪⎩

x = x((4 − 1.5e−1) − (1.25 − 0.5e−1)x − yx(1 − e−0.5x) − E),

y = y(−2 + x),

0 = E(x − 1) − v.

(4.1)

Obviously, the system (4.1) has a positive equilibrium point X0 = (x0, y0,E0) =(2,1,1). Besides, we can calculate that N1 = N2 = 1

1−e−1 , and then the positive bi-

furcation value is v0 = N1N2

= 1.In Theorem 2.1, we choose ε = 0.006, then by Theorems 2.1 and 3.1, the positive

equilibrium point X0(v) of the system (4.1) is locally asymptotically stable whenv = 0.995 < v0, as it is illustrated by computer simulations in Fig. 1; The periodicsolution occurs from X0(v) when v = 0.9999 < v0, as it is illustrated by computersimulations in Fig. 2; The center X0(v0) of the system (4.1) is an unstable focuswhen v = v0 = 1, as it is illustrated by computer simulations in Fig. 3; The positiveequilibrium point X0(v) of the system (4.1) is unstable when v = 1.002 > v0, as it isillustrated by computer simulations in Fig. 4.


Fig. 1 The positive equilibrium point X0(v) of the system (4.1) is locally asymptotically stable withv = 0.995 < v0 and the initial conditions x0 = 1.997, y0 = 0.998,E0 = 0.999

Fig. 2 The bifurcating periodic solution occurs from the positive equilibrium point X0(v) of the system(4.1) with v = 0.9999 < v0 and the initial conditions x0 = 1.997, y0 = 0.998,E0 = 0.999

5 Discussion

In reality, economic profit is a very important factor for even every citizen, and theharvested biological resources in the predator-prey systems are usually sold as com-modities in the market in order to achieve the economic interest. And then, modellingand qualitative analysis for bio-economic systems are necessary. We can see that theeconomic principle of Gordon [9] provides theoretical evidence for the establish-ment of bio-economic systems. The advantages of bio-economic systems are thatthe systems not only study the interaction mechanism in the predator-prey biological

W. Liu et al.

Fig. 3 The center X0(v0) of the system (4.1) is an unstable focus with v = v0 = 1 and the initial condi-tions x0 = 1.97, y0 = 0.97,E0 = 0.97

Fig. 4 The positive equilibrium point X0(v) of the system (4.1) is unstable with v = 1.002 > v0 and theinitial conditions x0 = 1.997, y0 = 0.997,E0 = 0.997

systems, but also investigate the effect of the harvesting effort on the predator-preysystems from an economic perspective. In this paper, we investigate the effects ofvarying the economic profit on the dynamics of the bio-economic system (1.7). Thetheoretical analysis of Theorem 2.1 shows that the positive economic profit can causea stable equilibrium point to become an unstable equilibrium point and even a switch-ing of stabilities. If the fishermen’s pursuit of economic profit increases beyond thepositive bifurcation value v0, the bio-economic system will undergo a Hopf bifur-cation, which will be harmful to the sustainable development of the predator-preybiological system. According to Theorem 2.1, we may come to the conclusion that


the increase of the positive economic profit destabilizes the bio-economic system.Consequently, with the purpose of maintaining the sustainable development of thebiological resources as well as keeping the economic profit of the harvesting effort atan ideal level, the fishermen should guarantee their positive economic profit less thanthe bifurcation value. Furthermore, as a complement to Theorem 2.1, when the posi-tive economic profit v equals the bifurcation value v0, i.e., when the positive equilib-rium point X0(v) of the bio-economic system (1.7) becomes the center X0(v0), The-orem 3.1 investigates the center stability of the bio-economic system with the helpof some Lyapunov functions. The results of Theorems 2.1 and 3.1 are theoreticallybeneficial to the management of ecological resources in the predator-prey biologi-cal system. In addition, stage structure, sudden perturbations, time delays, diffusioneffects, disease effects may be incorporated into our bio-economic system, whichwould make the bio-economic system exhibit much more complicated dynamics.

Acknowledgements The authors are greatly indebted to the anonymous referee and the editor for theircareful reading and insightful comments that led to truly significant improvement of the manuscript. Thework is supported by Nature Science Foundation of Jiangxi Province under Grant 20122BAB201006.

Appendix A

The local parameterization method in Ref. [4] is briefly presented below.A differential-algebraic system is also called a constrained system. We consider

the following constrained system with a one-dimensional parameter v:

{ ˙X = f (v, X),

0 = g(v, X),(I)

where v ∈ R, f : R × Rn → Rn, g : R × Rn → Rm, f and g are both four timescontinuously differentiable functions, f = (f1, f2, . . . , fn)

T , g = (g1, g2, . . . , gm)T ,n − m = 2. Let X0(v) be an equilibrium point of the system (I), which is assumed tosatisfy the following conditions:

(H1) rank DXg(v, X0(v)) = m;(H2) DXg(v, X0(v)) = (0,C)m×n, where C is a nonsingular matrix of dimension

m × m.

For the system (I), we employ the following local parameterization:

X = ψ(v,Y ) = X0(v) + U0Y + V0h(v,Y ), g(v,ψ(v,Y )

) = 0,

where U0 = (I20

)n×2, V0 = ( 0

Im

)n×m

, Y = (y1, y2)T ∈ R2, h is a continuous mapping

from R × R2 into Rm which is smooth with respect to Y . And then, the parametric

W. Liu et al.

system of the system (I) takes the form of

Y = UT0 f

(v,ψ(v,Y )

),

which is the Eq. (2.3) in this paper.

Appendix B

The system (1.7) is equivalent to the following constrained system near the equilib-rium point X0: ⎧⎪⎪⎪⎨

⎪⎪⎪⎩x = x(a − bx − ky

x(1 − e−cx) − E),

y = y(−d + x),

E = f3(x, y,E),

0 = E(px − s) − v,

(II)

where the concrete form of the function f3(x, y,E) is no need to write out. Obvi-ously, we can not directly employ the local parameterization (in Appendix A) forthe system (II), since the system (II) does not meet the condition (H2). However, bymaking the transformation X = QX for the system (II), we can obtain the followingequivalent constrained system:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩


(1 − e−cx) + pE0px0−s

x − E),

y = y(−d + x),˙E = f3,

0 = (− pE0px0−s

x + E)(px − s) − v,

(III)

where the concrete form of the function f3 is also no need to write out. It is clear thatthe equivalent system (III) satisfies the condition (H1). In addition, we can calculatethat

DXg(v, X) =(

−2p2E0x

px0 − s+ psE0

px0 − s+ pE,0,px − s

).

Substituting X0 = (x0, y0, E0) = (x0, y0,E0 + px0E0px0−s

) to the above equation, we have

DXg(v, X0) = (0,0,px0 − s).

Therefore, the equivalent system (III) also satisfies the condition (H2). Subsequently,we can employ the local parameterization in Appendix A for the equivalent con-strained system (III). And then, the parametric system of the system (III) takes theform of Eq. (2.3).


Appendix C

In Ref. [4], Chen et al. have proved that the parametric system (2.3) with v = v0 andX = X0 takes the form of⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

y1 = f1y1(v0, X0)y1 + f1y2(v0, X0)y2 + 12f1y1y1(v0, X0)y

21 + f1y1y2(v0, X0)y1y2

+ 12f1y2y2(v0, X0)y

22 + 1

6f1y1y1y1(v0, X0)y31 + 1

2f1y1y1y2(v0, X0)y21y2

+ 12f1y1y2y2(v0, X0)y1y

22 + 1

6f1y2y2y2(v0, X0)y32 + o(|Y |4),

y2 = f2y1(v0, X0)y1 + f2y2(v0, X0)y2 + 12f2y1y1(v0, X0)y

21 + f2y1y2(v0, X0)y1y2

+ 12f2y2y2(v0, X0)y

22 + 1

6f2y1y1y1(v0, X0)y31 + 1

2f2y1y1y2(v0, X0)y21y2

+ 12f2y1y2y2(v0, X0)y1y

22 + 1

6f2y2y2y2(v0, X0)y32 + o(|Y |4).

(C.1)To obtain the above parametric system, the coefficients of the system (C.1) need

to be determined. According to Ref. [4], the coefficients can be calculated as follows.By performing some simple calculations, we can obtain

DXf1(v, X) =(

a − 2bx − kcye−cx + 2pE0x

px0 − s− E,−k

(1 − e−cx

),−x

),

DXf2(v, X) = (y,−d + x,0), (C.2)

DXg(v, X) =(

pE + psE0

px0 − s− 2p2E0x

px0 − s,0,px − s

).

Moreover,

Dψ(v,Y ) = (Dy1ψ(v,Y ),Dy2ψ(v,Y )

) =(

DXg(v, X)

UT0

)−1 (0I2

)

=⎛⎜⎝pE + psE0

px0−s− 2p2E0x

px0−s0 px − s

1 0 00 1 0

⎞⎟⎠

−1 ⎛⎝0 0

1 00 1

⎞⎠

=⎛⎜⎝

1 00 1

1px−s

(2p2E0xpx0−s

− psE0px0−s

− pE) 0

⎞⎟⎠ . (C.3)

In view of Eqs. (C.2) and (C.3), we have

f1y1(v, X) = DXf1(v, X)Dy1ψ(v,Y )

= a − 2bx − kcye−cx + 2pE0x

px0 − s

− E − x

px − s

(2p2E0x

px0 − s− psE0

px0 − s− pE

), (C.4)

W. Liu et al.

f1y2(v, X) = DXf1(v, X)Dy2ψ(v,Y ) = −k(1 − e−cx

),

f2y1(v, X) = DXf2(v, X)Dy1ψ(v,Y ) = y,

f2y2(v, X) = DXf2(v, X)Dy2ψ(v,Y ) = −d + x.

Substituting v0, X0 into Eq. (C.4), we get

f1y1(v0, X0) = ky0

x0+ px0E0

px0 − s−

(bx0 + ky0e

−cx0

x0+ kcy0e

−cx0

)= 0,

f1y2(v0, X0) = −k(1 − e−cx0

), (C.5)

f2y1(v0, X0) = y0, f2y2(v0, X0) = −d + x0 = 0.

From Eq. (C.4), we can calculate that

DXf1y1(v, X)

=(

−2b + kc2ye−cx + 2pE0

px0 − s− psE

(px − s)2− 2p2E0x

(px0 − s)(px − s)

− ps2E0

(px0 − s)(px − s)2+ 2p2sE0x

(px0 − s)(px − s)2,−kce−cx,

s

px − s

), (C.6)

DXf1y2(v, X) = (−kce−cx,0,0),

DXf2y1(v, X) = (0,1,0), DXf2y2(v, X) = (1,0,0).

In view of Eqs. (C.3) and (C.6), we obtain

f1y1y1(v, X) = DXf1y1(v, X)Dy1ψ(v,Y )

= −2b + kc2ye−cx + 2pE0

px0 − s

− 2psE

(px − s)2− 2p2E0x

(px0 − s)(px − s)− 2ps2E0

(px0 − s)(px − s)2

+ 4p2sE0x

(px0 − s)(px − s)2,

f1y1y2(v, X) = DXf1y1(v, X)Dy2ψ(v,Y ) = −kce−cx, (C.7)

f1y2y2(v, X) = DXf1y2(v, X)Dy2ψ(v,Y ) = 0,



f2y2y2(v, X) = DXf2y2(v, X)Dy2ψ(v,Y ) = 0.


Substituting v0, X0 into Eq. (C.7), we have

f1y1y1(v0, X0) = −2b + kc2y0e−cx0 − 2psE0

(px0 − s)2,

f1y1y2(v0, X0) = −kce−cx0,(C.8)

f1y2y2(v0, X0) = 0, f2y1y1(v0, X0) = 0,

f2y1y2(v0, X0) = 1, f2y2y2(v0, X0) = 0.

From Eqs. (C.3) and (C.7), we derive

DXf1y1y1(v0, X0) =(

−kc3y0e−cx0 + 6p2sE0

(px0 − s)3, kc2e−cx0,− 2ps

(px0 − s)2

),

DXf1y1y2(v0, X0) = (kc2e−cx0 ,0,0

), DXf1y2y2(v0, X0) = (0,0,0),

(C.9)DXf2y1y1(v0, X0) = DXf2y1y2(v0, X0) = DXf2y2y2(v0, X0) = (0,0,0),

Dψ(v0,0) = (Dy1ψ(v0,0),Dy2ψ(v0,0)

) =⎛⎝1 0

0 10 0

⎞⎠ .

From Eq. (C.9), we can easily get

f1y1y1y1(v0, X0) = DXf1y1y1(v0, X0)Dy1ψ(v0,0) = −kc3y0e−cx0 + 6p2sE0

(px0 − s)3,

f1y1y1y2(v0, X0) = DXf1y1y1(v0, X0)Dy2ψ(v0,0) = kc2e−cx0 ,

f1y1y2y2(v0, X0) = DXf1y1y2(v0, X0)Dy2ψ(v0,0) = 0,

f1y2y2y2(v0, X0) = DXf1y2y2(v0, X0)Dy2ψ(v0,0) = 0,(C.10)




f2y2y2y2(v0, X0) = DXf2y2y2(v0, X0)Dy2ψ(v0,0) = 0.

Substituting Eqs. (C.5), (C.8) and (C.10) into Eq. (C.1), we can obtain the parametricsystem of the system (2.2) with v = v0 and X = X0, which takes the following form:

⎧⎪⎪⎨⎪⎪⎩

y1 = −k(1 − e−cx0)y2 + (kc2y0e

−cx0

2 − b − psE0(px0−s)2 )y2

1 − kce−cx0y1y2

+ (p2sE0

(px0−s)3 − kc3y0e−cx0

6 )y31 + kc2e−cx0

2 y21y2 + o(|Y |4),

y2 = y0y1 + y1y2.

(C.11)

W. Liu et al.

Compared with the normal form (2.6), we shall normalize the above parametricsystem (C.11) with the following nonsingular linear transformation:(

y1y2

)= P

(u1u2

),

where P = ( ω0 00 −y0

),U = (u1, u2)

T . For convenience, we use Y instead of U . Con-sequently, the normal form of the system (2.2) takes the form of Eq. (2.7).

References

1. Baek, H.K., Kim, S.D., Kim, P.: Permanence and stability of an Ivlev-type predator-prey system withimpulsive control strategies. Math. Comput. Model. 50, 1385–1393 (2009)

2. Brown, C.W., Kahoui, M.E., Novotni, D., Weber, A.: Algorithmic methods for investigating equilibriain epidemic modeling. J. Symb. Comput. 41, 1157–1173 (2006)

3. Chen, L.S.: Mathematical Models and Methods in Ecology. Science Press, Beijing (1988). (in Chi-nese)

4. Chen, B.S., Liao, X.X., Liu, Y.Q.: Normal forms and bifurcations for the differential-algebraic sys-tems. Acta Math. Appl. Sin. 23, 429–433 (2000). (in Chinese)

5. Dong, L.Z., Chen, L.S., Sun, L.H., Jia, J.W.: Ultimate behavior of predator-prey system with constantharvesting of the prey impulsively. J. Appl. Math. Comput. 22, 149–158 (2006)

6. Edneral, V.F.: An algorithm for construction of normal forms. Lect. Notes Comput. Sci. 4770, 134–142 (2007)

7. Errami, H., Seiler, W.M., Eiswirth, M., Weber, A.: Computing Hopf bifurcations in chemical reactionnetworks using reaction coordinates. Lect. Notes Comput. Sci. 7442, 84–97 (2012)

8. Feng, P.: Analysis of a delayed predator-prey model with ratio-dependent functional response andquadratic harvesting. J. Appl. Math. Comput. (2013). doi:10.1007/s12190-013-0691-z

9. Gordon, H.S.: Economic theory of a common property resource: the fishery. J. Polit. Econ. 62, 124–142 (1954)

10. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of VectorFields. Springer, New York (1983)

11. Hogarth, W.L., Norbury, J., Cunning, I., Sommers, K.: Stability of a predator-prey model with har-vesting. Ecol. Model. 62, 83–106 (1992)

12. Ivlev, V.S.: Experimental Ecology of the Feeding of Fishes. Yale University Press, New Haven (1961)13. Liu, C., Zhang, Q.L., Duan, X.D.: Dynamical behavior in a harvested differential-algebraic prey-

predator model with discrete time delay and stage structure. J. Franklin Inst. 346, 1038–1059 (2009)14. Liu, C., Zhang, Q.L., Zhang, X., Duan, X.D.: Dynamical behavior in a stage-structured differential-

algebraic prey-predator model with discrete time delay and harvesting. J. Comput. Appl. Math. 231,612–625 (2009)

15. Liu, W., Chen, Y., Fu, C.: Dynamic behavior analysis of a differential-algebraic predator-prey systemwith prey harvesting. J. Biol. Syst. 21, 13500 (2013), 23 pp. doi:10.1142/S0218339013500228

16. Lucas, W.F.: Modules in Applied Mathematics: Differential Equation Models. Springer, New York(1983)

17. Sauerbrei, S., Nascimento, M.A., Eiswirth, M., Varela, H.: Mechanism and model of the oscillatoryelectro-oxidation of methanol. J. Chem. Phys. 132, 154901 (2010)

18. Sturm, T., Weber, A., Abdel-Rahman, E.O., Kahoui, M.E.: Investigating algebraic and logical al-gorithms to solve Hopf bifurcation problems in algebraic biology. Math. Comput. Sci. 2, 493–515(2009)

19. Yodzis, P.: Predator-prey theory and management of multispecies fisheries. Ecol. Appl. 4, 51–58(1994)

20. Zhang, J.Y., Feng, B.Y.: Geometric Theory of Ordinary Differential Equations and Bifurcation. PekingUniversity Press, Beijing (2000). (in Chinese)

21. Zhang, Y., Zhang, Q.L.: Chaotic control based on descriptor bioeconomic systems. Control Decis. 22,445–452 (2007)

http://dx.doi.org/10.1007/s12190-013-0691-z

http://dx.doi.org/10.1142/S0218339013500228


22. Zhang, X., Zhang, Q.L.: Bifurcation analysis and control of a class of hybrid biological economicmodels. Nonlinear Anal. Hybrid Syst. 3, 578–587 (2009)

23. Zhang, X., Chen, L., Neumann, A.U.: The stage-structured predator-prey model and optimal harvest-ing policy. Math. Biosci. 168, 201–210 (2000)

24. Zhang, Y., Zhang, Q.L., Zhao, L.C.: Bifurcations and control in singular biological economical modelwith stage structure. J. Syst. Eng. 22, 232–238 (2007)

25. Zhang, X., Zhang, Q.L., Zhang, Y.: Bifurcations of a class of singular biological economic models.Chaos Solitons Fractals 40, 1309–1318 (2009)

qualitative analysis of an ivlev-type bio-economic system

Documents