Hopf Bifurcation Analysis of a Predator-Prey Biological Economic System with Nonselective Harvesting

A modified predator-prey biological economic system with nonselective harvesting is investigated. An important mathematical feature of the system is that the economic profit on the predator-prey system is investigated from an economic perspective. By using the local parameterization method and Hopf bifurcation theorem, we analyze the Hopf bifurcation of the proposed system. In addition, themodifiedmodel enriches the database for the predator-prey biological economic system. Finally, numerical simulations illustrate the effectiveness of our results.


Introduction
At present, the increasingly serious problem of environmental degradation and resource shortage makes the analysis and modeling of biological systems more interesting.From the perspective of human needs, the exploitation of biological resources and harvest of population are usually practiced in the fields of wildlife, fishery, and forestry management.It is well known that one of dominant themes in both ecology and mathematical ecology is the dynamic relationship between predators and their prey because of its universal existence and importance in population dynamics.Many authors [1][2][3][4][5][6][7][8][9][10][11][12] have studied the dynamics of predator-prey models with harvesting and obtained various dynamic behaviors, such as permanence, extinction, stability of equilibrium, Hopf bifurcation, and limit cycle.Most of these discussions on biological models are based on normal systems governed by differential equations or difference equations.
In these years, it has been shown that harvesting has a strong impact on dynamic evolution of a population; see [8][9][10][11][12][13][14][15][16].In fact, the harvesting should be a variable from real world view, because it may vary with seasonality, market demand, harvesting cost, and so on.On the other hand, economic profit is a very important factor for governments, merchants, and even every citizen, so it is necessary to research biological economic systems with economic profit, which can be described by differential-algebraic equations.
In particular, according to the economic principle in [5], Zhang et al. [11][12][13] put forward a class of modified predatorprey systems, which are established by differential-algebraic equations.The advantages of the systems proposed in [11][12][13] are that these models investigated the interaction mechanism in the predator-prey ecosystem and offered a new cognitive perspective for the harvested predator-prey biological system.That is, the harvest effort on the predator-prey system can be realized from an economic perspective.However, to our knowledge, the systems in most of the articles on this subject are with just one capture harvesting, such as the system with predator harvesting or the system with prey harvesting; so far there have been no attempts in the study of the bifurcation of the predator-prey biological economic system with nonselective harvesting.
The aim of this paper is to investigate the Hopf bifurcation of a predator-prey biological economic system with nonselective harvesting by using bifurcation theory in [17,18] and center manifold theory in [17][18][19].
The rest of the paper is arranged as follows: a predatorprey biological economic system with nonselective harvesting is established in Section 2. We investigate the Hopf bifurcation for this system in the closed positive cone  3 + in Section 3. Numerical simulations will be performed to illustrate the analytical results in Section 4. A brief discussion is given in Section 5.

Model
The basic model we consider is based on the following Lotka-Volterra predator-prey model with harvest: where x and ỹ denote prey and predator population densities at time t, respectively. > 0, d > 0 are the intrinsic growth rate of prey and the death rate of predator in the absence of food, respectively. > 0 is the carrying capacity of prey. > 0 and b > 0 measure the effect of the interaction of the two populations.Ẽ represents harvesting effort.Ẽ x and Ẽ ỹ indicate that the harvests for prey and predator population are proportional to their densities at time t.
Based on the model system (1) and the economic theory of fishery resource proposed by Gordon [5] in 1954, a differential-algebraic model which consists of two differential equations and an algebraic equation can be established as follows: where p and p are harvesting reward per unit harvesting effort for unit weight of prey and predator and c and c are harvesting cost per unit harvesting effort for prey and predator, respectively. > 0 is the economic profit per unit harvesting effort.For convenience, substituting these dimensionless variables in system (2), and then obtain the following biological economic system expressed by differential-algebraic equation: In this paper, we mainly discuss the effects of economic profit on the dynamics of the system (4) in the region  3 + = {(, , ) |  > 0,  > 0,  > 0}.(5)

Hopf Bifurcation
In this section, we will present some analytical criteria for the Hopf bifurcation of the bioeconomic system (4).In order to obtain the criteria, we need the following preparations.Now, we try to find all positive equilibrium points of the system (4).The positive equilibrium point  0 = ( 0 ,  0 ,  0 )  of system (4) satisfies the following equations: By computing, we can easily obtain that the system (4) has an equilibrium point  0 = ( 0 ,  0 ,  0 )  = ((1/) 0 + /, −(( + 1)/) 0 + ( − )/,  0 )  , where  0 satisfies the equation Obviously, The paper only concentrates on the positive equilibrium point of the system (4), since the biological meaning of the positive equilibrium point implies that the prey, the predator, and the harvest effort on prey all exist, which are relevant to our study.Therefore, throughout the paper, we assume that For the system (4), we consider the following local parameterization: where and ℎ is a continuous mapping from  ×  2 into  which is smooth with respect to .And then, we can deduce that the parametric system of the system (4) takes the form of Therefore, the Jacobian matrix () of the system ( 12) at  = 0 takes the form of where Therefore, the characteristic equation of the matrix () can be expressed as where Remark 1.The positive equilibrium point  0 of the system (4) corresponds to the equilibrium point  = 0 of the parametric system (12).For this reason, () can be considered as Jacobian matrix of the system (4) at  0 , which can be also determined by the method in [16].
In (14), letting  1 () = 0, we obtain the bifurcation value 14) has a pair of conjugate complex roots: By computing, we have Therefore, a phenomenon of Hopf bifurcation occurs at the bifurcation value  0 .
In order to calculate the Hopf bifurcation, according to [3,17], when  =  0 ,  =  0 , we need to lead the normal form of the system (4) as follows: where  0 = ( 0 ).And it can be proved that the parametric system (12) with  =  0 and  =  0 takes the form of In the following, we will calculate the coefficients of the above parametric system (18).By calculation, we derive Thus, Substituting  0 ,  0 into (20), From ( 20), we have According to (19) and ( 22), we obtain Substituting  0 ,  0 into (23), By (23) we get Substituting  0 ,  0 into ( 19) and ( 25), it is easy to compute that According to ( 18), ( 21), (24), and (26), we obtain the parametric system (4), which takes the form of where Compared with the normal form (17), we should normalize the parametric system (27) with the following nonsingular linear transformation: where  = ( − 12 0  11 − 0 ).For convenience, we use  instead of .Thus, the normal form of system (4) takes the form of where Theorem 2. For the system (4), there exist a positive constant  and two small enough neighborhoods of the positive equilibrium point  0 ():  and , where 0 <  ≪ 1,  ⊂ .
(1) If  0 > 0, that is, (i) when  0 <  <  0 + ,  0 () repels all the points in , and  0 () is unstable; (ii) when  0 − <  <  0 , there exists at least one periodic solution in , one of them repels all the points in  \ { 0 ()}, and there also exists one (may be the same one) that repels all the points in  \ , and  0 () is locally asymptotically stable.
(2) If  0 < 0, that is, (i) when  0 −  <  <  0 ,  0 () absorbs all the points in , and  0 () is locally asymptotically stable; (ii) when  0 <  <  0 +, there exists at least one periodic solution in , one of them absorbs all the points in \{ 0 ()}, and there also exists one (may be the same one) that absorbs all the points in  \ , and  0 () is unstable.Proof.From (17) According to the Hopf bifurcation theorem in [1,14], next we will judge the sign of the value  0 which is defined as follows: In what follows, there are two cases that ought to be discussed.That is,  0 > 0 and  0 < 0. And then the following process is similar to the proof of the Hopf bifurcation theorem in [1,14], so the process is omitted here.Remark 3. The local stability of  0 is equivalent to the local stability of  0 .

Numerical Simulations
In this section, we present some numerical simulations to illustrate our theoretical analysis.On the basis of condition (9), the coefficients of the system (4) are chosen as follows: Clearly, the system (35) has a positive equilibrium point  0 = ( 0 ,  0 ,  0 ) = (0.3484, 0.5290, 0.1226), and the bifurcation value  0 = 0.1657.In Theorem 2, we let  = 0.015, and then, by Theorem 2, the positive equilibrium point  0 () of the system (35) is locally asymptotically stable when  = 0.155 <  0 , which has been illustrated in Figure 1; the periodic solution occurs from  0 () when  = 0.164 <  0 , which has been illustrated in Figure 2; the positive equilibrium point  0 () of the system (35) is unstable when  = 0.171 >  0 , which has been illustrated in Figure 3.

Discussion
In this paper, we investigate the effects of the varying economic profit on the dynamics of the bioeconomic system (4).According to Theorem 2, we can see that if the fishermen's pursuit of economic profit  is equal to or larger than the bifurcation value  0 , then the status of preys, the predators, and the harvest effort will be unstable.Clearly, this is harmful to the predator-prey biological economic system.Therefore, in order to ensure the continuable and healthy development of the biological economic system as well as maintain the ideal income from the harvest effort, the fishermen ought to guarantee that their positive economic profit  is less than the bifurcation value  0 .
As we know, harvesting has a strong impact on the dynamic evolution of a population.And many works have been done for the predator-prey system with harvesting; see [7-9, 11-16, 20].Particularly, a class of biological economic systems is proposed in [8, 11-13, 15, 20].Some scholars analyze the system with predator harvesting and some scholars analyze the system with prey harvesting.Compared with the above researches, the main contribution of this paper lies in the following aspect.The predator-prey system we consider incorporates nonselective harvesting.In fact, predator harvest and prey harvest can also bring economic benefits for us, so the system with nonselective harvesting that we investigate is more realistic.And the analysis result in this paper will be more scientific.So the modified model in our paper enriches the database for the predator-prey biological economic system.
In addition, stage structure, time delays, diffusion effects, and disease effects may be incorporated into our bioeconomic system, which would make the bioeconomic system exhibit much more complicated dynamics.